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Inside PESSOA - A detailed description of the model
Vanda Almeida
Gabriela Castro
Ricardo M. Félix
Paulo Júlio
José R. Maria
October 2013
The analyses, opinions and findings of these papers represent
the views of the authors, they are not necessarily those of the
Banco de Portugal or the Eurosystem
Please address correspondence to
Banco de Portugal, Economics and Research Department
Av. Almirante Reis 71, 1150-012 Lisboa, Portugal
Tel.: 351 21 313 0000, email: [email protected]
BANCO DE PORTUGAL
Av. Almirante Reis, 71
1150-012 Lisboa
www.bportugal.pt
Edition
Economics and Research Department
Lisbon, 2013
ISBN 978-989-678-212-2
ISSN 2182-0422 (online)
Legal Deposit no. 3664/83
Inside PESSOA—A Detailed Description of the Model∗
Vanda Almeida
Gabriela Castro
Ricardo M. Félix
José R. Maria
Paulo Júlio
Economics and Research Department, Banco de Portugal†
September 25, 2013
Abstract
This article presents a detailed description of PESSOA—Portuguese Economy Structural
Small Open Analytical model. PESSOA is a dynamic general equilibrium model that can
be applied to any small economy integrated in a monetary union. The main theoretical
reference behind its structure is Kumhof, Muir, Mursula, and Laxton (2010). The model
features non-Ricardian characteristics, a multi-sectoral production structure, imperfect market competition, and a number of nominal, real, and financial rigidities. PESSOA has been
calibrated to match Portuguese and euro area data and used to illustrate a number of key
macroeconomic issues, ranging from the effects of structural reforms to alternative fiscal
policy options.
JEL classification: E62, F41, H62
Keywords: DSGE models; euro area; small-open economy.
∗
We thank all the support provided by Douglas Laxton, Michael Kumhof, and Antti Rippati. The views
expressed in this article are those of the authors and do not necessarily reflect the views of Banco de Portugal.
Any errors and mistakes are ours. Comments are welcome.
†
Address: Banco de Portugal, Departamento de Estudos Económicos, Rua Francisco Ribeiro, 2, 1150-165
Lisboa. Corresponding author: Ricardo Mourinho Félix. E-mail: [email protected]
Contents
1 Introduction
1
2 Overview of PESSOA
2
3 The model
6
3.1
Households: general features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.1.1
Overlapping generations scheme . . . . . . . . . . . . . . . . . . . . . . .
7
3.1.2
Utility function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.1.3
Labor productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Households with access to financial markets (type-A households) . . . . . . . . .
11
3.2.1
Budget constraint and maximization problem . . . . . . . . . . . . . . . .
11
3.2.2
Aggregate consumption and aggregate wealth . . . . . . . . . . . . . . . .
16
Households with no access to financial markets (type-B households) . . . . . . .
27
3.3.1
Budget constraint and maximization problem . . . . . . . . . . . . . . . .
27
3.3.2
Aggregate consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.4
Labor unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.5
Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.5.1
Capital goods producers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.5.2
Entrepreneurs and banks . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.5.3
Manufacturers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
3.5.4
Distributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.6
Fiscal and monetary policy authorities . . . . . . . . . . . . . . . . . . . . . . . .
69
3.7
Rest of the world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.8
Economic aggregates and market clearing conditions . . . . . . . . . . . . . . . .
75
3.2
3.3
4 The model without financial frictions
78
4.1
Households and labor unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.2
Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.3
Fiscal and monetary policy authorities . . . . . . . . . . . . . . . . . . . . . . . .
81
4.4
Market clearing conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
5 Shocks
82
6 Calibration
83
7 Economic analysis and policy simulations
85
i
Appendices
90
List of equations: the stationary model . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
List of equations: the steady-state model . . . . . . . . . . . . . . . . . . . . . . . . .
97
List of shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Calibration: key ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Calibration: main parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Calibration: financial sector parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 108
List of Figures
1
Flows of goods between agents in PESSOA . . . . . . . . . . . . . . . . . . . . .
3
2
Labor productivity of age group a . . . . . . . . . . . . . . . . . . . . . . . . . .
11
3
The cdf of a lognormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4
Fiscal stimulus based on G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
List of Tables
1
Key ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2
Main parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3
Financial sector parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
List of Boxes
1
The standard cost minimization problem. . . . . . . . . . . . . . . . . . . . . . .
10
2
Type-A households maximization problem. . . . . . . . . . . . . . . . . . . . . .
14
3
Generation a Euler equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
4
Type-A household budget constraint: successive forward substitution. . . . . . .
25
5
Unions maximization problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
6
Framework and notation related with capital utilization and accumulation. . . .
34
a(uJt ).
7
The adjustment cost function
. . . . . . . . . . . . . . . . . . . . . . . . .
37
8
The consequences of the idiosyncratic shock ωt+1 . . . . . . . . . . . . . . . . . . .
42
9
Banks and entrepreneurs capital earnings share. . . . . . . . . . . . . . . . . . . .
45
10
Entrepreneurs maximization problem. . . . . . . . . . . . . . . . . . . . . . . . .
46
11
Using the lognormal distribution in the financial accelerator set up. . . . . . . . .
50
12
Manufacturer maximization problem . . . . . . . . . . . . . . . . . . . . . . . . .
58
13
The distributor assemblage stage . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
14
Distributor maximization problem . . . . . . . . . . . . . . . . . . . . . . . . . .
68
15
The foreign distributor’s demand for export goods. . . . . . . . . . . . . . . . . .
75
16
Capital goods producers maximization problem . . . . . . . . . . . . . . . . . . .
80
ii
Be complete in everything, for to be complete in anything is to be right.
Fernando Pessoa
1
Introduction
Macroeconomics is nowadays inseparable from the formal rigor of analytical models, since they
allow for the analysis of the fundamental driving forces behind aggregate economic phenomena and for well-grounded diagnosis of economic performances. It is therefore natural that
an increasing number of organizations and policy-making institutions are using well-founded
macroeconomic models to aid policy recommendations and decision making.
A myriad of macroeconomic models emerged over the past 30 years, both in academia and
policy-making institutions, and a vast literature has been devoted to the development of richer
and more realistic frameworks to study a different number of issues. Some consensus has emerged
recently around New Keynesian general equilibrium models as a reference tool. These models rely
on strong theoretical microeconomic foundations—in the spirit of the seminal real business cycle
model pioneered by Kydland and Prescott (1977, 1982)—and comprise market imperfections as
well as a large number of nominal and real rigidities (Calvo, 1983; Rotemberg, 1982; King, 1991;
Rotemberg and Woodford, 1995; Erceg, Henderson, and Levin, 2000; Abel, 1990; Christiano,
Eichenbaum, and Evans, 2005) that mimic important evidence on macroeconomic dynamics.
New Keynesian general equilibrium models are able to replicate a satisfactory number of
stylized facts, and real rigidities and imperfect market competition create a role for economic
stabilization policies. Macro-financial linkages stemming from liquidity channels or from balance
sheet channels of borrowers or lenders—an operational constraint that can embed a financial
accelerator effect—are also a growing field in macroeconomics. Asymmetric information, as in
Bernanke, Gertler, and Gilchrist (1999b), and limited commitment problems, as in Kiyotaki and
Moore (1997), are two typical theoretical origins behind the modelling of credit frictions.
General equilibrium models are currently the most appealing tool for structural macroeconomic policy analysis, being widely used in policy-making institutions. Examples include the
IMF’s Global Integrated Monetary and Fiscal model to assess the impact of fiscal stimulus
in the context of the ongoing global crisis (Freedman et al., 2009a,b), the Bank of Sweden’s
RAMSES model to evaluate policy options and perform forecasting exercises (Adolfson et al.,
2007b), the Bank of Finland’s AINO model to analyze the macroeconomic impacts of aging
and aid in regular projection exercises (Kilponen and Ripatti, 2006), and the ECB’s New Area
Wide model used for a wide range of purposes within the Eurosystem (Christoffel, Coenen, and
Warne, 2008). The relative performance of seven widely used structural models to alternative
1
fiscal policies can be found in Coenen et al. (2010).
DSGE models are not free from caveats, facing a challenging trade-off between parsimony,
that allows for analytical and computational tractability, and complexity, required to capture
realistic macroeconomic dynamics. Parsimony is often achieved through a number of simplifying
assumptions. For instance, replacing rational expectations—a somewhat strong assumption—
by bounded rationality has led to a series of models that lack the ability to deal with policy
issues, due to their parsimoniousness in many other dimensions. Regardless of how the model
matches the data facts, one must be aware, when evaluating policy options or making policy
recommendations, that DSGE models are stylized representations of reality, and do not embrace
all the complex and relevant economic features.
This article contains a detailed description of a large-scale dynamic general equilibrium
model for a small economy integrated in a monetary union, developed at Banco de Portugal.
The structure of the domestic economy draws fundamentally from Kumhof and Laxton (2007,
2009a,b), Kumhof et al. (2009) and Kumhof et al. (2010). The model is calibrated to match
Portuguese and euro area data, and is termed PESSOA—Portuguese Economy Structural Small
Open Analytical model. It can however be applied to any small economy integrated in a
monetary union. PESSOA has been used in several policy evaluation and simulation exercises,
ranging from the effects of structural reforms to fiscal stimulus and consolidation (Almeida,
Castro, and Félix, 2009, 2010; Almeida et al., 2010a,b, 2013). The current version of PESSOA
contemplates financial frictions, which are increasingly important since the triggering of the
2008 financial crisis, along the lines of Bernanke, Gertler, and Gilchrist (1999b). The research
agenda includes plans to augment the model in several dimensions, for instance to have a more
prominent banking sector and a richer labor market.
This article has the following structure. Section 2 provides an overview of PESSOA and
its transmission mechanisms. Section 3 presents a complete description of the model with
financial frictions, alongside with a brief motivation of the framework. We derive the optimality
conditions for each type of agent, the dynamic equations describing the behavior of economic
aggregates, and the corresponding stationary and steady-state formulations. The key equations
that characterize the model’s solution are clearly identified throughout the text, and listed in the
appendix for convenience. Section 4 briefly outlines the version of the model without financial
frictions. Section 5 explains how shocks are introduced, turning the model into a powerful tool
for policy analysis and simulation. Section 6 presents the calibration. Finally, Section 7 provides
a brief overview of the major simulation exercises performed with PESSOA.
2
Overview of PESSOA
PESSOA is a New-Keynesian DSGE model for a small-open economy integrated in a monetary
union. Its basic structure builds mainly on the IMF’s large scale multi-country model, the Global
Integrated Monetary and Fiscal model (GIMF) Kumhof and Laxton (2007, 2009a,b); Kumhof
et al. (2009, 2010). PESSOA features non-Ricardian characteristics, a multi-sectoral production
structure, imperfect market competition, and a number of nominal, financial and real rigidities
that allow for realistic short-run dynamics and create room for welfare improving stabilization
policy. Figure 1 depicts the flows of goods between agents in PESSOA.
2
which enforces net wealth transfers from succumbing households to those that survive. This
setup generates a strong non-Ricardian outcome whereby households prefer debt financing to
tax financing of public expenditures, since future generations will bear some fraction of the
tax burden (Buiter, 1988). Households also face a declining lifetime productivity of labor which
adds to the non-Ricardian properties of the model, shifting their proneness towards paying taxes
latter, when labor income is lower, rather then sooner.
We consider two types of households: those who perform inter-temporal optimization by
accessing asset markets to smooth consumption, and those who do not have access to asset
markets and therefore can only optimize intra-temporally. These latter agents, termed handto-mouth households, further enhance the non-Ricardian features of the model, since they are
unable to shift consumption over time. Hence, any shock that impacts their budget constraint
or changes their share on total population has an immediate, direct, impact on macroeconomic
aggregates. Both household types derive utility from consumption and leisure, and are subject
to external habit formation.
Households rent labor services to heterogeneous labor unions, receiving a given wage and
paying the corresponding labor income taxes. Furthermore, the typical household receives its fair
share of labor union’s dividends, as well as lump-sum transfers from the government and from
abroad. In addition, financially unconstrained agents receive dividends from firms and earn/pay
interest on asset/debt holdings. They are also remunerated for services in the bankruptcy monitoring of firms, which they perform at no cost and with no effort whenever an entrepreneurial
firm goes bankrupt.
Labor unions rent labor services from households and sell them to manufacturers, charging a
markup over the wage paid to households. Market power arises from the fact that labor unions
supply differentiated, imperfectly substitutable labor services. Dividends are fully transferred to
households. This modeling strategy—widely used in DSGE models—implies that households are
rewarded for labor services in excess of their marginal rate of substitution between consumption
and leisure. Unions face adjustment costs on wage changes in order to mimic the dynamics of
sticky wage growth.
Manufacturers—the intermediate goods producer—combine capital, rented from entrepreneurs, and labor services, rented from labor unions, to produce two types of differentiated
goods, tradable and non-tradable. These are used by distributors as inputs in different stages of
their production process. We obtain staggered price adjustment by imposing adjustment costs
on price changes. Adjusting the fraction of time worked by households is also costly, so that
hours worked adjust sluggishly. Manufacturers pay social security taxes on their payroll and
capital income taxes on profits. After-tax profits are distributed to financially unconstrained
households as dividends.
Distributors—the final goods producer—produce four types of differentiated goods, each acquired by a unique type of costumer: consumption goods are acquired by households, investment
goods by capital goods producers, government consumption goods by the public sector, and export goods by foreign distributors. Final goods are produced in a two-stage process. In the
first stage, the distributor obtains assembled goods by combining domestic tradable goods with
imported goods. This stage determines also the imports of the domestic economy. In the second
4
stage, the distributor combines assembled goods with domestic non-tradable goods, obtaining
the final good. Analogously to manufacturers, price adjustment costs lead to price staggering. Distributors pay capital income taxes on profits, and distribute dividends to financially
unconstrained households.
The baseline model includes a financial transmission mechanism along the lines of Bernanke,
Gertler, and Gilchrist (1999a), Christiano, Motto, and Rostagno (2010), and Kumhof et al.
(2010), whereby financial frictions affect the after tax return on capital and therefore capital
demand. Before each production cycle, capital goods producers buy the undepreciated capital
stock from entrepreneurs, combining it with investment goods bought from distributors to produce new installed capital. New capital is then sold to entrepreneurs, which will own it during
the next production cycle.
Entrepreneurs do not have access to sufficient internal resources to finance desired capital
purchases, but can borrow the difference from banks at a cost. Entrepreneurs are however risky:
each faces an idiosyncratic shock that changes the value of the firm after decisions have been
made. All entrepreneurs have the same a priori expectation about the shock, but the posterior
distribution may be quite different. If hit by a severe shock, the value of assets collapse, and
the entrepreneur may be forced to declare bankruptcy, handing over the value of the firm to the
bank. Contrarily, if hit by a propitious shock, the value of entrepreneur’s assets rise, and net
worth increases as a result.
Banks operate in a perfectly competitive environment, making zero-profits at all times.
They are pure financial intermediaries, with the sole mission of borrowing funds from financially
unconstrained households, lending them to entrepreneurs. When lending to an entrepreneur
that goes bankrupt, the bank must pay monitoring costs to be able to recover the value of the
firm. Since entrepreneurs are risky, so are the loans of banks, who therefore charge a spread
over the risk free interest rate to cover for the losses incurred in the mass of entrepreneurs that
declares bankruptcy. The existence of identical a priori expectations on the idiosyncratic shock
implies that the credit spread is identical to all entrepreneurs. Albeit the riskness of individual
loans, the aggregate portfolio of banks is risk free, since each bank recovers through the credit
spread what is lost to the bankrupt entrepreneurs. Households loans are therefore risk free, and
thus they lend to banks at the risk free interest rate.
The financial accelerator mechanism magnifies economic fluctuations, by creating an extra
channel through which shocks are transmitted and propagated to the real economy. Any shock
decreasing aggregate demand also negatively impacts the price of capital, therefore increasing
the number of entrepreneurs in financial distress and reducing the value of net worth for those
that survive. As risk increases, so does the credit spread. With lower internal resources and
more expensive credit, entrepreneurs acquire less physical capital. Investment is reduced and
credit slumps, magnifying the fall in output and employment. Therefore, the model implies,
realistically, a countercyclical credit spread, and procyclical investment and stock market value,
in addition to procyclical consumption, inflation, and employment already featuring the model
without a financial sector. In the model version with no financial accelerator, capital goods
producers sell capital directly to manufacturers.
The government buys from distributors government consumption goods, and performs in-
5
come transfers across households. These activities are financed through direct and indirect
taxation—namely labor income, capital income, and consumption taxes—and also through
transfers from the monetary union. The government issues one-period bonds to finance excess expenditures, paying an interest rate on public debt that is assumed in general to be equal
to the monetary union’s financing cost. However, a differential is allowed to exist if an exogenous
risk premium on domestic bonds emerges in financial markets, for instance due to credibility
issues or to a financial turbulence. To ensure that debt follows an non-explosive path, we impose
a fiscal rule linking the government-debt-to-GDP ratio to a pre-determined target. Hence, deviations from that target are followed by tax adjustments or changes in public expenditures, that
restore the long-run government debt to a sustainable path. Government consumption goods
have no particular use, nor they provide any direct welfare benefit.
The foreign economy in PESSOA corresponds to the rest of the monetary union. The domestic economy interacts with the foreign economy via goods and financial markets. In the
goods market, domestic distributors buy imported goods from abroad to be used in the assemblage stage. Likewise for foreign distributors, who buy export goods to domestic distributors
for the same purpose. In the financial market, financially unconstrained domestic households
can trade assets with the foreign economy to smooth out consumption over time. Contrary to
most DSGE models that feature infinitely lived households, the Blanchard-Yaari framework of
PESSOA allows to pin down the net foreign asset position endogenously (refer to Frenkel and
Razin, 1996 and Harrison et al., 2005 for additional details).
Some parameters in PESSOA—mostly related with key economic outcomes such as labor
and capital shares or the labor allocation between the tradables and the non-tradables—are
calibrated to match Portuguese data from the national accounts, while others rely on the literature mainstream and on estimates for Portugal whenever available. Adding financial frictions
to the model does not change the model’s parameters or key ratios, with some notable exceptions. In particular, the net foreign position of the economy deteriorates when financial frictions
are incorporated. As households desire to hold exactly the same amount of assets (relative to
GDP) in both versions of the model, corporate bonds—which are introduced along with financial frictions—draw funds from abroad on a one-to one basis. Finally, physical capital stock is
also lower when financial frictions are incorporated, motivated by the existence of a steady-state
external finance premium.
3
The model
The model is set in discrete time. The domestic and the foreign economy experience both a
constant technological growth rate of g = Tt /Tt−1 ∀t, where Tt is the level of labor augmenting
technology. We follow the convention of representing time as subscripts, and any other identifier,
such as sector identifiers, as superscripts. Time subscripts refer to the period when the quantity
is used, and may differ from the period when production or consumption decisions take place.
The model’s real variables, say Xt , endowed with a steady-state growth rate of g, are converted
to stationary form after dividing by Tt . We denote these new variables by X̌t = Xt /Tt . The
real counterpart of nominal variables, say Zt , are given by zt = Zt /Pt , and their stationary form
by žt = Zt /(Pt Tt ), where Pt is the after tax price level of the consumption good, selected as
6
X
the numéraire of the economy. We use the notation pX
t = Pt /Pt to refer to X’s relative price
vis-à-vis the consumption good.
3.1
Households: general features
This section clarifies the overlapping generations scheme, the utility function, and the labor
productivity profile of households.
3.1.1
Overlapping generations scheme
All households evolve according to the overlapping generations scheme à la Blanchard (1985).
They are subject to stochastic finite lifetimes and face an instant probability of death of 1 − θ ∈
[0, 1], independent of age. The overall size of the population N is constant, implying that
N (1 − θ) households die in each period t and that the same number are born.
The household’s probability of dying at period t equals the probability of staying alive until
t − 1 times the probability of dying at t, θt−1 (1 − θ). The average life expectancy at any time
is constant at (1 − θ)−1
1(1 − θ)θ0 + 2(1 − θ)θ1 + 3(1 − θ)θ2 + . . . = (1 − θ)
∞
X
tθt−1 = (1 − θ)
t=1
∞
X
t=1
−1
= (1 − θ)(1 − θ)
θt−1
∞
X
θt−1
t=1
−1
(1 − θ)
= (1 − θ)−1
Instead of biological death, 1 − θ can also be interpreted as the relevant economic horizon
behind agents’ decisions, i.e. the probability of “economic death” or an indicator of the degree
of “myopia” (Blanchard, 1985; Frenkel and Razin, 1996; Harrison et al., 2005; Bayoumi and
Sgherri, 2006). In other words, the future can be seen as a period of lower economic relevance.
In this case, (1 − θ)−1 is interpreted as the “average planning horizon.”
Two types of households coexist in each and every period: asset holders, identified as type-A
households and assumed to represent N (1 − ψ) of total population; and “hand-to-mouth households,” termed type-B households and assumed to represent N ψ of total population. Type-A
households smooth out their lifetime consumption by trading assets, whilst type-B households
do not have access to financial markets and therefore consume all their income in each and every
period (they are hand-to-mouth households à la Galı́, López-Salido, and Vallés, 2007).
At time t, the new cohorts being born and the existing cohorts that die are consistent with
total population N . Thus, the size of a new generation of type-A households is N (1 − ψ)(1 − θ),
and of type-B households is N ψ(1 − θ). At age a, the size of these cohorts is N (1 − ψ)(1 − θ)θa
in the case of type-A households and N ψ(1 − θ)θa in the case of type-B households, implying
an aggregate figure of N (1 − θ)θa . At period t, total population is given by all those that were
born at t, t − 1, t − 2, . . . and are still alive
0
1
2
N (1 − θ)θ + N (1 − θ)θ + N (1 − θ)θ + . . . = N (1 − θ)
∞
X
s=0
7
θs = N
Each cohort is assumed to be large enough so that 1 − θ is also the rate at which the generation
size decreases through time. Thus, although each individual is uncertain about the time of
death, the size of a cohort declines deterministically through time.
3.1.2
Utility function
A representative household of type H ∈ {A, B} with age a derives utility from consumption,
H , and leisure, 1 − LH . The term LH stands for hours worked as a fraction of total time
Ca,t
a,t
a,t
endowment. The expected lifetime utility function is
Et
∞
X
H
(βθ)s Ua+s,t+s
(1)
s=0
where Et is the expectation operator and 0 ≤ β ≤ 1 stands for the standard discount factor. Instantaneous utility is given by the following Constant Relative Risk Aversion (CRRA)
specification
H
Ua,t

1−γ
!η H
H
1  Ca,t
1−η H 
=
(1 − LH
a,t )
1−γ
Hab H
t
where γ > 0 is the risk aversion coefficient, η H ∈ [0, 1] is a distribution parameter, and Hab H
t
stands for type-H households external habits. Aggregation across generations is feasible under
multiplicative habits; however, these generate lower consumption persistence as compared to
additive habits.
In equation (1), households attach an extra value to the present and over-discount the future,
as they account for the probability θ of being dead (Harrison et al., 2005), in addition to
the standard discount factor β. Households do not draw utility from holding money, which
is consistent with a cashless limit (Woodford, 2003). Moreover, they ignore intergenerational
transfers that could change utility levels of not-yet-born cohorts, i.e. there is no bequest motive.
H is a bundle of different varieties of consumption
Households aggregate consumption, Ca,t
H (c), obtained according to the following Constant Elasticity of Substitution (CES)
goods c, Ca,t
aggregator
H
Ca,t
=
Z
1
σ C −1
H
Ca,t
(c) σC dc
σC
σ C −1
0
where σtC ≥ 0 stands for the elasticity of substitution between different varieties of the consumption good. Consumption goods are bought from the final goods producers of consumption
H (c), is obtained by minimizing congoods—the distributors. The demand for each variety, Ca,t
H
sumption expenditure subject to the definition of Ca,t
Z
min
H (c)
Ca,t
0
1
H
PtC (c)Ca,t
(c)dc
s.t.
H
Ca,t
=
Z
0
8
1
σ C −1
H
Ca,t
(c) σC dc
σC
σ C −1
where PtC (c) is the price of variety c. The solution steps are presented in Box 1. This problem
implies that the aggregate price level before consumption taxes PtC is a combination of individual
prices
PtC
1
Z
=
C
PtC (c)1−σ dc
1
1−σ C
0
In addition, the demand for variety c is
H
Ca,t
(c)
=
PtC (c)
PtC
−σC
H
Ca,t
H ,
Consumption habits are a function of lagged type-H households aggregate consumption, Ct−1
and thus are not affected by current decisions or by the age of each cohort. In addition, they
are measured relative to the respective population size. More specifically
Hab A
t =
A
Ct−1
N (1 − ψ)
Hab B
t =
B
Ct−1
Nψ
!v
!v
where v ∈ [0, 1] parameterizes the degree of habit persistence.
3.1.3
Labor productivity
According to the life-cycle theory, households labor income increases throughout adulthood and
decreases at the retirement age. However, modeling such a profile makes aggregation unfeasible.
The life-cycle income profile used herein is based on a simpler pattern according to which labor
productivity declines over lifetime at a constant rate χ (Blanchard, 1985). More precisely, labor
productivity of a type-H household with age a is
Φa = kχa , ∀t
where χ ∈ [0, 1] is the rate of decay. This function is depicted in Figure 2.
Population average productivity is normalized to one, implying
(1 − θ)
∞
X
θa Φa = 1 ⇔ (1 − θ)k (1 − θχ)−1 = 1 ⇔ k =
a=0
1 − θχ
1−θ
Since the productivity profile is the same for both household types within the same cohort and
the population structure is constant through time, the average productivity of type-A and of
type-B households must also equal one. Recall that, at each period t, N (1 − ψ)(1 − θ) type-A
households are born and N (1 − ψ)(1 − θ)θa households of the same type with age a still live.
For the case of type-B households, the equivalent expressions are N ψ(1 − θ) and N ψ(1 − θ)θa ,
9
Box 1: The standard cost minimization problem.
A common problem in PESSOA is a minimization problem with the following structure
1
Z
PtC (c)Ca,t (c)dc
min
Ca,t (c)
1
Z
s.t.
Ca,t =
0
σ C −1
Ca,t (c) σC dc
σC
σ C −1
0
where PtC (c) is the price of variety c and is taken as given by households. Superscript H ∈ {A, B} has
been omitted for simplicity, as the problem is identical for both household types. Parameter σ C ≥ 0 is
the elasticity of substitution between varieties. The Lagrangian for this problem is
" Z
1
Z
PtC (c)Ca,t (c)dc − λt
L(·) =
0
1
σ C −1
Ca,t (c) σC dc
σC
σ C −1
#
− Ca,t
0
where λt is the Lagrange multiplier, which equals the marginal cost of acquiring one extra unit of Ca,t ,
i.e. λt = PtC . The following steps solve the problem in the context of households, but generalizing to
other agents is straightforward. Making ∂L(·)/∂Ca,t (c) = 0 yields
PtC (c)
−
σC
σC − 1
" Z
1
PtC
⇔ PtC (c) − PtC
σ C −1
Ca,t (c) σC dc
1
Z
1
σ C −1
0
σ C −1
Ca,t (c) σC dc
σC
σ C −1
1
σC
#
C
σ −1 −1
σC − 1
σC
C
(c)
=0⇔
a,t
σC
− 1C
σ
Ca,t (c)
=0⇔
0
⇔ PtC (c) − PtC · Ca,t
⇔ Ca,t (c) =
PtC (c)
PtC
1
σC
Ca,t (c)
− 1C
σ
=0⇔
−σC
Ca,t
(1.1)
The aggregate price index PtC is obtained after replacing equation (1.1) in the definition of Ca,t
"Z
1
Ca,t =
0
PtC (c)
PtC
!σ
−σC
C
−1
σC
Ca,t
#
dc
σC
σ C −1
⇔
C
"Z
⇔ Ca,t =
1
# Cσ
σC −1
σ −1
σC
−σ C
C
Pt (c)
dc
0
C
⇔ (PtC )−σ =
Z
1
PtC (c)
1−σ
C
dc
σC
σ C −1
Ca,t
⇔
(PtC )−σC
⇔
0
⇔
PtC
Z
=
1
C
PtC (c)1−σ dc
1
1−σ C
(1.2)
0
Equations (1.1) and (1.2) will be recalled throughout this technical appendix whenever a particular
agent—household, firm or union—faces an equivalent cost minimization problem.
respectively. The average productivity of type-A households is therefore
P
∞
a
X
N (1 − ψ)(1 − θ) ∞
a=0 θ Φa
= 1 ⇔ N (1 − ψ)(1 − θ)
θa Φa = N (1 − ψ)
N (1 − ψ)
a=0
10
(2)
Figure 2: Labor productivity of age group a
Φa
2
1.5
1
0.5
Φa = κχa
0
0
100
200
300
400
a
And for type-B households
P
∞
a
X
N ψ(1 − θ) ∞
a=0 θ Φa
= 1 ⇔ N ψ(1 − θ)
θ a Φa = N ψ
Nψ
(3)
a=0
Equations (2) and (3) imply that labor productivity of type-A and type-B households equals
the total number of individuals, respectively N (1 − ψ) and N ψ.
3.2
Households with access to financial markets (type-A households)
This section presents and solves the maximization problem of Overlapping Generation (type-A)
households, i.e. those with access to financial markets.
3.2.1
Budget constraint and maximization problem
On the expenditure side, a type-A household with age a buys a consumption bundle worth
A , where P = (1 + τ C )P C is the after-tax price of the consumption bundle, P C is the price
Pt Ca,t
t
t
t
t
paid to distributors, and τtC is an ad-valorem consumption tax. The after tax price, Pt , is taken
as numéraire, and thus the relative price of final consumption goods is
pCt =
1
1 + τtC
(4)
where pCt = PtC /Pt . This equation establishes the link between the price paid by households for
the final good and the price received by consumption goods distributors. In the steady state
pC =
1
1 + τC
(5)
Type-A households may buy domestic and foreign financial assets. There are two types of
11
domestic bonds: those issued by the national Government, Ba,t , and those issued by Banks,
which act as financial intermediaries by lending to entrepreneurs operating in the tradable
T , and in the nontradable sector, B N . For brevity, let B
ba,t ≡ Ba,t + Ba,t T + B N
sector, Ba,t
a,t
a,t
denote all domestically issued bonds held by type-A households with age a at time t. Foreign
∗ . Domestic government bonds cannot be held by foreigners, i.e. there
bonds are denoted by Ba,t
is a complete home bias in domestic government debt. Thus, markets are incomplete in this
model.
∗ could in principle
ba,t and Ba,t
All bonds are denominated in local currency, implying that B
be denominated in distinct domestic and foreign currency, respectively. In this case, foreign
assets could be converted into domestic currency through the indirect quotation of the nominal
∗ in domestic
exchange rate vis-á-vis the rest of the world, εt (and foreign assets would worth εt Ba,t
currency). In the special case of a Monetary Union, however, the nominal exchange rate equals
ba,t and B ∗ . Domestic bonds and
one, and there is no difference in the unit of account of B
a,t
foreign bonds held between period t and period t + 1 pay a gross nominal interest rate of it and
i∗t , respectively, at the beginning of period t + 1. The interest rate i is allowed to differ from
i∗ given that type-A households demand a risk premium Ψ for holding domestic assets. It is
common to set a steady-state value of Ψ = 1 in the case of a Monetary Union, which implies no
deviation between domestic and foreign interest rates.
A type-A household with age a has three sources of income: labor, capital, and interest.
The latter is associated with financial wealth derived from bond holdings.
Households supply labor services to labor unions, paying taxes to the government on their
labor income, and receiving an after-tax amount of (1 − τtL )Wt Φa LA
a,t , where Wt Φa is the
productivity-adjusted wage rate and τtL is the employees labor income tax. The wage rate
Wt is taken as given.
There exists a competitive life insurance company which guarantees that households do not
implement intergenerational wealth transfers. Given that there is individual uncertainty about
the time of death, in the absence of this insurance company households could die leaving either
unintended positive bequests, if they die as creditors, or unintended negative bequests, if they
die indebted. The gap between individual uncertainty and no aggregate uncertainty is filled by
the insurance company, who collects the wealth of 1 − θ agents who did not survive between
two consecutive periods, and distributes it to the θ agents who survived. More precisely, each
household with age a receives an additional (1 − θ)/θ for each unit of financial wealth. Thus,
financial wealth is multiplied by a factor 1 + (1 − θ)/θ = 1/θ > 1, becoming
1
∗
ba−1,t−1 + i∗t−1 Ψεt Ba−1,t−1
it−1 B
θ
The assumption of a large population turnover ensures that income receipts and the payout of
the insurance company match in every period.
Another source of revenue of type-A households is the remuneration for services in the
N
T
T
bankruptcy monitoring of firms. In real terms this equals rbr a,t = pN
t rbr a,t + pt rbr a,t , where
T
rbr N
a,t and rbr a,t are the remuneration for services in the nontradable and tradable sectors,
T
respectively, and pN
t and pt are the relative prices, measured against the numérarie of the
economy, Pt .
12
Type-A households also receive dividends from firm/union i operating in sector x, denoted as
A,x
Da,t
(i),
x ∈ {T , N , U, C, G, I, X , K, EP}. In the intermediate goods sector, dividends originate
from tradable (T ) and nontradable (N ) manufacturers. In the final goods sector, dividends
originate from distributors of private consumption (C), government consumption (G), investment
(I) and export (X ) goods. The remaining dividends originate from capital producers (K), unions
(U) and, in the financial accelerator context, from entrepreneurs (EP). Finally, households
A
receive lump-sum transfers from the government, TRG A
a,t , and from abroad, TRX a,t . The latter
is measured in local currency, implying that the amount effectively received is εt TRX A
a,t .
The nominal budget constraint embodying that expenditures cannot exceed revenues is
i
1h
A
∗
∗
ba,t + εt Ba,t
ba−1,t−1 + i∗t−1 Ψεt Ba−1,t−1
Pt Ca,t
+B
≤ it−1 B
θ
+ (1 − τtL )Wt Φa LA
a,t + Pt rbr a,t
X
+
x∈{T ,N ,U ,C,G
I,X ,K,EP}
Z
0
1
A,x
A
Da,t
(i)di + TRG A
a,t + εt TRX a,t
(6)
The optimization problem of a type-A household consists in maximizing expected lifetime utility
A
∗
T
N
∞
(1) with respect to {Ca+s,t+s
, LA
a+s,t+s , Ba+s,t+s , Ba+s,t+s , Ba+s,t+s , Ba+s,t+s }s=0 , subject to (6).
The problem and the first-order conditions are presented in detail in Box 2.
Combining equations (2.2) and (2.3) from Box 2, we obtain the consumption/labor supply
choice
A
Ca,t
=
⇔ 1 − LA
a,t =
ηA
1 − τtL wt Φa 1 − LA
a,t ⇔
A
1−η
A
Ca,t
1 − ηA
ηA
1 − τtL wt Φa
(7)
where wt = Wt /Pt is the real wage.
Aggregate consumption of type-A households at time t is
CtA = N (1 − ψ)(1 − θ)
∞
X
A
θa Ca,t
(8)
θa Φa LA
a,t
(9)
a=0
and effective labor supply is
LA
t = N (1 − ψ)(1 − θ)
∞
X
a=0
Aggregation takes into account the size of each cohort at the time of birth, N (1 − ψ)(1 − θ),
and the size of the remaining generations, N (1 − ψ)(1 − θ)θa . In equation (9), individual labor
supply is weighted by labor productivity to obtain the effective labor supply. Using (7) in (8)
CtA
= N (1 − ψ)(1 − θ)
∞
X
a=0
θ
a
ηA
1 − τtL wt Φa 1 − LA
a,t
A
1−η
13
Box 2: Type-A households maximization problem.
Type-A households maximize expected lifetime utility (1) with respect to
A
∞
∗
T
N
Ca+s,t+s , LA
a+s,t+s , Ba+s,t+s , Ba+s,t+s , Ba+s,t+s , Ba+s,t+s s=0
subject to (6). The Lagrangian for the maximization problem is


1−γ
ηA



∞
 1 
A
X

Ca+s,t+s

A
1−η A 

L(·) = Et
(1
−
L
)
(βθ)s


v
a+s,t+s
A

Ct+s−1

1−γ 

s=0

N (1−ψ)

+
"
λA
a+s,t+s 1 T
N
it+s−1 (Ba+s−1,t+s−1 + Ba+s−1,t+s−1
+ Ba+s−1,t+s−1
)
Pt+s
θ
+
∗
L
+ (1 − τt+s
i∗t+s−1 Ψεt+s Ba+s−1,t+s−1
)Wt+s Φa+s LA
a+s,t+s
+
Pt+s rbr a+s,t+s +
X
x∈{T ,N ,U ,C,G
I,X ,K,EP}
−
1
Z
A,x
A
(i)di + TRG A
Da+s,t+s
a+s,t+s + εt TRX a+s,t+s
0
A
T
N
∗
Pt+s Ca+s,t+s
− (Ba+s,t+s + Ba+s,t+s
+ Ba+s,t+s
) − εt+s Ba+s,t+s


#






A
where λA
a+s,t+s = Λa+s,t+s Pt+s is the marginal utility of an extra unit of the consumption bundle and
A
Λa+s,t+s the original lagrange multiplier.
Let
ηA

A

Ca,t

A
A
uA
v 
a,t Ca,t , La,t =  C A
1−η
(1 − LA
a,t )
A
(2.1)
t−1
N (1−ψ)
A
and notice that the first derivatives with respect to Ca,t
and LA
a,t are
∂uA
a,t
1−η A
= (1 − LA
a,t )
A
∂Ca,t

= − 
A
Ct−1
N (1 − ψ)
−vηA
A η
η A (Ca,t
)
A
−1
= uA
a,t
ηA
A
Ca,t
ηA

∂uA
a,t
∂LA
a,t
A
Ca,t
A
Ct−1
N (1−ψ)

v 
1 − ηA
−η A
1 − η A (1 − LA
= −uA
a,t )
a,t
1 − LA
a,t
The first-order conditions are as follows.
1. Optimal consumption
1−γ
∂uA
η A (uA
∂L(·)
a,t
a,t )
A
A −γ
A
=
0
⇔
λ
=
(u
)
⇔
λ
=
a,t
a,t
a,t
A
A
A
∂Ca,t
∂Ca,t
Ca,t
14
(2.2)
2. Labor supply
λA
∂uA
∂L(·)
a,t
a,t
−γ
=0⇔
(uA
⇔
(1 − τtL )Wt Φa =
a,t )
A
Pt
∂La,t
∂LA
a,t
L
A
⇔ λA
a,t (1 − τt )wt Φa = (1 − η )
⇔ λA
a,t =
1−γ
(uA
a,t )
⇔
1 − LA
a,t
1−γ
(uA
1 − ηA
a,t )
(1 − τtL )wt Φa 1 − LA
a,t
(2.3)
where wt = Wt /Pt is the real wage.
3. Optimal domestic government bond holdings
λA
λA
∂L(·)
it
a,t
a+1,t+1 it
A
=0⇔
= βθEt
⇔ λA
a,t = βEt λa+1,t+1
∂Ba,t
Pt
Pt+1 θ
πt+1
(2.4)
where πt+1 = Pt+1 /Pt corresponds to the (gross) inflation rate between period t and period t + 1.
4. Optimal foreign bond holdings
∗
λA
λA
∂L(·)
a,t
a+1,t+1 it Ψεt+1
=
0
⇔
=0⇔
ε
=
βθE
t
t
∗
∂Ba,t
Pt
Pt+1
θ
A
⇔ λA
a,t = βEt λa+1,t+1
it ∗ Ψ εt+1
εt
(2.5)
πt+1
5. Optimal private domestic bond holdings
T
The optimal conditions associated with ∂L(·)/∂Ba,t
∂L(·)/∂Ba,t = 0 and are therefore omitted.
N
= ∂L(·)/∂Ba,t
∞
∞
X
X
ηA
L
a
=
(1
−
τ
)w
N
(1
−
ψ)(1
−
θ)
θ
Φ
−
θ a Φ a LA
t
a
t
a,t
1 − ηA
a=0
=
ηA
1 − ηA
= 0 are identical to
!
a=0
(1 − τtL )wt [N (1 − ψ) − LA
t ]
The simplification in the last step is due to (2) and (9). The above equation can be rewritten as
CtA
ηA
=
(1 − τtL )wt
A
1 − ηA
N (1 − ψ) − Lt
(10)
which clarifies how changes in real wages wt or in the labor income tax τtL affect the consumption/labor supply choice and confirms the distortionary nature of labor taxes. In stationary
form (i.e. after rescaling trend variables by technology) this equation becomes
A
Čt
ηA
=
(1 − τtL )w̌t
A
1 − ηA
N (1 − ψ) − Lt
(11)
15
In the steady state
ηA
Č A
=
(1 − τ L )w̌
A
N (1 − ψ) − L
1 − ηA
(12)
Combining equations (2.4) and (2.5) results in the no-arbitrage condition in financial markets—
the uncovered interest rate parity
it = i∗t Ψ
εt+1
εt
(13)
where we ignored the expected value operator. This equation is already in stationary form, since
none of the variables has a deterministic trend. In the steady state
i = i∗ Ψ
3.2.2
(14)
Aggregate consumption and aggregate wealth
A key behavioral equation of type-A households is the aggregate consumption equation, linking
optimal aggregate consumption to wealth. To obtain this equation, one needs to establish
the optimal link between consumption and wealth for each generation a, and then aggregate
across generations. It is useful to break this exercise in five steps: (I) Euler equations for
each generation a; (II) the no-Ponzi game condition and the subjective discount factor ; (III)
human wealth, stemming from labor-based and from capital-based income; (IV) financial wealth,
associated with bond holdings; and (V) marginal propensity to consume out of wealth.
(I) Euler equations for each generation a.
An Euler equation for each generation a is
obtained by replacing (2.3) in (2.4) (we ignore the expected value operator for simplicity)
(ua,t )1−γ
(ua+1,t+1 )1−γ it
1 − ηA
1 − ηA
=
β
L )w
A
(1 − τtL )wt Φa 1 − LA
(1 − τt+1
t+1 Φa+1 1 − La+1,t+1 πt+1
a,t
(15)
where πt+1 = Pt+1 /Pt stands for the gross inflation rate of final consumer goods and ua,t is
defined in (2.1). This equation can be expressed as
A
A
Ca+1,t+1
= jt Ca,t
(16)
where the new object
jt =
CtA
A
Ct−1
!vηA
1− γ1
L
1 − τt+1
wt+1
L
1 − τt wt
!
1−η A
1− γ1
β
16
it
πt+1
1
γ
χ
1−η A
1− γ1
(17)
is introduced to simplify notation. The derivation is presented in Box 3. Notice that equation
(17) is not generation specific, a fact that allows aggregation across generations. In stationary
form jt becomes
ČtA g
A
Čt−1
jt =
!vηA
1− γ1
L
1 − τt+1
w̌t+1 g
L
w̌t
1 − τt
1−η A
1− γ1
β
it
πt+1
γ1
χ
1−η A
1− γ1
(18)
where g is the deterministic growth rate of the (labor-augmenting) technological progress. In
the steady state
j=g
1+η A (v−1)
1− γ1
γ1
i
β
χ
π
1−η A
1− γ1
(19)
(II) The no-Ponzi game condition and the subjective discount factor. The following
no-Ponzi game condition is assumed to hold
ba+s,t+s + εt+s B ∗
lim Et R̃t,s [B
a+s,t+s ] = 0, ∀a
s→∞
where R̃t,s is the nominal subjective discount factor
R̃t,s =

Qs
for s > 0
1
for s = 0
θ
l=1 it+l−1
(20)
This condition states that households cannot engage in Ponzi schemes, which would allow them
to attained infinite consumption by rolling-over debt over lifetime (recall that the no-Ponzi game
condition is a constraint that prevents overaccumulation of debt, whereas the transversality
condition is an optimality condition that rules out overaccumulation of wealth). Let
rt =
it
(21)
πt+1
denote the (gross) real interest rate. In the steady state
r=
i
π
(22)
Equation (20) for s > 0 can thus be rewritten as
R̃t,s =
s Y
l=1
θ
rt+l−1
−1
(πt+l )
= r̃t,s
s
Y
(πt+l )−1
l=1
17
(23)
Box 3: Generation a Euler equation.
A
A
To derive Ca+1,t+1
= jt Ca,t
, notice that equation (15) can be rewritten as
L
(1 − τt+1
)wt+1 Φa+1 πt+1 1
=
it β
(1 − τtL )wt Φa
ua+1,t+1
ua,t
1−γ
1 − LA
a,t
1 − LA
a+1,t+1
(3.1)
Using (2.1) and (7), the right-hand side simplifies to

η A
A
Ca+1,t+1
(1 −
 (CtA /N (1−ψ))v

 
CA
η A
a,t
v
A /N (1−ψ)
(Ct−1
)
=
A
A
Ca+1,t+1
/Ca,t
v
A
CtA /Ct−1
=
A
A
Ca+1,t+1
/Ca,t
v
A
CtA /Ct−1
=
A
A
Ca+1,t+1
/Ca,t
v
A
CtA /Ct−1
1−η A
LA
a+1,t+1 )
1−η A
(1 − LA
a,t )
!ηA (1−γ)
!ηA (1−γ)
1 − LA
a+1,t+1
1 − LA
a,t



!ηA (1−γ)
1−γ
1 − LA
a,t
1 − LA
a+1,t+1




!(1−ηA )(1−γ)−1
A
Ca+1,t+1
1−η A
η A (1−τ L )wt+1 Φa+1
t+1
A
Ca,t
1−η A
η A (1−τtL )wt Φa
(1−ηA )(1−γ)−1


A
Ca+1,t+1
(1 − τtL )wt Φa
A
L )w
Ca,t
(1 − τt+1
t+1 Φa+1
!(1−ηA )(1−γ)−1
Using this result, equation (3.1) can be expanded to
(1
(1−ηA )(1−γ)
L
− τt+1
)wt+1 Φa+1
(1 − τtL )wt Φa
⇔
L
(1 − τt+1
)wt+1 Φa+1
(1 − τtL )wt Φa
⇔
A
Ca+1,t+1
A
Ca,t
⇔
A
Ca+1,t+1
⇔
A
Ca+1,t+1
!−γ
=
CtA
A
Ct−1
πt+1
=
β · it
πt+1
=
β · it
vηA (1−γ) CtA
A
Ct−1
−vηA (1−γ)
L
(1 − τt+1
)wt+1 Φa+1
(1 − τtL )wt Φa
CtA
A
Ct−1
vηA (1− γ1 ) L
)wt+1 Φa+1
(1 − τt+1
(1 − τtL )wt Φa
CtA
A
Ct−1
vηA (1− γ1 ) L
(1 − τt+1
)wt+1
L
(1 − τt )wt
=
=
(1−ηA )(1−γ)
ηA (1−γ)
A
Ca+1,t+1
A
 Ca,t

 C A v 
t
A
Ct−1

The last step follows from Φa = kχa . Letting
A vηA 1− γ1 L
(1 − τt+1
)wt+1
Ct
jt =
A
Ct−1
(1 − τtL )wt
1−η A
18
A
Ca+1,t+1
A
Ca,t
(1−ηA )(1−γ)
!(1−ηA )(1−γ)−1
⇔
!−γ
⇔
πt+1
⇔
β · it
(1−ηA )(1− γ1 ) γ1
it
A
β
Ca,t
⇔
πt+1
(1−ηA )(1− γ1 ) γ1
A
1
it
A
β
χ(1−η )(1− γ ) Ca,t
πt+1
1− γ1
β
equation (16) follows immediately.
A
Ca+1,t+1
A
Ca,t
it
πt+1
γ1
χ
1−η A
1− γ1
where r̃t,s is the real subjective discount factor
r̃t,s =

Qs
for s > 0
1
for s = 0
θ
l=1 rt+l−1
(24)
For future reference, notice that, for s > 0
r̃t,s+1 =
s
θ Y θ
rt
rt+l
l=1
and
r̃t+1,s
s
Y
θ
=
rt+l
l=1
implying that
r̃t,s+1 =
θ
r̃t+1,s
rt
(25)
(III) Human wealth. Human wealth stems from labor and capital. Generation a’s contemporaneous nominal income at t is
Inc a,t = (1 −
X
τtL )Wt Φa +
1
Z
x∈{T ,N ,U ,C, 0
G,I,X ,K,EP}
A,x
Da,t
(i)di
+
TRG A
a,t
+
εt TRX A
a,t
+ Pt rbr a,t
The labor-based income is the value of households entire time endowment. Notice that households use a fraction 1 − LA
a,t of labor-based income to buy leisure, and the remaining to buy
consumption goods. Capital-based income comprises dividends, transfers, and the remuneration
from services in the bankruptcy monitoring of firms.
Lifetime human wealth of a household with age a in period t, HW a,t , is the present discounted
value of all future incomes. This can be split in a labor component and a capital component
HW a,t =
∞
X
R̃t,s Inc a+s,t+s =
∞
X
L
R̃t,s (1 − τt+s
)Wt+s Φa+s
s=0
s=0
|
+
∞
X
s=0
R̃t,s
XZ
x
1
A,x
Da+s,t+s
(i)di
{z
}
HW L
a,t
+
TRG A
a+s,t+s
+
εt+s TRX A
a+s,t+s
+ Pt+s rbr a+s,t+s
0
|
{z
}
HW K
a,t
Aggregating across generations yields
∞
∞
X
X
K
L
K
a
a
HW t = HW L
+
HW
=
N
(1
−
ψ)(1
−
θ)
θ
HW
+
N
(1
−
ψ)(1
−
θ)
θ
HW
t
t
a,t
a,t
a=0
(26)
a=0
Equation (26) can be expressed in real terms by dividing both sides by Pt , yielding hw t =
K
hw L
t + hw t . Further dividing by technological progress Tt one obtains aggregate lifetime human
19
wealth in stationary form
ˇ t = hw
ˇ L + hw
ˇ K
hw
t
t
(27)
In the steady state
ˇ = hw
ˇ L + hw
ˇ K
hw
(28)
ˇ L and hw
ˇ K . Nominal labor-based
It remains to determine the complete expressions for hw
t
t
aggregate lifetime human wealth simplifies to
HW L
t
= N (1 − ψ)(1 − θ)
∞
X
θa HW L
a,t
a=0
= N (1 − ψ)(1 − θ)
∞
X
θa
a=0
= N (1 − ψ)(1 − θ)
= N (1 − ψ)(1 − θ)
∞
X
∞
X
L
R̃t,s (1 − τt+s
)Wt+s Φa+s
s=0
θa
∞
X
a=0
s=0
X
∞
a
L
R̃t,s (1 − τt+s
)Wt+s kχa+s
a
θ kχ
a=0
= N (1 − ψ)
∞
X
X
∞
L
R̃t,s (1 − τt+s
)Wt+s χs
s=0
L
R̃t,s (1 − τt+s
)Wt+s χs
s=0
= N (1 − ψ)
∞
X
L
(1 − τt+s
)Wt+s χs r̃t,s
s=0
s
Y
(πt+l )−1
(29)
l=1
Equation (29) incorporates the relationship in (2), according to which (1 − θ)
P∞
a=0 θ
a kχa
= 1,
and the link between nominal and real subjective discount factors in (23). Dividing by Pt and
letting hw L
t denote real labor-based aggregate human wealth yields
hw L
t = N (1 − ψ)
∞
X
L
r̃t,s χs (1 − τt+s
)Wt+s
s=0
= N (1 − ψ)
= N (1 − ψ)
∞
X
s=0
∞
X
s
1 Y
(πt+l )−1
Pt
l=1
r̃t,s χs
(1 −
L )W
τt+s
t+s
Pt+s
L
r̃t,s χs (1 − τt+s
)wt+s
(30)
s=0
since Pt+s
hw L
t
Qs
−1
l=1 (πt+l )
= Pt . Equation (30) can be further decomposed as follows
= N (1 − ψ)(1 −
τtL )wt
+ N (1 − ψ)
∞
X
s=1
20
L
)wt+s
r̃t,s χs (1 − τt+s
= N (1 − ψ)(1 −
τtL )wt
+ N (1 − ψ)
∞
X
L
r̃t,s+1 χs+1 (1 − τt+s+1
)wt+s+1
s=0
∞
= N (1 − ψ)(1 −
τtL )wt
θ·χX
L
+ N (1 − ψ)
r̃t+1,s χs (1 − τt+1+s
)wt+1+s
rt
s=0
θ·χ L
hw t+1
= N (1 − ψ)(1 − τtL )wt +
rt
where we used the result in (25) stating that r̃t,s+1 = (θ/rt )r̃t+1,s . This equation has the
following stationary form
ˇ L = N (1 − ψ)(1 − τ L )w̌t + g θ · χ hw
ˇ L
hw
t
t
t+1
rt
(31)
In the steady state
ˇ L = N (1 − ψ)(1 − τ L )w̌ + g θ · χ hw
ˇ L
hw
r
and therefore
L
ˇ L = N (1 − ψ)(1 − τ )w̌
hw
1 − g θ·χ
r
(32)
According to these equations, agents discount future labor-based income streams at a higher
rate when the probability of death or the productivity decay rate are higher.
Nominal capital-based aggregate lifetime human wealth is
HW K
t =N (1 − ψ)(1 − θ)
×
∞
X
∞
X
θa HW K
a,t = N (1 − ψ)(1 − θ)
R̃t,s
x
s=0
θa ×
a=0
a=0
1
X Z
∞
X
A,x
A
Da+s,t+s
(i)di + TRG A
a+s,t+s + εt+s TRX a+s,t+s + Pt+s rbr a+s,t+s
0
All firms within a given sector are identical and thus all pay equal dividends in equilibrium,
R 1 A,x
A,x
(i)di = Da,t
. Assuming that dividends, remuneration payments for the
implying that 0 Da,t
bankruptcy monitoring of firms, and net transfers received by each type-A household are the
A
same for all cohorts, irrespective of their age, and letting DtA , rbr t , TRG A
t and TRX t denote
aggregate figures at time t, the previous expression simplifies to
HW K
t
=
=
∞
X
s=0
∞
X
s=0
R̃t,s
X
A,x
Dt+s
+
TRG A
t+s
+
εt+s TRX A
t+s
+
TRG A
t+s
+ Pt+s rbr t+s
x
r̃t,s
s
Y
l=1
(πt+l )
−1
X
A,x
Dt+s
+
εt+s TRX A
t+s
+ Pt+s rbr t+s
x
where we used the result in (23). Dividing by Pt and letting hw K
t denote real capital-based
21
aggregate human wealth yields
hw K
t
∞
X
=
=
=
r̃t,s
s
Y
(πt+l )−1 X
s=0
l=1
∞
X
1
r̃t,s
X
s=0
Pt+s
∞
X
X
r̃t,s
Qs
l=1 (πt+l )
−1
+
TRG A
t+s
+
εt+s TRX A
t+s
+ Pt+s rbr t+s
x
A,x
Dt+s
+
TRG A
t+s
+
εt+s TRX A
t+s
+ Pt+s rbr t+s
x
dA,x
t+s
+
trg A
t+s
x
s=0
since Pt+s
Pt
A,x
Dt+s
∗
εt+s Pt+s
trx A
+
t+s + rbr t+s
Pt+s
A
A
= Pt . Following our convention, dA,x
t , trg t , and trx t represent respec-
tively the real value of dividends, net transfers from the government, and net transfers from
abroad for type-A households. Net transfers from abroad at t are deflated by the foreign price
Pt∗ . Let
t =
εt Pt∗
Pt
(33)
denote the period t real exchange rate (thus, an increase in t represents a real depreciation).
Real capital-based aggregate lifetime human wealth can be further decomposed as
hw K
t
=
dA,x
t
X
+
trg A
t
+
t trx A
t
+ rbr t +
x
=
X
X
r̃t,s
X
A
dA,x
+ trg A
t
t + t trx t + rbr t +
∞
X
r̃t,s+1
X
x
+
trg A
t+s
+
t+s trx A
t+s
+ rbr t+s
A
A
dA,x
t+s+1 + trg t+s+1 + t+s+1 trx t+s+1 + rbr t+s+1
x
s=0
A
dA,x
+ trg A
t
t + t trx t + rbr t +
dA,x
t+s
x
s=1
x
=
∞
X
θ
hw K
rt t+1
since, from equation (25), r̃t,s+1 = (θ/rt )r̃t+1,s . The previous equation has the following stationary form
ˇ K=
hw
t
X
x∈{T ,N ,U ,C,G
I,X ,K,EP}
θ ˇ K
ˇ
ˇ A
ˇ A
dˇA,x
+ trg
hw
t
t + t trx t + rbr t + g
rt t+1
(34)
In the steady state,
ˇ K=
hw
X
x∈{T ,N ,U ,C,G
I,X ,K,EP}
ˇ + g θ hw
ˇ K
ˇ A + trx
ˇ A + rbr
dˇA,x + trg
r
and therefore
ˇ K=
hw
1
1 − g θr
X
ˇ
ˇ A + trx
ˇ A + rbr
dˇA,x + trg
x∈{T ,N ,U ,C,G
I,X ,K,EP}
22
(35)
Equations (34) and (35) were derived under the deliberate omission of the exact functional forms
for dividends and transfers. In practice they can take several forms. A useful possibility that
can be used is to allow for income transfers from type-A to type-B households.
(IV) Financial wealth. Financial wealth of a type-A household with age a at time t is
FW a,t =
i
1h
ba−1,t−1 + i∗ Ψεt B ∗
it−1 B
t−1
a−1,t−1
θ
Summing up across all type-A households yields nominal aggregate financial wealth
FW t = N (1 − ψ)(1 − θ)
∞
X
a=0
θ
a1
θ
∗
ba−1,t−1 + i∗t−1 Ψεt Ba−1,t−1
it−1 B
∞
∞
X
X
a−1 ∗
∗
a−1 b
θ Ba−1,t−1
θ Ba−1,t−1 + it−1 Ψεt
= N (1 − ψ)(1 − θ) it−1
a=0
a=0
bt−1 + i∗ Ψεt B ∗
= it−1 B
t−1
t−1
(36)
bt−1 and B ∗ are nominal aggregate values of domestic and foreign bond holdings,
where B
t−1
respectively. Aggregation takes into account that the wealth of 1 − θ agents that did not survive
is distributed to the θ agents that did survive. Furthermore, with no heirs and no bequests,
b−1,t−1 = 0. In real terms equation (36) becomes
B
∗
Pt−1
1
1
εt
∗
∗
∗
∗
b
b
fw t =
it−1 bt−1 + it−1 Ψεt
b
=
it−1 bt−1 + it−1 Ψ
t−1 bt−1
πt
Pt−1 t−1
πt
εt−1
bt /Pt and b∗ = B ∗ /P ∗ . The above equation uses the fact Pt = πt Pt−1 and the
where bbt = B
t
t
t
definition of real exchange rate in (33). Employing the equivalence bbt ≡ bt + bTt + bN
t , its
stationary form is
ˇ =
fw
t
1
εt
T
N
∗
∗
it−1 (b̌t−1 + b̌t−1 + b̌t−1 ) + it−1 Ψ
t−1 b̌t−1
g · πt
εt−1
(37)
In the steady state
ˇ =
fw
1 i b̌ + b̌T + b̌N + i∗ Ψb̌∗
g·π
(38)
(V) Marginal propensity to consume out of wealth. The condition that associates consumption with wealth is derived from the nominal budget constraint of a representative OLG
household with age a, given in (6). Rewrite the after-tax wage income of a type-A household as
L
L
A
(1 − τtL )Wt Φa LA
a,t = (1 − τt )Wt Φa − (1 − τt )Wt Φa (1 − La,t )
23
= (1 − τtL )Wt Φa −
(1 − η A )
A
Pt Ca,t
ηA
The simplification in the last step is due to (7). The type-A household budget constraint can
thus be restated as
A
ba,t + εt B ∗ ≤ −
Pt Ca,t
+B
a,t
i
1h
(1 − η A )
A
∗
∗
b
P
C
+
i
B
+
i
Ψε
B
t a,t
t−1 a−1,t−1
t a−1,t−1
t−1
ηA
|θ
{z
}
FW a,t
Z
X
+ (1 − τtL )Wt Φa +
x∈{T ,N ,U ,C,G
I,X ,K,EP}
1
0
|
A,x
A
Da,t
(i)di + TRG A
a,t + εt TRX a,t + Pt rbr a,t
{z
}
Inc a,t
or equivalently
A
A
ba,t + εt B ∗ = − 1 − η Pt C A + FW a,t + Inc a,t ⇔
Pt Ca,t
+B
a,t
a,t
ηA
A
Pt Ca,t
∗
ba,t + εt Ba,t
+B
= FW a,t + Inc a,t
⇔
ηA
(39)
We now need to rewrite (39) in terms of human and financial wealth. This requires successive
forward substitution of this equation and the use of the no-arbitrage condition, yielding
∞
X
R̃t,s
s=0
A
Pt+s Ca+s,t+s
= HW a,t + FW a,t
ηA
(40)
The detailed derivation is presented in Box 4. Using the definition of r̃t,s in (24), the relationship
Q
in (16) according to which Ca+1,t+1 = jt Ca,t , and the fact Pt+s sl=1 (πt+l )−1 = Pt , the left-hand
side of equation (40) can be expressed as
∞
X
s=0
∞ Y
s
∞ Y
s
A
X
X
Pt+s Ca+s,t+s
r̃t,s Pt+s
Pt+s
A
A
R̃t,s
= Ca,t
jt+l−1 R̃t,s A = Ca,t
jt+l−1
A
η
η
πt+l η A
=
s=0 l=1
∞ Y
s
X
r̃t,s
A
Pt Ca,t
jt+l−1 A
η
s=0 l=1
s=0 l=1
A
= Θt Pt Ca,t
where the new variable Θt is defined as
Θt =
∞ Y
s
X
s=0 l=1
jt+l−1
r̃t,s
ηA
Equation (40) thus becomes
A
Θt Pt Ca,t
= HW a,t + FW a,t
(41)
The expression for aggregate consumption at time t as a function of lifetime human wealth and
24
Box 4: Type-A household budget constraint: successive forward substitution.
Let us omit the expected value operator. Writing equation (39) for period t + 1 yields
A
i
Pt+1 Ca+1,t+1
1h b
∗
∗
ba+1,t+1 + εt+1 Ba+1,t+1
+B
it Ba,t + i∗t Ψεt+1 Ba,t
⇔
= Inc a+1,t+1 +
A
η
θ
"
#
A
θ Pt+1 Ca+1,t+1
1 ∗
∗
∗
b
b
⇔ Ba,t =
+ Ba+1,t+1 + εt+1 Ba+1,t+1 − Inca+1,t+1 − it Ψεt+1 Ba,t
it
ηA
θ
(4.1)
Replacing (4.1) in (39) and re-arranging
A
A
i i∗
Pt Ca,t
θ Pt+1 Ca+1,t+1
θ hb
t
∗
∗
∗
+
+
Ψεt+1 Ba,t
+ εt Ba,t
B
+
ε
B
a+1,t+1
t+1
a+1,t+1 −
ηA
it
ηA
it
it
θt
= Inc a,t + Inc a+1,t+1 + FW a,t
it
(4.2)
∗
∗
The no-arbitrage condition in (13) implies that −(i∗t /it )Ψεt+1 Ba,t
+ εt Ba,t
= 0. Writing (39) for t + 2
and inserting into (4.2) yields
"
A
A
A
Pt+2 Ca+2,t+2
Pt Ca,t
θ Pt+1 Ca+1,t+1
θ θ
∗
ba+2,t+2 + εt+2 Ba+2,t+2
− Inc a+2,t+2
+
+
+B
ηA
it
ηA
it it+1
ηA
1 ∗
θ
θ
∗
∗
− it+1 Ψεt+2 Ba+1,t+1 + εt+1 Ba+1,t+1
= Inc a,t + Inc a+1,t+1 + FW a,t
θ
it
it
This simplifies to
A
A
A
Pt Ca,t
θ Pt+1 Ca+1,t+1
θ2 Pt+2 Ca+2,t+2
θ2
∗
b
+
+
+
Ba+2,t+2 + εt+2 Ba+2,t+2
ηA
it
ηA
it it+1
ηA
it it+1
θ
θ2
=Inc a,t + Inc a+1,t+1 +
Inc a+2,t+2 + FW a,t
it
it it+1
(4.3)
where we used the fact
θ2
θ
1 ∗
∗
∗
−
it+1 Ψεt+2 Ba+1,t+1
+ εt+1 Ba+1,t+1
=0
it it+1 θ
it
due to the no-arbitrage condition it+1 = i∗t+1 Ψεt+2 /εt+1 . Using successive forward substitution, one
can express equation (4.3) as
Y
T
∞ Y
s
A
A
X
Pt Ca,t
Pt+s Ca+s,t+s
θ
θ
∗
b
+
+ lim
Ba+T,t+T + εt+T Ba+T,t+T
T →∞
ηA
i
ηA
it+l−1
s=1 l=1 t+l−1
l=1
∞ Y
s
X
θ
= Inc a,t +
Inc a+s,t+s + FW a,t
(4.4)
it+l−1
s=1
l=1
Using the definition of the nominal subjective discount factor in (20) and the no-Ponzi game condition
∗
ba+s,t+s + εt+s Ba+s,t+s
lim R̃t,s [B
]=0
s→∞
equation (4.4) reduces to
∞
X
s=0
R̃t,s
∞
A
X
Pt+s Ca+s,t+s
=
R̃t,s Inc a+s,t+s + FW a,t
ηA
s=0
= HW a,t + FW a,t
25
financial wealth can be obtained by aggregating (41) over different age groups
Θt Pt CtA = HW t + FW t
In real terms, this equation has the following stationary form
ˇ t + fw
ˇ )
ČtA = (Θt )−1 (hw
t
(42)
In the steady state
ˇ + fw
ˇ)
Č A = Θ−1 (hw
(43)
The key behavioral equation of type-A households linking optimal aggregate consumption to
aggregate wealth is therefore
ˇ L + hw
ˇ K + fw
ˇ )
ˇ t + fw
ˇ = mpc (hw
ČtA = mpc t hw
t
t
t
t
t
where mpc t ≡ (Θt )−1 is the time-varying marginal propensity to consume of out of wealth.
Finally, one can write Θt recursively as
∞
Θt =
∞
s
s
XY
XY
r̃t,s
r̃t,s+1
1
1
j
+
=
+
j
jt+1+l−1 A
t
t+l−1
ηA
ηA
ηA
η
s=1 l=1
=
s=0 l=1
∞ s
· jt X Y
1
θ
+
ηA
rt
s=0 l=1
jt+1+l−1
r̃t+1,s
ηA
yielding
Θt =
1
θ · jt
+
Θt+1
ηA
rt
(44)
In the steady state Θt = Θt+1 = Θ and therefore
Θ=
1/η A
1 − (θ · j)/r
(45)
Using (11), aggregate effective labor supply for type-A households is
LA
t
= N (1 − ψ) − (Θt )
−1
ˇ L + hw
ˇ K + fw
ˇ )
· (hw
t
t
t
26
1 − ηA
1
A
η
(1 − τtL )w̌t
3.3
Households with no access to financial markets (type-B households)
This section presents and solves the maximization problem of hand-to-mouth (type-B) households, i.e. those with no access to financial markets.
3.3.1
Budget constraint and maximization problem
On the expenditure side, a type-B household with age a buys a consumption bundle worth
B . These households are hired by labor unions, receiving an after-tax labor income of
Pt Ca,t
B
(1 − τtL )Wt Φa LB
a,t . They also receive transfers from the government, TRG a,t , and from abroad,
B,U
TRX B
a,t , and dividends from unions, Da,t (i). The nominal budget constraint is
B
Pt Ca,t
≤ (1 −
τtL )Wt Φa LB
a,t
Z
+
0
1
B,U
B
Da,t
(i)di + TRG B
a,t + εt TRX a,t
(46)
The optimization problem of a type-B household consists in maximizing expected lifetime utility
B
∞
(1) with respect to {Ca+s,t+s
, LB
a+s,t+s }s=0 , subject to (46). The first-order conditions are similar
to those from a type-A household and are therefore omitted. The consumption/labor supply
choice is similar to (7), with B replacing A
ηB
L
B
1
−
τ
w
Φ
1
−
L
⇔
t
a
t
a,t
1 − ηB
B
Ca,t
1 − ηB
=
ηB
1 − τtL wt Φa
B
Ca,t
=
⇔ 1 − LB
a,t
(47)
Aggregation is also immediate. The counterpart of equation (10) for type-B households is
CtB
ηB
=
(1 − τtL )wt
B
B
1
−
η
N ψ − Lt
In stationary form this equation becomes
ηB
ČtB
=
(1 − τtL )w̌t
1 − ηB
N ψ − LB
t
(48)
In the steady state
Č B
ηB
=
(1 − τ L )w̌
N ψ − LB
1 − ηB
3.3.2
(49)
Aggregate consumption
Type-B households aggregate consumption is solely a function of contemporaneous aggregate
income, since they are unable to undertake any type of intertemporal smoothing. Using the
27
budget constraint in (46), real aggregate consumption of type-B households is
CtB
= N ψ(1 − θ)
∞
X
B
θa Ca,t
a=0
∞
X
"
= N ψ(1 − θ)
θa (1 − τtL )wt Φa LB
a,t +
a=0
∞
X
B
B
θa dB,U
a,t + trg a,t + t trx a,t
#
a=0
B,U
B
= (1 − τtL )wt LB
+ trg B
t + dt
t + t trx t
(50)
The term LB
t corresponds to the effective labor supply
LB
t
= N ψ(1 − θ)
∞
X
θa Φa LB
a,t
a=0
As for type-A, we assume that dividends and net transfers received by each type-B household
are the same for all cohorts, irrespective of age. Equation (50) has the following stationary form
ˇB,U + trg
ˇ B
ˇ B
ČtB = (1 − τtL )w̌t LB
t + dt
t + t trx t
(51)
In the steady state
ˇ B + trx
ˇ B
Č B = (1 − τ L )w̌LB + dˇB,U + trg
(52)
As in the cases of (34) and (35), equations (51) and (52) were obtained under the deliberate
omission of exact functional forms for dividends and transfers. In a more general setup, type-B
households could receive income transfers from type-A households.
Equations (51) and (52) can be further detailed by using the consumption/labor supply
choice.
Plugging equation (47) in the budget constraint, and simplifying (note that
L
L
B
(1 − τtL )Wt Φa LB
a,t = (1 − τt )Wt Φa − (1 − τt )Wt Φa (1 − La,t )), one obtains
B
Ca,t
= η B (1 − τtL )wt + η B
Z
0
1
B
B
dB,U
a,t (i)di + trg a,t + t trx a,t
Aggregating across a and dividing by technological progress yields
B
B
ˇ
ˇ
ČtB = η B N ψ(1 − τtL )w̌t + η B dˇB,U
+
trg
+
trx
t
t
t
t
The term η B captures the fact that type-B households use the extra dividends and transfers to
increase consumption by a fraction η B . The remaining fraction 1 − η B is spent on leisure. Using
(48), aggregate effective labor supply for type-B households follows immediately
LB
t
= Nψ −
ČtB
1 − ηB
1
B
η
(1 − τtL )w̌t
28
3.4
Labor unions
Labor unions hire labor services from households and sell them to manufacturers operating in
the intermediate goods market. Labor unions are perfectly competitive in the input market and
monopolistically competitive in the output market—they charge a markup to manufacturers,
therefore creating a wedge between the wage paid by these firms and the wage received by
households. Unions’ profits are distributed to households in the form of dividends.
More specifically, there exists a continuum h ∈ [0, 1] of labor unions supplying labor to a
continuum j ∈ [0, 1] of manufacturers operating in the tradable sector (T ) and to an identical
continuum of manufacturers operating in the nontradable sector (N ). Intermediate goods sectors are indexed by J ∈ {T , N }. Each labor union supplies a specific variety of labor. Each
manufacturer j operating in sector J demands some quantity of each labor variety from union
h, UtJ (h, j), and aggregate varieties to form an homogeneous labor input, UtJ (j), according to
the following CES specification
Z
UtJ (j) =
1
σ U −1
UtJ (h, j) σU dh
σU
σ U −1
0
where σ U ≥ 0 is the elasticity of substitution between labor varieties.
Manufacturers select the demand for each labor variety h by minimizing the cost of acquiring
each one of them, subject to the fact that they must attain an overall labor quantity of UtJ (j).
Letting Vt (h) denote the wage charged by union h, each manufacturer solves
1
Z
Vt (h)UtJ (h, j)dh
min
UtJ (h,j) 0
s.t.
UtJ (j) =
Z
1
σ U −1
UtJ (h, j) σU dh
σU
σ U −1
0
The Lagrange multiplier of this problem corresponds to the manufacturer marginal cost of
acquiring an extra unit of the labor input, i.e. the Lagrange multiplier equals Vt . The solution
steps are similar to those presented in Box 1, yielding
UtJ (h, j)
=
Vt (h)
Vt
1
Z
Vt =
−σU
1−σ U
Vt (h)
UtJ (j)
(53)
1U
1−σ
dh
0
The demand for labor variety h, Ut (h), is obtained by integrating (53) over j and then summing
across J, yielding
Ut (h) =
Vt (h)
Vt
−σU
Ut
(54)
where Ut is aggregate labor demand.
We obtain a sluggish wage adjustment capturing the short-run dynamics present in the data
29
by imposing quadratic adjustment costs with the form
2

Vt (h)
φ
V
(h)
U
ΓVt (h) =
− 1
Tt Ut  t−1
Vt−1
2
(55)
Vt−2
where φU is a sector specific scaling factor. The level of technology Tt enters as an economy-wide
scaling factor so that costs do not become insignificant over time. Since all labor unions are
identical, they solve the same optimization problem and set the same pricing rule in equilibrium
(see Box 5). Hence Vt (h) = Vt , ∀h. Letting πtV = Vt /Vt−1 denote the (gross) rate of wage
inflation, equation (55) becomes in stationary form
Γ̌Vt =
φU
Ut
2
2
πtV
−
1
V
πt−1
(56)
In the steady state wage inflation is constant and therefore
Γ̌V = 0
(57)
Let v̌t denote the relative price charged by labor unions scaled by productivity, v̌t = Vt /(Pt Tt ).
Scaling by productivity is necessary—which was already the case with w̌t —since the relative
price of labor grows at rate g in the steady state. This contrasts with the remaining relative
prices, namely those charged by intermediate and final goods producers, which are stationary
and not scaled by productivity. Wage inflation can thus be written as
πtV =
Pt Tt
v̌t
v̌t
πt =
πt g
v̌t−1 Pt−1 Tt−1
v̌t−1
or equivalently
v̌t
v̌t−1
=
πtV
πt g
(58)
In the steady-state v̌t = v̌t−1 and hence
πV = π · g
(59)
Equation (59) shows that the steady-state wage inflation π V is determined by the final consumer
price inflation π and by the technological growth rate g. This contrasts with inflation rates of
intermediate and final goods, which are solely a function of the final consumer price inflation.
30
The dividends of labor union h are
DtU (h) = (1 − τtL ) Vt (h) − Wt Ut (h) − Pt ΓVt (h)
(60)
Notice that nominal adjustment costs are measured in terms of the numéraire. In equilibrium
the indexer h can be dropped, since all unions behave identically. Dividing (60) by Pt Tt yields
real dividends in stationary form
L
V
dˇU
t = (1 − τt ) (v̌t − w̌t )Ut − Γ̌t
(61)
In the steady state
dˇU = (1 − τ L )(v̌ − w̌)U
(62)
Labor unions select the wage profile {Vt+s (h)}∞
s=0 that maximizes the present discounted value
of the dividends stream, subject to the constraints imposed by demand and adjustment costs.
This problem is presented and solved in Box 5.
Equation (5.1), which defines unions optimal pricing rule, mapping wages paid to households
wt to wages charged by unions vt , has the following stationary form
φU
σU
w̌t − v̌t = U
σU − 1
σ −1
V
V V
L
π
1 − τt+1
πtV
θ · g Ut+1 πt+1
πt
− 1 t+1
−1 V −
V
πt−1
πt−1
1 − τtL rt Ut
πtV
πtV
(63)
In the steady state wage inflation is constant and thus
v̌ =
σU
w̌
−1
(64)
σU
This equation makes clear that, in the steady state where adjustment costs are zero, the wage
charged by unions is simply a markup σ U /(σ U − 1) over the marginal cost. Outside the steady
state one must add adjustment costs to the pricing rule.
3.5
Firms
This section describes capital goods producers, entrepreneurial firms (also denominated as entrepreneurs), banks, manufacturers, and distributors.
The intermediate good sector is composed by tradable goods and non-tradable goods manufacturers. Each manufacturer combines labor services, rented from labor unions, with capital,
rented from entrepreneurs. Intermediate goods are sold to distributors to be combined with
imported goods, yielding four types of differentiated final goods: private consumption (C), investment goods (I), government consumption (G) and export goods (X ).
31
Box 5: Unions maximization problem.
The labor union selects the wage profile {Vt+s (h)}∞
s=0 that maximizes the present discounted value of
the dividends stream, subject to the constraints imposed by demand in (54) and adjustment costs in
(55). The objective function for the union problem is
L(·) = Et
∞
X
L
R̃t,s (1 − τt+s
)
s=0
×



Vt+s (h) − Wt+s


Vt+s (h)
Vt+s

−σU
Ut+s − Pt+s Tt+s Ut+s
φU 
2
Vt+s (h)
Vt+s−1 (h)
Vt+s−1
Vt+s−2
2 


− 1


Dropping the expected value operator, the first-order condition yields

−σU
U
Vt (h) −σ −1 Ut
∂L(·)
L  Vt (h)
U
= 0 ⇔R̃t,0 (1 − τt )
Ut − σ Vt (h) − Wt
∂Vt (h)
Vt
Vt
Vt

− Pt Tt Ut φU 
Vt (h)
Vt−1 (h)
Vt−1
Vt−2

− 1 
Vt (h)
Vt−1 (h)
Vt−1
Vt−2

L
+ R̃t,1 (1 − τt+1
)Pt+1 Tt+1 Ut+1 φU 



1 
Vt (h)
Vt+1 (h)
Vt (h)
Vt
Vt−1

− 1 
Vt+1 (h)
Vt (h)
Vt
Vt−1


1
=0
Vt (h)
This equation can be simplified by recalling that R̃t,0 = 1 and R̃t,1 = θ/it . Moreover, all labor unions
solve the same problem in equilibrium, and so Vt (h) = Vt , ∀h. The previous equation thus collapses to
"
V
V #
Ut
Pt
πt
πt
L
U
(1 − τt ) Ut − σ (Vt − Wt ) −Tt Ut φU
−1
V
V
Vt
Vt πt−1
πt−1
V
V
πt+1
θ · φU Pt+1 πt+1
L
+(1 − τt+1
)Tt+1 Ut+1
−
1
=0
it
Vt
πtV
πtV
Multiplying by Vt /(Ut Pt ) we obtain
V
V Vt
V t − Wt
πt
πt
(1 − τtL )
− σU
−Tt φU
−
1
V
V
Pt
Pt
πt−1
πt−1
+(1 −
θ
L
τt+1
)
· φU Pt+1 Tt+1 Ut+1
it
Pt
Ut
V
πt+1
−1
πtV
V
πt+1
=0
πtV
Using the definitions πt = Pt /Pt−1 and it = rt · πt+1 , the previous expression simplifies to
vt − σ U (vt − wt ) − Tt φU
V
V
V
L
1 − τt+1
πt+1
πt
πtV
θ · φU Tt+1 Ut+1 πt+1
−
1
+
−
1
=0
V
V
L
V
rt
Ut
πt−1
πt−1
1 − τt
πt
πtV
Rearranging
" V
V
V #
L
1 − τt+1
πt+1
σU
φU
πtV
πt
θ Tt+1 Ut+1 πt+1
w t − vt = U
Tt
−1
−
−1
V
V
L
V
U
σ −1
σ −1
Ut
πt−1
πt−1
1 − τt rt
πt
πtV
32
(5.1)
Capital is produced by capital goods producers and acquired by entrepreneurial firms, who
are the owners of the capital stock during each production cycle. Entrepreneurial firms have
insufficient internal resources to finance capital acquisitions, but are allowed to borrow additional
funds from banks at a given cost. In each period, entrepreneurial firms face an idiosyncratic risk
shock changing the value of the capital stock (or equivalently the return on the capital stock)
and making their activity risky. Banks operate in a perfectly competitive environment. Each
bank borrows resources from households, lending them to many entrepreneurs, implying a zero
ex-post profit at all times. They charge to entrepreneurs a spread over the risk free rate—which
depends on the entrepreneurs’ probability of default, leverage, return on assets, and monitoring
costs incurred by banks—on the funds borrowed, to compensate for the losses incurred on those
entrepreneurs who default on their debt.
Banks and capital goods producers are perfectly competitive in both input and output markets. All remaining firms operate in a monopolistically competition environment in the output
market—therefore charging a markup over the marginal cost—and are perfectly competitive in
the input market.
3.5.1
Capital goods producers
Capital is exclusively produced by capital goods producers. It is thereafter bought by entrepreneurs and rented afterwards to manufacturers for usage in the production process. Box 6
summarizes the relevant framework and notation used to characterize the capital goods sector.
Capital goods producers are price takers in input and output markets.
More specifically, there is a continuum of capital goods producers indexed by i ∈ [0, 1], for
each manufacturing sector J ∈ {T , N }, where T stands for the Tradable sector and N stands for
the Non-tradable sector. In each period, capital goods producers supplying sector J purchase
e J (i) and investment goods from investment
previously installed capital from entrepreneurs K
t
e J (i), according to the following law
goods producers I J (i) to produce new installed capital K
t
t+1
of motion
J
e t+1
e tJ (i) + ζtI ItJ (i)
K
(i) = K
(65)
where ζtI is an investment efficiency shock. We impose a sluggish pattern for investment, consistent with the data, by assuming quadratic adjustment costs with the form
ΓIJ
t (i)
φIJ J
=
I
2 t
!2
ItJ (i)/g
−1
J (i)
It−1
(66)
where ItJ is sector J’s overall investment at period t. Since the problem faced by each capital goods producer is identical, the indexer i can be dropped when referring to equilibrium
conditions. Thus, in stationary form (66) becomes
Γ̌IJ
t
φIJ ˇJ
=
I
2 t
!2
IˇtJ
−1
IˇJ
(67)
t−1
33
Box 6: Framework and notation related with capital utilization and accumulation.
J
K̄t+1
–
Represents the total stock of physical capital that can be used at period t + 1. This
quantity is produced by capital goods producers and bought by entrepreneurs at the end
J
of period t. It respects the standard law of motion K̄t+1
= (1 − δ J )K̄tJ + ζtI ItJ , where
J
I
δ is the depreciation rate, ζt is an investment efficiency shock, and ItJ represents the
investment level, which is set by capital goods producers.
KtJ
–
Represents the stock of capital that is actually used by sector J’s manufacturers in period
t, KtJ = uJt K̄tJ . This quantity is influenced by capital utilization uJt , which is optimally
selected by entrepreneurs.
eJ
K
t+1
–
Represents “new installed capital” at period t. This quantity is set by capital goods
eJ = K
e tJ + ζtI ItJ .
producers, and evolves according to the following law of motion: K
t+1
J
J
J
e
The quantity K̃t+1 is sold to entrepreneurs at the end of period t, and thus K
t+1 = K̄t+1 .
e tJ
K
–
Represents “previously installed capital” at period t. This quantity is bought by capital
goods producers from entrepreneurs at the end of period t. Since capital depreciates
e J = (1 − δ J )K̄ J . The quantity K
e J can be seen as the
during the production cycle, K
t
t
t
“undepreciated fraction of physical capital that has been used in the period t production
cycle” (Christiano, Motto, and Rostagno, 2010).
According to these conventions, at the end of period t entrepreneurs receive back from manufacturers
utilized capital KtJ = uJt K̄tJ . During period t, the existing stock of capital, K̄tJ , faces a depreciation
of δtJ , regardless of whether it has been used in the production cycle. Therefore, physical capital
e tJ = (1 − δ J )K̄tJ . This amount, termed “previously installed capital,” is
at the end of period t is K
sold to capital goods producers, who combine them with investment goods to produce “new installed
eJ = K
e tJ +ζtI ItJ . At the end of period t, entrepreneurs buy at price PtK the new installed
capital,” K
t+1
J
e J , and rent it, partially or entirely, to manufacturers, receiving an unitary rent
capital, K̄t+1
=K
t+1
KJ
of Rt .
In the steady state investment in stationary form is constant and therefore
Γ̌IJ = 0
(68)
Dividends of capital goods producer i supplying sector J are
J
e t+1
e tJ (i) − PtI ItJ (i) + ΓIJ
DtKJ (i) = PtKJ K
(i) − PtKJ K
t (i)
J
e t (i) + ζtI ItJ (i) − PtKJ K
e tJ (i) − PtI ItJ (i) + ΓIJ
= PtKJ K
t (i)
= PtKJ ζtI ItJ (i) − PtI ItJ (i) + ΓIJ
t (i)
(69)
The second step uses the law of motion of capital in (65). The term PtKJ is simultaneously
the price of previously and newly installed capital at the end of period t, since the marginal
rate of transformation between the two is assumed to be one. From equation (69) follows that
capital goods producers select the investment level ItJ , which adds to the existing capital stock
e J to yield new installed capital K
e J . In the process they pay for investment goods plus the
K
t
t+1
adjustment cost. Taxes on capital are paid by entrepreneurs, since they are the capital holders
in this economy. The stationary specification of (69) is obtained after dropping the indexer i
34
and dividing by Pt Tt
I ˇJ
I ˇJ
IJ
dˇKJ
= pKJ
t
t ζt It − pt It + Γ̌t
(70)
where pKJ
= PtKJ /Pt is the relative price of sector J’s capital goods and pIt = PtI /Pt is the
t
relative price of investment goods. In the steady state adjustment costs are zero and shocks are
absent. Thus
dˇKJ = pKJ − pI IˇJ
(71)
J (i)}∞ that maximizes the net
Capital goods producers select the intertemporal profile {It+s
s=0
present value of the dividends stream, subject to adjustment costs in (66), and taking PtKJ and
PtI as given. The problem is
max Et
J (i)
It+s
∞
X
(
KJ I
J
I
R̃t,s Pt+s
ζt+s It+s
(i) − Pt+s
s=0
2 )
J (i)
I
φ
IJ
t+s
J
It+s
(i) +
−1
IJ
J
2 t+s g · It+s−1
(i)
Using the fact R̃t,1 = θ/it , the first-order condition defining optimal investment demand is (we
drop the expected value operator)
"
PtKJ ζtI
=
PtI
1 + φIJ
#
J
J
I
It+1 (i)
It+1
(i)
ItJ (i)
ItJ
θ · φIJ Pt+1
J
−1
−
I
−1
2
J (i)
J (i)
it
g · It−1
g · It−1
PtI t+1 g · ItJ (i)
g · ItJ (i)
Dropping the indexer i, using the definition of real interest rate rt = it /πt , and dividing by Pt
we obtain

I
I
pKJ
t ζt = pt 1 + φIJ
ItJ
−1
J
g · It−1
!
ItJ
θ · φIJ pIt+1
−
J
g · rt pIt
g · It−1
J
It+1
−1
g · ItJ
!
J
It+1
ItJ
!2 

(72)
In stationary form condition (72) becomes

I
I
pKJ
1 + φIJ
t ζt = pt
!
IˇtJ
IˇtJ
g · θ · φIJ pIt+1
−
1
−
J
J
rt
pIt
Iˇt−1
Iˇt−1
!
J
Iˇt+1
−1
IˇtJ
J
Iˇt+1
IˇtJ
!2 

(73)
In the steady state shocks and adjustment costs are absent, and therefore
pKJ = pI
(74)
Hence, the price of capital equals the marginal cost (there is no markup) and dividends equal
zero in equilibrium. This result follows directly from the fact that capital goods producers are
35
perfectly competitive in input and output markets. Outside the steady state there is a wedge
between the price of capital and the price of investment goods, induced by investment efficiency
shocks and adjustment costs.
3.5.2
Entrepreneurs and banks
There is a continuum of infinitely lived entrepreneurial firms l ∈ [0, 1] for each manufacturing sector J ∈ {T , N }. At the beginning of period t + 1 entrepreneurs have an aggregate
physical capital stock of (we use the expressions “entrepreneurs” and “entrepreneurial firms”
interchangeably)
J
K̄t+1
= (1 − δtJ )K̄tJ + ζtI ItJ
(75)
where δtJ is sector J’s capital depreciation rate (which may also be subject to a capital destroying
shock), K̄tJ represents the total stock of physical capital at t, and ζtI is the time t investment
efficiency shock previously introduced. Dividing (75) by Tt yields the stationary specification
for capital accumulation
ˇ J = 1 (1 − δ J )K̄
ˇ J + ζ I IˇJ
K̄
t+1
t
t
t t
g
(76)
In the steady state this equation simplifies to
1
J
J ˇJ
J
ˇ
ˇ
K̄ =
(1 − δ )K̄ + I
g
which can be written as
IˇJ
= g + δJ − 1
ˇ
J
K̄
(77)
Thus, in the steady-state the investment to capital ratio equals the net growth rate plus the
depreciation rate. Below, we clarify the idiosyncratic risky environment in which entrepreneurs
select capital purchases and characterize the renting activity. For convenience, we solve the
model backwards, i.e. we first present the renting activity and only thereafter characterize
capital purchases. However, the ordering is irrelevant since these two stages are independent.
Capital renting
The entrepreneurial firm l operating in sector J selects the capital utilization rate, uJt (l), after
observing the idiosyncratic shock ωt (l). To ease the exposition, we drop the indexer l from the
idiosyncratic shock variable ωt . We adopt the following functional form for the costs of capital
utilization
a
uJt (l)
J
J
1 J J J 2
J
J
J σa
= φa σa ut (l) + φa 1 − σa ut (l) + φa
−1
2
2
36
(78)
Box 7: The adjustment cost function a(uJt ).
Let us drop the indexer l. As in Christiano, Motto, and Rostagno (2010), the adjustment cost function
a(uJt ) takes the form
J
2
1
σa
−1
a uJt = φJa σaJ uJt + φJa 1 − σaJ uJt + φJa
2
2
where φJa and σaJ are parameters that govern the behavior of a(uJt ). In particular, φJa > 0 is selected to
ensure that capital utilization costs equal one in the steady state, whereas σaJ > 0 controls the degree
of convexity. The first and second order derivatives with respect to uJt are
da
= φJa σaJ uJt + φJa 1 − σaJ
J
dut
d2 a
J J
2 = φa σa > 0
J
d ut
The sign of the first derivative depends on the magnitude of σaJ . In the steady-state uJt = 1 implying
that adjustment costs are zero
J
1
σa
a(1) = φJa σaJ + φJa (1 − σaJ ) + φJa
− 1 = φJa σaJ + φJa − φJa σaJ − φJa = 0
2
2
With financial frictions, capital utilization rate uJt is selected by entrepreneurs, whereas without financial frictions it is selected by capital goods producers. In both cases, the first-order condition of the
optimization problem is the same, determining the real rental rate of capital paid by manufacturers in
equilibrium
rtKJ =
da
= φJa σaJ uJt + φJa 1 − σaJ
J
dut
In the steady state with uJt = 1
rKJ = φJa σaJ + φJa (1 − σaJ ) = φJa
Hence, φJa pins down rKJ uniquely in the steady state. Finally, notice that
φJa σaJ
d2 a/duJt =
= σaJ
φJa σaJ + φJa (1 − σaJ )
da/duJt uJt =1
Thus, σaJ establishes the curvature of a(uJt ). An increase in the curvature of the adjustment cost
function (an increase in σaJ ) implies higher costs for a given change in the capital utilization rate.
where φJa > 0 is calibrated to ensure that the capital utilization rate equals one in the steady
state and σaJ > 0 is a parameter that controls the curvature of a(uJt ). The parameter φJa plays
a key role in (78), since it pins down the real rental rate of capital in the steady state. The
function a(uJt ) is analyzed in detail in Box 7. In equilibrium all entrepreneurs set the same
utilization rate and thus adjustment costs are
J
1
2
σa
a uJt = φJa σaJ uJt + φJa 1 − σaJ uJt + φJa
−1
2
2
37
(79)
In the steady state uJt = 1 (since φJa > 0 is calibrated accordingly), and
a(1) = 0
(80)
The capital stock effectively used in production is, in stationary form
ˇJ
ǨtJ = uJt K̄
t
(81)
In the steady state
ˇJ
Ǩ J = K̄
(82)
The after-tax profits of entrepreneur l operating in sector J arising from the rental activity are
(l) =(1 − τtK ) RtKJ uJt (l) − Pt a uJt (l) ωtJ K̄tJ (l)
ΠEPJ,RA
t
(83)
where τtK is the capital income tax rate and RtKJ is the nominal rental rate of capital charged to
intermediate goods producers. Entrepreneurs are price takers, and thus RtKJ is taken as given.
The component ωtJ is the time t idiosyncratic shock that changed physical capital from K̄tJ into
ωtJ K̄tJ . Entrepreneurs select the capital utilization rate profile {uJt+s (l)}∞
s=0 that maximizes the
present discounted value of after-tax profits related with the rental activity. Since there are no
intertemporal interactions (contrary to other firms), this is equivalent to maximize (83) in each
period. The first-order condition yields
dΠEPJ,RA
(l)
da
da
K
KJ
t
= 0 ⇔ (1 − τt ) Rt − Pt J
ωtJ K̄tJ (l) = 0 ⇔ RtKJ = Pt J
J
dut (l)
dut (l)
dut (l)
The idiosyncratic shock plays no role in the entrepreneur decision. Moreover, the utilization rate
of capital is independent of the capital stock. In stationary form and using the result presented
in Box 7, the first-order condition can be restated as
rtKJ = φJa σaJ uJt + φJa (1 − σaJ )
(84)
Thus, the real rental rate of capital equals the cost of using one additional unit of installed
capital. In the steady state the parameter φJa pins down uniquely the real rental rate of capital
rKJ = φJa
(85)
38
The resource cost associated with capital utilization—a key element of the intermediate goods
market clearing condition as it measures real resources that were not used in production—is
RCU Jt = Pt a(uJt )K̄tJ
It is useful to evaluate this resource cost in terms of PtJ and in stationary form
rcu
ˇ Jt =
ˇJ
a(uJt )K̄
t
pJt
(86)
Dividing by PtJ is crucial for the market clearing condition, as the resource cost becomes measured in the same unit of intermediate goods. In the steady state a(1) = 0 and thus
rcu
ˇ J =0
(87)
Idiosyncratic risky environment and optimal capital purchases
J (l) from
At the end of period t each entrepreneur buys a nominal amount of capital PtKJ K̄t+1
J (l) represents the total quantity of capital that can
capital goods producers. The element K̄t+1
be used in the period t + 1 production cycle. To purchase the new amount of capital, the
entrepreneur has to respect the following balance sheet
J
PtKJ K̄t+1
(l) = BtJ (l) + NtJ (l)
where BtJ (l) is an external finance component (bank loans) and NtJ (l) is an internal finance
component (net worth), both evaluated at the end of period t.
Bank loans represent an important liability with potentially large macroeconomic effects,
since the entrepreneur is forced to declare bankruptcy if he is unable to paid them back. The
balance sheet equation can be rearranged to focus on external finance needs
J
BtJ (l) = PtKJ K̄t+1
(l) − NtJ (l)
(88)
This equation can be expressed in terms of the leverage ratio BtJ (l)/NtJ (l)
J (l)
PtKJ K̄t+1
BtJ (l)
=
−1
NtJ (l)
NtJ (l)
Aggregating the balance sheet equation across all entrepreneurs and dividing by Pt Tt yields
J
ˇJ
b̌Jt = pKJ
t K̄t+1 · g − ňt
(89)
39
The steady-state version is
ˇ J · g − ňJ
b̌J = pKJ K̄
(90)
After purchasing new capital from capital goods producers (but before selecting the utilization
J
rate), entrepreneurs experience an idiosyncratic shock ωt+1
that changes the capital stock from
J (l) to ω J K̄ J (l). This shock takes place at the beginning of period t + 1, creating
K̄t+1
t+1 t+1
a risky environment. In the extreme case of a severe shock, the entrepreneur experiences a
loss that makes impossible to meet his debt obligations, forcing him to declare bankruptcy.
J ,
More specifically, there exists an endogenous threshold level for the idiosyncratic shock, ω̄t+1
J
J
separating two distinct outcomes: if ωt+1
≥ ω̄t+1
the entrepreneur is able to payoff his debts and
J
J
is therefore solvent, whereas if ωt+1
< ω̄t+1
the entrepreneur cannot meet his debt obligations
J
and is forced to declare bankruptcy. The value for the threshold ω̄t+1
is optimally set by the
entrepreneur when selecting optimal capital purchases.
The idiosyncratic shock is directly observed by the entrepreneur at no cost. On the contrary,
banks can only observe the shock if they pay a unitary monitoring cost µt+1 over the firm value.
This asymmetric information setup is in line with the “costly state verification” literature,
pioneered by Townsend (1979). Monitoring costs include all bankruptcy costs (auditing costs,
accounting and legal costs, asset liquidation, business interruption effects, among others). The
monitoring activity is undertaken by type-A households at no cost and with no effort, and thus
the remuneration for monitoring work performance paid by banks corresponds to a pure income
effect.
J
follows a log-normal distribution with a mean of unity,
The random variable ωt+1
ln ωtJ
1 EJ 2
EJ 2
∼ N − σt
, σt
2
J
distributed independently over time and across entrepreneurs. Let FJt (x) = Pr[ωt+1
< x] denote
the cumulative distribution function and ft (x) the corresponding probability density function of
J . The standard deviation σ EJ is itself a stochastic process, known to entrepreneurs when
ωt+1
t
deciding on capital purchases (contrary to ωtJ , which is only observed afterwards). Figure 3
depicts the unit mean cumulative log-normal distribution F(ω).
Entrepreneurs ex-ante after-tax return to capital, Ret KJ
t , is
Ret KJ
t
K ) RKJ uJ
J
J
KJ
K
J
KJ
(1 − τt+1
t+1 t+1 − Pt+1 a(ut+1 ) + (1 − δt+1 )Pt+1 + τt+1 δt+1 Pt+1
= Et
PtKJ
K ) r KJ uJ
J
J
KJ
K
J
KJ
(1 − τt+1
t+1 t+1 − a(ut+1 ) + (1 − δt+1 )pt+1 + τt+1 δt+1 pt+1
= Et
KJ
Pt
Pt
Pt Pt+1
K ) r KJ uJ
J
J
KJ
K
J
KJ
(1 − τt+1
t+1 t+1 − a(ut+1 ) + (1 − δt+1 )pt+1 + τt+1 δt+1 pt+1
= Et πt+1
pKJ
t
This element is identical for all entrepreneurs, since in equilibrium uJt+1 (l) = uJt+1 , ∀l. The
40
Box 8: The consequences of the idiosyncratic shock ωt+1 .
J
The cut-off level ω̄t+1
must satisfy the following condition
J
KJ J
BJ
J
ω̄t+1
Ret KJ
t Pt K̄t+1 (l) = it+1 (l)Bt (l)
J
J
which states that, if ωt+1
= ω̄t+1
, the value of the firm is just enough to keep up with financial
obligations, implying a net worth value of zero. For a given draw ωt+1 , this condition bears the
following consequences


J
KJ J
the bank pays monitoring cost µt+1 ωt+1
Ret KJ


t Pt K̄t+1 (l)






KJ
J
KJ
J

and receives (1 − µt+1 )ωt+1 Ret t Pt K̄t+1 (l)


ωt+1 < ω̄t+1 ⇒












the entrepreneur receives 0 and goes bankrupt

If



J

the bank receives iBJ


t+1 (l)Bt (l) from the entrepreneur









ω
≥
ω̄
⇒

t+1
t+1

J
KJ J
BJ
J


the entrepreneur receives ωt+1
Ret KJ


t Pt K̄t+1 (l) − it+1 (l)Bt (l) ≥ 0,




which is the quantity to be subject to a maximization procedure
goes bankrupt. The key point of this framework is that, if the idiosyncratic shock is above the
J , the entrepreneur is able to honor the debt contract paying to the bank
threshold value ω̄t+1
J
iBJ
t+1 (l)Bt (l), whereas in the opposite case he is forced to declare bankruptcy. The cut-off level
J
ω̄t+1
has to satisfy the following condition
J
KJ J
BJ
J
ω̄t+1
Ret KJ
t Pt K̄t+1 (l) = it+1 (l)Bt (l)
(93)
J
J , the value of the entrepreneurial firm is just enough to keep
which states that, if ωt+1
= ω̄t+1
up with financial obligations, implying an ex-post net worth value of zero. Given the decision
J (l) at the end of period t, equation (93) sets liquidity needs, and
to buy capital worth PtKJ K̄t+1
J , Ret KJ and iBJ (l).
establishes a critical link between ω̄t+1
t
t+1
If the entrepreneur goes bankrupt, the bank keeps only a fraction 1 − µt+1 of the value of
the entrepreneurial firm (the recovery rate)
J
KJ J
(1 − µt+1 )ωt+1
Ret KJ
t Pt K̄t+1 (l)
The remaining fraction µt+1 represent monitoring costs paid to type-A households for monitoring
services. If the entrepreneur is able to satisfy the debt contract, the bank receives the borrowed
J
amount iBJ
t+1 (l)Bt (l), while the entrepreneur, as the residual claimant, obtains
J
KJ J
BJ
J
ωt+1
Ret KJ
t Pt K̄t+1 (l) − it+1 (l)Bt (l)
J
KJ J
J
KJ KJ J
=ωt+1
Ret KJ
t Pt K̄t+1 (l) − ω̄t+1 Ret t Pt K̄t+1 (l)
J
J
KJ J
= ωt+1
− ω̄t+1
Ret KJ
t Pt K̄t+1 (l)
where we used equation (93). All these possible outcomes are summarized in Box 8.
42
Each entrepreneur l maximizes
Z
∞
EPJ
J
J
et+1
N
(l)f(ωt+1
)dωt+1
0
where
(
EPJ
et+1
N
(l) =
KJ KJ J
J − ω̄ J
J
J
ωt+1
t+1 Ret t Pt K̄t+1 (l) , if ωt+1 ≥ ω̄t+1
0
, otherwise
is the t + 1 net worth before dividends are paid out and prior to aggregate shocks. This is
e EPJ (l) over the non-default region ω J ≥ ω̄ J
naturally equivalent to maximize N
t+1
Z
∞
J
ω̄t+1
t+1
t+1
EPJ
J
J
et+1
N
(l)f(ωt+1
)dωt+1
(94)
We now turn to the restrictions faced by entrepreneurs, which include the terms of the debt
contract and account for the environment where banks operate.
Banks are assumed to make zero ex-ante and ex-post profits at all times. Their sole activity
is to borrow resources from households in order to finance entrepreneurs capital acquisitions. We
make the simplifying assumption that households are exclusively involved in risk-free activities,
and thus they only lend to banks if they are indifferent against receiving the non-state contingent
rate of return it . That is, the terms of the contract must state that households receive a noncontingent gross amount of it+1 BtJ (l) at t + 1, given a loan of BtJ (l) at t, regardless of the shock
J . Hence, entrepreneurial activity and bank loans are risky, but household loans are not—the
ωt+1
major role of banks is to transform risky borrowing from entrepreneurs into riskless lending from
households. To guarantee that households contractual interest rate is effectively verified, the
debt contract has to be state contingent to account for all unexpected perturbations, implying
J
J
that both ω̄t+1
and iBJ
t+1 (l) have to be functions of time t + 1 idiosyncratic shocks ωt+1 . The
debt contract has therefore to satisfy the following participation constraint
J
Z ω̄t+1
J
J
J
J
KJ J
J
J
[1 − F(ω̄t+1
)]iBJ
+
(1
−
µ
)
(l)B
(l)
ωt+1
Ret KJ
t+1
t+1
t
t Pt K̄t+1 (l)f(ωt+1 )dωt+1 = it Bt (l)
0
|
{z
}|
{z
} | {z } |
{z
} | {z }
No default
probability
Bank revenues
in case of
no default
Recovery
rate
Average value of capital
J
J
in case of default ωt+1
<ω̄t+1
(95)
risk-free
amount
This condition implies that households always receive a gross amount of it BtJ (l) for lending to
J . Banks, in turn, receive iBJ (l)B J (l) if no default
entrepreneur l, regardless of the shock ωt+1
t
t+1
J —if a default occurs. Hence, the bank
occurs, and the value of the firm—which depends on ωt+1
demands a risk premium to the entrepreneur whenever the probability of default is positive, i.e.
iBJ
t+1 (l) ≥ it . Moreover, a bank can make a profit or loss for a given entrepreneur (for a given
J ), even though on expectation profits are zero. In addition, we assume that
realization of ωt+1
J ), so that
each bank lends to many entrepreneurs (each facing a different idiosyncratic shock ωt+1
ex-post profits are also zero by the law of large numbers.
As discussed by Levin, Natalucci, and Zakrajsek (2004), Ret KJ
= it if µt+1 = 0 (the frict
tionless case), but iBJ
t+1 − it > 0 as long as the above probability of default is positive. In any
case, the degree of financial frictions is always conditional on µt+1 . Bernanke, Gertler, and
43
Gilchrist (1999b) defined the ratio of default costs to quantity borrowed, reflecting the premium
for external finance, as
EF Pt =
µt+1
J
R ω̄t+1
0
J Ret KJ P KJ K̄ J (l)f(ω J )dω J
ωt+1
t
t
t+1
t+1
t+1
J
Bt (l)
(96)
This quantity EF Pt will be our measure of the “External Finance Premium”. Plugging equation
(93) into (95) yields
1−
J
F(ω̄t+1
)
J
KJ J
ω̄t+1
Ret KJ
t Pt K̄t+1 (l)
J
ω̄t+1
Z
+ (1 − µt+1 )
J
KJ J
J
J
J
ωt+1
Ret KJ
t Pt K̄t+1 (l)f(ωt+1 )dωt+1 = it Bt (l)
0
Notice that the interest rate required by the bank increases to compensate for the higher
J . Using equation (88) and
probability of default following an increase in the threshold ω̄t+1
rearranging
"
[1 −
J
J
F(ω̄t+1
)]ω̄t+1
Z
J
ω̄t+1
+ (1 − µt+1 )
#
J
J
J
ωt+1
f(ωt+1
)dωt+1
KJ J
Ret KJ
t Pt K̄t+1 (l)
0
J
= it PtKJ K̄t+1
(l) − NtJ (l)
(97)
J , K̄ J (l)} that maximizes the period t + 1 net worth in
In each period, entrepreneurs select {ω̄t+1
t+1
(94), subject to the terms of the debt contract summarized in the banks participation constraint.
This problem can be restated as (see Box 9)
J
KJ J
max 1 − Γ(ω̄t+1
) Ret KJ
t Pt K̄t+1 (l)
(98)
J
ω̄t+1
J (l)
K̄t+1
KJ J
J
J
KJ J
J
s.t. Γ(ω̄t+1
) − µt+1 G(ω̄t+1
) Ret KJ
t Pt K̄t+1 (l) = it Pt K̄t+1 (l) − Nt (l)
where
Z
J
Γ(ω̄t+1
)
≡
J
ω̄t+1
J
J
J
ωt+1
f(ωt+1
)dωt+1
+
J
ω̄t+1
Z
0
| {z }
|
Banks capital
earnings gross share
J
Γ(ω̄t+1
)
1−
{z
|
}
Entrepreneurs capital
earnings net share
{z
}
J
J
If ωt+1
<ω̄t+1
Z
≡
0
{z
}
J
J
If ωt+1
<ω̄t+1
J
ω̄t+1
J
J
f(ωt+1
)dωt+1
{z
|
}
J
J
If ωt+1
≥ω̄t+1
∞
+
J
ω̄t+1
|
∞
J
J
J
J
(ωt+1
− ω̄t+1
)f(ωt+1
)dωt+1
|
{z
J
J
If ωt+1
≥ω̄t+1
}
and
J
G(ω̄t+1
)≡
Z
J
ω̄t+1
J
J
J
ωt+1
f(ωt+1
)dωt+1
0
J )−µ
J
Banks capital earnings net share is Γ(ω̄t+1
t+1 G(ω̄t+1 ).
J
The entrepreneur problem has been reduced so far to selecting K̄t+1 (l) and a schedule ω̄t+1
(as a function of the realized values of Ret KJ
t ). The distribution of aggregate and idiosyncratic
risks, the price of capital PtKJ , and net worth NtJ (l), are taken as given. The solution to the
44
Box 9: Banks and entrepreneurs capital earnings share.
The entrepreneurs maximization problem
Z ∞
J
J
KJ J
J
J
max
(ωt+1
− ω̄t+1
)Ret KJ
t Pt K̄t+1 (l)f(ωt+1 )dωt+1
J
ω̄t+1
J
K̄t+1
(l)
s.t.
(97)
J
ω̄t+1
can be simplified by noticing that capital earnings are shared between banks and entrepreneurs. Ignoring time subscripts and the superscript J, total density can be decomposed as
Z ∞
Z ω̄
ωf(ω)dω
+
(9.1)
ωf(ω)dω
1 =
| ω̄ {z
}
}
| 0 {z
Region associated with
the default area
Region associated with
the non-default area
Equation (9.1) can be manipulated to account for alternative outcomes
ω̄
Z
1
Z
=
∞
ωf(ω)dω +
0
ω̄
ω̄
Z
=
|0
Z
The entrepreneur
goes bankrupt
|
ω̄
|
∞
ω̄
Gross share of banks
(before paying monit. costs)
=
Γ(ω̄)
f(ω)dω
∞
+
f(ω)dω
{z
}
∞
ω̄
Z
| ω̄
The entrepreneur
does not go bankrupt
{z
Z
f(ω)dω − ω̄
ω̄
Z
ωf(ω)dω +
{z
}
∞
ωf(ω)dω + ω̄
(ω − ω̄)f(ω)dω
{z
}
The entrepreneur
does not go bankrupt
}
|
{z
}
Share of entrepreneurs
+
1 − Γ(ω̄)
(9.2)
R∞
Equation (9.2) splits total density 0 ωf(ω)dω = 1 in banks gross share, Γ(ω̄), and entrepreneurs gross
share, 1 − Γ(ω̄). Entrepreneurs gross share equals their net share, since they observe ω at no cost. On
the contrary, banks have to pay monitoring costs µG(ω̄) if the entrepreneur goes bankrupt, where
Z
G(ω̄) ≡
ω̄
ωf(ω)dω
0
Hence, the banks net share is Γ(ω̄) − µG(ω̄).
The entrepreneurs maximization problem can therefore be restated as
KJ J
J
max 1 − Γ(ω̄t+1
) Ret KJ
t Pt K̄t+1 (l)
J
ω̄t+1
J
K̄t+1
(l)
KJ J
J
J
KJ J
J
s.t. Γ(ω̄t+1
) − µt+1 G(ω̄t+1
) Ret KJ
t Pt K̄t+1 (l) = it Pt K̄t+1 (l) − Nt (l)
entrepreneur problem is presented in Box 10. Notice that, in the current setup, entrepreneurs
face an additional balance sheet effect taht is absent from models without financial frictions: a
marginal increase in capital raises the entrepreneur leverage, and therefore the costs of external
finance. Thus, ceteris paribus, entrepreneurs set a relatively lower level of physical capital.
Even though all entrepreneurs face the same decision rule, they may have different net worth
levels and set distinct capital purchases. Dropping the indexer l when aggregating for the overall
45
Box 10: Entrepreneurs maximization problem.
J
J
The entrepreneur selects the profile {ω̄t+1
, K̄t+1
(l)} that solves problem (98). The Lagrangian is
J
KJ J
L(·) = 1 − Γ(ω̄t+1
) Ret KJ
t Pt K̄t+1 (l)
h
i
J
J
KJ J
KJ J
J
+λt Γ(ω̄t+1
) − µt+1 G(ω̄t+1
) Ret KJ
t Pt K̄t+1 (l) − it Pt K̄t+1 (l) − Nt (l)
where λt is the Lagrange multiplier. Let
J
ΓJt+1 ≡ Γ(ω̄t+1
)
with
ΓJt+1
0
≡
dΓJt+1
J
= 1 − F(ω̄t+1
)
J
dω̄t+1
J
GJt+1 ≡ G(ω̄t+1
)
with
GJt+1
0
≡
dGJt+1
J
J
= ω̄t+1
f(ω̄t+1
)
J
dω̄t+1
The first-order conditions are as follows.
J
1. The optimal cutoff ω̄t+1
0
0
0 ∂L(·)
KJ J
KJ J
= 0 ⇔ ΓJt+1 Ret KJ
ΓJt+1 − µt+1 GJt+1 Ret KJ
t Pt K̄t+1 (l) = λt
t Pt K̄t+1 (l) ⇔
J
∂ ω̄t+1
0
ΓJt+1
(10.1)
⇔λt =
0
0
ΓJt+1 − µt+1 GJt+1
2. Optimal capital purchases
∂L(·)
KJ
+ λt
= 0 ⇔ 1 − ΓJt+1 Ret KJ
t P
J
∂ K̄t+1 (l)
ΓJt+1
−
µt+1 GJt+1
KJ
Ret KJ
t P
−
it PtKJ
=0
Dividing by it PtKJ and noting that
Ret KJ
Ret KJ
ret KJ
t
t
t
=
=
it
rt πt+1
rt
yields
1 − ΓJt+1
ret KJ
t
+ λt
rt
ΓJt+1 − µt+1 GJt+1
ret KJ
t
−1 =0
rt
(10.2)
J
which implies that all entrepreneurs set the same cutoff ω̄t+1
Finally, notice that nominal capital purchases can be expressed as
J
PtKJ K̄t+1
(l) =
it NtJ (l)
KJ
J
J )−µ
it − Γ(ω̄t+1
t+1 G(ω̄t+1 ) Ret t
and therefore each entrepreneur sets capital purchases according to his net worth. Nevertheless, all
entrepreneurs select the same leverage ratio
J
J
) − µt+1 G(ω̄t+1
) Ret KJ
Γ(ω̄t+1
BtJ (l)/NtJ (l)
t
=
it
1 + BtJ (l)/NtJ (l)
economy and pugging (10.1) into (10.2) yields
1−
ΓJt+1
ret KJ
t
+
rt
ΓJt+1
ΓJt+1
0
0
− µt+1 GJt+1
!
0
46
ΓJt+1 − µt+1 GJt+1
ret KJ
t
−1 =0
rt
(99)
This equation—which is already in stationary form—establishes the link between ret KJ
and
t
J . The steady-state version is
ω̄t+1
ret KJ
+
1−Γ
r
ΓJ
J
ΓJ
0
0
− µ GJ
!
ΓJ − µGJ
0
ret KJ
−1 =0
r
(100)
The constraint in problem (98) defining the zero-profit condition can be written in terms of
J /N and of the corporate risk spread Ret KJ /i . Dropping
the assets-to-net worth ratio PtKJ K̄t+1
t
t
t
the indexer l and dividing by it Nt yields
J
KJ J
J
Ret KJ
P KJ K̄t+1
t Pt K̄t+1
= t
−1
Γt+1 − µt+1 GJt+1
it
Nt
Nt
In stationary form
J
Ret KJ
t
Γt+1 − µt+1 GJt+1
rt πt+1
J
PtKJ K̄t+1
Pt Tt+1
Nt
Pt Tt g
=
J
PtKJ K̄t+1
Pt Tt+1
Nt
Pt Tt g
−1⇔
ˇJ g
KJ ˇ J
ret KJ
pKJ K̄
t+1
t pt K̄t+1 g
⇔ ΓJt+1 − µt+1 GJt+1
= t
−1
rt
ňt
ňt
(101)
This equation states that ex-ante bank profits are zero, but is silent about ex-post events.
However, given that each bank lends to many entrepreneurs, ex-post profits are also zero. Banks
will thus make positive profits on some entrepreneurs and negative profits on others, but on
average profits are zero. Following Kumhof et al. (2010), the period t ex-post version of (101)
becomes
ˇ Jg
KJ ˇ J
J
ret KJ
pKJ K̄
m1,t pt−1 K̄t g
= t−1 t − 1
Γt − µt GJt
rm1,t
ňt−1
ňt−1
(102)
where rm1,t is the ex-post real interest rate
rm1,t =
it−1
πt
(103)
and ret KJ
m1,t is the ex-post after-tax return on capital
KJ
ret KJ
m1,t = pt−1
−1 h
i
K J KJ
(1 − τtK ) uJt rtKJ − a(uJt ) + (1 − δtJ )pKJ
+
τ
δ
p
t
t t t
(104)
KJ
Notice that, even though ret KJ
m1,t has the same formulation as ret t−1 , they are conceptually
different. The former is the ex-post return on capital at t, after the shock ωtJ is known. The
47
latter is the ex-ante return on capital at t − 1, before the shock ωt is known. For latter reference,
KJ
notice that Ret KJ
m1,t = πt ret m1,t . The steady-state version of (102) is
ˇ Jg
KJ ˇ J
J
ret KJ
K̄ g
pKJ K̄
m1 p
Γ − µGJ
=
−1
rm1
ň
ň
(105)
and of (103) and (104) are respectively
rm1 =
i
π
(106)
and
KJ
ret KJ
m1 = p
−1 h
i
(1 − τ K ) uJ rKJ − a(uJ ) + (1 − δ J )pKJ + τ K δ J pKJ
(107)
Since ωtJ follows a unit mean log-normal distribution, the terms ΓJt , GJt , (ΓJt )0 , and (GJt )0 can
be computed by resorting to the following auxiliary variable
z̄tJ =
ln(ω̄tJ ) 1 EJ
+ σt
2
σtEJ
Solving for ω̄t yields
2
1
ω̄tJ = exp z̄tJ σtEJ − σtEJ
2
(108)
In the steady state
2
1
ω̄ J = exp z̄ J σ EJ − σ EJ
2
(109)
Let Φ(x) denote the cumulative distribution function of the standard normal distribution. The
following results hold in stationary form
ΓJt = Φ z̄tJ − σtEJ + ω̄tJ 1 − Φ z̄tJ
(110)
GJt = Φ z̄tJ − σtEJ
(111)
48
ΓJt
0
= 1 − Φ z̄tJ
0
= ω̄tJ f ω̄tJ
GJt
(112)
(113)
where
f ω̄tJ = √
1
exp
2π ω̄tJ σtEJ
1
− (z̄tJ )2
2
(114)
The derivation is clarified in Box 11. The steady-state versions are respectively
ΓJ = Φ z̄ J − σ EJ + ω̄ J 1 − Φ z̄ J
(115)
GJ = Φ z̄ J − σ EJ
(116)
(117)
ΓJ
0
= 1 − Φ z̄ J
0
= ω̄ J f ω̄ J
GJ
(118)
where
f ω̄
J
1
=√
exp
2π ω̄ J σ EJ
2
1
− z̄ J
2
(119)
To close the entrepreneurial sector one needs to describe in more precise terms how aggregate net
KJ J
worth evolves over time. Ex-post aggregate returns to capital are Ret KJ
m1,t Pt−1 K̄t . The difference
KJ J
BJ J
between this amount net of bankruptcy costs µJt Gt Ret KJ
m1,t Pt−1 K̄t and the gross amount it−1 Bt−1
paid for bank loans goes entirely to entrepreneurs, accumulating over time possibly up to the
point where external finance is no longer needed. If this is the case, leverage converges to
zero, and the financial accelerator setup collapses to the benchmark case with no financial
frictions. To prevent this, we impose some restrictions on the time path of aggregate net worth.
49
Box 11: Using the lognormal distribution in the financial accelerator set up.
The random variable ωtJ follows a lognormal distribution
ln ωtJ ∼ N (−1/2(σtEJ )2 , (σtEJ )2 )
Consequently
E(ωtJ ) = 1
Var(ωtJ ) = σtEJ
2
The density function of ωtJ is
f(ωtJ ) = √
2 1 ln(ωtJ ) 1 EJ
1
exp
−
+
σ
2
2 t
σtEJ
2πωtJ σtEJ
Let us omit the time subscript and the sector superscript. Furthermore, consider the following auxiliary
variables and results
z̄ =
y=
ln(ω̄) 1 E
+ σ
σE
2

E

ω = exp yσ −
ln(ω) 1 E
+ σ ⇒

σE
2
y − σ E =
ln(ω)
σE
1
2
σE
2 − 12 σ E
⇒ dω = σ E exp yσ E −
⇒ (y − σ E )2 =
ln(ω)
σE
2
1
2
σE
2 dy
− ln(ω) +
1 E 2
2σ
R∞
We first show that ω̄ f(ω)dω = 1 − Φ(z̄), where Φ(·) is the cumulative distribution function of the
standard normal distribution
2 Z ∞
Z ∞
1 ln(ω) 1 E
1
√
+ σ
dω
f(ω)dω =
exp −
2
σE
2
2πωσ E
ω̄
ω̄
Z
∞
√
=
z̄
Z
∞
1 2
1
E
E 2
√
exp − (y − 2yσ + σ ) dy
2
2πω
∞
1
1
√ exp − ln(ω) exp − (y − σ E )2 dy
2
2π
∞
2 1
1 ln(ω) 1 E
√ exp − ln(ω) −
−
σ
dy
2
σE
2
2π
∞
2
2 1
1
ln(ω)
1 E
√ exp −
σ
dy
+
2
ln(ω)
−
ln(ω)
+
2
σE
2
2π
∞
2 Z ∞
1
1 ln(ω) 1 E
1
1 2
√ exp −
√
+
σ
dy
=
exp
−
y
dy
2
σE
2
2
2π
2π
z̄
=
z̄
Z
=
z̄
Z
=
z̄
Z
=
z̄
Z
2
1
1
1
exp − y 2 σ E exp yσ E − σ E
dy
2
2
2πωσ E
=
z̄
Z
z̄
=1−
−∞
1
1 2
√ exp − y dy = 1 − Φ(z̄)
2
2π
50
Consider now the following auxiliary variables and results
z̃ =
ỹ =
ln(ω̄) 1 E
− σ ⇒ z̃ = z̄ − σ E
σE
2

E

ω = exp ỹσ +
ln(ω) 1 E
− σ ⇒

σE
2
ỹ + σ E =
We show that
Z
R∞
ω̄
ln(ω)
σE
1
2
σE
2 + 12 σ E
⇒ dω = σ E exp ỹσ E +
⇒ (ỹ + σ E )2 =
ln(ω)
σE
2
1
2
σE
2 dỹ
+ ln(ω) +
1 E 2
2σ
ωf(ω)dω = 1 − Φ(z̄ − σ E )
∞
Z
∞
ωf(ω)dω =
ω̄
ω̄
Z
2 ω
1 ln(ω) 1 E
√
exp −
+ σ
dω
2
σE
2
2πωσ E
∞
√
=
z̃
Z
∞
1
1 2
1 E 2
E
E 2
E
√ exp − (ỹ + 2ỹσ + σ ) + ỹσ + σ
dỹ
2
2
2π
∞
Z z̃
1
1
1
1
√ exp − ỹ 2 dỹ = 1 −
√ exp − ỹ 2 dỹ
2
2
2π
2π
−∞
=
z̃
Z
2
1
1
1
exp − (ỹ + σ E )2 σ E exp ỹσ E + σ E
dỹ
2
2
2πσ E
=
z̃
= 1 − Φ(z̃) = 1 − Φ(z̄ − σ E )
Replacing these results in the definitions of ΓJt , GJt , (ΓJt )0 , and (GJt )0 yields
ΓJt =
Z
ω̄tJ
ωtJ f(ωtJ )dωtJ + ω̄tJ
0
GJt
Z
=
ω̄tJ
Z
∞
ω̄tJ
f(ωtJ )dωtJ = Φ(z̄tJ − σtEJ ) + ω̄tJ 1 − Φ(z̄tJ )
ωtJ f(ωtJ )dωtJ = Φ(z̄tJ − σtEJ )
0
ΓJt
0
GJt
0
= ω̄tJ f(ω̄tJ ) +
Z
∞
ω̄tJ
f(ωtJ )dωtJ − ω̄tJ f(ω̄tJ ) = 1 − Φ(z̄tJ )
= ω̄tJ f(ω̄tJ )
More specifically, we assume that entrepreneurs pay out dividends to households on a regular
basis, worth DtJ = PtJ div Jt (sector J’s dividends are measured in real units of output of the
respective sector). Additionally, aggregate net worth accumulation may also be hindered by
output–destroying shocks, valued at PtJ StJ,y . Sector J’s entrepreneurial firms net worth therefore
evolves according to
KJ J
J
KJ
KJ J
J
J
J
J J,y
K̄
−
i
B
NtJ = Ret KJ
P
K̄
−
µ
G
Ret
P
t
t−1
m1,t t−1 t
t
m1,t t−1 t
t−1 − Pt div t − Pt St
J,y KJ J
J
J
J
J
= Ret KJ
P
K̄
(1
−
µ
G
)
−
i
B
t−1 t−1 − Pt div t + St
m1,t t−1 t
t t
KJ J
J
KJ J
J
= Ret KJ
P
(1
−
µ
G
)
−
i
P
−
N
− PtJ div Jt + StJ,y
K̄
K̄
t−1
m1,t t−1 t
t t
t−1 t
t−1
=
J
it−1 Nt−1
+
KJ J
Pt−1
K̄t
Ret KJ
m1,t (1
−
µJt Gt )
51
− it−1
− PtJ div Jt + StJ,y
J
KJ K̄ J − N J . In stationary form this
where we used the aggregate version of (88), Bt−1
= Pt−1
t
t−1
equation becomes
ňJt
=
rm1,t J
KJ ˇ J
KJ
J
ˇ J + Š J,y
ňt−1 + pt−1 K̄t ret m1,t (1 − µt Gt ) − rm1,t − pJt div
t
t
g
(120)
The steady-state version is
g
J,y KJ ˇ J
KJ
J
J ˇ J
ň =
p K̄ ret m1 (1 − µ G) − rm1 − p div + Št
g − rm1
J
(121)
ˇ J evolve according to a structure present in Kumhof et al. (2010). First, let
Dividends div
t
ˇ Jt be defined as a fraction StJ,d of net worth gross return
entrepreneurial income pJt inc
h
i
ˇ J + Š J,y
ˇ Jt = StJ,d ňJt + pJt div
pJt inc
t
t
=
StJ,d
rm1,t J
KJ
J
ˇJ
ňt−1 + pKJ
t−1 K̄t ret m1,t (1 − µt Gt ) − rm1,t
g
The element StJ,d is a dividend related net worth shock (with Et StJ,d typically in a range between
0 and 0.05) affecting the share of gross returns on net worth that is distributed to households.
Hence, a shock to StJ,d originates a pure redistribution effect between entrepreneurs and houseˇ J,ma
holds, without direct resource implications. Moreover, let inc
and ňJ,ma
denote moving
t
t
ˇ Jt and ňJt , respectively (moving averages are assumed to contain only lagged
averages of inc
ˇ J evolve according to
terms). Dividends div
t
J
ˇ J = inc
ˇ J,ma
+ θnw
(ňJt − ňJ,ma
)
div
t
t
t
(122)
J is a parameter measuring the change in dividends if net worth raises or decreases
where θnw
against its long-run value. This parameter is typically set between 0 and 0.05 (Kumhof et al.,
2010). In the steady state ňJt = ňJ,ma
, and the above equation becomes
t
ˇ J = inc
ˇ J,ma
div
(123)
This setup deserves some clarification. To prevent aggregate net worth from increasing indefinitely over time, we assume that entrepreneurs face output destroying shocks and pay regular
dividends to households. The remaining amount are simply retained earnings, used to finance
entrepreneurs activity in the following period (in complement with external finance). Dividends,
in turn, are a function of current and past income of entrepreneurs, but also depend on the deviation of current and past net worth value from the long-run value. Therefore, dividends are
lower if net worth is below the steady-state value and vice-versa. Moving averages enhance the
52
dynamic properties of the model, allowing adjustments to occur smoothly through time.
J ) of entrepreneurs declares bankruptcy and goes out of
In each period t, a fraction F(ω̄t+1
business. To ensure that entrepreneurial firms have a mass of one at all times, we assume that
the same fraction of new businesses starts in every period, viz the most successful entrepreneurs
use a portion of their net worth to set a new business. This setup implies that aggregate net
worth is not affected by new entrants, thus ensuring consistency between the path for aggregate
net worth and individual decisions. Moreover, given that the decision rule is the same for
all entrepreneurs, they all set the same leverage ratio in equilibrium, even though they have
different net worth values and therefore purchase different amounts of capital. Thus, aggregate
figures evolve according to a pre-determined path, even though in equilibrium entrepreneurs face
distinct outcomes.
Finally, sector J’s real bankruptcy monitoring costs paid by banks are
ˇ J ret KJ pKJ µJ GJ
ˇ J = 1 K̄
rbr
t
m1,t t−1 t t
pJt t
(124)
This is not a physical resource cost but a remuneration for monitoring work performance, paid
to type-A households in a lump-sum fashion. In the steady state
ˇ J ret KJ pKJ µJ GJ
ˇ J = 1 K̄
rbr
m1
pJ
3.5.3
(125)
Manufacturers
Manufacturers combine labor services (hired from unions) with capital goods (rented from entrepreneurs) to produce an intermediate good, which is then sold to distributors. They are
perfectly competitive in the input market and monopolistically competitive in the output market, charging a markup to distributors. Profits are distributed to households in the form of
dividends.
More specifically, there is a continuum of manufacturing firms j ∈ [0, 1] in each sector
J ∈ {T , N }, where T stands for the Tradable sector and N stands for the Non-tradable sector.
Each firm produces a specific variety of the intermediate good, which is bought by a continuum of
distributor firms f ∈ [0, 1] operating in sectors F ∈ {C, G, I, X }. Let ZtJF (j, f ) stand for the time
t quantity of variety j produced by firm j operating in sector J and purchased by distributor f
operating in sector F . Distributors buy intermediate goods from many manufacturers, bundling
them together in a homogeneous intermediate good of type J, ZtJF (f ), to be used in the final
goods production. The bundling technology is given by the familiar CES aggregator
ZtJF (f ) =
Z
1
σ J −1
ZtJF (j, f ) σJ dj
σJ
σ J −1
0
where σ J ≥ 0 is the elasticity of substitution between varieties of sector J intermediate good.
53
The demand for intermediate goods faced by firm j results from the standard cost minimization problem of distributers, in which they select ZtJF (j, f ) so as to minimize the costs of
acquiring intermediate goods, given that the bundle ZtJF (f ) must be achieved. The problem is
1
Z
PtJ (j)ZtJF (j, f )dj
min
ZtJF (j,f )
s.t.
ZtJF (f ) =
0
1
Z
σ J −1
ZtJF (j, f ) σJ dj
σJ
σ J −1
0
where PtJ (j) is the price charged by the intermediate goods producer j operating in sector J.
Since distributors are perfectly competitive in the input market, they take the price PtJ (j) as
given. As usual, the Lagrange multiplier corresponds to the buyers’ marginal cost of acquiring
an extra unit of input, i.e., it equals the aggregate input price, in this case PtJ . Following the
same steps as in Box 1 we obtain
ZtJF (j, f )
PtJ
Z
=
1
=
PtJ (j)
PtJ
−σJ
J
PtJ (j)1−σ dj
ZtJF (f )
(126)
1
1−σ J
0
Integrating equation (126) over all distributors f and summing across F yields the demand for
variety j
ZtJ (j)
=
PtJ (j)
PtJ
−σJ
ZtJ
(127)
where ZtJ is the aggregate demand for sector J intermediate good.
Each manufacturing firm j operating in sector J combines labor services UtJ (j) with capital
KtJ (j) according to the following production function
ZtJ (j) ≡ FtJ KtJ (j), UtJ (j) =
ξJ
1
ξJξ−1
1
ξJξ−1 ξJ −1
J ξJ
J
J
J
J
= 1 − αU
Kt (j) J + αU ξJ Tt At Ut (j) J
(128)
where ξJ ≥ 0 is the elasticity of substitution between capital an labor for a firm operating
J ≤ 1 is a distribution parameter. The element AJ is a temporary
in sector J and 0 ≤ αU
t
labor-augmenting technology shock specific to sector J. The marginal productivity of labor and
capital are respectively
1
1
ξJ −1 −1
∂FtJ
J ξJ
ZtJ (j) ξJ Tt AJt UtJ (j) ξJ
= αU
Tt AJt = Tt AJt
J
∂Ut (j)
ξ1
1
ξJξ−1 −1
∂FtJ
J ξJ
J
J
J
J
=
1
−
α
Z
(j)
K
(j)
=
t
U
t
∂KtJ (j)
J Z J (j)
αU
t
Tt AJt UtJ (j)
ξ1
J
!1
J Z J (j) ξJ
1 − αU
t
KtJ (j)
Since all manufacturing firms in sector J solve the same optimization problem, the equilibrium
54
is symmetric and one can drop the indexer j when solving for the equilibrium. Equation (128)
in stationary form boils down to
ŽtJ
≡
F̌tJ
ǨtJ , UtJ
=
1−
J
αU
1−
J
αU
ξ1
J
ǨtJ
ξ1
J
ξJξ−1
J
+
J
αU
+
J
αU
ξ1
J
AJt UtJ
ξJξ−1
ξ ξJ−1
ξJξ−1
ξ ξJ−1
J
J
(129)
In the steady state
J
Ž ≡ F̌
J
J
Ǩ , U
J
=
J
Ǩ
ξJξ−1
J
ξ1
J
J
A U
J
J
J
(130)
Moreover, in a symmetric equilibrium, the stationary form of labor and capital marginal productivities are
0
F̌tUJ
∂ F̌tJ
≡
= AJt
∂UtJ
0
F̌tKJ
∂ F̌tJ
=
≡
∂ ǨtJ
J J
αU
Žt
AJt UtJ
ξ1
J
(131)
J ! ξ1J
J
1 − αU
Žt
(132)
ǨtJ
The steady-state versions are
F̌
UJ 0
∂ F̌ J
≡
= ĀJ
∂U J
F̌
KJ 0
∂ F̌ J
≡
=
∂ Ǩ J
J J
Ž
αU
J
Ā U J
ξ1
J
(133)
J ! ξ1J
J
1 − αU
Ž
(134)
Ǩ J
where ĀJ is the steady-state value of the labor-augmenting technology shock. Following Ireland (2001) and Laxton and Pesenti (2003), price inflation of intermediate goods is subject to
Rotemberg-type quadratic adjustment costs, which generate inflation persistence

ΓPJ
t (j) =
PtJ (j)
J (j)
φPJ J  Pt−1
Zt  J
Pt−1
2
J
Pt−2
2

− 1
(135)
where φPJ is a sector specific scaling factor determining the magnitude of price adjustment costs
55
for firms operating in sector J. In stationary form equation (135) becomes
Γ̌PJ
t
φPJ J
=
Ž
2 t
πtJ
−1
J
πt−1
2
(136)
J
where πtJ = PtJ /Pt−1
is sector J (gross) inflation rate. In the steady state prices are constant
and therefore
Γ̌PJ = 0
(137)
On the real side, a sluggish adjustment of hours worked is also ensured through quadratic
adjustment costs
φUJ J
ΓUJ
U
t (j) =
2 t
!2
UtJ (j)
−1
J (j)
Ut−1
(138)
where φUJ is a sector specific scaling factor determining the magnitude of adjustment costs. In
stationary form and after dropping the indexer j equation (138) collapses to
ΓUJ
t
φUJ J
U
=
2 t
UtJ
−1
J
Ut−1
2
(139)
In the steady state hours worked are constant and therefore
ΓUJ = 0
(140)
Capital is accumulated by entrepreneurs and rented to manufacturers at a unitary nominal
rental rate of RtKJ . Thus, taxes on capital holdings are paid by entrepreneurs. Manufacturers
pay a profit tax τtK , identical to the one faced by entrepreneurs, and a social security tax, τtSP .
Dividends of manufacturer j operating in sector J are therefore
DtJ (j)
=(1 −
τtK )
PtJ (j)ZtJ (j) − RtKJ KtJ (j)
−(1 +
τtSP )Vt
UtJ (j)
+
ΓUJ
t (j)
−
PtJ
ΓPJ
t (j)
+ Tt $
J
(141)
where PtJ Tt $J is a quasi-fixed cost paid by manufacturers that counteracts the steady-state
economic profits arising from monopolistic competition and faciliates the calibration of the
model (particularly the depreciation rate and the ratio of investment to GDP).
Let pJt = PtJ /Pt denote the relative price of sector J intermediate good and rtKJ denote
the real rental rate of capital. The stationary form of equation (141) in real terms (i.e. after
56
dropping the indexer j and dividing by Pt Tt ) is
dˇJt = 1 − τtK
J J
KJ J
SP
J
UJ
J
PJ
J
pt Žt − rt Ǩt − (1 + τt )v̌t Ut + Γt
− pt Γ̌t + $
(142)
In the steady state adjustment costs are zero and thus
dˇJ = 1 − τ K
J J
KJ J
SP
J
J J
p Ž − r Ǩ − (1 + τ )v̌ · U − p $
(143)
The manufacturer operating in sector J and producing variety j sets labor demand UtJ (j),
J (j), and the price P J (j) in each period in order to maximize the present
capital demand Kt+1
t
discounted value of the dividends stream, subject to variety j demand in (127), to the production
technology in (128), and to adjustment costs in (135) and (138) (notice that at date t capital
KtJ (j) is already installed and cannot be changed). The problem is presented and solved in Box
12.
A useful relationship in what follows is the ratio of sector J’s inflation to final consumer
goods’ inflation
πtJ
pJ
= Jt
πt
pt−1
(144)
In the steady state relative prices are constant and the relationship becomes
πJ = π
(145)
Dividing equation (13.1) by 1 − τtK
1 − σ J ZtJ and using the definition of real interest rate
rt = it /πt+1 one obtains the optimal pricing rule
φPJ
σ J λJt
−1= J
J
σ − 1 pJt
σ −1
"
!
K
J ZJ
1 − τt+1
πtJ
θ πt+1
πtJ
t+1
−
1
−
J
J
πt−1
πt−1
1 − τtK rt πt+1 ZtJ
J
πt+1
−1
πtJ
!
J
πt+1
πtJ
#
The term λJt captures the real cost of producing one additional unit of the intermediate good.
Using (144), the stationary form of this equation becomes
σ J λJt
φPJ
−1= J
σ J − 1 pJt
σ −1
"
πtJ
−1
J
πt−1
K
J
1 − τt+1
πtJ
θ · g pJt+1 Žt+1
−
J
πt−1
1 − τtK rt pt ŽtJ
57
J #
J
πt+1
πt+1
−1
(146)
πtJ
πtJ
Box 12: Manufacturer maximization problem
J
J
J
The manufacturer selects the triplet {Pt+s
(j), Ut+s
(j), Kt+s+1
(j)}∞
s=0 that maximizes the present discounted value of the dividends stream, subject to variety j demand in (127), to the production technology in (128), and to adjustment costs in (135) and (138). The Lagrangian is
L(·) =Et
∞
X
(
K
τt+s
R̃t,s 1 −
J
Pt+s
(j)
s=0
−(1 +
SP
τt+s
)Vt+s
J
−Pt+s
J
Ut+s
(j)
φPJ J
Z
2 t+s
"
−Pt+s λJt+s (j)
J
Pt+s
(j)
J
Pt+s
φUJ J
+
U
2 t+s
−σJ
J
KJ
J
Zt+s
− Rt+s
Kt+s
(j)
2 !
J
Ut+s
(j)
−1
J
Ut+s−1
(j)
!
2
J
J
Pt+s−1
Pt+s
(j)
J
÷ J
− 1 + Tt+s $
J
Pt+s−1
(j) Pt+s−2
J
Pt+s
(j)
J
Pt+s
−σJ
#)
J
Zt+s
−
J
Ft+s
J
J
Kt+s
(j), Ut+s
(j)
where Pt+s λJt+s (j) reflects the (nominal)marginal cost of producing an additional unit of the interJ
J
J
(j) is the production function defined in (128). The first-order
(j), Ut+s
Kt+s
mediate good and Ft+s
conditions are as follows (we drop the expected value operator).
1. Optimal pricing rule


∂L(·)
= 0 ⇔ 1 − τtK  1 − σ J
J
∂Pt (j)
+Pt λJt (j)
1−
τtK
σ
J
PtJ (j)
PtJ
−σJ


ZtJ − φPJ ZtJ 
J
Pt−1
J
Pt−2


− 1 
PtJ
J (j)
Pt−1
J
Pt−1
J
Pt−2



−σJ −1
PtJ (j)
ZtJ
(PtJ )−σJ
 PJ
+
PtJ (j)
J (j)
Pt−1
J
θ
K
J 
1 − τt+1
Pt+1 φPJ Zt+1
it
t+1 (j)
PtJ (j)
PtJ
J
Pt−1

J
  Pt+1
(j)
2
J
 (Pt (j)) 
− 1  P J  = 0
t
J
Pt−1
where we used the fact R̃t,1 = θ/it . As the equilibrium is symmetric the indexer j can be dropped and
the previous expression collapses to
"
J
J #
J
πt
πt
λJt
K
J
J
1 − τt
1 − σ Zt −φPJ Zt
−
1
+
1 − τtK σ J ZtJ
J
J
J
πt−1
πt−1
pt
+
J
θ
K
J
1 − τt+1
πt+1 φPJ Zt+1
it
J
J
πt+1
πt+1
−
1
=0
πtJ
πtJ
(13.1)
J
where we used the facts pJt = PtJ /Pt and πtJ = PtJ /Pt−1
.
2. Labor demand
J
Ut (j)
UtJ
∂FtJ
J
1+
Vt 1 + φUJ
−1
− Pt λt (j) J
J (j)
J (j)
Ut−1
Ut−1
∂Ut (j)
J
J
Ut+1 (j)
Ut+1 (j)
θ
K
SP
J
+
1 − τt+1
1 + τt+1
Vt+1 Ut+1
φUJ
−1
2 = 0
J
it
Ut (j)
UtJ (j)
∂L(·)
= 0 ⇔ − 1 − τtK
∂UtJ (j)
τtSP
58
since R̃t,1 = θ/it . In equilibrium the above expression can be rearranged to yield
J
J J
Ut
Ut
K
SP
K
J ∂Ft
− 1 − τt
1 + τt
Vt 1 + φUJ
−
1
+
1
−
τ
P
λ
t
t
t
J
J
Ut−1
Ut−1
∂UtJ
+
θ
K
SP
1 − τt+1
1 + τt+1
Vt+1 φUJ
it
J 2
J
Ut+1
Ut+1
−
1
=0
J
Ut
UtJ
(13.2)
3. Capital demand
J
∂Ft+1
∂L(·)
K
KJ
J
=
0
⇔
−(1
−
τ
=0
)
R
−
P
λ
(j)
t+1 t+1
t+1
t+1
J (j)
J (j)
∂Kt+1
∂Kt+1
KJ
⇔ Rt+1
= Pt+1 λJt+1 (j)
J
∂Ft+1
J (j)
∂Kt+1
K
given that τt+1
< 1. In equilibrium
KJ
Rt+1
= Pt+1 λJt+1
J
∂Ft+1
J
∂Kt+1
(13.3)
In the steady state adjustment costs are zero and thus
pJ =
σJ
λJ
−1
(147)
σJ
These equations stress that sector J prices are a markup over the marginal cost. Outside the
steady state, the markup is influenced by price adjustment costs.
Dividing (13.2) by 1 − τtK 1 + τtSP Vt , and using the facts rt = it /πt+1 , vt = Vt /Pt , and
Pt+1 = πt+1 Pt , one obtains the equation defining labor demand
0
λJt
FtUJ − 1 =φUJ
SP
1 + τt
vt
UtJ
−1
J
Ut−1
!
UtJ
J
Ut−1
K 1 + τ SP
θ · φUJ 1 − τt+1
t+1 vt+1
−
K
rt
1 − τt 1 + τtSP vt
J
Ut+1
−1
UtJ
!
J
Ut+1
UtJ
!2
where (FtUJ )0 ≡ ∂FtJ /∂UtJ . Using (58) leaded one period we obtain the following stationary
form for labor demand
"
0
J
J
J 2 #
K
SP
V
λJt · F̌tUJ
Ut+1
1 + τt+1
πt+1
Ut+1
UtJ
Ut
θ 1 − τt+1
−1 = φUJ
−1
−
−1
(148)
J
J
K
SP
J
SP
Ut−1
Ut−1 rt 1 − τt 1 + τt πt+1
Ut
UtJ
1 + τt
v̌t
In the steady state adjustment costs are zero and the previous equation collapses to
0
1 + τ SP v̌ = λJ F̌ UJ
(149)
59
This equation states that labor demand is a function of its price, 1 + τ SP v̌, the intermediate
goods marginal cost, λJ , and the intermediate goods demand, captured implicitly by (F̌ UJ )0 .
An increase in the relative price of labor vis-á-vis the marginal cost must be compensated by
an increase in labor marginal productivity, which is achieved by substituting capital for labor.
Outside the steady state one must also take into account quantity adjustment costs. Finally,
the stationary form of equation (13.3) defining manufacturers capital demand is
rtKJ = λJt F̌tKJ
0
(150)
The steady-state version is
rKJ = λJ F̌ KJ
0
(151)
The interpretation of these two equations are analogous to that of (148) and (149), except that
outside the steady state manufacturers do not face capital adjustment costs.
3.5.4
Distributors
Distributors produce four types of final goods: Consumption goods (C), Investment goods (I),
Government consumption goods (G), and Export goods (X ). For each type of final good F ∈
{C, G, I, X } there is a continuum of distributors f ∈ [0, 1], each producing a specific variety of
the good. Distributors use the same technology irrespective of the sector where they operate.
Production is divided into a two stage process. In the first stage, a distributor producing variety
f of the final good F combines domestic tradable goods ZtT F (f ) with a bundle of differentiated
imported goods MtF (f ), through a CES technology. This stage yields a differentiated tradable
good YtAF (f ), henceforth termed assembled good. In the second stage, the same distributor
combines the assembled good with domestic nontradable goods ZtN F (f ) to obtain the variety
f of type-F final good, YtF (f ). Distributors are perfectly competitive in the input market and
monopolistic competitive in the output market, charging a markup to final costumers. Profits
are distributed to households in the form of dividends.
Each type of final good is demanded by a unique type of costumer: private consumption goods
are demanded by households, government consumption goods by the government, investment
goods by capital goods producers, and export goods by foreign agents. For each costumer type
E ∈ {C, G, I, X } there is a continuum of agents e ∈ [0, 1], demanding many varieties of the
good. Let YtF (f, e) stand for the time t quantity of variety f from the final good F purchased
by costumer e. Each agent bundles the different varieties of the final good together to form an
homogeneous consumption good YtF (e) according to the CES specification
YtF (e) =
Z
1
σ F −1
YtF (f, e) σF df
σF
σ F −1
(152)
0
60
where σ F ≥ 0 is the elasticity of substitution between varieties of type-F final good. The
demand for each variety by a particular costumer, YtF (f, e), is obtained by minimizing the costs
of acquiring final goods, subject to the fact that the bundle in (152) must be achieved
1
Z
PtF (f )YtF (f, e)df
min
YtF (f,e)
s.t.
YtF (e) =
0
Z
1
σ F −1
YtF (f, e) σF df
σF
σ F −1
0
The element PtF (f ) is the price charged by the distributer of variety f operating in sector F .
Following the same steps as in Box 1 yields
YtF (f, e)
PtF
=
1
Z
=
PtF (f )
PtF
−σF
1−σ F
PtF (f )
df
YtF (e)
(153)
1
1−σ F
0
This same optimization problem was already solved for households, although with a different
H (c) corresponds to the consumption of variety c by a type-H household
notation. Therein, Ca,t
with age a at time t, which is exactly equivalent to YtC (f, e). In this case, f = c, and e indexes
the household.
Integrating (153) over e yields the demand for variety f faced by a distributor operating in
sector F
YtF (f )
=
PtF (f )
PtF
−σF
YtF
(154)
where YtF is the aggregate demand for sector F final good.
The production process is divided in two interdependent stages: the assemblage stage and
the distribution stage. In the assemblage stage, domestic tradable goods are combined with
imported goods, yielding a bundle of assembled goods. In the second stage, assembled goods
are combined with domestic nontradable goods to obtain the final good.
The assemblage stage.
domestic tradable goods
In this stage, the distributor uses a CES technology to combine
ZtT F (f )
with differentiated imported goods MtF (f ). This process yields
a composite differentiated good—the assembled good.
We obtain a realistic pattern for the import content by imposing quadratic adjustment costs
with the following specification
ΓAF
t (f )
2
AAF
φAF
t (f ) − 1
=
2 1 + AAF (f ) − 1 2
t
(155)
where φAF is a sector specific scaling factor and
AAF
t (f ) =
MtF (f )/YtAF (f )
F /Y AF
Mt−1
t−1
61
Given that all distributors in sector F are identical and face the same decision problem (see Box
13), the equilibrium is symmetric. Dropping the indexer f and dividing trend variables by the
level of technology yields adjustment costs in stationary form
Γ̌AF
t
2
ǍAF
−1
φAF
t
=
2 1 + ǍAF − 1 2
t
(156)
with ǍAF
given by
t
ǍAF
=
t
M̌tF /Y̌tAF
F /Y̌ AF
M̌t−1
t−1
(157)
These equations state that adjustment costs at t increase with the change in the share of the
import content in the assembled good. In the steady state the import share is constant and
therefore
Γ̌AF = 0
(158)
ǍAF
=0
t
(159)
Each distributor f operating in sector F combines domestic tradable goods with imported goods
according to the following production function
YtAF (f ) ≡FtAF ZtT F (f ), MtF (f ), ΓAF
t (f ) =
+ 1−
AF
αZ
1
ξAF
AF
αZ
F
ξAξF −1
AF
Mt (f ) 1 − ΓAF
t (f )
1
ξAF
ZtT F (f )
ξAF −1
ξAF
ξAF
ξAF −1
+
(160)
where ξAF ≥ 0 is the elasticity of substitution between domestic tradable goods and imported
AF ≤ 1 is the home bias parameter. In
good for a distributor operating in sector F and 0 ≤ αZ
stationary form this equation boils down to
Y̌tAF
=
AF
αZ
ξ1
AF
ŽtT F
ξAξF −1
AF
+ 1−
AF
αZ
+ 1−
AF
αZ
ξ1 AF
M̌tF
1−
Γ̌AF
t
ξAξF −1
ξ ξAF−1
ξAξF −1
ξ ξAF−1
AF
AF
(161)
In the steady state
Y̌
AF
=
AF
αZ
ξ1
AF
Ž
TF
ξAξF −1
AF
ξ1 AF
62
M̌
F
1 − Γ̌
AF
AF
AF
(162)
Each distributor selects the quantities {ZtT F (f ), MtF (f )} in each period so as to minimize the
cost of producing the assembled good, subject to adjustment costs in (155) and to the production
technology in (160). The problem is presented and solved in Box 13.
Dividing equation (14.1) by Tt and dropping the indexer f we obtain the demand for domestic
tradable goods in stationary form
AF
ŽtT F = αZ
pTt
λAF
t
−ξAF
Y̌tAF
(163)
where pTt = PtT /Pt is the relative price of the tradable good and λAF
stands for the real
t
marginal cost. This equation states that the demand for tradable domestic goods is a function
of the relative price of this input vis-á-vis the marginal cost. The steady-state version is
Ž
TF
=
AF
αZ
pT
λAF
−ξAF
Y̌ AF
(164)
A similar procedure for equation (14.2) yields the demand for imported goods in stationary form
M̌tF
1−
Γ̌AF
t
= 1−
AF
αZ
et
λAF
· ι̌AF
t
t
−ξAF
Y̌tAF
(165)
where the element ι̌A,F is
ι̌AF
t
=1−
Γ̌AF
t
− 1 ǍAF
ǍAF
t
t
− φAF h
2 i2
1 + ǍAF
−1
t
(166)
The interpretation of (165) is analogous to that of (163), except that one must also take into
account the costs of adjusting the import content of the assembled good. In the steady state
AF
M̌ F = 1 − αZ
e −ξAF AF
Y̌
λAF
(167)
and
ι̌AF = 1
(168)
The distribution stage. In the second stage, the distributor combines assembled goods
YtAF (f ) with nontradable goods ZtN F (f ) bought from manufacturers to produce a final good
63
Box 13: The distributor assemblage stage
The distributor selects the quantities {ZtT F (f ), MtF (f )} in each period that minimize the cost of producing a given amount of the assembled good, subject to adjustment costs in (155) and to the production
technology in (160). Let
AF
AF
F
AF
AF
ΓAF
A
and
AAF
(f
)
=
Γ
M
(f
),
Y
(f
)
MtF (f ), YtAF (f )
t
t
t
t
t
t (f ) = At
The Lagrangian is
"
L(·) =PtT ZtT F (f ) + Pt∗ εt MtF (f ) − Pt λAF
t (f )
+ 1−
AF
αZ
ξ1 F
ξAξF −1
AF
AF
Mt (f ) 1 − ΓAF
t (f )
AF
αZ
!
ξ1
AF
ξAF −1
ZtT F (f ) ξAF
ξAF
ξAF −1
#
−
YtAF (f )
where Pt∗ is the price of the imported good in foreign currency and λAF
t (f ) stands for the real cost of
producing an extra unit of type-F assembled good. In what follows, we treat ΓAF
t (f ) separately.
For later reference we compute the marginal product of domestic tradable goods and imported goods
1
1
ξAF −1 −1
1
∂FtAF
AF ξAF
AF ξAF
= YtAF (f ) ξAF αZ
ZtT F (f ) ξAF
= αZ
T
F
∂Zt (f )
YtAF (f )
ZtT F (f )
ξ1
AF
1 F
ξAξF −1 −1 AF
ξ1
∂FtAF
AF ξAF
AF
AF
AF
AF
ιt (f )
1
−
α
M
(f
)
1
−
Γ
(f
)
=
Y
(f
)
Z
t
t
t
∂MtF (f )
! ξ1
AF
AF
1
(f
)
Y
t
AF ξAF
= 1 − αZ
ιAF
t (f )
MtF (f ) 1 − ΓAF
t (f )
where
ιAF
t (f )
=1−
ΓAF
t (f )
−
MtF (f )
∂ΓAF
t
∂MtF (f )
−1
F
AF
F
Noticing that ∂AAF
, the derivative of adjustment costs with respect
t /∂Mt (f ) = At (f ) · Mt (f )
to imported goods is
AF
−1 h
2 i
AF
F
AF
A
(f
)
−
1
A
(f
)
·
M
(f
)
1
+
A
(f
)
−
1
AF
t
t
t
t
∂Γt
=φAF
h
2 i2
∂MtF (f )
1 + AAF
t (f ) − 1
2
AF
−1
F
AAF (f ) − 1 AAF
t (f ) − 1 At (f ) · Mt (f )
− t
h
2 i2
1 + AAF
t (f ) − 1
AF
AAF
φAF
t (f ) − 1 At (f )
= F
h
2 i 2
Mt (f )
1 + AAF
(f
)
−
1
t
64
The first-order conditions of the optimization problem are as follows.
1. Domestic tradable goods demand
∂FtAF
∂FtAF
pT
∂L(·)
= 0 ⇔ PtT − Pt λAF
=0⇔
= AFt
⇔
t (f )
T
F
T
F
T
F
∂Zt (f )
∂Zt (f )
∂Zt (f )
λt (f )
AF
1
−ξAF
1
Yt (f ) ξAF
pTt
pTt
AF ξAF
TF
AF
⇔ αZ
YtAF (f )
= AF
⇔ Zt (f ) = αZ
ZtT F (f )
λt (f )
λAF
(f
)
t
(14.1)
where pTt = PtT /Pt is the relative price of the tradable good.
2. Imported goods demand
∂L(·)
∂FtAF
∂FtAF
et
= 0 ⇔ εt Pt∗ − Pt λAF
=0⇔
= AF
⇔
t (f )
F
F
∂Mt (f )
∂Mt (f )
∂MtF (f )
λt (f )
! ξ1
AF
1
YtAF (f )
et
AF ξAF
⇔ 1 − αZ
ιAF
t (f ) = AF
F
AF
λt (f )
Mt (f ) 1 − Γt (f )
We used the definition of real exchange rate et = εt Pt∗ /Pt . The first-order condition can be further
simplified to
1
− 1
et
AF − ξAF
1 − αZ
YtAF (f ) ξAF ⇔
AF
· ιt (f )
−ξAF
et
AF
⇔ MtF (f ) 1 − ΓAF
(f
)
=
1
−
α
YtAF (f )
t
Z
AF
λAF
t (f ) · ιt (f )
F
− ξ 1
AF =
Mt (f ) 1 − ΓAF
t (f )
λAF
t (f )
(14.2)
YtF (f ), according to the following CES technology
YtF (f ) =FtF YtAF (f ), ZtN F (f )
ξF
1
1
ξF −1
ξF −1 ξF −1
F ξF
F ξF
AF
NF
ξF
ξF
= 1 − αZ
+ αZ
Yt (f )
Zt (f )
(169)
where ξF ≥ 0 is the elasticity of substitution between assembled goods and nontradable goods
F ≤ 1 is the nontradable goods bias parameter. The final good Y F (f ) is thereafter
and 0 ≤ αZ
t
sold at the price PtF (f ). In stationary form this production technology boils down to
Y̌tF
=
1−
F
αZ
ξ1
F
Y̌tAF
ξFξ−1
F
+
F
αZ
+
F
αZ
ξ1
F
ŽtN F
ξ1
NF
ξFξ−1
ξFξ−1
ξF
ξF −1
F
(170)
The steady-state version is
Y̌
F
=
1−
F
αZ
ξ1
F
Y̌
AF
ξFξ−1
F
F
Ž
F
ξF
ξF −1
(171)
Distributors must pay adjustment costs when updating prices, according to the following quadratic
65
specification

ΓPF
t (f ) =
PtF (f )
F (f )
φPF F  Pt−1
Yt  F
Pt−1
2
F
Pt−2
2
(172)

− 1
where φPF is a sector specific scaling factor determining the magnitude of price adjustment costs
for firms operating in sector F . These adjustment costs create a nominal rigidity that leads to
price stickiness in final goods. In stationary form adjustment costs are
Γ̌PF
=
t
φPF F
Y̌
2 t
2
πtF
−
1
F
πt−1
(173)
F is sector F (gross) inflation rate. In the steady state final goods inflation
where πtF = PtF /Pt−1
is constant and therefore
Γ̌PF = 0
(174)
Analogously to manufacturers, distributors pay a quasi-fixed cost totaling PtF Tt $F , which ensures that economic profits arising from monopolistic competition are zero in the steady state.
Dividends of distributor f operating in sector F are
h
AF
DtF (f ) = (1 − τtD ) PtF (f )YtF (f )−Pt λAF
(f )
t (f )Yt
F
−PtN ZtN F (f ) − PtF ΓPF
t (f ) + Tt $
i
(175)
Notice that distributors pay for the assembled good that they produced in the assemblage stage
and for the nontradable good bought from manufacturers—in addition to adjustment and quasiF
F
fixed costs. The real marginal cost of the assembled good corresponds to λAF
t . Let pt = Pt /Pt
N
denote the relative price of sector F final good and recall that the element pN
t = Pt /Pt ,
introduced in Section 3.5.3, is the relative price of the nontradable good. The stationary form
of (175) in real terms (i.e. after dropping the indexer f and dividing by Pt Tt ) thus becomes
F F
D
AF AF
NF
PF
dˇF
− pN
− pF
+ $F
t = (1 − τt ) pt Y̌t − λt Y̌t
t Žt
t Γ̌t
(176)
In the steady state adjustment costs are zero and hence
dˇF = (1 − τ D ) pF Y̌ F − λAF Y̌ AF − pN Ž N F − pF $F
(177)
Distributor f operating in sector F sets assembled goods utilization YtAF (f ), nontradable goods
demand ZtN F (f ), and final goods price PtF (f ) in each period in order to maximize the present
66
discounted value of the dividends stream, subject to the constraints imposed by variety f demand
in (154), by the production technology in (169), and by adjustment costs in (172). The problem
is presented and solved in Box 14.
A useful relationship in what follows is the ratio of sector F inflation to final consumer goods
inflation
πtF
pF
= Ft
πt
pt−1
(178)
In the steady state prices are constant and the relationship becomes
πF = π
(179)
Dividing equation (15.1) by 1 − τtD
1 − σ F YtF and using the definition of real interest rate
rt = it /πt+1 one obtains the optimal pricing rule
σ F λFt
φPF
−1= F
σ F − 1 pFt
σ −1
"
!
F YF
D
1 − τt+1
πtF
πtF
θ πt+1
t+1
−
1
−
F
F
πt−1
πt−1
1 − τtD rt πt+1 YtF
F
πt+1
−1
πtF
!
F
πt+1
πtF
#
The term λFt captures the real cost of producing one additional unit of the final good. Using
(178), the stationary form of this equation becomes
φPF
σ F λF
t
−1= F
σ F − 1 pF
σ
−1
t
"
πtF
−1
F
πt−1
D
F
1 − τt+1
πtF
θ · g pF
t+1 Y̌t+1
−
F
πt−1
1 − τtD rt pt Y̌tF
F
πt+1
−1
πtF
F
πt+1
πtF
#
(180)
In the steady state adjustment costs are zero and thus
pF =
σF
λF
−1
(181)
σF
These equations stress that sector F prices are a markup over the marginal cost. Outside the
steady state, the markup is influenced by price adjustment costs.
Equation (15.2) in stationary form becomes
Y̌tAF
= 1−
F
αZ
λAF
t
λF
t
−ξF
Y̌tF
(182)
This equation states that the assembled good utilization is a function of the relative (marginal)
F
F
cost of the assembled good vis-à-vis the final good, λAF
t /λt , and the final goods demand Y̌t .
An increase in the relative cost of the assembled good leads to a substitution effect towards
67
Box 14: Distributor maximization problem
F
AF
NF
The distributor selects the triplet {Pt+s
(f ), Yt+s
(f ), Zt+s
(f )}∞
s=0 that maximizes the present discounted value of the dividends stream, subject to variety h demand in (154), to technology in (169),
and to adjustment costs in (172). The Lagrangian can be written as
L(·) =Et
∞
X
(
D
τt+s
R̃t,s 1 −
F
Pt+s
(f )
s=0
N
NF
−Pt+s
Zt+s
(f )
−
"
−Pt+s λF
t+s (f )
φPF F
Y
2 t+s
F
Pt+s
F
Pt+s
(f )
F
Pt+s
−σF
F
Pt+s
(f )
F
Pt+s
−σF
F
AF
Yt+s
− Pt+s λAF
t+s (f )Yt+s (f )
!
2
F
F
Pt+s−1
Pt+s
(f )
F
÷ F
− 1 + Tt+s $
F
Pt+s−1
(f ) Pt+s−2
#)
F
Yt+s
−
F
Ft+s
AF
NF
Yt+s
(f ), Zt+s
(f )
where Pt+s λF
t+s (f ) reflects the (nominal) marginal cost of producing an additional unit of the final
F
AF
NF
good and Ft+s
(Yt+s
(f ), Zt+s
(f )) is the production function defined in (169). For latter reference we
compute the marginal product of assembled goods and nontradable goods
! ξ1
F
F
1 − αZ
Yt (f ) F
YtAF (f )
∂FtF
=
∂YtAF (f )
∂FtF
=
∂ZtN F (f )
F F
Yt (f )
αZ
N
Zt F (f )
ξ1
F
The first-order conditions are as follows (we drop the expected value operator).
1. Optimal pricing rule


∂L(·)
= 0 ⇔ 1 − τtD  1 − σ F
∂PtF (f )
+Pt λF
t (f )
1−
τtD
σ
F
PtF (f )
PtF
−σF


YtF − φPF YtF 
F
Pt−1
F
Pt−2


− 1 
PtF
F (f )
Pt−1
F
Pt−1
F
Pt−2



−σF −1
PtF (f )
YtF
σ F
F
Pt
 PF
+
PtF (f )
F (f )
Pt−1
F
θ
D
F 
1 − τt+1
Pt+1 φPF Yt+1
it
t+1 (f )
PtF (f )
PtF
F
Pt−1

F
  Pt+1
(f )
2
F
 (Pt (f )) 
− 1  P F
=0
t
F
Pt−1
where we used the fact that R̃t,1 = θ/it . In equilibrium the indexer f can be dropped and the previous
expression collapses to
"
F
F #
F
πt
λF
πt
t
D
F
F
−
1
+
1 − τtD σ F YtF
1 − τt
1 − σ Yt −φPF Yt
F
F
F
πt−1
πt−1
pt
+
F
θ
D
F
1 − τt+1
πt+1 φPF Yt+1
it
F
F
F
F
where we used the facts pF
t = Pt /Pt and πt = Pt /Pt−1 .
68
F
πt+1
−1
πtF
F
πt+1
=0
πtF
(15.1)
2. Assembled good utilization
! ξ1
F
F
1 − αZ
Yt (f ) F
∂L(·)
∂FtF
λAF
t (f )
AF
F
= 0 ⇔ λt (f ) = λt (f )
⇔ F
=
⇔
∂YtAF (f )
∂YtAF (f )
λt (f )
YtAF (f )
AF
−ξF
λt (f )
AF
F
YtF (f )
⇔ Yt (f ) = (1 − αZ )
λF
t (f )
In equilibrium
F
YtAF = (1 − αZ
)
λAF
t
λF
t
−ξF
YtF
(15.2)
3. Nontradable good demand
∂LF
∂FtF
pN
t
= 0 ⇔ PtN = Pt λF
⇔ Ft
=
t (f )
N
F
N
F
∂Zt (f )
∂Zt (f )
λt (f )
N −ξF
pt
NF
F
⇔ Zt (f ) = αZ
YtF (f )
F
λt (f )
F F
αZ
Yt (f )
N
Zt F (f )
ξ1
F
⇔
In equilibrium
F
ZtN F = αZ
pN
t
λF
t
−ξF
YtF
(15.3)
nontradable goods. In the steady state
F
Y̌ AF = 1 − αZ
λAF
λF
−ξF
Y̌ F
(183)
Equation (15.3) in stationary form is
F
ŽtN F = αZ
pN
t
λF
t
−ξF
Y̌tF
(184)
The interpretation of (184) is analogous to that of (182). In the steady state
Ž
3.6
NF
=
F
αZ
pN
λF
−ξF
Y̌ F
(185)
Fiscal and monetary policy authorities
The government buys from distributors a particular consumption good, Gt , and performs lumpsum transfers to households, TRG t . To finance expenditures, the government levies taxes on
labor income, τtL and τtSP , households’ consumption, τtC , and firms’ dividends, τtK and τtD ; and
receives transfers from abroad, TRE t . The government issues debt to finance expenditures not
69
covered by revenues, according to a fiscal rule that binds debt to a stationary path.
Government consumption operates as a pure inefficient good that does not affect agent
decisions or welfare. This simplifying setup naturally ignores many of the government roles in
the economy. First, the role as an employer is neglected. Second, the government is not allowed
to invest in public goods or in capital goods. The latter might complement private capital,
interacting with the production function of manufacturers. Third, the absence of unemployment
precludes the existence of unemployment benefits. These are important simplifications to keep
the model tractable, but they suffice to generate important debt dynamics and to analyze the
response of public debt to shocks.
Several government variables—namely government consumption, lump-sum transfers to households and tax rates—follow autoregressive processes with similar specifications.
Naturally,
net government transfers to both household types must add total government transfers, i.e.
ˇB
ˇ t = trg
ˇA
trg
t + trg t . It should be noted that PESSOA may consider transfers from type-A to
type-B households.
Government revenues arise from three major sources: consumption, labor, and dividends.
The real revenue accruing to the government from consumption taxes is, in stationary form
rv
ˇ Ct = τtC · (pCt Čt )
(186)
and in the steady state
rv
ˇ C = τ C · (pC · Č)
(187)
Revenues associated with employees labor taxes arise from two sources: households and labor
unions. The households’ tax base is w̌t Ut , whereas the unions’ tax base is (v̌ − w̌)Ut − Γ̌Vt .
Adding these two amounts yields
L
V
rv
ˇL
t = τt · (v̌t Ut − Γ̌t )
(188)
Since labor unions’ profits are distributed to households, the tax base equals the overall wage
received by households for their labor supply Ut net of adjustment costs paid by unions. In the
steady state with zero adjustment costs the employees labor tax revenue is
rv
ˇ L = τ L · (v̌ · U )
(189)
70
Labor taxes also include manufacturers’ social security contributions
rv
ˇ SP
= τtSP · (v̌t Ut )
t
(190)
The steady-state version is
rv
ˇ SP = τ SP · (v̌ · U )
(191)
In stationary form, tax revenues accruing from capital taxes on manufacturers are
τtK
h
X
i
J
KJ J
SP
J
UJ
−
$
−
r
Ǩ
−
(1
+
τ
)v̌
U
+
Γ
pJt ŽtJ − Γ̌PJ
t
t
t
t
t
t
t
J∈{T ,N }
and from entrepreneurs
τtK
h
X
ˇ Ji
rtKJ ǨtJ − a(uJt ) + δtJ pKJ
K̄t
t
J∈{T ,N }
This yields a revenue from taxes on capital of
K
rv
ˇK
t = τt
X
h
ˇ Ji
J
J
pJt ŽtJ − Γ̌PJ
− (1 + τtSP )v̌t UtJ + ΓU
− a(uJt ) + δtJ pKJ
K̄t (192)
t −$
t
t
J∈{T ,N }
In the steady state with zero adjustment costs
rv
ˇ K = τK
X
h
ˇJ
pJ (Ž J − $J ) − (1 + τ SP )v̌ · U J − δtJ pKJ K̄
t
i
(193)
J∈{T ,N }
Finally, from equation (176) it follows that tax revenues accruing from distributors are
D
rv
ˇD
t = τt
X
h
i
F
PF
F
T TF
F
N NF
pF
Y̌
−
Γ̌
−
$
−
p
Ž
−
M̌
−
p
Ž
t
t
t
t
t
t
t
t
t
(194)
F ∈{C,I,G,X }
The steady-state version is
rv
ˇ D = τD
X
h
pF (Y̌ F − $F ) − pT Ž T F − M̌ F − pN Ž N F
i
(195)
F ∈{C,I,G,X }
Transfers from abroad, TRE t , are assumed fully exogenous. Government’s revenue can therefore
71
be expressed in stationary form as
X
rv
ˇt =
ˇt
rv
ˇ xt + tre
(196)
x∈{C,L,
SP,K,D}
and in the steady state
X
rv
ˇ =
ˇ
rv
ˇ x + tre
(197)
x∈{C,L,
SP,K,D}
The government’s budget constraint states that the amount of government expenditures not
covered by revenues must be financed by issuing debt. In nominal terms this can be represented
by
Bt = it−1 Bt−1 + PtG Gt + TRG t − RV t
Dividing by Pt Tt yields the government’s budget constraint in stationary form
b̌t =
it−1
ˇ t − rv
b̌t−1 + pGt Ǧt + trg
ˇt
πt · g
(198)
Using i = π · r, the steady-state version can be expressed as
b̌ =
r
ˇ t − rv
b̌t−1 + pGt Ǧt + trg
ˇt
g
(199)
To ensure that public debt follows a non-explosive path, we introduce a fiscal rule, imposing a
ˇ t , and consequently of the debt–
convergence of the government surplus–to–GDP ratio, sg
ˇ t /GDP
ˇ t , to a predetermined (target) steady-state level. The fiscal rule implies
to–GDP ratio, b̌t /GDP
that at least one fiscal instrument must adjust endogenously. This choice is purely arbitrary
and any variable fully controlled by the government can play this role, but the literature tends
to set the labor income tax as the endogenous fiscal instrument (Harrison et al., 2005; Kilponen
and Ripatti, 2006; Kumhof and Laxton, 2007; Kumhof et al., 2010). In line with Kumhof and
Laxton (2009a), the following fiscal rule in stationary form is considered
sg
ˇt
=
ˇ t
GDP
sg
ˇt
ˇ t
GDP
target
+ d1
rv
ˇ t − rv
ˇ ss
t
ˇ t
GDP
+ d2
b̌t
−
ˇ t
GDP
b̌t
ˇ t
GDP
target !
+ d3 ln
GDPt
GDPt?
(200)
where rv
ˇ ss
t is the structural overall tax revenue—collected when tax bases stand at their steadyˇ t and GDP
ˇ ss
state levels—in stationary form; GDP
t are respectively observed and steady-state
GDP levels in stationary form, and GDPt? represents potential output, defined as a moving
72
target
average of past realizations. In the steady state with rv
ˇ t = rv
ˇ ss
, and GDPt = GDPt?
t , b̌t = b̌t
the fiscal rule collapses to
sg
ˇ
=
ˇ
GDP
sg
ˇ
ˇ
GDP
target
(201)
The fiscal policy rule takes into consideration government debt stabilization and business cycle
smoothness, adjusting automatically the endogenous tax rate to bind debt to a stationary path.
The parameter d1 reflects how the government adjusts the surplus-to-GDP ratio to cyclical
economic fluctuations. For instance, if d1 = 1, the fiscal rule has an intrinsic structural nature.
During a boom, when tax revenues are above the long run level, the government increases surplus
above target, paying off debt. During a slump, debt increases to allow for a government surplus
below the long run value. In this case, the fiscal rule mimics the automatic stabilizer mechanism,
minimizing the variability of the tax instrument and reducing output fluctuations throughout
the cycle vis-á-vis a balance budget rule (d1 = 0). Setting d1 < 0 yields a counter-cyclical fiscal
policy rule (i.e. a pro-cyclical government deficit), in which the government decreases surplus
during booms and vice-versa. On the opposite direction, setting d1 > 1 yields an aggressive
stabilization policy in which surplus is adjusted to minimize cyclical fluctuations. The parameter
d2 reflects the degree of aversion to deviations of the debt-to-GDP ratio from the target value.
Finally, parameter d3 determines the aggressiveness of the fiscal rule to the output gap.
Monetary policy is fully exogenous, since interest rates are set by an external monetary
authority and the home economy is assumed small enough to affect macroeconomic aggregates
in a MU.
3.7
Rest of the world
The home country imports differentiated goods MtF at the price Pt∗ , to be used in the distribu∗ , is assumed to follow
tor’s assemblage stage. The imported goods’ inflation rate, πt∗ = Pt∗ /Pt−1
an exogenous process.
In domestic currency the distributor pays t = p∗t εt for each unit of the imported good
relative to the numeráire of the economy, which equals the real exchange rate by definition.
Thus, the growth rate of imported goods’ inflation can be expressed as
t
t−1
=
p∗t εt
p∗t−1 εt−1
or equivalently
t
t−1
=
πt∗ εt
πt εt−1
(202)
73
In the steady-state
π∗ = π
(203)
Exports are essentially determined by foreign demand and price competitiveness. In a multicountry model, exports demand is endogenously determined, depending on relative comparative
advantages amongst countries. In a single country small open economy model the rest of the
world is not explicitly modeled, and thus one must find alternate strategies to determine exports
demand. We follow closely Adolfson et al. (2007a) and assume that in the rest of the world there
exists a continuum of distributors m ∈ [0, 1], who demand YtX (m) units of the final good from
domestic distributors operating in the exports sector, to be used in their assemblage stage. This
good is thereafter combined with foreign tradable goods ZtT ∗ (m) according to the following
production function
YtA∗ (m)
≡F
ZtT ∗ , YtX (m)
1
ξA∗ −1
= αYA∗ ξA∗ YtX (m) ξA∗
+ 1−
αYA∗
1
ξA∗
ξA∗ −1
ZtT ∗ (m)
ξA∗
ξA∗
ξA∗ −1
(204)
yielding YtA∗ (m) units of the assembled good. The home economy is assumed to be sufficiently
small so that domestic shocks do not affect the rest of the world. The element ξA∗ is the elasticity
of substitution between rest of the world tradable goods and domestic exports, and αA∗ is the
rest of the world home bias parameter.
Each foreign distributor selects the quantities {YtX (m), ZtT ∗ (m)} in each period so as to
minimize the cost of producing the assembled good, subject to the production function in (204).
The problem is solved in Box 15, yielding the following demand for domestic goods in stationary
form
YtX = αYA∗
pX
t
t
−ξA∗
YtA∗
(205)
Y A∗
(206)
In the steady state
Y X = αYA∗
pX
−ξA∗
An important condition resulting from interactions with the rest of the world is the net foreign
asset position
∗
εt Bt∗ = i∗t−1 Ψt−1 εt−1 Bt−1
+ PtX Xt − εt Pt∗ Mt + εt TRE t + TRX t
74
Box 15: The foreign distributor’s demand for export goods.
The foreign distributor selects the quantities {YtX (m), ZtT ∗ (m)} in each period that minimize the cost
of producing a given amount of the assembled good
min
YtX (m),ZtT ∗ (m)
PtT ∗ ZtT ∗ (m) +
PtX X
Y (m)
εt t
subject the production technology in (204). The Lagrangian is
"
ξA∗ −1
1
PtX X
A∗
T∗ T∗
Yt (m)−λt (m)
αYA∗ ξA∗ YtX (m) ξA∗
L(·) = Pt Zt (m) +
εt
ξA∗ #
1
ξAξ∗ −1 ξA∗ −1
A∗ ξA∗
T∗
+ 1 − αY
Zt (m) A∗
where λA∗
t (m) stands for foreign distributer’s marginal cost of producing an extra unit of the assembled
good. We assume that foreign distributors work under perfect competition, and thus the marginal cost
∗
equals the price they charge for their product, i.e. λA∗
t (m) = P , ∀m. The first-order condition of the
X
optimization problem with respect to Yt (m) is
1
PtX
∂L(·)
=
0
⇔
= Pt∗ αYA∗ ξA∗
X
εt
∂Yt (m)
YtA∗ (m)
YtX (m)
ξ1
A∗
⇔
YtX (m)
=
αYA∗
PtX
εt Pt∗
−ξA∗
YtA∗ (m)
The stationary form of the net foreign asset position is
t b̌∗t =
i∗t−1 Ψt−1 t−1 ∗
ˇ
ˇ
b̌t−1 + pX
t X̌t − t M̌t + t (tre t + trx t )
πt · g
(207)
and the steady-state version is
b̌∗ =
3.8
ˇ + trx)
ˇ
pX X − M̌ + (tre
i∗ Ψ
1 − π·g
(208)
Economic aggregates and market clearing conditions
We close the model by presenting economic aggregates and market clearing conditions.
Aggregate consumption. To obtain aggregate consumption, we add the respective optimal
quantities of the different consumer types
Čt = ČtA + ČtB
(209)
75
In the steady state
Č = Č A + Č B
(210)
Aggregate labor supply and demand. Similarly to aggregate consumption, aggregate labor
supply is
B
Lt = LA
t + Lt
(211)
In the steady state
L = LA + LB
(212)
In turn, aggregate labor demand is determined by manufacturers operating in the tradable and
non-tradable sectors. Outside the steady state, one as to take into consideration manufacturers’
adjustment costs, which are measured in terms of worked hours, even though they do not affect
production (examples include training and lay-offs), and unions’ adjustment costs
T
N
Ut = UtT + UtN + ΓU
+ ΓU
+ Γ̌Vt
t
t
(213)
In the steady state adjustment costs are zero and thus
U = UT + UN
Labor market clearing condition.
(214)
The labor market clearing condition is
Lt = Ut
(215)
which in the steady state becomes
L=U
(216)
76
Intermediate goods market clearing conditions.
In the intermediate goods’ market, the
output produced by manufacturers net of adjustment and fixed costs, and of the entrepreneurs’
output destroying net worth shock, must equal the demand from distributors
J
ŽtJ − Γ̌P
− rcu
ˇ Jt − $J − StJ,y = ŽtJC + ŽtJI + ŽtJG + ŽtJX , J ∈ T , N
t
(217)
Taking into account adjustment costs is necessary, since they correspond to real resources that
are consumed in the production process. Likewise for the resource cost associated with capital
utilization and the entrepreneurs’ output destroying net worth shock. In the steady state
Ž J − $J − S J,y = Ž JC + Ž JI + Ž JG + Ž JX , J ∈ T , N
(218)
where S J,y is the steady-state value for the entrepreneurs’ output destroying net worth shock.
Final goods market clearing conditions.
In the final goods sector, the output produced
by distributors net of adjustment costs must equal costumers’ demand
F
I
Y̌tF − ΓP
− 1I (F ) ΓTt I + ΓN
− $F = 1C (F )Čt + 1I (F )Iˇt + 1G (F )Ǧt + 1X (F )X̌t
t
t
(219)
where F ∈ {C, I, G, X }, I = I T + I N , and 1x (F ) is an indicator function which takes the value
of 1 if F ∈ x and 0 otherwise. Recall that capital goods producers face specific investment
adjustment costs, whereas the remaining agents do not face similar costs. In the steady state
Y̌tF − $F = 1C (F )Č + 1I (F )Iˇ + 1G (F )Ǧ + 1X (F )X̌, F ∈ {C, I, G, X }
(220)
Aggregate imports. To obtain total imports, we add the corresponding figures respecting
final goods distributors
M̌t = M̌tC + M̌tI + M̌tG + M̌tX
(221)
In the steady state
M̌ = M̌ C + M̌ I + M̌ G + M̌ X
(222)
77
Final output. GDP on the expenditure side, measured in terms of Pt , can be simply defined
as
ˇ t = Čt + pGt Ǧt + pIt Iˇt + pX
GDP
t X̌t − t M̌t
(223)
In the steady state, this equation becomes
ˇ = Č + pG Ǧ + pI Iˇ + pX X̌ − M̌
GDP
(224)
The definition of GDP is however not unique and alternative weighting schemes can be
followed, for instance using Fisher weights.
4
The model without financial frictions
This section pinpoints the major differences between the model with and the model without
financial accelerator and financial frictions. Without financial frictions, entrepreneurs and banks
can be drop altogether, implying the following adjustments to the model.
4.1
Households and labor unions
Type-A households do not lend money to banks, and therefore b̌Tt = b̌N
t = 0, ∀t. Additionally,
there are no bankruptcy monitoring services and entrepreneurs’ dividends are absent, implying
ˇ t = dˇA,EP = 0, ∀t.
that rbr
t
Labor unions behavior is identical to the model with financial frictions.
4.2
Firms
Taxes for holding capital are transferred to capital goods producers, who now produce, accumulate and rent capital to manufacturers. The remaining firms behave identically.
In addition to investment, capital goods producers now also decide capital utilization. As
before, they are perfectly competitive in input and output markets, but now they pay the
capital-based income tax, which was previously paid by entrepreneurs. In each period, capital
goods producers supplying sector J purchase previously installed capital from manufacturing
firms K̄tJ and investment goods from investment goods producers ItJ to produce new installed
J (i), according to the same aggregate law of motion presented in (75). In equilibrium,
capital K̄t+1
capital accumulation is given by (76), and in the steady state by (77).
Capital goods producers face the same investment adjustment costs as before, presented in
equation (66). In stationary form, these are given by (67), and in the steady state by (68).
Since they also select the capital utilization rate, they face the same costs of capital utilization
as entrepreneurs in the version with financial frictions, given by (78). In stationary form and
in the steady state these are given respectively by (79) and (80). The capital stock effectively
used in production is given by (81) in stationary form, and by (82) in the steady state.
78
The after-tax dividends of capital goods producers operating in sector J are defined as the
difference between operational cashflows and net operational profits
DtKJ (i) = RtKJ uJt (i) − Pt a uJt (i) K̄tJ (i) − PtI ItJ (i) + ΓIJ
t (i)
{z
}
|
Operational cashflows
n
o
−τtK RtKJ uJt (i) − Pt a uJt (i) − δtJ PtKJ K̄tJ (i) − PtI ΓIJ
(i)
t
|
{z
}
(225)
Net operational profits
where τtK is the capital income tax rate, PtI is the price of investment goods, and PtKJ is the price
of capital. The price of old and new capital is the same, since the marginal rate of transformation
between the two is assumed to be one.
The expression in (225) is designed to mimic the Portuguese framework. Operational cashflows include the overall revenue minus the overall expenditure. However, for fiscal purposes,
the relevant concept is net operational profits, which take into consideration that investment
is not considered for the tax base and that depreciation is tax deductible. In real terms, the
stationary form of equation (225) is
ˇJ
I IJ
ˇJ
dˇKJ
= (1 − τtK ) rtKJ uJt − a(uJt ) K̄
− pIt IˇtJ + τtK δtJ pKJ
t
t − p t Γt
t K̄t
(226)
In the steady-state, with uJt = 1 and a(1) = ΓIJ
t = 0, dividends become
ˇ J − pI IˇJ + τ K δ J pKJ K̄
ˇJ
dˇKJ = (1 − τ K )rKJ K̄
(227)
J (i), K̄ J
J
∞
Capital goods producers select the intertemporal profile {It+s
t+s+1 (i), ut (i)}s=0 that max-
imizes the present discounted value of the dividends stream, subject to investment adjustment
costs in (66), to the law of motion of capital in (75), and to capital utilization costs in (78). The
problem is presented and solved in Box 16.
The stationary form of equation (17.1) is
pKJ
=
t
θ
rt
K
1 − τt+1
KJ J
rt+1
ut+1 − a uJt+1
J
K
+ pKJ
1
−
δ
1
−
τ
t+1
t+1
t+1
(228)
where we used the definition of real interest rate rt = it /πt+1 and the expressions for relative
prices. In the steady state uJt+1 = 1 and a(1) = 0. Additionally, investment goods inflation
equals total inflation. Therefore, equation (228) simplifies to
p
KJ
θ
θ
J
K
1−
1−δ 1−τ
= 1 − τ K rKJ
r
r
79
Box 16: Capital goods producers maximization problem
J
J
The capital goods producer select the triplet {It+s
(i), K̄t+s+1
(i), uJt (i)}∞
s=0 that maximizes the present
discounted value of the dividends stream, subject to investment adjustment costs in (66), to the law of
motion of capital in (75), and to capital utilization costs in (78). The Lagrangian can be written as
(
(
∞
X
J
K
KJ J
J
L(·) =Et
R̃t,s (1 − τt+s ) Rt+s
ut+s (i)K̄t+s
(i) − Pt+s a uJt+s (i) K̄t+s
(i)
s=0
"
I
−Pt+s
φIJ J
I
2 t+s
−λJt+s
J
K̄t+s+1
(i)
2 # )
J
It+s
(i)/g
I
J
K
J
KJ
−1
− Pt+s
It+s
(i) + τt+s
δt+s
Pt+s
K̄t+s (i)−
J
It+s−1
(i)
)
− (1 −
J
J
δt+s
)K̄t+s
(i)
−
I
J
ζt+s
It+s
(i)
where the Lagrange multiplier λJt+s corresponds to the marginal cost of buying an extra unit of capital,
KJ
i.e. λJt+s = Pt+s
. The first-order conditions are as follows (we drop the expected value operator).
1. Capital supply
KJ J
∂Lt (·)
θ
K
) Rt+1
ut+1 (i) − Pt+1 a uJt+1 (i)
= 0 ⇔ PtKJ = (1 − τt+1
J
it
∂ K̄t+1 (i)
K
J
KJ
KJ
J
+τt+1
δt+1
Pt+1
+ Pt+1
(1 − δt+1
)
(17.1)
KJ
.
where we used the facts R̃t,1 = θ/it and λJt+s = Pt+s
2. Investment demand
ItJ (i)/g
1/g
−1
J (i)
J (i)
It−1
It−1
!
J
J
I
I
(i)/g
(i)/g
θ
t+1
t+1
K
I
J
−1
− (1 − τt+1
)Pt+1
φIJ It+1
2
it
ItJ (i)
ItJ (i)
∂Lt (·)
= 0 ⇔ PtKJ ζtI =PtI + (1 − τtK )PtI φIJ ItJ
∂ItJ (i)
(17.2)
3. Optimal capacity utilization
∂Lt (·)
∂a
= 0 ⇔ RtKJ = Pt J
⇔ RtKJ = Pt φJa σaJ uJt (i) + φJa (1 − σaJ )
J
∂ut (i)
∂ut (i)
(17.3)
where we used the expression for da/duJt presented in Box 7.
and hence
p
KJ
1 − τ K rKJ
=
r/θ − 1 − δ J (1 − τ K )
(229)
Equation (17.2) in stationary form becomes
I
pKJ
t ζt
= 1 + (1 − τtK )φIJ
pI
!
IˇtJ
−1
J
Iˇt−1
IˇtJ
J
Iˇt−1
!
πI
θ·g
K
−
(1 − τt+1
)φIJ t+1
rt
πt+1
80
J
Iˇt+1
−1
IˇtJ
!
J
Iˇt+1
IˇtJ
!2
(230)
where we used the relationship in (178) forwarded one period. Investment is constant in the
steady state and hence
pK = pI
(231)
The investment demand equation relates the real price of capital with investment specific shocks
and with investment adjustment costs. As pKJ
deviates from pIt , capital goods producers become
t
more prone to adjust capital and face the corresponding adjustment costs. More specifically,
I
a value of pKJ
t /pt above (below) unity means that new capital goods are relatively cheaper
(more expensive) vis-á-vis existing capital, and thus capital investment—and consequently the
investment to capital ratio—should increase (decrease).
Equation (17.3) in stationary form becomes
rtKJ = φJa σaJ uJt + φJa (1 − σaJ )
(232)
Thus, the rate of return on capital equals the marginal cost of using an additional unit of the
installed capital. In the steady-state uJ = 1 and
rKJ = φJa
(233)
Finally, the resource cost associated with variable capital utilization is given by (86) in stationary
form, and by (87) in the steady state.
4.3
Fiscal and monetary policy authorities
The government’s behavior is identical to the model with financial friction, with the exception
of capital tax revenues, which now incorporate investment adjustment costs faced by capital
goods producers
K
rv
ˇK
t = τt
h
X
i
ˇJ
J
J
pJt ŽtJ − Γ̌PJ
−(1+τtSP )v̌t UtJ +ΓU
− a(uJt )+δtJ pKJ
K̄t −pIt ΓIJ
(234)
t −$
t
t
t
J∈{T ,N }
In the steady state with zero adjustment costs revenues collapse to
rv
ˇ K = τK
X
h
ˇJ
pJ (Ž J − $J ) − (1 + τ SP )v̌ · U J − δtJ pKJ K̄
t
J∈{T ,N }
which is identical to (193).
81
i
(235)
Figure 4: Fiscal stimulus based on G
G
GSS + ∆
Permanent
GSS
Temporary
t0
4.4
t1
Time
Market clearing conditions
Market clearing conditions are identical to the model without financial frictions, with the exception that output destroying net worth shocks, StJ,y , are absent and hence they must be dropped
from equations (217) and (218).
5
Shocks
Any exogenous parameter of a dynamic general equilibrium model can be turned into a stochastic
variable, through an appropriate exogenous process. One possibility behind the law of motion
of a shock is to consider autoregressive formulations. In this case we allow for two different
possibilities, depending on the parameter. The first is based on the level of the variable, say xt
xt = (1 − ρ)x̄ + ρxt−1 + ext
where x̄ is the steady-state value, ρ is the persistence parameter, and ext stands for a time t i.i.d.
zero-mean innovation. This specification is used when variables are expressed as percentages
(e.g. tax rates), or when innovations are better interpreted as shocks to the level variable (e.g.
elasticities of substitution or entrepreneurs’ riskiness). The second is based on the logarithm of
the variable, say log xt
log xt = (1 − ρ) log x̄ + ρ log xt−1 + ext
This specification is preferred for variables expressed in levels where innovations should be interpreted as percentage changes (e.g. labour augmenting technological shocks). In the appendix
we present the specific formulation for several shocks that can be introduced in PESSOA.
Another possibility defining the exogenous process of a shock is to consider pre-determined
paths that are not captured by autoregressive formulations. This is exemplified in Figure 4. In
this case, we consider both a temporary and permanent increase in government consumption
G. The starting date t0 is the period when the shock is initiated; t1 is the ending date if the
stimulus is temporary; GSS is the steady-state level of government consumption; and ∆ is the
82
amplitude of the shock. The macroeconomic impacts of these exogenous processes have been
discussed in Almeida, Castro, Félix, and Maria (2010a).
6
Calibration
PESSOA is calibrated to match Portuguese and euro-area economies. Some parameters are
exogenously set by taking into consideration common options in the literature, available historical data or empirical evidence. Others are endogenously determined within the model, with
the objective of matching desired features, for instance the consumption- or investment-to-GDP
ratios. Table 1 in the appendix reports the main steady-state ratios, whilst Table 2 and Table
3 present the model’s non-financial and financial key parameters, respectively. A comparison
with actual data is included when available.
The annual growth rate of the labor-augmenting productivity is set to 2 percent, which is a
plausible estimate for potential output growth in both Portugal and the euro area (Almeida and
Félix, 2006; Musso and Westermann, 2005; Proietti and Musso, 2007). Steady-state inflation
stands at 2 percent per year and the euro area nominal interest rate at 4.5 percent (Coenen,
McAdam, and Straub, 2007). The risk premium is nil, implying Ψ = 1. The elasticity of
substitution between rest of the world tradable goods and domestic exports is assumed to be
large.
Households parameters are largely based on Fagan, Gaspar, and Pereira (2004), Harrison
et al. (2005), Kumhof and Laxton (2007) and Kumhof et al. (2010). Consumption shares, η A and
η B , are calibrated to ensure a unitary elasticity of labor supply to real wage. The intertemporal
elasticity of substitution is set to 0.5, implying a coefficient of relative risk aversion of 2. The
degree of habit persistence is 0.7 and the discount factor roughly 0.99. The instant probability
of death, θ, and the productivity decay rate, χ, are assumed to be identical, implying an average
lifetime and an expected working life of 25 years. The share of hand-to-mouth households is 40
percent, broadly in line with the estimates for Portugal presented in Castro (2006).
The steady-state wage markup of labor unions is 25 percent, which is relatively high for the
euro area, but in line with the figures for Portugal. Wage adjustment costs imply that wages
take roughly 5 quarters to adjust to the new equilibrium, a value in line with euro area estimates
published in Coenen, McAdam, and Straub (2007).
The depreciation rate of capital is assumed to be identical in the tradable and non-tradable
sectors, and is calibrated by taking into account actual data on the investment-to-GDP ratio.
The unitary elasticity of substitution between capital and labor in the production function
takes into account the actual labor income share. The steady-state price markup of tradable
and non-tradable goods is calibrated using OECD product market regulation indicators, as
well as the correlation between tradable and non-tradable goods markups and product market
regulation indicators found in Høj et al. (2007). The price markup of the non-tradable goods
is set at 20 percent, to capture low competition levels in this sector, whilst for the tradable
sector the price markup is 10 percent. Investment adjustment costs and labor adjustment costs
are parameterized so as to ensure plausible investment and labor dynamics, respectively, while
adjustment costs in price changes are calibrated to match reasonable average price adjustment
time spans. In particular, price adjustment in the non-tradable goods sector (4.5 quarters) is
83
slightly slower than in the tradable goods sector (roughly 3 quarters). Production functions
are calibrated in line with the National Accounts import contents and the non-tradable goods
content of each type of final good.
The elasticity of substitution between domestic tradable goods and imported goods is assumed to be identical across firms and set above unity (Coenen, McAdam, and Straub, 2007;
Harrison et al., 2005; Erceg, Henderson, and Levin, 2000; Kumhof et al., 2010). The degree
of monopolistic competition amongst distributors is lower than among manufacturers, with the
steady-state markup being set to 5 percent, except in the case of exporters, where fiercer competition justifies a lower markup. In terms of price stickiness, price adjustment costs imply an
adjustment to equilibrium in roughly 2 quarters for consumption goods and investment goods.
Prices in the export sector adjust faster, in slightly more than 1 quarter, while prices of government consumption goods take around 4 quarters to adjust. With the exception of government
goods, where the sizable wage share justifies a slower adjustment, these parameterizations lie
within the plausible range of price adjustment costs reported in Keen and Wang (2007). Adjustment costs in import contents are set to ensure that they move plausibly with real exchange rate
fluctuations. The combination of assembled goods with non-tradable goods in the production
of final goods is assumed to feature a low substitutability, in line with Mendoza (2005) and
Kumhof et al. (2010).
Steady-state tax rates, transfers from the rest of the euro area, government consumption, and
government transfers are calibrated to match actual data. The fiscal policy rule is parameterized
to ensure smooth adjustments—the exact calibration depending on the simulation (see Almeida
et al., 2013 for alternative configurations). The target debt is set to 60 percent of GDP, in line
with fiscal targets set in the European Union. This implies a steady-state fiscal balance of -2.4
percent.
In the financial sector of the model, tradable and non-tradable sectors have identical calibrations. The leverage ratio of entrepreneurs, B/N , the probability of default, F(ω̄), and the return
on capital, RetK , are approximated with aggregate Portuguese historical features. The leverage
ratio is 100 percent. The same value is used in Bernanke, Gertler, and Gilchrist (1999b) or
Kumhof et al. (2010). The probability of default—8 percent—is relatively close to the exit rates
reported in Mata, Antunes, and Portugal (2010), and in line with the value found in Kumhof
et al. (2010). The return on capital is approximately 9 percent, a spread of 4 percentage points
over the risk free rate. These figures imply a non-default loan rate, iB , of 6.5 percent, which is
close to the “debt cost ratio” published by the Banco de Portugal. This figure embodies a spread
of 1.7 percentage points over the risk free rate. The steady-state monitoring cost parameter is
11 percent, implying that on aggregate banks recover 89 percent of the value of bankrupt firms.
The monitoring cost level is relatively close to the “average unconditional loss” estimated by
Bonfim, Dias, and Richmond (2012) for the 1995–2008 period. Finally, the standard deviation
of idiosyncratic risk shock of 0.32 stands between the benchmark figures used by Christiano,
Motto, and Rostagno (2013) and Bernanke, Gertler, and Gilchrist (1999b), of 0.26 and 0.53
respectively.
One must stress that the model’s parameters reflect our best assessment of the economic
environment. They are naturally subject to uncertainty and may be updated if necessary to
84
reflect new or better information, or changes in the underlying environment. Introducing new
extensions into the model may also lead to parameter revisions. For instance, the incorporation
of financial frictions into the model leads to a sharp deterioration of the net foreign position
position of the economy if calibration remains unchanged. As households desire to hold exactly
the same amount of assets (relative to GDP), corporate bonds—which are introduced along
with financial frictions—draw funds from abroad on a one-to-one basis. Structural reforms may
justify also a re-calibration of the model. An effective labor market reform, for instance may
imply a lower steady-state wage markup in the medium run. The model’s parameters may also
be conditional on the specific issue that is being addressed. Castro et al. (2013) consider two
distinct alternative states of the economy, viz “normal times” and “crisis times.” The former
is based on the standard calibration of the model, while the latter embodies some changes in
parameters that may better reflect the underlying economic environment.
7
Economic analysis and policy simulations
PESSOA has been used to illustrate a few number of issues, most of them for the Portuguese
economy. In Almeida, Castro, and Félix (2009), the model is used to assess the impact of a
number of shocks that played an important role in Portuguese economic developments, namely:
a slowdown in total factor productivity, a decline in interest rates and a reduction in liquidity
constraints, a decrease of exports non-price competitiveness, and a short-lived fiscal boom and
subsequent fiscal consolidation.
In Almeida, Castro, and Félix (2010), the authors provide well-grounded support for structural reforms in Portugal, by showing that substantial increases in competition in the nontradable goods sector and in the labor market could induce important international competitiveness gains. These structural reforms could be valuable instruments in promoting necessary
adjustments for a small open economy integrated in a monetary union.
Almeida, Castro, Félix, and Maria (2010a) evaluate the impact of fiscal stimulus in a small
open economy within a monetary union, and conclude that increases in permanent government
expenditures should be avoided due to their negative welfare effects, as opposed to temporary
stimulus. This result creates room for welfare improving stabilization policies. Nevertheless, a
temporary stimulus which triggers a hike in the country’s risk premium due to high indebtedness
levels may have a negligible effect on welfare.
In Almeida, Castro, Félix, and Maria (2013), PESSOA is used to show that a fiscal consolidation strategy based on a permanent reduction in Government expenditure increases the
long-run level of output, private consumption and welfare, at the cost of short-run welfare losses
and output reduction.
Finally, Castro, Félix, Júlio, and Maria (2013) evaluate the size of short-run fiscal multipliers
associated with fiscal consolidation under two distinct alternative scenarios, viz “normal times”
and “crisis times.” The authors find that first year fiscal multipliers can be around 60-70
percent larger in crisis times vis-à-vis normal times for a government consumption-based fiscal
consolidation, and around 40-60 percent larger for a revenue-based fiscal consolidation.
85
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89
Appendices
List of Equations: the stationary model
1
Distributors. See equation (4).
pCt =
2
1
1+τtC
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Type-A households. See equation (11).
A
Čt
N (1−ψ)−LA
t
3
4
=
ηA
(1
1−η A
16
Type-A households. See equation (18).
vηA 1− 1 1−ηA
γ
L
A
1−τt+1
Čt g
w̌t+1 g
jt = Č A
w̌t
1−τ L
. . . . . . .
17
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1− γ1
t
it
β πt+1
1
γ
χ
1−η A
1− γ1
Type-A households. See equation (21).
it
πt+1
Type-A households. See equation (27).
ˇ t = hw
ˇ L + hw
ˇ K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
hw
t
t
7
9
20
Type-A households. See equation (31).
ˇ L . . . . . . . . . . . . . . . . . . . . . . . . . .
ˇ L = N (1 − ψ)(1 − τ L )w̌t + g θ·χ hw
hw
t+1
t
t
rt
8
15
it = i∗t Ψ εt+1
εt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
rt =
6
− τtL )w̌t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Type-A households. See equation (13).
t−1
5
11
21
Type-A households. See equation (34); x ∈ {T , N , C, G, I, X , K, EP}.
θ ˇ K
ˇ K = P dˇA,x + trg
ˇ
ˇ A
ˇ A
hw
t
t + t trx t + rbr t + g rt hw t+1 . . . . . . . . . . . . . . . . . . . .
x t
22
Type-A households. See equation (37).
h
i
ˇ = 1 it−1 (b̌t−1 + b̌T + b̌N ) + i∗ Ψ εt t−1 b̌∗
fw
. . . . . . . . . . . . . . . . .
t
t−1
t−1
t−1
t−1 εt−1
g·πt
23
10 Type-A households. See equation (42).
ˇ t + fw
ˇ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ČtA = (Θt )−1 (hw
t
26
11 Type-A households. See equation (44).
Θt =
1
ηA
+
θ·jt
rt Θt+1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
12 Type-B households. See equation (48).
ČtB
N ψ−LB
t
=
ηB
(1
1−η B
− τtL )w̌t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 Type-B households. See equation (51).
90
27
ˇB,U + trg
ˇ B
ˇ B
ČtB = (1 − τtL )w̌t LB
t + t trx t . . . . . . . . . . . . . . . . . . . . . . . . .
t + dt
28
14 Labor Unions. See equation (56).
2
V
π
Γ̌Vt = φ2U Ut πVt − 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
t−1
15 Labor Unions. See equation (58).
v̌t
v̌t−1
πtV
πt g
=
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
16 Labor Unions. See equation (61).
dˇUt = (1 − τtL ) (v̌t − w̌t )Ut − Γ̌Vt
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17 Labor Unions. See equation (63).
πtV
πV
φU
σU
− 1 πVt −
w̌ − v̌t = σU −1
σ U −1 t
πV
V
πt+1
− 1 πV
. . . . . . . . .
31
18 Capital goods producers. See equation (67).
2
IˇtJ
φIJ ˇJ
IJ
Γ̌t = 2 It IˇJ − 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
t−1
t−1
L
1−τt+1
θ·g Ut+1
1−τtL rt Ut
31
V
πt+1
πtV
t
t−1
19 Capital goods producers. See equation (70).
IJ
I ˇJ
I ˇJ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
= pKJ
dˇKJ
t ζt It − pt It + Γ̌t
t
20 Capital goods producers. See equation (73).
"
IˇtJ
IˇJ
KJ
I
I
IJ
pt ζt = pt 1 + φIJ IˇJ − 1 IˇJt − g·θ·φ
rt
pI
t+1
pIt
J
Iˇt+1
IˇtJ
−1
J
Iˇt+1
IˇtJ
35
2 #
. . . . . . . .
35
21 Entrepreneurs. See equation (76).
1
ˇ
ˇ
J
J
J
I
J
ˇ
K̄t+1 = g (1 − δt )K̄t + ζt It
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
t−1
t−1
22 Entrepreneurs. See equation (79).
uJt
2
uJt
+
φJa
1−
σaJ
uJt
+
φJa
σaJ
2
. . . . . . . . . . . . . . . . . . .
37
ˇJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ǨtJ = uJt K̄
t
38
a
=
1 J J
2 φa σa
−1
23 Entrepreneurs. See equation (81).
24 Entrepreneurs. See equation (84).
rtKJ = φJa σaJ uJt + φJa (1 − σaJ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
25 Entrepreneurs. See equation (86).
rcu
ˇ Jt =
ˇJ
a(uJ
t )K̄t
pJ
t
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
26 Entrepreneurs. See equation (89).
ˇJ
J
b̌Jt = pKJ
t K̄t+1 · g − ňt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
39
27 Entrepreneurs. See equation (91).
i
−1 h
K ) r KJ uJ
J
J
KJ
K
J
KJ
ret KJ
= pKJ
(1 − τt+1
. . .
t
t
t+1 t+1 − a(ut+1 ) + (1 − δt+1 )pt+1 + τt+1 δt+1 pt+1
41
28 Entrepreneurs. See equation (99).
!
0
−1 =0 . . . . . .
47
−1 . . . . . . . . . . . . . . . . . . . . . . . . . .
47
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
31 Entrepreneurs. See equation (104).
i
h
KJ −1 (1 − τ K ) uJ r KJ − a(uJ ) + (1 − δ J )pKJ + τ K δ J pKJ
ret KJ
=
p
. . . . . . . .
t
t
t
t
t
t
t
t
t
m1,t
t−1
47
32 Entrepreneurs. See equation (108).
2
1
EJ
J
J
EJ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ω̄t = exp z̄t σt − 2 σt
48
1−
ret KJ
ΓJt+1 rtt
ΓJ
t+1
+
ΓJ
t+1
0
−µt+1 GJ
t+1
0
ΓJt+1
−
ret KJ
µt+1 GJt+1 rtt
29 Entrepreneurs. See equation (102).
KJ ˇ J
J
ret KJ
m1,t pt−1 K̄t g
Γt − µt GJt rm1,t
=
ňt−1
ˇJ
pKJ
t−1 K̄t g
ňt−1
30 Entrepreneurs. See equation (103).
rm1,t =
it−1
πt
33 Entrepreneurs. See equation (110).
ΓJt = Φ z̄tJ − σtEJ + ω̄tJ 1 − Φ z̄tJ
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
34 Entrepreneurs. See equation (111).
GJt = Φ z̄tJ − σtEJ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
35 Entrepreneurs. See equation (112).
ΓJt
0
= 1 − Φ z̄tJ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
36 Entrepreneurs. See equation (113).
GJt
0
= ω̄tJ f ω̄tJ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
37 Entrepreneurs. See equation (114).
1
1
J
J
2
f ω̄t = √2πω̄J σEJ exp − 2 (z̄t )
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
38 Entrepreneurs. See equation (120).
rm1,t J
ˇ
J
KJ
J
KJ
J
ˇ J + Š J,y . . . . . . .
ňt =
− pJt div
t
t
g ňt−1 + pt−1 K̄t ret m1,t (1 − µt Gt ) − rm1,t
52
t
t
39 Entrepreneurs. See equation (122).
J (ňJ − ňJ,ma ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ˇ J = inc
ˇ J,ma
div
+ θnw
t
t
t
t
40 Entrepreneurs. See equation (124).
92
52
ˇ J=
rbr
t
1 ˇJ
KJ J J
K̄t ret KJ
m1,t pt−1 µt Gt
pJ
t
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
41 Manufacturers. See equation (129).
J
ŽtJ ≡ F̌tJ ǨtJ , Ut
=
J
1 − αU
ξJ − 1
J
1
Ǩt
J
ξJ −1
J
AJt Ut
ξJ
ξJ −1
. . . . . . .
55
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
ξJ
ξJ
+ αU
1
ξJ
ξJ
42 Manufacturers. See equation (131).
F̌tUJ
0
≡
∂ F̌tJ
∂UtJ
= AJt
J
αJ
U Žt
J
AJ
U
t t
ξ1
J
43 Manufacturers. See equation (132).
0
F̌tKJ
∂ F̌tJ
∂ ǨtJ
(1−αJU )ŽtJ
1
ξJ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
44 Manufacturers. See equation (136).
J
2
πt
φPJ J
Γ̌PJ
=
Ž
−
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
t
t
2
πJ
56
45 Manufacturers. See equation (139).
2
J
Ut
φUJ J
−
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
U
ΓUJ
=
t
t
2
UJ
56
46 Manufacturers. See equation (142).
J J
J
K
KJ
J
SP
J
UJ
J
PJ
J
ˇ
dt = 1 − τt
pt Žt − rt Ǩt − (1 + τt )v̌t Ut + Γt
− pt Γ̌t + $
. . . . . .
57
≡
=
ǨtJ
t−1
t−1
47 Manufacturers. See equation (144).
πtJ
πt
=
pJ
t
pJ
t−1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48 Manufacturers. See equation (146).
J
J
πtJ
πt
φPJ
σ J λt
−
1
=
−
1
−
J
J
J
J
σ −1 p
σ −1
π
πJ
t
t−1
t−1
K
J
J
1−τt+1
θ·g pt+1 Žt+1
K
1−τt rt pt ŽtJ
49 Manufacturers. See equation (148).
"
0
J
J
UJ
λJ
Ut
Ut
t · F̌t
−
1
=
φ
−
1
−
UJ
J
J
Ut−1
Ut−1
(1+τtSP )v̌t
πJ
K
SP
V
θ 1−τt+1 1+τt+1 πt+1
rt 1−τtK 1+τtSP πt+1
t+1
πtJ
πJ − 1 πt+1
. . . . . . .
J
UJ
t+1
UtJ
57
57
t
−1
U J 2
t+1
UtJ
#
. . .
59
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
50 Manufacturers. See equation (150).
rtKJ = λJt F̌tKJ
0
51 Distributors. See equation (156).
2
Γ̌AF
=
t
AF
φAF (Ǎt −1)
2 1+(ǍAF −1)2
t
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
52 Distributors. See equation (157).
ǍAF
=
t
M̌tF /Y̌tAF
F /Y̌ AF
M̌t−1
t−1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53 Distributors. See equation (161).
93
62
Y̌tAF =
AF
αZ
1
ξAF
ξAF −1
TF
Žt
AF
+ 1 − αZ
1
ξAF −1
F
ξAF
M̌t 1 − Γ̌AF
t
ξAF
ξAF −1
. . . . . .
62
Y̌tAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
ξAF
ξAF
54 Distributors. See equation (163).
AF
ŽtT F = αZ
pT
t
λAF
t
−ξAF
55 Distributors. See equation (165).
et −ξAF AF
AF
M̌tF 1 − Γ̌AF
= 1 − αZ
Y̌t
. . . . . . . . . . . . . . . . . . . . . .
t
λAF ·ι̌AF
t
63
t
56 Distributors. See equation (166).
AF
ǍAF
t −1 Ǎt
AF
AF
ι̌t = 1 − Γ̌t − φAF h
2 i2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
1+ ǍAF
t −1
57 Distributors. See equation (170).
Y̌tF
F
αZ
1
Y̌tAF
ξF −1
+
F
αZ
1
ŽtN F
ξF −1
ξF
ξF −1
. . . . . . . . . . . . . . . .
65
58 Distributors. See equation (173).
F
2
πt
φPF F
Γ̌PF
=
Y̌
−
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
t
t
2
πF
66
=
1−
ξF
ξF
ξF
ξF
t−1
59 Distributors. See equation (176).
AF − pN Ž N F − pF Γ̌PF + $ F
. . . . . . . . . . . . .
dˇFt = (1 − τtD ) pFt Y̌tF − λAF
t
t
t
t
t Y̌t
66
60 Distributors. See equation (178).
πtF
πt
=
pF
t
pF
t−1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61 Distributors. See equation (180).
F
F
πt
πtF
φPF
σ F λt
−
1
=
−
1
−
F
F
F
F
σ −1 p
σ −1
π
πF
t
D
F
F
1−τt+1
θ·g pt+1 Y̌t+1
D
1−τt rt pt Y̌tF
πF
t+1
πtF
t+1
. . . . . . .
67
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
t−1
t−1
−1
πF 67
πtF
62 Distributors. See equation (182).
F
Y̌tAF = 1 − αZ
−ξF
λAF
t
λF
t
Y̌tF
63 Distributors. See equation (184).
F
ŽtN F = αZ
pN
t
λF
t
−ξF
Y̌tF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
64 The government. See equation (186).
rv
ˇ Ct = τtC · (pCt Čt ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
65 The government. See equation (188).
L
V
rv
ˇL
t = τt · (v̌t Ut − Γ̌t ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
66 The government. See equation (190).
rv
ˇ SP
= τtSP · (v̌t Ut ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
t
94
71
67 The government. See equation (192).
h
ˇ Ji
P
K
J
J
PJ
J − (1 + τ SP )v̌ U J + ΓU J − a(uJ ) + δ J pKJ K̄
rv
ˇK
71
t
t = τt
t
t
t
t
t t
t
J∈{T ,N } pt Žt − Γ̌t − $
68 The government. See equation (194).
h
i
P
D
F
F
P F − $ F − pT Ž T F − M̌ F − pN Ž N F . . . . . .
rv
ˇD
t t
t = τt
t t
t
t
F ∈{C,I,G,X } pt Y̌t − Γ̌t
71
69 The government. See equation (196); Z ≡ {C, L, SP, K, D}.
P
rv
ˇt=
ˇ xt
x∈{Z} rv
ˇt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+ tre
72
70 The government. See equation (198).
it−1
πt ·g b̌t−1
b̌t =
ˇ t − rv
ˇt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+ pGt Ǧt + trg
72
71 The government. See equation (200).
sg
ˇt
ˇ t
GDP
=
sg
ˇ t target
ˇ t
GDP
+ d1
rv
ˇ t −rv
ˇ ss
t
ˇ t
GDP
+ d2
b̌t
ˇ t
GDP
−
b̌t
ˇ t
GDP
target + d3 ln
GDPt
GDPt?
. . .
72
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
72 Rest of the World. See equation (202).
t
t−1
=
πt∗ εt
πt εt−1
73 Rest of the World. See equation (205).
YtX = αYA∗
pX
t
t
−ξA∗
YtA∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
74 Rest of the World. See equation (207).
t b̌∗t =
i∗t−1 Ψt−1 t−1 ∗
b̌t−1
πt ·g
ˇ
ˇ
+ pX
t X̌t − t M̌t + t (tre t + trx t ) . . . . . . . . . . . . . . . . . .
75
75 Economic aggregates and market clearing conditions. See equation (209).
Čt = ČtA + ČtB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
76 Economic aggregates and market clearing conditions. See equation (211).
B
Lt = LA
t + Lt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
77 Economic aggregates and market clearing conditions. See equation (213).
Ut = UtT + UtN + ΓUt T + ΓUt N + Γ̌Vt . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
78 Economic aggregates and market clearing conditions. See equation (215).
Lt = Ut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
79 Economic aggregates and market clearing conditions. See equation (217).
ŽtJ − Γ̌Pt J − rcu
ˇ Jt − $T − StJ,y = ŽtJC + ŽtJI + ŽtJG + ŽtJX , J ∈ T , N . . . . . . . .
77
80 Economic aggregates and market clearing conditions. See equation (219); F ∈ {C, I, G, X }.
I − T $ F = 1 (F )Č + 1 (F )Iˇ + 1 (F )Ǧ + 1 (F )X̌
Y̌tF − ΓPt F − 1I (F ) ΓTt I + ΓN
t
t
t
t
t
C
I
G
X
t
81 Economic aggregates and market clearing conditions. See equation (221).
95
77
M̌t = M̌tC + M̌tI + M̌tG + M̌tX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
82 Economic aggregates and market clearing conditions. See equation (223).
ˇ t = Čt + pG Ǧt + PtI Iˇt + PtX X̌t − t M̌t . . . . . . . . . . . . . . . . . . . . . . . .
GDP
t
78
83 Capital goods producers. Model without financial frictions. See equation (226).
ˇJ
ˇJ . . . . . . . . . .
I IJ − pI IˇJ + τ K δ J pKJ K̄
dˇKJ
= (1 − τtK ) rtKJ uJt − a(uJt ) K̄
t
t − pt Γt
t t
t t t
t
79
84 Capital goods producers. Model without financial frictions. See equation (228).
KJ J
θ
KJ
K
J
KJ
J
K
pt = rt 1 − τt+1 rt+1 ut+1 − a ut+1 + pt+1 1 − δt+1 1 − τt+1
. . . . . . . . .
79
85 Capital goods producers. Model without financial frictions. See equation (230).
2
I
J
J
I
πt+1
Iˇt+1
Iˇt+1
pKJ
IˇtJ
IˇtJ
θ·g
K
K
t ζt
= 1 + (1 − τt )φIJ IˇJ − 1
− rt (1 − τt+1 )φIJ πt+1 IˇJ − 1
80
pI
IˇJ
IˇJ
t−1
t
t−1
t
86 Capital goods producers. Model without financial frictions. See equation (232).
rtKJ = φJa σaJ uJt + φJa (1 − σaJ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
87 The government. See equation (234).
h
ˇJ
P
J
PJ
J
SP
K
J
J
U J − a(uJ )+δ J pKJ K̄
rv
ˇK
=
τ
t
t
t
t t
t −
J∈{T ,N } pt Žt −Γ̌t −$ −(1+τt )v̌t Ut +Γt
i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
pIt ΓIJ
t
81
96
List of Equations: the steady-state model
1
Distributors. See equation (5).
pC =
2
1
1+τ C
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Type-A households. See equation (12).
Č A
N (1−ψ)−LA
3
4
5
=
ηA
(1
1−η A
− τ L )w̌ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i = i∗ Ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Type-A households. See equation (19).
1 1−ηA 1− 1
1+η A (v−1) 1− γ1
i γ
γ
βπ χ
. . . . . . . . . . . . . . . . . . . . . . . .
j=g
17
Type-A households. See equation (22).
i
π
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
Type-A households. See equation (32).
ˇ L=
hw
8
17
Type-A households. See equation (28).
ˇ = hw
ˇ L + hw
ˇ K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
hw
7
16
Type-A households. See equation (14).
r=
6
11
N (1−ψ)(1−τ L )w̌
1−g θ·χ
r
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Type-A households. See equation (35); x ∈ {T , N , C, G, I, X , K, EP}.
A
A
ˇ K = 1 θ P dˇA,x + trg
ˇ
ˇ
ˇ
hw
+
trx
+
rbr
. . . . . . . . . . . . . . . . . . . . . . .
x
1−g
22
r
9
Type-A households. See equation (38).
ˇ =
fw
1
g·π
i b̌ + b̌T + b̌N + i∗ Ψb̌∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
10 Type-A households. See equation (43).
ˇ + fw
ˇ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Č A = Θ−1 (hw
26
11 Type-A households. See equation (45).
Θ=
1/η A
1−(θ·j)/r
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
12 Type-B households. See equation (49).
Č B
N ψ−LB
=
ηB
(1
1−η B
− τ L )w̌ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
13 Type-B households. See equation (52).
ˇ B + trx
ˇ B
Č B = (1 − τ L )w̌LB + dˇB,U + trg
. . . . . . . . . . . . . . . . . . . . . . . . .
14 Labor Unions. See equation (57).
97
28
Γ̌V = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
15 Labor Unions. See equation (59).
πV = π · g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
16 Labor Unions. See equation (62).
dˇU = (1 − τ L )(v̌ − w̌)U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
17 Labor Unions. See equation (64).
v̌ =
σU
w̌
σ U −1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
18 Capital goods producers. See equation (68).
Γ̌IJ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
19 Capital goods producers. See equation (71).
dˇKJ = pKJ − pI IˇJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
20 Capital goods producers. See equation (74).
pKJ = pI
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
21 Entrepreneurs. See equation (77).
IˇJ
ˇJ
K̄
= g + δJ − 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
22 Entrepreneurs. See equation (80).
a(1) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
23 Entrepreneurs. See equation (82).
ˇJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ǩ J = K̄
38
24 Entrepreneurs. See equation (85).
rKJ = φJa
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
25 Entrepreneurs. See equation (87).
rcu
ˇ J =0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
26 Entrepreneurs. See equation (90).
ˇ J · g − ňJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
b̌J = pKJ K̄
40
27 Entrepreneurs. See equation (92).
i
−1 h
ret KJ = pKJ
(1 − τ K )rKJ + (1 − δ J )pKJ + τ K δ J pKJ
. . . . . . . . . . . . . . . .
41
28 Entrepreneurs. See equation (100).
98
ΓJ
KJ
1 − ΓJ retr +
ΓJ
0
!
0
−µ GJ
−1 =0 . . . . . . . . . . . . . .
47
−1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
ΓJ
0
−
KJ
µGJ retr
29 Entrepreneurs. See equation (105).
J
ret KJ
ˇJg
KJ K̄
m1 p
=
Γ − µGJ rm1
ň
ˇJg
pKJ K̄
ň
30 Entrepreneurs. See equation (106).
i
π
rm1 =
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
31 Entrepreneurs. See equation (107).
i
h
KJ −1 (1 − τ K ) uJ r KJ − a(uJ ) + (1 − δ J )pKJ + τ K δ J pKJ
ret KJ
=
p
. . . . . . . . .
m1
48
32 Entrepreneurs. See equation (109).
h
2 i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ω̄ J = exp z̄ J σ EJ − 12 σ EJ
48
33 Entrepreneurs. See equation (115).
ΓJ = Φ z̄ J − σ EJ + ω̄ J 1 − Φ z̄ J
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
34 Entrepreneurs. See equation (116).
GJ = Φ z̄ J − σ EJ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
35 Entrepreneurs. See equation (117).
ΓJ
0
= 1 − Φ z̄ J
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
36 Entrepreneurs. See equation (118).
GJ
0
= ω̄ J f ω̄ J
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
37 Entrepreneurs. See equation (119).
2
f ω̄ J = √2πω̄1J σEJ exp − 12 z̄ J
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
38 Entrepreneurs. See equation (121).
J
g
ˇ
J
KJ
J
KJ
J
J
J,y
ˇ
ň = g−rm1 p K̄ ret m1 (1 − µ G) − rm1 − p div + Š
. . . . . . . . . . . .
52
39 Entrepreneurs. See equation (123).
ˇ J = inc
ˇ J,ma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
div
52
40 Entrepreneurs. See equation (125).
ˇ J=
rbr
1 ˇJ
KJ µJ GJ
K̄ ret KJ
m1 p
pJ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
41 Manufacturers. See equation (130).
Ž J ≡ F̌ J Ǩ J , U
J
=
J
1 − αU
1
ξJ
Ǩ
ξJ − 1
J
ξJ
99
J
+ αU
1
ξJ
AJ U
ξJ −1
J
ξJ
ξJ
ξJ −1
. . . . . . .
55
42 Manufacturers. See equation (133).
F̌ UJ
0
≡
∂ F̌ J
∂U J
= ĀJ
J
αJ
U Ž
AJ U J
ξ1
J
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
43 Manufacturers. See equation (134).
0
F̌ KJ
≡
∂ F̌ J
∂ Ǩ J
=
(1−αJU )Ž J
1
ξJ
Ǩ J
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
44 Manufacturers. See equation (137).
Γ̌PJ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
45 Manufacturers. See equation (140).
ΓUJ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
46 Manufacturers. See equation (143).
J J
J
K
KJ
J
SP
J
J
J
ˇ
d = 1−τ
p Ž − r Ǩ − (1 + τ )v̌ · U − p $ . . . . . . . . . . . . . . . .
57
47 Manufacturers. See equation (145).
πJ = π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
48 Manufacturers. See equation (147).
pJ =
σJ
λJ
σ J −1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
49 Manufacturers. See equation (149).
0
1 + τ SP v̌ = λJ F̌ UJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
50 Manufacturers. See equation (151).
rKJ = λJ F̌ KJ
0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
51 Distributors. See equation (158).
Γ̌AF = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
52 Distributors. See equation (159).
ǍAF
=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
t
62
53 Distributors. See equation (162).
Y̌
AF
=
AF
αZ
1
ξAF
Ž T F
ξAF −1
+ 1−
AF
αZ
1
F
ξAF −1
ξAF
M̌ 1 − Γ̌AF
ξAF
ξAF −1
. . . . . .
63
Y̌ AF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
ξAF
ξAF
54 Distributors. See equation (164).
AF
Ž T F = αZ
pT
λAF
−ξAF
55 Distributors. See equation (167).
100
AF
M̌ F = 1 − αZ
−ξAF
e
λAF
Y̌ AF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
56 Distributors. See equation (168).
ι̌AF = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
57 Distributors. See equation (171).
Y̌ F =
1
F
Y̌
ξF −1
AF
1
F
+ αZ
Ž
ξF −1
NF
ξF
ξF −1
. . . . . . . . . . . . . . . .
65
Γ̌PF = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
1 − αZ
ξF
ξF
ξF
ξF
58 Distributors. See equation (174).
59 Distributors. See equation (177).
dˇF = (1 − τ D ) pF Y̌ F − λAF Y̌ AF − pN Ž N F − pF $F
. . . . . . . . . . . . . . . . . .
66
60 Distributors. See equation (179).
πF = π
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
61 Distributors. See equation (181).
pF =
σF
λF
σ F −1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
62 Distributors. See equation (183).
F
Y̌ AF = 1 − αZ
λAF −ξF
λF
Y̌ F
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
63 Distributors. See equation (185).
F
Ž N F = αZ
pN
λF
−ξF
Y̌ F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
64 The government. See equation (187).
rv
ˇ C = τ C · (pC · Č) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
65 The government. See equation (189).
rv
ˇ L = τ L · (v̌ · U ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
66 The government. See equation (191).
rv
ˇ SP = τ SP · (v̌ · U ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
67 The government. See equation (193).
h
i
P
ˇJ . . . . . . . . . . .
rv
ˇ K = τ K J∈{T ,N } pJ (Ž J − $J ) − (1 + τ SP )v̌ · U J − δtJ pKJ K̄
t
71
68 The government. See equation (195).
h
i
P
. . . . . . . . . .
rv
ˇ D = τ D F ∈{C,I,G,X } pF (Y̌ F − $F ) − pT Ž T F − M̌ F − pN Ž N F
71
69 The government. See equation (197); Z ≡ {C, L, SP, K, D}.
101
rv
ˇ =
P
ˇ
x∈{Z} rv
x
ˇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+ tre
72
70 The government. See equation (199).
ˇ t − rv
ˇt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
b̌ = gr b̌t−1 + pGt Ǧt + trg
72
71 The government. See equation (201).
target
sg
ˇ
sg
ˇ
=
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ˇ
ˇ
GDP
GDP
73
72 Rest of the World. See equation (203).
π ∗ = πt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
73 Rest of the World. See equation (206).
Y X = αYA∗
pX
−ξA∗
Y A∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
74 Rest of the World. See equation (208).
b̌∗ =
ˇ trx)
pX X−M̌ +(tre+
ˇ
∗
Ψ
1− iπ·g
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
75 Economic aggregates and market clearing conditions. See equation (210).
Č = Č A + Č B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
76 Economic aggregates and market clearing conditions. See equation (212).
L = LA + LB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
77 Economic aggregates and market clearing conditions. See equation (214).
U = UT + UN
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
78 Economic aggregates and market clearing conditions. See equation (216).
L=U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
79 Economic aggregates and market clearing conditions. See equation (218).
Ž J − $J − S J,y = Ž JC + Ž JI + Ž JG + Ž JX , J ∈ T , N
. . . . . . . . . . . . . . . .
77
80 Economic aggregates and market clearing conditions. See equation (220).
Y̌tF − $F = 1C (F )Č + 1I (F )Iˇ + 1G (F )Ǧ + 1X (F )X̌, F ∈ {C, I, G, X } . . . . . . . . .
77
81 Economic aggregates and market clearing conditions. See equation (222).
M̌ = M̌ C + M̌ I + M̌ G + M̌ X
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
82 Economic aggregates and market clearing conditions. See equation (224).
ˇ = Č + pG Ǧ + pI Iˇ + pX X̌ − M̌
GDP
. . . . . . . . . . . . . . . . . . . . . . . . . . .
78
83 Capital goods producers. Model without financial frictions. See equation (227).
ˇ J − pI IˇJ + τ K δ J pKJ K̄
ˇJ
dˇKJ = (1 − τ K )rKJ K̄
102
. . . . . . . . . . . . . . . . . . . . . . .
79
84 Capital goods producers. Model without financial frictions. See equation (229).
K r KJ
1−τ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
pKJ =
J
K
r/θ− 1−δ (1−τ )
80
85 Capital goods producers. Model without financial frictions. See equation (231).
pK = pI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
86 Capital goods producers. Model without financial frictions. See equation (233).
rKJ = φJa
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
87 The government. See equation (235).
h
i
P
ˇJ . . . . . . . . . . .
rv
ˇ K = τ K J∈{T ,N } pJ (Ž J − $J ) − (1 + τ SP )v̌ · U J − δtJ pKJ K̄
t
81
103
List of shocks (including steady-state values)
1
Population size.
log Nt = (1 − ρN ) log N̄ + ρN log Nt−1 + eN
t
In the steady state: Nt = N̄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
83
Share of liquidity constrained households.
log ψt = (1 − ρψ ) log ψ̄ + ρψ log ψt−1 + eψ
t
In the steady state: ψt = ψ̄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
83
Elasticity of substitution between varieties of the consumption good.
C + eσC
σtC = (1 − ρσC )σ̄ C + ρσC σt−1
t
In the steady state: σtC = σ̄ C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
83
Elasticity of substitution between varieties of the intermediate good.
J + eσJ
σtJ = (1 − ρσJ )σ̄ J + ρσJ σt−1
t
In the steady-state: σtJ = σ̄ J
5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
Elasticity of substitution between varieties of the final good.
F + eσF
σtF = (1 − ρσF )σ̄ F + ρσF σt−1
t
In the steady-state: σtF = σ̄ F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
83
Elasticity of substitution between varieties of labor.
U + eσU
σtU = (1 − ρσU )σ̄ U + ρσU σt−1
t
In the steady-state σtU = σ̄ U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
83
Risk premium on domestic bonds.
log Ψt = (1 − ρΨ ) log Ψ̄ + ρΨ log Ψt−1 + eΨ
t
In the steady-state: Ψt = Ψ̄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
83
Investment efficiency shock.
I + eI
ζtI = (1 − ρI ) + ρI ζt−1
t
In the steady state: ζ I = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
83
Labor augmenting thehnology shock.
J
log AJt = (1 − ρAJ ) log ĀJ + ρAJ log AJt−1 + eA
t
In the steady-state: AJt = ĀJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
104
83
10 Government consumption.
ˇ + ρ log Ǧ
G
log Ǧt = (1 − ρG ) log Ḡ
t−1 + et
G
ˇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In the steady state: Ǧt = Ḡ
83
11 Lump-sum transfers to households.
ˇ
ˇ t = (1 − ρtrg ) log trg
¯ + ρtrg log trg
ˇ t−1 + etrg
log trg
t
ˇ
ˇ t = trg
¯ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In the steady state: trg
83
12 Tax rates; z ∈ {C, SP, K, D}.
z + e τz
τtz = (1 − ρτz )τ̄ z + ρτz τt−1
t
In the steady state: τ̄tz = τ̄ z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
13 Imported goods’ inflation rate.
∗
log πt∗ = (1 − ρπ∗ ) log π̄ ∗ + ρπ∗ log πt−1
+ eπt
∗
In the steady state: πt∗ = π̄ ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
14 Entrepreneurs’ riskiness.
EJ + eσ
σtEJ = (1 − ρσ ) log σ̄ EJ + ρσ log σt−1
t
In the steady state: σtEJ = σ̄ EJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
15 Bankruptcy monitoring costs.
µJt = (1 − ρµ ) log µ̄J + ρµ log µJt−1 + eµt
In the steady state: µJt = µ̄J
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
16 Entrepreneurs’ dividend net worth shock.
J,d
StJ,d = (1 − ρSd ) log S̄ J,d + ρSd log St−1
+ eSt d
In the steady state: StJ,d = S̄ J,d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
83
Calibration: key ratios
Table 1: Key ratios
Model
Data
0.60
0.23
0.21
0.29
0.33
0.64
0.22
0.21
0.29
0.37
0.57
0.57
Tradable sector
Non-tradable sector
0.55
0.59
0.54
0.58
Capital (ratio to GDP)
2.33
NA
Tradable sector
Non-tradable sector
2.51
2.21
NA
NA
0.60
-0.02
0.43
0.46
0.57
-0.05
0.40
0.44
Expenditure (ratio to GDP)
Private consumption
Government consumption and GFCF
Private investment
Exports
Imports
Labor income share (ratio to overall income)
Government (ratio to GDP)
Public debt
Fiscal surplus
Total revenues
Total expenditure
External account (as a % of GDP)
Net foreign assets
Without Financial accelerator
With Financial accelerator
Current and capital accounts
Without Financial accelerator
With Financial accelerator
Trade balance
-0.50
-0.34
-1.29
-0.06
-0.01
-0.05
-0.04
Sources: Banco de Portugal quarterly database, National accounts data, and authors’ own calculations.
Notes: The figures for Portugal are for the 1999–2007 period.
106
-0.08
Calibration: main parameters
Table 2: Main parameters
Monetary union parameters
Interest rate (A)
Labor-augmenting prod. growth (A)
Inflation target (A)
Risk Premium
EoS bt. ROW tradable goods and domestic exports
Households and Unions
Discount rate
Intertemporal elasticity of substitution
Instant probability of death (A)
Productivity decay rate (A)
Habit persistence
Consumption share—Type-A households
Consumption share—Type-B households
Share of type-B households
Wage markup (SS)
Wage adjustment cost
Manufacturers and capital goods producers
EoS bt. capital and labor
Price markup—tradable sector (SS)
Price markup—non-tradable sector (SS)
Labor adjustment cost
Price adjustment cost
Quasi labor income share—tradable sector
Quasi labor income share—non-tradable sector
Depreciation rate (A)
Investment adjustment cost
Distributors
EoS bt. domestic tradable and imported goods
EoS bt. assembled and non-tradable goods
Price markup of distributors (SS)
Price markup of exporters (SS)
Adjustment cost of import content
Price adjustment cost (except for government goods)
Price adjustment cost for government goods
Government
Consumption tax rate
Capital income tax rate
Dividends tax rate
Employers’ payroll tax rate
Sensibility to deviations from target
Revenues
Debt-to-GDP ratio
GDP
Parameter
Value
i∗
g
π∗
Ψ
ξA∗
1.045
1.02
1.02
1.00
2.50
β
1/γ
1−θ
1−χ
ν
ηA
ηB
ψ
σ U /(σ U − 1)
φU
0.996
0.50
0.04
0.04
0.70
0.73
0.65
0.40
1.25
125
ξJ
σ T /(σ T − 1)
σ N /(σ N − 1)
φU J
φP T ; φP N
αUT
αUN
δJ
φIJ
0.99
1.10
1.20
5.00
125
0.56
0.60
0.09
10.0
ξAF
ξF
σ F /(σ F − 1), F ∈
/ {X }
σ X /(σ X − 1)
φAF
φAC ; φAI ; φAX
φAG
1.50
0.50
1.05
1.03
2.00
125
400
τC
τK
τD
τ SP
0.31
0.17
0.15
0.19
d1
d2
d3
0.0
0.0
0.5
Sources: Banco de Portugal quarterly database, National accounts data, several studies on the Portuguese and euro area
economies, and authors’ own calculations.
Notes: A—Annualized; EoS—Elasticity of Substitution; SS—Steady state; ROW—Rest of the World; J ∈ {T , N };
F ∈ {C, G, I, X }. The labor income tax rate is the default endogenous instrument in the fiscal rule and is thus omitted.
The reported parameters di , i = 1, 2, 3, represent default values, however they can take any alternative configuration. The
model is quarterly and parameters are not annualized, unless otherwise indicated.
107
Calibration: financial sector parameters
Table 3: Financial sector parameters
Parameter
Leverage (%)
Probability of default (%)
Return on capital (%)
Return on cap. spread (pp)
Non-default loan rate (%)
Loan rate spread (pp)
Borrowers Riskiness
Monitoring costs
B/N
F(ω̄)
RetK
RetK − i
iB
B
i −i
σ
µ
Model
100.0
8.0
9.1
4.0
6.5
1.7
0.32
0.11
Data, PT
Interval
Average
95.0–130.2
8.7–10.6
8.0–10.0
n.a.
5.9–8.5
n.a.
n.a.
n.a.
104.9
9.4
9.4
n.a.
6.7
n.a.
n.a.
n.a.
Sources: Banco de Portugal, Eurostat, Mata, Antunes, and Portugal (2010), and authors’ own calculations.
Notes: Average figures for the 2002–2008 period, except the probability of default, which is for the 1996–1998 period.
Leverage is defined as the debt-to-equity ratio of non-financial corporations, where debt includes debt securities issued by
non-financial corporations as well as loans (consolidated data). The values reported for the probability of default consider
the total exit rate for the Portuguese economy, as defined in Mata, Antunes, and Portugal (2010). More recent data from
Quadros de Pessoal for the 2002–2007 period shows an exit rate comprised between 8.1 and 9.2 percent. The return on
capital is calibrated to match the definition of “return on investment” used by the Banco de Portugal, the ratio between
ordinary profit plus interest costs, against other shares and other equity plus financial debt. The non-default loan rate is
proxied by the “debt cost ratio” used by Banco de Portugal, the ratio between interest costs and financial debt.
108
WORKING PAPERS
2010
1/10 MEASURING COMOVEMENT IN THE TIME-FREQUENCY SPACE
— António Rua
2/10 EXPORTS, IMPORTS AND WAGES: EVIDENCE FROM MATCHED FIRM-WORKER-PRODUCT PANELS
— Pedro S. Martins, Luca David Opromolla
3/10 NONSTATIONARY EXTREMES AND THE US BUSINESS CYCLE
— Miguel de Carvalho, K. Feridun Turkman, António Rua
4/10 EXPECTATIONS-DRIVEN CYCLES IN THE HOUSING MARKET
— Luisa Lambertini, Caterina Mendicino, Maria Teresa Punzi
5/10 COUNTERFACTUAL ANALYSIS OF BANK MERGERS
— Pedro P. Barros, Diana Bonfim, Moshe Kim, Nuno C. Martins
6/10 THE EAGLE. A MODEL FOR POLICY ANALYSIS OF MACROECONOMIC INTERDEPENDENCE IN THE EURO AREA
— S. Gomes, P. Jacquinot, M. Pisani
7/10 A WAVELET APPROACH FOR FACTOR-AUGMENTED FORECASTING
— António Rua
8/10 EXTREMAL DEPENDENCE IN INTERNATIONAL OUTPUT GROWTH: TALES FROM THE TAILS
— Miguel de Carvalho, António Rua
9/10 TRACKING THE US BUSINESS CYCLE WITH A SINGULAR SPECTRUM ANALYSIS
— Miguel de Carvalho, Paulo C. Rodrigues, António Rua
10/10 A MULTIPLE CRITERIA FRAMEWORK TO EVALUATE BANK BRANCH POTENTIAL ATTRACTIVENESS
— Fernando A. F. Ferreira, Ronald W. Spahr, Sérgio P. Santos, Paulo M. M. Rodrigues
11/10 THE EFFECTS OF ADDITIVE OUTLIERS AND MEASUREMENT ERRORS WHEN TESTING FOR STRUCTURAL BREAKS
IN VARIANCE
— Paulo M. M. Rodrigues, Antonio Rubia
12/10 CALENDAR EFFECTS IN DAILY ATM WITHDRAWALS
— Paulo Soares Esteves, Paulo M. M. Rodrigues
13/10 MARGINAL DISTRIBUTIONS OF RANDOM VECTORS GENERATED BY AFFINE TRANSFORMATIONS OF
INDEPENDENT TWO-PIECE NORMAL VARIABLES
— Maximiano Pinheiro
14/10 MONETARY POLICY EFFECTS: EVIDENCE FROM THE PORTUGUESE FLOW OF FUNDS
— Isabel Marques Gameiro, João Sousa
15/10 SHORT AND LONG INTEREST RATE TARGETS
— Bernardino Adão, Isabel Correia, Pedro Teles
16/10 FISCAL STIMULUS IN A SMALL EURO AREA ECONOMY
— Vanda Almeida, Gabriela Castro, Ricardo Mourinho Félix, José Francisco Maria
17/10 FISCAL INSTITUTIONS AND PUBLIC SPENDING VOLATILITY IN EUROPE
— Bruno Albuquerque
Banco de Portugal
| Working Papers
i
18/10 GLOBAL POLICY AT THE ZERO LOWER BOUND IN A LARGE-SCALE DSGE MODEL
— S. Gomes, P. Jacquinot, R. Mestre, J. Sousa
19/10 LABOR IMMOBILITY AND THE TRANSMISSION MECHANISM OF MONETARY POLICY IN A MONETARY UNION
— Bernardino Adão, Isabel Correia
20/10 TAXATION AND GLOBALIZATION
— Isabel Correia
21/10 TIME-VARYING FISCAL POLICY IN THE U.S.
— Manuel Coutinho Pereira, Artur Silva Lopes
22/10 DETERMINANTS OF SOVEREIGN BOND YIELD SPREADS IN THE EURO AREA IN THE CONTEXT OF THE ECONOMIC
AND FINANCIAL CRISIS
— Luciana Barbosa, Sónia Costa
23/10 FISCAL STIMULUS AND EXIT STRATEGIES IN A SMALL EURO AREA ECONOMY
— Vanda Almeida, Gabriela Castro, Ricardo Mourinho Félix, José Francisco Maria
24/10 FORECASTING INFLATION (AND THE BUSINESS CYCLE?) WITH MONETARY AGGREGATES
— João Valle e Azevedo, Ana Pereira
25/10 THE SOURCES OF WAGE VARIATION: AN ANALYSIS USING MATCHED EMPLOYER-EMPLOYEE DATA
— Sónia Torres,Pedro Portugal, John T.Addison, Paulo Guimarães
26/10 THE RESERVATION WAGE UNEMPLOYMENT DURATION NEXUS
— John T. Addison, José A. F. Machado, Pedro Portugal
27/10 BORROWING PATTERNS, BANKRUPTCY AND VOLUNTARY LIQUIDATION
— José Mata, António Antunes, Pedro Portugal
28/10 THE INSTABILITY OF JOINT VENTURES: LEARNING FROM OTHERS OR LEARNING TO WORK WITH OTHERS
— José Mata, Pedro Portugal
29/10 THE HIDDEN SIDE OF TEMPORARY EMPLOYMENT: FIXED-TERM CONTRACTS AS A SCREENING DEVICE
— Pedro Portugal, José Varejão
30/10 TESTING FOR PERSISTENCE CHANGE IN FRACTIONALLY INTEGRATED MODELS: AN APPLICATION TO WORLD
INFLATION RATES
— Luis F. Martins, Paulo M. M. Rodrigues
31/10 EMPLOYMENT AND WAGES OF IMMIGRANTS IN PORTUGAL
— Sónia Cabral, Cláudia Duarte
32/10 EVALUATING THE STRENGTH OF IDENTIFICATION IN DSGE MODELS. AN A PRIORI APPROACH
— Nikolay Iskrev
33/10 JOBLESSNESS
— José A. F. Machado, Pedro Portugal, Pedro S. Raposo
2011
1/11 WHAT HAPPENS AFTER DEFAULT? STYLIZED FACTS ON ACCESS TO CREDIT
— Diana Bonfim, Daniel A. Dias, Christine Richmond
2/11 IS THE WORLD SPINNING FASTER? ASSESSING THE DYNAMICS OF EXPORT SPECIALIZATION
— João Amador
Banco de Portugal
| Working Papers
ii
3/11 UNCONVENTIONAL FISCAL POLICY AT THE ZERO BOUND
— Isabel Correia, Emmanuel Farhi, Juan Pablo Nicolini, Pedro Teles
4/11 MANAGERS’ MOBILITY, TRADE STATUS, AND WAGES
— Giordano Mion, Luca David Opromolla
5/11 FISCAL CONSOLIDATION IN A SMALL EURO AREA ECONOMY
— Vanda Almeida, Gabriela Castro, Ricardo Mourinho Félix, José Francisco Maria
6/11 CHOOSING BETWEEN TIME AND STATE DEPENDENCE: MICRO EVIDENCE ON FIRMS’ PRICE-REVIEWING
STRATEGIES
— Daniel A. Dias, Carlos Robalo Marques, Fernando Martins
7/11 WHY ARE SOME PRICES STICKIER THAN OTHERS? FIRM-DATA EVIDENCE ON PRICE ADJUSTMENT LAGS
— Daniel A. Dias, Carlos Robalo Marques, Fernando Martins, J. M. C. Santos Silva
8/11 LEANING AGAINST BOOM-BUST CYCLES IN CREDIT AND HOUSING PRICES
— Luisa Lambertini, Caterina Mendicino, Maria Teresa Punzi
9/11 PRICE AND WAGE SETTING IN PORTUGAL LEARNING BY ASKING
— Fernando Martins
10/11 ENERGY CONTENT IN MANUFACTURING EXPORTS: A CROSS-COUNTRY ANALYSIS
— João Amador
11/11 ASSESSING MONETARY POLICY IN THE EURO AREA: A FACTOR-AUGMENTED VAR APPROACH
— Rita Soares
12/11 DETERMINANTS OF THE EONIA SPREAD AND THE FINANCIAL CRISIS
— Carla Soares, Paulo M. M. Rodrigues
13/11 STRUCTURAL REFORMS AND MACROECONOMIC PERFORMANCE IN THE EURO AREA COUNTRIES: A MODELBASED ASSESSMENT
— S. Gomes, P. Jacquinot, M. Mohr, M. Pisani
14/11 RATIONAL VS. PROFESSIONAL FORECASTS
— João Valle e Azevedo, João Tovar Jalles
15/11 ON THE AMPLIFICATION ROLE OF COLLATERAL CONSTRAINTS
— Caterina Mendicino
16/11 MOMENT CONDITIONS MODEL AVERAGING WITH AN APPLICATION TO A FORWARD-LOOKING MONETARY
POLICY REACTION FUNCTION
— Luis F. Martins
17/11 BANKS’ CORPORATE CONTROL AND RELATIONSHIP LENDING: EVIDENCE FROM RETAIL LOANS
— Paula Antão, Miguel A. Ferreira, Ana Lacerda
18/11 MONEY IS AN EXPERIENCE GOOD: COMPETITION AND TRUST IN THE PRIVATE PROVISION OF MONEY
— Ramon Marimon, Juan Pablo Nicolini, Pedro Teles
19/11 ASSET RETURNS UNDER MODEL UNCERTAINTY: EVIDENCE FROM THE EURO AREA, THE U.K. AND THE U.S.
— João Sousa, Ricardo M. Sousa
20/11 INTERNATIONAL ORGANISATIONS’ VS. PRIVATE ANALYSTS’ FORECASTS: AN EVALUATION
— Ildeberta Abreu
21/11 HOUSING MARKET DYNAMICS: ANY NEWS?
— Sandra Gomes, Caterina Mendicino
Banco de Portugal
| Working Papers
iii
22/11 MONEY GROWTH AND INFLATION IN THE EURO AREA: A TIME-FREQUENCY VIEW
— António Rua
23/11 WHY EX(IM)PORTERS PAY MORE: EVIDENCE FROM MATCHED FIRM-WORKER PANELS
— Pedro S. Martins, Luca David Opromolla
24/11 THE IMPACT OF PERSISTENT CYCLES ON ZERO FREQUENCY UNIT ROOT TESTS
— Tomás del Barrio Castro, Paulo M.M. Rodrigues, A.M. Robert Taylor
25/11 THE TIP OF THE ICEBERG: A QUANTITATIVE FRAMEWORK FOR ESTIMATING TRADE COSTS
— Alfonso Irarrazabal, Andreas Moxnes, Luca David Opromolla
26/11 A CLASS OF ROBUST TESTS IN AUGMENTED PREDICTIVE REGRESSIONS
— Paulo M.M. Rodrigues, Antonio Rubia
27/11 THE PRICE ELASTICITY OF EXTERNAL DEMAND: HOW DOES PORTUGAL COMPARE WITH OTHER EURO AREA
COUNTRIES?
— Sónia Cabral, Cristina Manteu
28/11 MODELING AND FORECASTING INTERVAL TIME SERIES WITH THRESHOLD MODELS: AN APPLICATION TO
S&P500 INDEX RETURNS
— Paulo M. M. Rodrigues, Nazarii Salish
29/11 DIRECT VS BOTTOM-UP APPROACH WHEN FORECASTING GDP: RECONCILING LITERATURE RESULTS WITH
INSTITUTIONAL PRACTICE
— Paulo Soares Esteves
30/11 A MARKET-BASED APPROACH TO SECTOR RISK DETERMINANTS AND TRANSMISSION IN THE EURO AREA
— Martín Saldías
31/11 EVALUATING RETAIL BANKING QUALITY SERVICE AND CONVENIENCE WITH MCDA TECHNIQUES: A CASE
STUDY AT THE BANK BRANCH LEVEL
— Fernando A. F. Ferreira, Sérgio P. Santos, Paulo M. M. Rodrigues, Ronald W. Spahr
2012
1/12 PUBLIC-PRIVATE WAGE GAPS IN THE PERIOD PRIOR TO THE ADOPTION OF THE EURO: AN APPLICATION
BASED ON LONGITUDINAL DATA
— Maria Manuel Campos, Mário Centeno
2/12 ASSET PRICING WITH A BANK RISK FACTOR
— João Pedro Pereira, António Rua
3/12 A WAVELET-BASED ASSESSMENT OF MARKET RISK: THE EMERGING MARKETS CASE
— António Rua, Luis C. Nunes
4/12 COHESION WITHIN THE EURO AREA AND THE U. S.: A WAVELET-BASED VIEW
— António Rua, Artur Silva Lopes
5/12 EXCESS WORKER TURNOVER AND FIXED-TERM CONTRACTS: CAUSAL EVIDENCE IN A TWO-TIER SYSTEM
— Mário Centeno, Álvaro A. Novo
6/12 THE DYNAMICS OF CAPITAL STRUCTURE DECISIONS
— Paula Antão, Diana Bonfim
7/12 QUANTILE REGRESSION FOR LONG MEMORY TESTING: A CASE OF REALIZED VOLATILITY
— Uwe Hassler, Paulo M. M. Rodrigues, Antonio Rubia
Banco de Portugal
| Working Papers
iv
8/12 COMPETITION IN THE PORTUGUESE ECONOMY: AN OVERVIEW OF CLASSICAL INDICATORS
— João Amador, Ana Cristina Soares
9/12 MARKET PERCEPTION OF FISCAL SUSTAINABILITY: AN APPLICATION TO THE LARGEST EURO AREA ECONOMIES
— Maximiano Pinheiro
10/12 THE EFFECTS OF PUBLIC SPENDING EXTERNALITIES
— Valerio Ercolani, João Valle e Azevedo
11/12 COLLATERAL REQUIREMENTS: MACROECONOMIC FLUCTUATIONS AND MACRO-PRUDENTIAL POLICY
— Caterina Mendicino
12/12 WAGE RIGIDITY AND EMPLOYMENT ADJUSTMENT AT THE FIRM LEVEL: EVIDENCE FROM SURVEY DATA
— Daniel A. Dias, Carlos Robalo Marques, Fernando Martins
13/12 HOW TO CREATE INDICES FOR BANK BRANCH FINANCIAL PERFORMANCE MEASUREMENT USING MCDA
TECHNIQUES: AN ILLUSTRATIVE EXAMPLE
— Fernando A. F. Ferreira, Paulo M. M. Rodrigues, Sérgio P. Santos, Ronald W. Spahr
14/12 ON INTERNATIONAL POLICY COORDINATION AND THE CORRECTION OF GLOBAL IMBALANCES
— Bruno Albuquerque, Cristina Manteu
15/12 IDENTIFYING THE DETERMINANTS OF DOWNWARD
CONSIDERATIONS AND NEW EMPIRICAL EVIDENCE
WAGE
RIGIDITY:
SOME
METHODOLOGICAL
— Daniel A. Dias, Carlos Robalo Marques, Fernando Martins
16/12 SYSTEMIC RISK ANALYSIS USING FORWARD-LOOKING DISTANCE-TO-DEFAULT SERIES
— Martín Saldías
17/12 COMPETITION IN THE PORTUGUESE ECONOMY: INSIGHTS FROM A PROFIT ELASTICITY APPROACH
— João Amador, Ana Cristina Soares
18/12 LIQUIDITY RISK IN BANKING: IS THERE HERDING?
— Diana Bonfim, Moshe Kim
19/12 BANK SIZE AND LENDING SPECIALIZATION
— Diana Bonfim, Qinglei Dai
2013
01/13 MACROECONOMIC FORECASTING USING LOW-FREQUENCY FILTERS
— João Valle e Azevedo, Ana Pereira
02/13 EVERYTHING YOU ALWAYS WANTED TO KNOW ABOUT SEX DISCRIMINATION
— Ana Rute Cardoso, Paulo Guimarães, Pedro Portugal
03/13 IS THERE A ROLE FOR DOMESTIC DEMAND PRESSURE ON EXPORT PERFORMANCE?
— Paulo Soares Esteves, António Rua
04/13 AGEING AND FISCAL SUSTAINABILITY IN A SMALL EURO AREA ECONOMY
— Gabriela Castro, José R. Maria, Ricardo Mourinho Félix, Cláudia Rodrigues Braz
05/13 MIND THE GAP! THE RELATIVE WAGES OF IMMIGRANTS IN THE PORTUGUESE LABOUR MARKET
— Sónia Cabral, Cláudia Duarte
06/13 FOREIGN DIRECT INVESTMENT AND INSTITUTIONAL REFORM: EVIDENCE AND AN APPLICATION TO PORTUGAL
— Paulo Júlio, Ricardo Pinheiro-Alves, José Tavares
Banco de Portugal
| Working Papers
v
07/13 MONETARY POLICY SHOCKS: WE GOT NEWS!
— Sandra Gomes, Nikolay Iskrev, Caterina Mendicino
08/13 COMPETITION IN THE PORTUGUESE ECONOMY: ESTIMATED PRICE-COST MARGINS UNDER IMPERFECT
LABOUR MARKETS
— João Amador, Ana Cristina Soares
09/13 THE SOURCES OF WAGE VARIATION: A THREE-WAY HIGH-DIMENSIONAL FIXED EFFECTS REGRESSION MODEL
— Sonia Torres, Pedro Portugal, John T. Addison, Paulo Guimarães
10/13 THE OUTPUT EFFECTS OF (NON-SEPARABLE) GOVERNMENT CONSUMPTION AT THE ZERO LOWER BOUND
— Valerio Ercolani, João Valle e Azevedo
11/13 FISCAL MULTIPLIERS IN A SMALL EURO AREA ECONOMY: HOW BIG CAN THEY GET IN CRISIS TIMES?
— Gabriela Castro, Ricardo M. Felix, Paulo Julio, Jose R. Maria
12/13 SURVEY EVIDENCE ON PRICE AND WAGE RIGIDITIES IN PORTUGAL
— Fernando Martins
13/13 CHARACTERIZING ECONOMIC GROWTH PATHS BASED ON NEW STRUCTURAL CHANGE TESTS
— Nuno Sobreira, Luis C. Nunes, Paulo M. M. Rodrigues
14/13 CATASTROPHIC JOB DESTRUCTION
— Anabela Carneiro, Pedro Portugal, José Varejão
15/13 OUTPUT EFFECTS OF A MEASURE OF TAX SHOCKS BASED ON CHANGES IN LEGISLATION FOR PORTUGAL
— Manuel Coutinho Pereira, Lara Wemans
16/13 INSIDE PESSOA - A DETAILED DESCRIPTION OF THE MODEL
— Vanda Almeida, Gabriela Castro, Ricardo M. Félix, Paulo Júlio, José R. Maria
Banco de Portugal
| Working Papers
vi