Download Market and Underground Activities in a Two

Market and Underground Activities in a
Two-Sector Dynamic Equilibrium Model∗
Francesco Busato†
Columbia University
Bruno Chiarini
University of Naples
This version: May 24, 2002
Abstract
In this paper a two sector dynamic general equilibrium model
is developed in order to evaluate the implications of the underground economy from a business cycle perspective. There are
three main results. First, introducing an underground sector
improves the fit of the model to the data, especially along several important labor market dimensions. Second, the model
produces substantial internal propagation of temporary shocks.
Third, it is shown that underground activities offer risk sharing
opportunities by allowing households to smooth income through
a proper labor allocation between the two sectors.
Journal of Economic Literature Classification Numbers: E320,
E260, J22, H200. Keywords: Two-sector Dynamic General Equilibrium Model, Underground Economy, Propagation of Shocks,
Taxation.
∗
We have benefited from the comments and suggestions of John Donaldson.
We thank also Paolo Siconolfi, Edmund Phelps, Domenico Tosato and the
participants in the seminars at various Universities. We also thank David
Giles and Stefano Pisani for providing useful information on the underground
data, Francesca Caponi for the comments and the information concerning
the legal and fiscal aspects involved in the calibration, and Glenn Williams
for the research assistance. Chiarini acknowledges financial support from the
Ateneo Research fund of the University of Rome, La Sapienza, Dinamiche
dell’integrazione europea e scelta di politica economica. All errors are ours.
†
Department of Economics, Columbia University, Mail Code 3308, 420 W
118th st, New York, NY 10027. E-mail: [email protected]. Fax number:
+1-212- 854-8059 or +39-06-86216758.
1
Introduction
In this paper we present a dynamic general equilibrium model that
includes an explicit modelling of the ”underground economy”. Shadow,
non-market or underground activities are a fact in many countries,
and there are significant indications that this phenomenon is large and
increasing.1 The estimated average size of underground sector (as a
percentage of total GDP) over 1996-97 in developing countries is 39
percent, on transition countries 23 percent, and in OECD countries
about 17 percent.2 Table 1, drawn from a recent survey by Schneider
and Enste (2000), shows that the size of underground economic activity
is quite large and increasing in many European countries.
Table 1
Even though the precise definition of the phenomenon often varies
depending on the chosen method of measurement, it is worth noticing that different measurement techniques provide similar approximate
magnitudes of the size and development of the underground economy
across countries.3
Since these unreported activities make a sizeable contribution to
national production and income, we will argue that it is difficult to
understand the business cycles without some knowledge of the fluctuations of this relevant component of the aggregate economy. In fact,
1
There is no universal agreement on what defines the underground economy.
Most recent studies use one of more of the following definitions: (a) unrecorded
economy (failing to fully or properly record economic activity, such as hiring workers
off-the-book); (b) unreported economy (legal activity meant to evade the tax code);
(c) illegal economy (trading in illegal goods and services). Obviously, the difficulty
in defining the sector extends to the estimation of its size. We are concerned with
the size of the underground economy as encompassing those activities which are
otherwise legal but go unreported or unrecorded.
2
Note that there exist several synonyms for describing what underground activities are: underground or non-market activities, shadow or hidden economy. On the
other hand we denote official activities as the official sector, or as market activities.
3
There exist several methods of estimating the size of the underground economy.
A detailed survey of the most widely methods used to measure the hidden activity
are discussed in Schneider and Enste 2000. See also Feige (1989) and Thomas (1992;
1999) among others.
1
what is most interesting from the business cycle perspective is that the
estimated data reveals that the non-market produced GDP presents
cyclical features significantly different from those of the market sector.
Figure 1 encapsulates the graphs of the Hodrick-Prescott filtered series for the market and the underground components of GDP for Italy,
New Zealand, the United States and the United Kingdom. A casual
glance at these graphs suggests that official and underground activities
give rise to two distinct cycles. This stylized fact seems peculiar to
underground activities, and robust across counties: although the figures
are based on different data sets and different estimation methods, they
all emphasize the countercyclical nature of underground production.
Figure 1
This is an interesting and striking feature suggesting that standard
models omit important aspects of the real world. For example, in models with underground sectors, firms and consumers may be more willing
to shift resources out of market activity in response to productivity and
policy disturbances than in models without such sector. Intuitively,
in an underground model, although these shocks may not affect total
hours worked, they may affect how hours are allocated between the
market and underground sectors. Institutional features, furthermore,
can have significant effects: the structure of the parameters that drive
the agent’s willingness to shift resources between sectors, such as the
burden of tax and social security contributions, is relevant for determining aggregate fluctuations. In this model, firms and households,
like their real world counterparts, are affected by work and revenue
incentives, and undertake intra-temporal allocations of their resources
in addition to intertemporal substitutions. Augmenting the standard
stochastic growth model with this sector, can therefore go quite far in
accounting for aggregate fluctuations in countries with reasonably sized
non-market activities.
In this paper we aim to replicate these features, and to show that
this model formulation provides a new degree of freedom for enriching
2
the analysis of important macroeconomic phenomena, thereby producing a better understanding of business cycle dynamics, as well as presenting important policy implications. While this analysis is mostly
directed to European countries like Belgium, Portugal, Spain, Greece
and Denmark, as Table 1 confirms, we specifically seek to match the
Italian stylized facts, because Italy present a larger underground sector,
and it allows to better appreciate its impact on the economy.
We focus on three major issues. First, we review some basic business cycle facts for the Italian economy and show how the introduction
of an underground sector into a dynamic general equilibrium model
can not only account for these facts, but also resolve some heretofore
unsatisfactory results such as the employment volatility puzzle and
productivity puzzle generated by standard Real Business Cycle (RBC)
models.4 Second, we show how this model, augmented to include this
second sector, improves the internal propagation of stochastic disturbances. Third, we assess the implication sectoral and aggregate shocks
in terms of resource reallocation between the market and the underground sector. We show how underground activities help to mitigate
recessions and the cost of high tax burden by allowing the household
to smooth consumption through a proper labor allocation between the
two sectors.5
The paper is structured as follows: Section 2 presents the structure
of the model, and characterize the equilibrium for this economy, while
Section 3 compare our model with home production class of models.
Then Section 4 discusses calibration, and Section 5 present numerical
simulation results, together with impulse response functions analysis.
Section 6 offers concluding comments.
4
Although previous papers (e.g. Baxter and King, 1993 and Braun, 1994) have
shown that including distortionary taxation improves the ability of the standard
RBC models to match some key observations, the economic mechanism and the
driving forces in a model with underground economy turn out to be quite different.
5
These results imply that the Real Business Cycle contributions that include only
tax disturbances may lead to misleading predictions when underground activities are
neglected.
3
2
Structure of the Model.
There are three agents in the model: the firm, the consumer-workerinvestor, and the government. In addition there are two sectors: the
market and the underground sector. Finally, there is a homogenous
consumption good.
Each firm produces final output by using two different technologies,
one associated with the market, and the other with the non-market sector. We could imagine that the same firm produces in the market economy in the day, while in the underground economy by night. Following
Prescott and Mehra (1980), we assume that each firm solves a myopic
profit maximization problem, on a period-by-period basis, subject to a
technological constraint, and to the possibility that it may be discovered producing in the unofficial economy, convicted of tax evasion and
subject to a penalty surcharge.
The consumers choose consumption, investment, and hours to work
at each date and in each sector (official and unofficial) to maximize the
expected discounted value of utility, subject to a sequence of budget
constraints, a proportional tax rate on the market wage, and the law
of motion for capital stock.
Finally, government levies proportional taxes on revenues and incomes, and balances its budget at each point in time. We assume that
government expenditure on goods and services does not contribute to
either production or to household utility.
2.1
The Firms.
Suppose there exists a continuum of firms, uniformly distributed over
a unit interval. Each firm i ∈ [0, 1] produces a homogenous good with
two different technologies, one used in market sector, and the other
i ,
used in underground sector. Denote market-produced output as ymt
i , and define total production equal
non-market-produced output as yut
4
i + yi = yi
6
to ymt
ut
tot,t .
Technologies are specified as follows:
¡ ¢α ¡ i ¢1−α
i
= Mt kti
ymt
nmt
and
i
yut
= Zt niut .
(1)
i , is the result of capital, k i , and market labor,
The market output, ymt
t
nimt , applied to a Cobb-Douglas technology. The non-market output,
i , is produced with a production function which uses only non-market
yut
labor, niut . Finally, Mt and Zt denote sectoral stochastic productivity
shocks.7
Remark 1 This technology specification is equivalent to a more general set-up where both production functions use capital and labor, for
¡ ¢ ¡
¢1−α
¡ ¢ ¡ ¢
i = M k i α ni
i = Z k i β ni 1−β . From
and yut
example ymt
t
t
t
mt
ut
ut
Uzawa (1965) and Lucas (1988) if β < α we can set the smaller elasticity to zero without loss of any generality. Since underground activities
are labor intensive, we can simplify the model, and preserving the main
economic intuition, by assuming that underground sector produces using
only labor. We anticipate that in a Rational Expectations Equilibrium
(REE) firms use both technologies (this claim is formally proved below).
We next assume that each firm allocates a share, θti , of the total
labor demand, nit , to market production (therefore nimt = θti nit ) and
¢
¡
the remainder, 1 − θti , to the other sector (and niut = 1 − θti nit ).8
Normalizing Nti to unity, we rewrite (1) as
¡ ¢α ¡ i ¢1−α
i
θt
ymt
= Mt kti
and
¡
¢
i
yut
= Zt 1 − θti .
(2)
6
To further clarify the notation, the subscripts m and u refer to the sector from
which production originates.
7
Section (5 presents some possible interpretations for a shock on the underground
sector productivity.
8
The use of a share for labor is also consistent both with the fact that labor supply
per person is approximately stationary in many economies although the real wage
grows, and with the utility function, homogenous in consumption, that we adopt to
model the household preferences. The aim is, therefore, to analyze the movement of
resources between the two sectors, to understand how agents reallocate inputs out
of the market and into the underground sectors.
5
It will turn out that the reallocation of hours from market to informal sector rather than exclusively from leisure to labor, increases
the volatility of the official labor input for a given technology shock.
Lemma 1 characterizes the properties of the aggregate production technology.
Lemma 1 (Production Technology) Production function is well behaved.
Proof. Appendix A
2.1.1
Firms’ Revenues
i , are taxed at the stochastic
Market-produced revenues, qtm (1 − tt )ymt
corporate rate tt , where qtm denotes the price of market-produced good.
The process for tt is specified below, in Section 3.1.4. Firms do not pay
i , where q u is the price of
taxes on non-market produces revenues, qtu yut
t
non-market-produced commodity. Firms, however, may be discovered
evading, with probability p ∈ (0, 1), and forced to pay the stochastic
tax rate, tt , increased by a surcharge factor, s > 1, applied to the
standard tax rate. Note that since the market-produced and the nonmarket produced goods are identical, in a REE they must have the
same price. Lemma 2 proves formally this claim.
Lemma 2 (Price Vector) Denote a price vector for this economy as
hqtm , qtu , wt , rt i, where wt denotes labor wage, and rt is price of capital
(see below). In a REE we have qtm = qtu ≡ qt . Normalizing commodity
price qt to unity, the normalized price vector supporting the equilibrium
equals h1, wt∗ , rt∗ i, where wt∗ and rt∗ denote equilibrium prices.
Proof. Appendix A
Since qt = 1 holds in the equilibrium, we can impose it along the
solution. In the first case (firm is discovered,with probability p), revi , are:
enues, denoted as yD,t
i
i
i
yD,t
= (1 − tt )ymt
+ (1 − stt )yut
6
In the second case (firm is not discovered, with probability 1 − p),
revenues equal:
i
i
i
yN
D,t = (1 − tt )ymt + yut
To compute total expected revenues, we apply linear projection,
¢
¡
i
i
+ (1 − p) yN
and we have E yti |It = pyD,t
D,t . Simplifying, we finally
¡ i ¢
i
i
have E yt |It = (1 − tt )ymt + (1 − pstt )yut .
2.1.2
Firms’ Costs
The cost of renting capital equals its marginal productivity rt , net of
capital depreciation, δ. The cost of market labor is represented by the
wage paid for hours worked, augmented by social security stochastic
tax rate, tt , which, for simplicity, is assumed equal to social security
tax rate.9 We denote the former as wtm = (1+tt )wt , where wt is pre-tax
wage, while the cost of non-market labor equals the pre-tax wage, i.e.
wtu = wt .
Total costs for i − th firm are defined as follows:
¡
¢
CO θti , kti = (1 + tt )wt θti + wt (1 − θti ) + rt kti .
(3)
The structure of costs is consistent with the nature of underground
activities. Even though it may be feasible, it would be highly unusual to
see people working full time in the non-market sector. It is customary
that employees work a certain amount of hours under a regular contract
(for example, during the day) , while additional hours and extra-hours
are done without any formal agreement (for example, by night) . From
a firm’s perspective, it means that a worker’s cost is augmented by
social security contributions only for the regular working time, while
there is no tax wedge on his remaining hidden hours. k
9
Note that this assumption is not strong. Social Security tax rate ranges from
21% to 22%, while Corporate tax rate from 19% to 36%. Even though a complete
representation of tax system would require to define a third tax rate for Social
Security Contributions, we are convinced that the tax structure we are using is
enough rich, and it seems to be a good approximation to actual data.
7
2.1.3
Profit Maximization
At each date t, firm i maximizes period expected profits10
¡
¢
¡
¢
max E yti |It − CO θti , kti ,
(θti ,kti )≥0
and its behavior is characterized by the following first order necessary
and sufficient conditions:11
¡ ¢α−1 ¡ i ¢1−α
θt
0 = (1 − tt )Mt α kti
− rt
¡ i ¢α ¡ i ¢−α
− (1 − tt ps)Zt − wt tt .
0 = (1 − tt )Mt (1 − α) kt
θt
2.1.4
(4)
(5)
Productivity and Tax Disturbances.
Finally, we formalize productivity disturbances and tax rates as a
stochastic vector of variables that follow a univariate AR(1) processes
in log:
At+1 = ΩAt +²t
where At is a vector [ln Mt , ln Zt , ln tt , ln τt ]0 containing the productivity
shocks, ln Mt , ln Zt , the stochastic corporate tax rate, ln tt , and the
stochastic personal income tax rate ln τt . Ω =diag (ρi ) , where i =
m, z, t, τ, is a 4 × 4 matrix describing the autoregressive components of
the disturbances relative to each of the four shocks. The innovation,
²0t = [εm , εz , εt , ετ ] , is a vector of i.i.d. normal random variables.
10
Note that in this context firms solve a myopic optimization problem. Given
the nature of the model, it would be interesting to generalize the firms’ behavior
incorporating an inter-temporal optimization problem as in Danthine and Donaldson
(2001), for example.
11
i
i
To derive (4) and (5) compute period expected profits, pyD,t
+ (1 − p) yN
D,t =
i
i
i
i
p((1 − tt )ymt + (1 − stt )yut ) + (1 − p)((1 − tt )ymt + yut ), and then take derivatives
with respect θti and kti .
8
2.2
The Consumer-Worker-Investor.
Suppose there exist a continuum of consumers, uniformly distributed
over a unit interval, supplying labor to the market and the underground sectors.12 Consumer j ∈ [0, 1] has preference over sequences
of consumption and labor, and maximizes expected utility as summa∞
P
β t u(cjt , θtj ), where Et
rized by the lifetime utility function U0j = E0
t=0
is the mathematical expectations operator conditional on information
available at time t, and β is a subjective discount factor.
To represent consumer behavior in this environment, we take a cue
from Cho and Rogerson’s (1988) and Cho and Cooley (1994) family
labor supply model. They distinguish labor supply with regard to an
intensive (the hours worked), and an extensive margin (the employment
margin). In our model we reinterpret these two dimensions as representing worker’s labor supply in the regular and in the underground
sectors. Hence we specify the momentary utility function as follows:
u(cjt , θtj ) =
(θj )1+γ
(cjt )1−q − 1
(1 − θtj )1−η
−h t
(1 − θtj ) − f
,
1−q
1+γ
1−η
(6)
where cjt denotes consumption profile of consumer j, θtj her market
labor supply, and 1 − θtj her non-market labor supply. This function is
separable between consumption and labor and allows to study how an
household allocates its labor between the market and the underground
sectors.13 To have a well behaved utility function, we assume that
12
We imagine that the same employee works some hours under a regular labor
contract, and, in addition, some others under a private agreement with the employer.
This working time represents consumer’s underground labor supply.
13
This specification is adapted from Cho and Rogerson (1988) and Cho and Cooley
(1994). Unlike the extreme cases of indivisible labor, where all the fluctuations
occur on the extensive margin, and the divisible labor in which the fluctuations take
place on the intensive margin, in this formulation of preferences households may
allocate their time along both margins (intensive-hours and extensive-employment
margin). Cho and Rogerson achieve this feature by introducing heterogeneity into
the opportunity sets of household decision makers, and Cho and Cooley introduce
some fixed costs of going to work that are not explicitly modelled. This allow
us to capture changes in labor in both the market and the underground sector
9
h, f ≥ 0, γ > −1, η ∈ (0, 1) and that all the parts of the momentary
utility function are twice differentiable and well behaved.14 The second
(θtj )1+γ
j
1+γ (1−θt ), represents the overall disutility of working, while
(1−θtj )1−η
last term, f 1−η
, reflects the idiosyncratic cost of working in
term, h
the
the underground sector. In particular, this cost may be associated with
the lack of any social and health insurance in the underground sector.
The representative household faces the following budget constraint:15
wt (1 − θtj τt ) + rt ktj = cjt + xjt .
(7)
where xt denotes investment at time t . Finally, investment increases
the capital stock according to the following state equation:
j
kt+1
= (1 − δ)ktj + xjt
(8)
where δ denotes the depreciation rate.
With the period utility function defined as in equation (6) the value
function Jt (ktj , Kt , At ) of the representative agent satisfies:
Jt (ktj , Kt , At ) = max
j
kt+1
,θtj
h
n
io
j
, Kt+1 , At+1 ) | It
u(cjt , θtj ) + βE Jt+1 (kt+1
subject to budget constraint (7), and the law of motion for the household capital stock (8).
Lemma 3 (Existence and Uniqueness of the Policy Function)
simultaneously, and it is consistent with the data, where we observe substantial
variations in both the markets.
14
To have a well defined dynamic optimization problem, we want the objective
function to obey well know conditions (see Stokey and Lucas, 1989). More precisely,
we want that (i) the household discount factor (β) ∈ (0, 1), that the objective
function is (ii) continuous and (iii) strictly increasing. Note that in this model the
third requirements is not as trivial as it could seem. Condition (iii) might not hold
for some parameterizations of the model. For this reason, Lemma 3 in Appendix A
shows that for the parameterization presented in section 4 condition (iii) holds.
15
More precisely, the right hand side summarizes the expenditure, and it is fairly
standard. The left hand side represents the income and equals wt (1 − τt )θtj + wt (1 −
θtj ) + Rt ktj . To derive (7), we factorize out wt and we simplify.
10
The policy function for this problem exists, and it is unique.
Proof. Appendix A
After deriving and manipulating the necessary and sufficient first
order conditions, we obtain the Euler equation (9), and the intratemporal consumption-labor allocation, condition (10):
1 = βE((
cjt+1
cjt
)−q Rt+1 | It )
0 = −wt τt (cjt )−q − (θtj )γ + h
(9)
2 + γ j 1+γ
− f (1 − θtj )−η (10)
(θ )
1+γ t
¡ i ¢α−1 ¡ i ¢1−α
where (1+(1−tt+1 )Mt+1 α Kt+1
θt+1
−δ) = (1 + rt+1 − δ) ≡
Rt+1 from firm profit maximization (see section 2.1.3).
2.3
The Government.
Finally, the flow government budget constraint is:
wt τt θt + (pstt ) yut + tt ymt = Gt
(11)
where Gt = Ḡ.16
2.4
Equilibrium
Equilibrium for our model is described as a Variant on a Recursive
Competitive Equilibrium (RCE) of Prescott and Mehra (1980) notion.
Since we are not aware of other dynamic general equilibrium models
explicitly incorporating an underground sector, we describe the equilibrium in its details.
A RCE for the decentralized economy with underground sector consists of:
16
Notice that the Government balances its budget only in expectation, since with
probability 1 − p some firms and workers are evading. Hence equation (11) will not
be satisfied on a state by state basis.
11
+
1. a set of continuous price functions, w(At , kt , Kt ) : <+
2 7−→ < , and
+
+
r(At , kt , Kt ) : <2 7−→ < ;
+
2. a value function Jt (At , kt , Kt ) : <+
3 7−→ < ;
3. and policy functions cj (At , kt , Kt ), θj (At , kt , Kt ), and xj (At , kt , Kt )
+
all from <+
3 7−→ <
such that:
1. firms are maximizing profits at the prevailing prices, i.e. (4) and
(5) are satisfied;
2. cj (kt , Kt , λt ) , θj (kt , Kt , λt ) solve (9) and (10), i.e. consumer-workerinvestors are maximizing utility at the prevailing prices;
3. government balances its budget in expectation, i.e. equation (11)
holds on a period by period base;
4. market clearing conditions hold for each market. Specifically,
R
R
for labor service in market sector, θj (At , kt , Kt ) dj = θi (At , kt ) di =
θt (At , kt , Kt ) ≡ Θt
R
R
for labor service in underground sector, (1−θj (At , kt , Kt ))dj = (1−
θi (At , kt ))di = 1 − θt (At , kt , Kt ) ≡ 1 − Θt
R
R
for consumption cj (At , kt , Kt ) dj = ci (At , kt ) di = ct (At , kt , Kt ) ≡
Ct
R
R
for investment z j (At , kt , Kt ) dj = z i (At , kt ) di = zt (At , kt , Kt ) ≡
Zt
R
R
for capital k j (At , kt , Kt ) dj = k i (At , kt ) di = kt (At , kt , Kt ) ≡ Kt
The economy satisfies conditions for existence and uniqueness of
the Equilibrium as detailed in Prescott and Mehra (1980), to which we
refer for details. To complete the characterization of the equilibrium,
Proposition 1 proves that we have an interior equilibrium.
Proposition 1 (Interior Solution) Here we show that a Rational
Expectation Equilibrium for this model involves interior solutions.
Proof. There are two possible corner solutions, the first θt = 0 is
discussed in Claim 1, and the second θt = 1 in Claim 2. We prove both
by contrapositive argument.
12
Claim 1: θt = 0 is not a REE. Suppose not, i.e. θt = 0 is a REE.
If this is the case, it must satisfy profit maximization (equations (4)
and (5)), utility maximization (equations (10) and (9)), and market
clearing conditions. From (5) we have that as long as θt → 0 then
wt → ∞. Substituting this into equation (10), the only value of θ
coherent with this price is θ = 1. Labor market does not clear: hence
θt = 0 cannot be a REE.
Claim 2: θt = 1 is not a REE. Suppose not, i.e. θt = 1 is
a REE. We use the same argument of Claim 1. From equation (10)
when θt = 1 then wt → −∞ and, from equation (5) there are no
values of θt ∈ (0, 1) that satisfy this condition. Again, market clearing
conditions are not satisfied for labor services, and thus θt = 1 cannot
be a REE.
Substituting the Equilibrium conditions and the Government budget constraint into the first order conditions, we obtain a non-linear
system of stochastic difference equations that defines the full set of
feasible equilibrium allocations.
Being highly non-linear, the system has no closed form solution.
To study its stochastic properties we apply the well known procedure
developed by King Plosser and Rebelo (1988a, b). In other words, we
assume certainty equivalence, we linearize the system around its steady
state, and we solve it applying linear approximations (e.g. Campbell
1994; Uhlig 1999).
3
Underground Economy Vs Home Production
Before proceeding to the analysis, it is interesting to compare the theoretical structure of our model with household (home) production models.17 We focus on four, selected, issues: the commodities’ number and
17
Home production has been part of standard labor paradigm. Fundamental
references include Becker (1965), Pollak and Watcher (1975), and Gronau (1977)
and (1986)). Only recently has been introduced into macro models. However, the
literature is quite large: see Benhabib, Rogerson and Wright (1991) for a survey, or
among the many Rios Rull (1993), McGrattan, Rogerson and Wright (1992), Fisher
13
their substitutability, the financing of capital investment, the insurance
opportunities offered by the second sector, and the different cyclical
properties between home production and underground activities.
First, consider the number of consumption goods and their substitutability. In the home production class of models there exist two
goods, denoted as market and non-market commodities, each of which
is produced with a sector specific technology. In addition, the preference specification allows for different degrees of substitutability between
market and non-market goods.18 In contrast, in the model with underground sector there exist only one homogenous good, which, however, is
produced using two different technologies: one associated with market
sector, and the other with underground sector. In this environment
it is natural to focus on the case of perfect substitutability between
market-produced final output and underground-produced one.
The second difference concerns the financing of investments. In
home production models only market-produced goods can be consumed
and invested, either into market capital or into non-market capital.
There are no uses for home production output other than consumption
- it cannot be sold or transformed into capital, for example, the way
that market-produced output can. In the underground economy model,
however, there exists only one capital stock (invested in the market sector), but market and non-market-produced output can be transformed
into market capital, without any adjustment cost. The underground
sector offers an additional channel for financing capital stock accumulation, and an additional dimension along which firms can employ the
available labor supply.19 Summarizing, while home production model is
a legitimate two sector model, the underground economy model could
be more appropriately defined as a two technology model, since the
same good is produced using two different technologies. Alternatively,
(1992), and Fung (1992).
18
It is customary, in this literature, to consider the version with perfect substitutability as the benchmark simulated economy.
19
Technically speaking, the specification of consumer intertemporal feasibility constraint, equation (7), incorporates this feature.
14
we could define it as a 2-cycle model, given the cyclical properties of
the market and the underground sectors.
In addition, an underground sector offers profit smoothing opportunities for firms, and insurance opportunities for consumers. More
precisely, firms can smooth their profits by a proper allocation of labor
demand between the two sectors, on a period by period base. In addition, consumers can smooth not only consumption, by substituting
over time consumption and investments, but also income, by allocating
their labor supply across sectors, on a period by period base.20 In the
model with underground sector consumers have two sources of income,
which, being countercyclical, offer insurance against bad times (section
5.3 offers a deeper discussion). This mechanism is absent in models
with home production.
Finally, Ingram et al. (1997) find that hours spent in home production are acyclical. It is very important to notice that this implies that
during recessions home production models predict that workers may
adjust by switching into leisure, whereas a model with underground activities predicts a switch into underground activities. Difference is that
in our class of model, non-market income increases during recessions,
mitigating slumps, by offering insurance opportunities to household.
Again this mechanism is not present in home production models.
4
Model Calibration
The model is calibrated for the Italian economy because it presents a
large underground sector, and it allows to better appreciate its impact
on the overall economy. Needless to say, this analysis is addressed
to European countries like Belgium, Denmark, Greece , Portugal and
Spain, as Table 1 confirms. Our calibration is based on seasonally
adjusted ISTAT series from 1970:1 to 1996:4, expressed in constant
1995 prices.
20
Needless to say, income smoothing is a device to smooth consumption over time.
15
Consumption is approximated by households’ final expenditures,
and output definitions are approximated by the corresponding GDP
definitions. Since Italian aggregate GDP, as well as GDP of the other
European countries, contains an estimate of a hidden sector, we decompose it into the market and the underground components by using the
series presented in Bovi (1999). Finally, all variables are transformed
in logarithms, and detrended using the Hodrick-Prescott filter.
The system of equations we use to compute the dynamic equilibria
of the model depends on a set of 12 parameters. Six pertains to household preferences, (q, h, f, η, γ, β), four to the structural-institutional
context (the probability of a firm being detected p, the surcharge factor s, the equilibrium income and corporate tax rates t and τ ), and
the remaining two parameters to technology (the capital elasticity α,
and the capital depreciation rate δ). The fact that the data on the
underground economy is difficult to obtain substantially complicates
the calibration. Because we are not aware of other studies which calibrate the parameters of a general equilibrium model augmented with a
non-market sector, we precisely detail our calibration procedure below.
A starred parameter denotes the precise calibrated value.
1. The probability of being detected, p. We calibrate this
parameter by estimating the unconditional mean of the ratio
of number of inspected firms to their total number, i.e. p̃t =
Inspected F irms at t
21
T otal N umber of F irms at t .
For Italy, as well as for the major-
ity of countries, only a portion of this data is publicly available.
For the Italian economy, the Ministry of Labor reports that the
number of inspected firms has been 118,119 in 2000, 106,307 in
1999 and 95,676 in 1998. The overall number of firms in the Italian economy has been 4,639,393 in 2000, 4,472,375 in 1999 and
4,311,369 in 1998. As suggested above, we first compute the probability of being detected in each year, p̃t , and then we estimate
21
Note that an inspected firm is not necessarily convicted of evasion and therefore
fined. Since inspections are based either on private information of Institutions, or
randomly, it may happen that behavior of a perfectly honest firm will be inspected.
16
the aggregate probability as p∗ =
omy
p∗
=
0.03.22
1
T
PT
t=1 p̃t .
For the Italian econ-
Even though this is not an efficient estimate, it
represents the best possible calibration for this parameter, given
the available data.
2. The surcharge factor s, the income tax rate t, and the
corporate tax rate τ . The parameter s represents the surcharge
on the standard tax rate that a firm, detected employing workers
in non-market sector, must pay. According to the Italian Tax
Law (Legislative Decree 471/97, Section 13, paragraph 1) the
surcharge equals 30 percent of the statutory tax rate if the firm
pays the fine when detected, or 200 percent when the firm refuses
to pay.23 We present results for s∗ = 1.30.
In Italy, corporations are subject to a progressive tax rate. A tax
rate of 19 percent is applied to the share of profits that represents 7 percent of the firm’s capitalization; the remaining portion
is then subjected to an increased tax rate of 36 percent. We
calibrate the steady state value of the corporate tax rate as the
average of these two numbers, i.e. t∗ = 0.275.
The personal income tax system is more complex, since we have
five tax rates, spanning from 18.5 percent to 45.5 percent.24 The
calibration of the income tax rate may be undertaken in two
ways. It may be estimated as the average tax rate, weighted by
the relative share of population in each income class. It may also
be estimated as the tax rate associated with the average income
of the working population (Adults 15-64 years old). We rely on
22
These data are available at the web site of the Italian Ministry of Labor, at the
URL http://www.minlavoro.it/Personale/div7-conferenzastampa 01032001.htm.
23
In this case the firm will prosecuted under Criminal Law perspective, and if
condemned pay 200 percent.
24
More precisely, the structure of the tax rates is the following as of 2001. For
incomes less than 10,331 Euros tax rate is 18.5 percent, for incomes between 10,331
Euros and 15,496 Euros tax rate is 25.5 percent, for incomes between 15,496 Euros
and 30,992 Euros tax rate is 33.5 percent, for incomes between 30,992 Euros and
63,283 Euros tax rate is 39.5 percent and, eventually, for incomes above 63,283
Euros tax rate is 45.5 percent. More details at the web-sites www.finanze.it or
www.tesoro.it.
17
the second procedure and since the average income equals 18,246
Euros we estimate the income tax rate at 33.5 percent.
3. The steady state value of non-market sector share, 1 −
θ. To calibrate this parameter we refer to Schneider and Enste
(2000) who estimate the share of the non-market sector for a panel
of OECD countries. The value for the Italian Economy, 1 − θ∗ =
0.265, is also consistent also with Mare’s (1996) estimates.
4. The preference parameters, q and β, the capital share, α,
and the capital depreciation rate δ. These parameters are
set to values commonplace in this literature (e.g. Fiorito and
Kollintzas, 1994, or Censolo and Onofri, 1993). More precisely,
we set q ∗ = 1, β ∗ = 0.98 and δ ∗ = 0.025.
5. Stochastic Shocks autocorrelation coefficients, ρm , ρu , ρt , ρτ
and innovation amplitudes, σm , σu , σt , στ . The ρ0 s are set to
.90 and the σ 0 s to 0.003. As we stress in next section, these val-
ues are much lower than the standard ones (see King and Rebelo,
1999). This means that the model has a particularly efficient
amplification mechanism which allows us to employ very small
shocks (Section 5.3 offer more details).
6. Preference parameters h, f, η and γ. The calibration of these
parameters is a not easy (see Cho and Cooley, 1994). We select them to match four moments: the ratio between standard
deviation of total output σ(yttot ), and the standard deviation of
total consumption, σ(ctot
t ), the correlation between total output
tot
and total consumption ρ(ctot
t , yt ), the correlation between unu
derground production and total consumption ρ(ctot
t , yt ), and the
correlation between market production and total consumption
m
∗
∗ = 1.99,
ρ(ctot
t , yt ). The calibrated values are h = 0.55, f
η ∗ = 0.40, γ ∗ = 3.00.25
25
Cho and Cooley (1994) calibrate these parameters using a similar procedure
for the United States, and choose h = 6.0, f = 0.87, η = 0.62, γ = 2.00. Note,
however, that their formulation of the model addresses issues different from matching
18
5
Simulation Results
Here we describe how well our model accounts for aggregate fluctuations, and we compare its performances with selected alternative approaches (section 5.1). Next, the economic mechanisms and the driving
forces operating in our model are analyzed and interpreted by studying
the Impulse Response Functions (IRF) to the sectoral and the aggregate shocks (section 5.2). Eventually, we focus our attention on the
model propagation mechanism (section 5.3).
5.1
Numerical Simulations
Table 2 illustrates how adding the underground sector implies a much
better fit to the data. The tables displays the relative ability of three
different models in order to match the major stylized facts characterizing the cyclical behavior of Italian economy over the sample 1970:11996:4.26 Its first part reports the actual stylized facts of the economy, while the following blocks present the corresponding business cycle properties, as generated by a Hansen (1985) indivisible labor version
of the stochastic growth model calibrated for Italian economy, a model
incorporating a monopolistic union (Chiarini and Piselli, 2001), and
our two sector model. Statistics are generated from 100 simulated time
series of length 104 quarters. A star denotes simulated moments, while
a hat moments estimated from actual data.
Table 2
Among the reported statistics, it is interesting to note how market
GDP is much more volatile (σ̂ym = 2.27) than the total and underground counterparts (σ̂yu = 1.11 and σ̂ytot = 1.47 respectively). Further, consumption is positively correlated with the market and the aggregate components of GDP (ρ̂(c, ym ) = 0.95 and ρ̂(c, ytot ) = 0.80),
moments for the market and the non-market sectors. Specifically, they study the
implications of this kind of utility function for the volatility of hours, employment
and productivity in the United States.
26
The comparison is particularly interesting, because, all the selected models incorporate some mechanism that substantially amplifies productivity shocks.
19
while it presents a negative correlation with the underground part
(ρ̂(c, yu ) = −0.47).
It is also interesting to compare our model’s performance with actual data along the so called consumption volatility puzzle, the productivity puzzle and the employment variability puzzle.27 First, notice
that our two-sector model replicates efficiently consumption volatility (precisely, σ̂ctot = 1.24 and σc∗tot = 1.17 (0.15)). This is an interesting improvement upon the indivisible labor version, and the union
model.28 We argue that this improvement is due to the introduction of
the underground sector. In particular, in the model there are two income sources (market and non-market income), which are both highly
volatile, and negatively correlated. This implies that, on the one hand,
consumers can smooth more easily consumption, by a proper labor
supply allocation between sectors, but, on the other hand, that they
are subject to two sources of fluctuations. For the parameterization
presented in section 5, the second effect dominates.
Moreover, our model matches, as well as the Chiarini and Piselli
(2001) model, the correlation between market labor and productivity (precisely, ρ̂(nt , πt ) = 0.19 and ρ∗ (nt , πt ) = 0.04 (0.16)).
The economic mechanism, however, differs significantly between the
two approaches. To understand the economic implications, Figure 2
describes the labor market response after an unexpected increase in
market productivity, Mt .
27
The employment variability puzzle refers to the fact that employment (or total
hours worked) is almost as variable as output, and strictly procyclical, something
which is difficult to replicate in a standard neoclassical model. The productivity
puzzle looks at the correlation between labor productivity and GDP. If productivity
shocks drive the cycle, by construction the productivity will be highly correlated with
the employment. The puzzle is that productivity and employment are negatively
correlated for most economies. As reported by Stadler (1994) this correlation is
negative of zero for almost all the countries. Finally, the consumption volatility
puzzle (Cochrane, 2001) refers to the fact that consumption volatility generated by
stochastic growth models is often too small relative to the data.
28
Precisely, the former does fairly well in matching the employment volatility,
but it is limited in replicating other business cycle facts. The latter has a better
fit compared to Hansen version, but it still generates a low consumption volatility
nion
= 0.29).
(precisely, σctot = 1.24 and σ̂cUtot
20
Figure 2
To begin, notice that the labor market of our model is characterized
by two pairs of demand and supply schedules, one for the market sector, and the other for the underground economy. After the productivity
shock, market labor demand shifts out (from point M0 to M1 ), and labor supply increases, initially along its schedule. This shock does not
affect underground labor demand, while it does affect the non-market
labor supply schedule, which shifts to the left (from point U0 to U1 ).
This step is crucial. Since the two labor supply schedules move in
opposite directions, the market labor supply schedule shifts out (from
point M1 to M2 ), further increasing employment in market sector and
reducing the wage. This happens because households allocate a share,
θ, of total labor supply to the market sector, and its complement, 1−θ,
to the underground economy. In other words, when the underground
labor supply share decreases, the market sector counterparts has to
increase. This mechanism reduces the correlation between labor productivity and GDP, and thus improves the empirical performances of
our model along the productivity puzzle. In the next section we provide further evidences on labor reallocation by using Impulse Response
Functions to the sectoral and aggregate disturbances.
Finally, while each of the three models generate sufficient employment volatility, only our two-sector model produces adequate labor
productivity volatility (precisely, σ̂π = 2.78 and σπ∗ = 2.94 (0.17)).
Since labor productivity volatility is related to labor volatility itself,
the outcome produced by our model should be related to the fact that
labor enters linearly into underground sector production function (see
equation (1)). This structure is meant to represent an aspect peculiar
to underground activities, where labor is highly flexible and elastic.
As a results it produces volatility in labor services, and consequently,
to labor productivity. It is interesting to underline the difference between this mechanism and Hansens’s (1985): the former uses a linearity
into the production function, that is on the labor demand side, while
the latter makes, roughly speaking, labor entering linearly into utility
21
function, that is on the supply side.
Summarizing, the introduction of an underground sector is relevant,
because the economic mechanisms and the driving forces we present
here are generally absent in the standard Real Business Cycle models.
It is likely, that for economies with an empirically relevant unofficial
sector, standard RBC model overlooks the relationship between the
market and the underground sectors generated by innovations in the
sectoral and the aggregate shocks, as we explain more in details in the
next section.
5.2
Sectoral and Aggregate Shocks
In the model there are two sectoral shocks, the market productivity
shock (Mt ) and the underground technology shock (Zt ); in addition,
there are two aggregate shocks that increase income and corporate
tax rates (τt and tt ). In order to understand the dynamics of the
economy when the equilibrium is disturbed by these exogenous shocks,
we analyze the IRF generated both by the sectoral and the aggregate
disturbances.
Figure 3 describes the model response when either the market or
the underground sector productivities increase exogenously by a one
standard deviation innovation (Panel A and Panel B, respectively).
Figure 3
Consider first Panel A. The marginal productivity of input factors increases (decreases) in market (underground) economy, attracting (transferring) resources from (to) underground (market) sector. As
there is more employment and more capital in this sector, and a decrease in the employment in the underground one, a market productivity shock implies an increase of the aggregate output. Note, however,
that the impact on aggregate production is partially offset by a reduction in the underground-produced output. Since market sector is
relatively larger in size than its underground counterpart, households
22
have more resources, which are allocated between consumption and
investments.
The previous picture, however, is reversed when the economy is hit
by a one-standard deviation innovation on underground sector productivity (Panel B). This shock may be motivated as representing several
inputs that often are left out of the model for convenience, such as
management skills, creativity, better workplace organization, etc. Elements that exist in the underground activities, and that are capable to
rise corresponding labor productivity.29 After the shock, the employment share in the underground economy rises, pulling down the share
of workers employed in the market sector. Notice, that as market employment and capital fall, the employment increase in the underground
sector is too small to compensate its reduction in the market economy.30 Here the capital stock reduction is driven by a drop in the
marginal productivity of invested capital, caused by the reallocation
of labor inputs to the underground sector. Figure 3 also shows that
sectoral shocks have asymmetric aggregate effects on output and labor
shares. Consider, for example, the impact response of aggregate output
after the sectoral shocks: an increase in market productivity raises it
29
More precisely, the productivity shock idiosyncratic to the underground sector
can be tied, for example, to the following three arguments. First, since labor in
the underground sector is very flexible, and the worked hours are voluntary, we
argue that the employee’s motivation is stronger. Second, a significant part of the
underground labor force is made of immigrants, which try to be as much productive
as they can, to be hired as a regular workers. Notice that immigrants usually
enter western European countries with temporary visas, which are converted into
permanent ones only when they prove to be regularly employed. The waves of
immigrants can be interpret as temporary shocks to underground sector. Finally,
there are young pensioners entering the underground labor market. These workers
have a high productivity, but choose to work in the underground sector for earning
an additional income, keeping at the same time their pensions. Needless to say, this
would not be possible if they were hired under a regular labor contract.
30
This is a direct consequence of the structure of the labor market. Indeed, in
equation (1) we normalize the aggregate employment to unity, which implies that a
sectoral shock generates, for what concern labor services, only a reallocation between
the two sectors. All resources take away from a sector are transferred to the other;
in this context the relative size of the two economies has an important role in
shaping aggregate fluctuations. A development of the actual model could generalize
the actual set-up introducing the aggregate employment as an additional control
variable.
23
1.8 percent above its steady state, while it falls below its steady state
level by less than 1 percent after a shock in the underground economy.
Since the rise of the burden of taxes and social security contributions
is one of the most cited causes of the growth of underground economy,
government taxation plays a relevant role in the allocation of output
and labor input between these sectors. Our interest in this analysis
is motivated by the desire to assess the cyclical properties of taxation
policies, and their implications in term of resource reallocation. More
precisely, we investigate how an unexpected increase in corporate and
personal income taxes affects production and labor allocation between
the market and the underground sectors. Figure 4 encapsulates the
IRF after a one-standard deviation increase in the corporate tax rate
tt (Panel A), and in the income tax rate τt (Panel B).
Figure 4
A temporary rise in the corporate tax rate, tt , reduces production and labor input in the market sector, enhancing employment and
output in the underground sector. The official labor market share is instantaneously below its steady state level by 6 percent, while the share
of underground labor rises almost symmetrically. Investment falls below its steady state pre-shock level by 22 percent, and the production
activity in the official economy drops by about 3 percent, while aggregate output falls below its steady state by 2 percent.31 Notice that the
rise in the unreported enterprisers thwarts to some degree the recession effects of the rise in taxation. The negative impact of the higher
taxation on output and income induces firms and households to work
more in the underground sector, highlighting a strong reallocation effect between the two sectors. An aggregate shock on corporate tax
pushes the economy into a recession, which is, however, mitigated by
the existence of the underground sector that offers an insurance or risk
31
The volatility of the investment seems higher than the standard RBC models:
see, for example Baxter and King (1993), or Ramey and Shapiro (1997).
24
sharing opportunity, through the labor reallocation toward that sector. We could think to the underground sector as offering an insurance
channel, alternative to financial markets, available to people facing,
for example, liquidity constraints. Individuals involved in underground
activities usually face restricted market participation.
The reaction of the economic system after a raise of personal income
tax, τt , does not differ, qualitatively speaking, from the previous scenario. Notice that the impact responses of the variables are just diminished in size, compared with Panel A, and with a standard RBC model,
too (e.g. King, Plosser and Rebelo, 1988a). In our simulation, investment’s peak impact response reaches a decline of about 3.5 percent, and
consumption drops by about 1 percent. That is because the consumers
can reallocate consumption and labor, not only inter-temporally, but
also intra-temporally between the two sectors, reducing the income loss
generated by the fiscal policy. More precisely, by shifting resources to
the underground sector, consumers are able to reduce their taxable income, as they keep constant their effective income. This intra-temporal
smoothing opportunity is reflected into the IRF as a softened response
to stochastic disturbances. Comparing this outcome with the previous
one (Figure 3), we argue that consumers reallocate resources intertemporarly more easily than firms do. That happens because the latter
ones invests capital in the market sector, which, given a concave production function (see equation 1), slows down the labor reallocation
mechanism. In addition the probability of being discovered, and fined,
reduces firms’ incentives to move large amounts of resources out of the
market sector.
This analysis leads us to conclude that aggregate RBC models for
economies with a significant underground sector cannot disregard these
activities for understanding fiscal policy effects and labor reallocation
over the cycle. Indeed, if we were to ignore this sector, we would abstract from some important aspects, from a theoretical and empirical
perspective. The actual RBC research shows that the introduction of
additional sources of variation, such as exogenous fiscal policy shocks,
25
helps to improve the theoretical models predictions (Christiano and
Eichenbaum, 1992; Braun, 1994; McGrattan, 1994; Edelberg, Eichenbaum, Fisher, 1999). But assessment and comparisons between these
models and the data may often be limited by the lack of untaxed activities. Hence, overlooking the underground sector may conduct to a
misleadingly representation of the vastly complex tax effects over the
cycle.
5.3
A Better Propagation Mechanism
King and Rebelo (1999), in their well known survey on Real Business
Cycle models, discuss extensively the central role of productivity shocks
in driving the business cycle.32 They also stress how the model performances rely on large and highly persistent technology shocks. To generate macroeconomic series consistent with the US and the European
data, the RBC models require a considerable variability in productivity, and a serial correlation parameter of the stochastic component of
productivity near one. But, at the same time, a well known shortcoming of the standard dynamic general equilibrium model is its weak
propagation mechanism, as pointed out by Cogley and Nason (1995), or
more recently by Chang, Gomes, Schorfheide (2001) among the others.
King and Rebelo (1999) identify four mechanisms that substantially
amplify productivity shocks, and lead to stronger comovements of the
main variables. The first one makes output respond more elastically to
productivity shocks, the second and the third ones rely on a larger elasticity either of the labor demand or the labor supply, while the fourth
is based on the non-separability between consumption and leisure (or
labor) into the utility function.
The propagation generated by our model is not based on any of
the listed, but it depends on an original mechanisms, peculiar to un32
Note, however, that the notion that technological shocks are related with business cycles was called into question by several contributions. Using different methodologies, Gali (1999), Shea (1998), and Basu, Kimball and Fernald (1999), and more
recently Nevill and Ramey (2002), present similar results: positive technological
shocks lead to decline in labor inputs.
26
derground activities. Indeed, its introduction into a stochastic growth
model does not affect either the slope of labor demand and supply
schedule, or the elasticity of output to shocks, but introduces a propagation mechanism triggered by the reallocation of labor services between the two sectors. In the light of the results stylized in Figure 2,
and presented in Figure 3 and 4, when a shock hits one of the two
sectors, it is transmitted to the other, which returns an additional,
smoothed, impulse to the former sector. By this end, the impact of a
shock is propagated within the economy, which necessitates of smaller
stochastic disturbances to generate series consistent with the actual
data.
It is particularly welcome that we obtain these results even if we
use a logarithmic utility function for consumption, and small shocks.
Following Cogley and Nason (1995) we define an improvement in the
propagation mechanism of a stochastic growth model in the sense of necessitating a lower autocorrelation coefficient for the process of stochastic disturbances, and a smaller standard deviation of the innovations
(defined in Section 2.1.4) for replicating business cycle facts. To highlight this feature, we compare, in Table 3, our model parameterization
with the one used in the standard benchmark model (Hansen, 1985),
with the parameterization used in the so-called high substitution class
models (see King and Rebelo, 1999), with the parameterization of a
model which uses a utility function like ours (see Cho and Cooley,
1994), and with a home production model (Benhabib, Greenwood and
Wright, 1996).
Table 3
It is interesting to note that while the standard RBC models need
significantly larger shocks (standard deviation of the innovation process, σ = 0.00712) and a high autocorrelation coefficient for the shock
(ρ ' 0.99), the high substitution economies reduce the first of these
magnitudes to about, σ = 0.0012 but still need a high AR(1) coefficient (ρ ' 0.99). The latter class of models also requires a high risk
27
aversion parameter (q = 3). The Cho and Cooley (1994) model does
not require a high relative risk aversion coefficient, and even if it has a
smaller autocorrelation coefficient (ρ ' 0.95) , it still needs a very high
standard deviation for the innovation process, σ = 0.0102. Our model
is, instead, able to match properly the data with a small amplitude
of the innovation process (standard deviation of the innovation process, σ = 0.003), with a better amplification of the shock across time
(ρ ' 0.90), and without assuming a high risk aversion (q = 1). The last
2
column of Table 3 present the variance of the stochastic shock σShock
2
(which differs from that of innovation process, σInnovation
), as gener2
=
ated by the different models. It is computed as σShock
2
σInnovation
1−ρ
and
offers an additional perspective on the model comparison.
6
Conclusions
In this paper, we use a general equilibrium production model with an
underground economy to generate quarterly data on unobservable variables. The responses of goods produced in the underground sector and
labor engaged in the underground market, along with the counterpart
variables in the official market, are analyzed under several productivity
and taxation stochastic shocks. We have four major findings.
First, the mechanism of reallocation of labor and production between the official and the underground sectors over the cycle can resolve several labor market puzzles, displaying more realistic real wageemployment and productivity-output correlations without calling for a
high degree of intertemporal substitution over the business cycle.
Second, this reallocation behavior is influenced by the productivity
shocks and taxation policies. Large fluctuations in the sectors are associated with restrictive taxation policies on income, output and social
security payments.
Third, investment experiences large variability, without necessity
of large shocks. In addition, consumption response is smooth, under a
logarithmic utility function. In other words, our model requires smaller
28
stochastic shocks than those required by the standard RBC models
for matching selected moments of the actual economies. This means
the model’s structure is characterized by a higher amplification of the
productivity shock by the model’s structure.
Finally, we feel that the properties of the constructed two-sector
model deserve a particular attention. In fact, new and interesting results arise, when we deal explicitly with cyclical behavior that distinguishes between market and underground labor input. Unlike the
standard RBC model, or economies with home production, here we
generate two distinct cycles of the activity, each affecting consumption
and investment outcomes.
References
[1]Becker G. S., 1965, A Theory of Allocation of Time, Economic Journal,
75, 493-517
[2]Basu S., K. Miles and J Fernald, Are Technology Improvements
Contractionary? 1999 manuscript.
[3]Baxter M. and R. King, 1993, Fiscal policy in general equilibrium,
American Economic Review 83, 315-334.
[4]Benhabib J., R. Rogerson and R. Wright R., 1991, Homework in
Macroeconomics: Household Production and Aggregate Fluctuations,
Journal of Political Economy 99, 1166-1187.
[5]Bovi M., 1999, Un miglioramento del metodo di Tanzi per la stima
dell’economia sommersa in Italia, ISTAT, Quaderni di Ricerca 2, 5-51.
[6]Braun R.A., 1994, Tax Disturbances and Real economic Activity in the
Postwar United States, Journal of Monetary Economics 33, 441-62.
[7]Burnside C., M. Eichembaum and S.T.Rebelo, 1995, Capital Utilization
and Returns to Scale, NBER Macroeconomics Annual, MIT Press,
67-110.
[8]Campbell J.Y.,1994, Inspecting the mechanism, Journal of Monetary
Economics 33, 463-506.
[9]Chang, Gomes, Schorfeide, 2001, Learning by Doing as Propagation
Mechanism, forthcoming American Economic Review
[10]Chiarini B. and P. Piselli, 2000, Aggregate Fluctuation in a Unionized
Labor Market, Working Paper 2, University of Naples Istituto
29
Universitario Navale, http://economia.uninav.it/ise/wp.htm
[11]Cochrane, J.H., 2001, Asset Pricing, Princeton University Press
[12]Christiano L. and M. Eichembaum, 1992, Current Real Business Cycle
theory and aggregate labour market fluctuations, American Economic
Review 82, 430-450.
[13]Cho J.O. and T. F. Cooley, 1995, The Business Cycle with Nominal
Contracts, Economic Theory 6, 13-33.
[14]Cho J.O. and T.F. Cooley, 1994, Employment and Hours over the
Business Cycle, Journal of Economics, Dynamics and Control 18, 411-432.
[15]Cho J.O. and R. Rogerson, 1988, Family Labour Supply and Aggregate
Fluctuations, Journal of Monetary Economics 21, 233-245.
[16]Cogley T. and J.M. Nason, 1995, Output Dynamics in Real Business
Cycle Models, American Economic Review 85, 492-511.
[17]Danthine J.P. and J.B. Donaldson, 2001, Output Dynamics in Real
Business Cycle Models, Decentralizing the Stochastic Growth Model,
mimeo, Columbia University
[18]Edelberg W, M. Eichenbaum, J. Fisher, 1999, Understanding the Effects
of a Shock to Government Purchases, Review of Economic Dynamics, 2,
1, 166-206
[19]Feige E.L., 1989, The Underground Economies. Tax Evasion and
Information Distortion, (Cambridge University Press; Cambridge).
[20]Fiorito R. and T. Kollintzas, 1994, Stylized facts of Business Cycles in
the G7 from a Real Business Cycles Perspective, European Economic
Review 38, 235-269.
[21]Fisher J.D.M., 1992, Relative Prices, Complementarities and
Comovements, Manuscript, Northwester University Reproduced
[22]Fung S., 1992, Inflation, Taxation and Home Production in a Real
Business Cycle Model, manuscript University of Western Ontario.
[23]Gali J., 1999, Technology, Employment, and the Business Cycle: Do
Technology Shocks Explain Aggregate Fluctuations, American Economic
Review, 89, 249-271.
[24]Giles D, 1999, Measuring the Hidden Economy: Implications for
Econometric Modelling, Economic Journal, 109, 456, 370-80.
[25]Greenwood J. and Z. Hercowitz, 1991, The Allocation of Capital and
Time over the Business Cycle, Journal of Political Economy 99,
1188-1214.
[26]Gronau, R, 1977, Leisure, Home Production, and Work-The Theory of
the Allocation of Time Revisited, Journal of Political Economy, 85, 6,
1099-1123
30
[27]Gronau, R, 1986, Home Production-A Survey, Handbook of labor
economics, 1, 273-304
[28]King R.G., C.I. Plosser and S.T. Rebelo, 1988a, Production, Growth
and Business Cycles: I. The Basic Neoclassical Model, Journal of
Monetary Economics, 21, 195-232.
[29]King R.G., C.I. Plosser and S.T. Rebelo, 1988b, Production, Growth
and Business cycles: II New directions, Journal of Monetary Economics
21, 309-342.
[30]King R.G. and S.T. Rebelo, 1999, Resuscitating Real Business Cycles, in
Taylor J.B. and Woodford M. (eds.), Handbook of Macroeconomics Vol.1
(Elsevier Science, Amsterdam).
[31]Hansen G.D., 1985, Indivisible Labour and Supply and the Business
Cycle, Journal of Monetary Economics 16, 309-327.
[32]Hodrick. R. J and E. Prescott, 1997, Postwar U.S. Business Cycles: An
Empirical Investigation, Journal of Money, Credit, and Banking, 29, 1,
1-16.
[33]Ingram B.F., N.R. Kocherlakota and N.E. Savin, 1994, Explaining
Business Cycles. A Multiple-Shock Approach, Journal of Monetary
Economics 34, 415-428.
[34]Lucas R.E.Jr., 1988, On the Mechanics of Development Planning,
Journal of Monetary Economics 22, 3-42.
[35]Nevill F. and V. Ramey (2002), Is the Technological Driven Business
Cycle Hypothesis Dead?, NBER Working Paper, 8726
[36]Mare’ M., 1996, L’evasione in Italia e nei paesi OECD: evidenze
empiriche, determinanti ed effetti economici, Moneta e Credito 49,
393-443.
[37]McGrattan E.R., 1994, The Macroeconomic Effects of Distortionary
Taxation, Journal of Monetary Economics 33, 559-71.
[38]McGrattan E.R., R. Rogerson and R. Wright, 1997, An Equilibrium
Model of the Business Cycle with Household Production and Fiscal
Policy, International Economic Review 38, 267-90.
[39]Olson M., 1982, Environmental Indivisibilities and Information Costs:
Fanaticism, Agnosticism, and Intellectual Progress, The American
Economic Review, Papers and Proceedings of the Ninety-Fourth Annual
Meeting of the American Economic Association, , 72, 2, 262-266.
[40]Pollak R.A, M.L. Watcher, 1975, The Relevance of the Household
Production and its Implications for the Allocation of Time, Journal of
Political Economy, 83, 255-77
[41]Prescott E. C. and R. Mehra, 1980, Recursive Competitive Equilibrium:
the Case of Homogenous Household, Econometrica, 1365-79
31
[42]Ramey V.A. and M.D. Shapiro, 1997, Costly Capital Reallocation and
the Effects of Government Spending, NBER Working Paper Series,
http://www.nber.org/papers/w6283
[43]Rios-Rull, J-V, Working in the Market, Working at Home, and the
Acquisition of Skills: A General-Equilibrium Approach, American
Economic Review, 83, 4, 893-907
[44]Shea J., 1998, What Do Technology Shocks Do?, NBER
Macroeconomics Annual, 275- 310.
[45]Stadler G.W., 1994, Real Business Cycle, Journal of Economics
Litterature XXXII, 1750-1783.
[46]Schneider F. and D. Enste, 2000, Shadow Economies: Size, Causes, and
Consequences, Journal of Economic Literature XXXVIII, 77-114.
[47]Spiro P. 1994, The Underground Economy: Toward a More Balanced
View of Alternative Methodologies Canadian Business Economics, 2, 4,
18-21.
[48]Stokey N. and R.E. Lucas with E.C. Prescott, 1989, Recursive Methods
in Economics Dynamics, (Harvard University Press; Cambridge MA).
[49]Tanzi V., 1983, The Underground Economy in the United States:
Annual Estimates, 1930-80, International Monetary Fund Staff Papers,
30, 2, 283-305.
[50]Tanzi V., 1980, Inflationary Expectations, Economic Activity, Taxes,
and Interest Rates, The American Economic Review, 70, 1, 12-21.
[51]Thomas J.J., 1992, Informal Economic Activity in LSE Handbooks in
Economics, (Harvester Wheatsheaf; London).
[52]Thomas J.J., 1999, Quantifying the Black Economy: ’Measurement
Without Theory’ Yet Again?, Economic Journal 109, 381-89.
[53]Uhlig H., 1999, A Toolkit for Analysing Nonlinear Dynamic Stochastic
Models Easily, in R. Marimon and A. Scott (eds) Computational Methods
for the Study of Dynamic Economies, (Oxford University Press; Oxford).
[54]Uzawa H., 1965, Optimal Technical Change in an Aggregative Model of
Economic Growth, International Economic Review 6, 18-31.
32
Appendix A
1
Proof and Derivations
Proof of Lemma 1 (Production set).
Here we show that production
function (2) satisfies a list of commonly assumed properties of production sets,
denoted by Y . Precisely, they are: (I) Y is non empty, (II) Y is closed, (III)
No-free lunch, (IV) Convexity.
(I) trivial.
(II) trivial.
i
(III) To prove that (2) satisfies no-free lunch notice that total output, ytot
,
i
i
dependes on capital, k , and labor, n , which is normalized to unity. Notice
that θ and 1 − θ denotes the share of total labor allocated to market
and non-market sector, respectively. Rewrite production function without
¡
¢˙
¡ ¢α ¡ i i ¢1−α
i
normalizing total labor: ytot
nθ
= M ki
+Z(ni 1 − θti ).
Hence
i
k i = ni = 0 ⇔ ytot
= 0.
(IV) easy.
Proof of Lemma 2 (Price Vector). Suppose not, i.e. qtm 6= qtu . Assume,
WLOG, that qtm > qtu . In this case only market-produced good will be produced:
all resources are shifted to corresponding sector, labor included. This implies
θ = 1. By proposition 1 we know that solution lies in the interior of the state
space, and thus in a REE we cannot have qtm > qtu . An analogous argument
rules out the case qtm < qtu .
Proof of Lemma 3 (Existence and Uniqueness of the Solution). To
prove the claim we rely on the key theorem in infinite dynamic programming,
that is the contraction mapping theorem applied to the Bellman Equation. The
critical theorem is due to Denardo (1967), and we report it for convenience.
Theorem 1 (Denardo (1967)) If state-space is compact, β < 1, and payoff
function u(k, K, A) is bounded above and below, than the map
T J(k, K, A) = sup u(k, K, A) + βE(J(k 0 , K 0 , A0 ))
is monotone in J, is a contraction mapping with modulus β in the space of
bounded functions, and has a unique fixed point.
In our model the state-space is compact, and β < 1. We just need to show
(cj )1−q −1
(θtj )1+γ
(1−θtj )1−η
that payoff function u(cjt , θtj ) = t 1−q
− h 1+γ
(1 − θtj ) − f 1−η
is
1
bounded above and below. Assume that cjt = θtj = 0 and compute u(0, 0) =
f
. To prove that function is bounded above we show that it is increasing
− 1−η
and concave. For this utility function this conditions is satisfied for a restricted
set of parameters, which include our parametrization. First, we compute the
Gradient:
·
¸ ·
¸
c−q
uc
J =
=
2+γ
−hθγ + hθ1+γ 1+γ
− f (1 − θ)−η
uθ
where uc and uθ are the partial derivatives with respect to consumption, c, and
market labor share, θ. Note, that, for c positive, uc > 0. Therefore, to have the
Gradient positive we need uθ > 0.
uθ = −hθγ + hθ1+γ
Next, we compute the Hessian:
·
¸ ·
ucc ucθ
−qc−q−1
H=
=
0
uθc uθθ
2+γ
− f (1 − θ)−η > 0.
1+γ
(A.1)
0
−hγθγ−1 + hθγ (2 + γ) + ηf (1 − θ)−η−1
¸
where ucc and uθθ are second derivative with respect to consumption and market
labor, respectively. uθc and ucθ are the mixed second derivative of the momentary utility function. Note, that, for q, c ≥ 0, ucc = −qc−q−1 ≤ 0, uθc = ucθ = 0
η−1
≶ 0. Hence, to have the
and that uθθ = −hγθγ−1 + 2hθγ + hθγ γ − ηf (1 − θ)
Hessian definite semi-negative, we want uθθ to be non positive:
−η−1
uθθ = −hγθγ−1 + 2hθγ + hθγ γ + ηf (1 − θ)
≤ 0.
(A.2)
When (A.1) and (A.2) the utility function satisfies the conditions for a unique
maximum. Note that condition (A.1) and condition (A.2) are both satisfied for
the parameters used in the baseline simulation (Table 2), and in the experiment
with higher steady state tax rates. More precisely, the value of uθ is 1.0027 and
the value of uθθ equals −1.5864.
2
Appendix B: Figures
Figure 1: Cyclical properties of
Market and Underground components
0.04
0.08 0.08
0.02
0.06
0.02
0.04
0.01
0.00
0.00
0.02
-0.04
0.04
0.00
0.02
-0.01
0.00
-0.02
0.02
0.04
-0.08
-0.03
0.04
Market Comp.
0.06
60
62
64
66
68
70
Underground Comp.
72
74
76
78
80
Market Comp.
-0.12 0.06
82
70
U.S. Panel, Tanzi (1983)
75
80
Underground Comp.
85
90
-0.04
95
Italy Panel, Bovi (1999)
0.20
0.08 0.10
0.6
0.15
0.4
0.04
0.05
0.10
0.2
0.00
0.05
0.00
0.00
0.0
-0.04
-0.2
0.05
-0.08
-0.05
-0.4
0.10
Market Comp.
0.15
78
80
82
84
Undeground Comp.
86
88
90
92
-0.12 -0.10
94
Market Comp.
Underground Comp.
-0.6
60 62 64 66 68 70 72 74 76 78 80 82 84
New Zealand Panel, Giles (1999)
U.K. Panel, Mare’ (1996)
Figure 1. Series are presented in a dual scale plot, and are all detrended using Hodrick-Prescott
(1997) …lter. U.S. Panel, Tanzi (1983): the market component is the GDP, while the underground
component is the ratio of estimated underground economy to the total GDP. Italy Panel, Bovi (1999):
the market component is the GDP, and the underground component is an estimated ”underground
GDP”. New Zealand Panel, Giles (1999): the market component is the ratio of Taxation to total
GDP, while the underground component is the ratio of estimated underground GDP to total GDP.
UK Panel, Mare’ (1996): the market component is the GDP, while the underground component is the
ratio of estimated underground economy to the total GDP
1
Figure 2: Labor Market Dynamics
wage
(net of
taxes)
NSmkt(0)
NSu(1)
NSu(0)
NSmkt(1)
M1
M0
Wage
(gross of
taxes)
M2
U1
U0
NDmkt(0) NDmkt(1)
NDu(0)= NDu(1)
Market and Underground
Employment
Figure 2. The schedules on the LHS (grey) depicts underground labor market, while the schedules
on the RHS (black) describe o¢cial labor market. Since underground demand and supply depend on
the net of tax wage, while market schedules are function of gross of tax wage, we use a dual scale plot.
The left scale refers to underground labor market, while the right one to o¢cial labor market. The
arrows describe the dynamics of the two markets after a positive and temporary productivity shock
in the market sector. U0 and M0 characterize the labor market equilibria before the shock, while
(U1 ; M2 ) the equilibria during the shock. Since it is a temporary shock, in the long run the system
converges back to (U0 ; M0 ).
2
Fig. 3: Impulse Response Functions after a sectoral productivity shock
Panel A: Market Productivity Shock (M)
Market Labor (*), Underground Labor (x) - shock to M
2
2
1.5
1.5
1
1
0.5
-0.5
-1
5
10
15
20
Quarters After Shock
25
30
-1.5
35
6
0
-0.5
0
7
0.5
0
-1
Consumption (+), Investment (x), Capital Stock (o) - shock to M
8
% deviation from S.S.
2.5
% deviation from S.S.
% deviation from S.S.
Market Output (*), Underground Output (x), Total Output (o) - shock to M
5
4
3
2
1
0
5
10
15
20
Quarters After Shock
25
30
0
35
0
5
10
15
20
Quarters After Shock
25
30
35
Panel B: Underground Productivity Shock (Z)
Market Output (*), Underground Output (x), Total Output (o) - shock to Z
Market Labor (*), Underground Labor (x) - shock to Z
1.5
Consumption (+), Investment (x), Capital Stock (o) - shock to Z
3
0
2.5
1
-0.5
2
0
-0.5
% deviation from S.S.
% deviation from S.S.
% deviation from S.S.
-1
1.5
0.5
1
0.5
0
-0.5
-1.5
-2
-2.5
-3
-1
-1
-3.5
-1.5
-1.5
0
5
10
15
20
Quarters After Shock
25
30
35
-2
0
5
10
15
20
Quarters After Shock
25
30
35
-4
0
5
10
15
20
Quarters After Shock
25
Figure 3. The …gure shows the …rst 32 quarter response of market output, underground output, total
output, market and underground labor, to a one standard deviation innovation to market productivity,
Mt (Panel A), and to underground productivity, Zt (Panel B). The curves are the quarterly percentage
deviations from a baseline scenario where all innovations are set to zero.
3
30
35
Fig. 4: Impulse Response Functions after a tax shocks
Panel A: Corporate Tax Shock (t)
Market Output (*), Underground Output (x), Total Output (o) - shock to t
Market Labor (*), Underground Labor (x) - shock to t
4
Consumption (+), Investment (x), Capital Stock (o) - shock to t
8
3
0
6
-5
1
0
-1
% deviation from S.S.
4
% deviation from S.S.
% deviation from S.S.
2
2
0
-10
-15
-2
-2
-20
-4
-3
-4
0
5
10
15
20
Quarters After Shock
25
30
-6
35
0
5
10
15
20
Quarters After Shock
25
30
-25
35
0
5
10
15
20
Quarters After Shock
25
30
35
Panel B: Income Tax Shock (tau)
Market Output (*), Underground Output (x), Total Output (o) - shock to tau
Market Labor (*), Underground Labor (x) - shock to tau
0.4
Consumption (+), Investment (x), Capital Stock (o) - shock to tau
2
0
-0.2
0.3
1.5
-0.4
0
-0.1
-0.2
-0.6
1
% deviation from S.S.
0.1
% deviation from S.S.
% deviation from S.S.
0.2
0.5
0
-0.3
-1
-1.2
-1.4
-1.6
-0.5
-0.4
-0.5
-0.8
-1.8
0
5
10
15
20
Quarters After Shock
25
30
35
-1
0
5
10
15
20
Quarters After Shock
25
30
35
-2
0
5
10
15
20
Quarters After Shock
25
Figure 4. The …gure shows the …rst 32 quarter response of market output, underground output,
total output, market and underground labor, to a one standard deviation innovation to corporate tax
rates, tt (Panel A), and in income tax rate, ¿ t (Panel B). The curves are the quarterly percentage
deviations from a baseline scenario where all innovations are set to zero.
4
30
35
Appendix C: Tables
Table 1: Underground Economy
for Selected OECD Countries
Countries
Australia
Belgium
Denmark
France
Greece
Germany
Great Britain
Italy
Portugal
Spain
USA
Average OECD
1994-95
13.8
21.5
17.8
14.5
25.0
13.5
12.5
26.0
22.1
22.4
9.20
16.0
1996-97
13.9
22.2
18.2
14.8
26.0
14.8
13.0
27.2
23.0
23.0
8.8
16.9
Table 1. Sources: Schneider and Enste (2000). Numbers in the columns represent the average size
(over the speci…ed) sample of the underground sector relative to the GDP.
5
Table 2: Business Cycle Properties of Italian Economy: Comparison across Models
Actual Data
variable
¾
^ variable
¾
^ variable
¾
^ ytot
½(ytot ; variable)
^
^(n; variable)
½
Indivisible Labori
variable
¾?variable
¾?
variable
¾?
ytot
?
½ (ytot ; variable)
?
ym
2.27
0.89
yu
1.11
-0.45
ytot
1.44
1.00
1.00
c
1.24
0.85
0.80
x
3.70
2.53
0.78
n
1.39
0.95
0.63
¼
2.78
1.90
0.36
0.19
ym
-
yu
-
ytot
1.45
1.00
c
0.29
0.20
x
3.44
2.37
n
1.27
0.88
¼
0.29
0.20
-
-
1.00
0.67
0.99
0.99
0.99
0.09
ym
-
yu
-
ytot
1.42
1.00
c
0.30
0.21
x
5.82
4.09
n
1.28
0.90
¼
0.41
0.29
-
-
1.00
0.38
1.00
0.96
0.49
0.04
ym
2.07
(0.22)
1.86
yu
1.94
(0.25)
1.33
ytot
1.45
(0.14)
1.00
c
1.17
(0.25)
0.80
x
9.63
(0.05)
6.64
n
1.60
(0.13)
1.10
¼
2.94
(0.17)
2.00
0.95
(0.01)
-0.96
(0.01)
1.00
-
0.69
(0.11)
0.98
(0.01)
0.73
(0.13)
0.08
(0.04)
0.04
(0.05)
½ (n; variable)
Union Modelii
variable
¾?variable
¾?
variable
¾?
ytot
?
½ (ytot ; variable)
?
½ (n; variable)
Our Modeliii
variable
¾?variable
¾?
variable
¾?
ytot
?
½ (ytot ; variable)
½? (n; variable)
Table 2. In the table ytot represents aggregate output, ym market output, and yu underground output;
c denotes total consumption, x aggregate investment, n market employment, and ¼ average labor productivity. Finally, ½(variabile; ytot ) represents correlation between ytot and variable, ½(n; variabile)
correlation between n and variable, and ¾ variable denotes standard deviation of variable. Number is
parenthesis represent the small sample standard deviations. The summary statistics are generated by
one-hundred simulations, and the series are of length 104 quarters. References: (i) see Hansen (1985)
e Chiarini e Piselli (2000), for (ii) see Chiarini and Piselli (2000); …nally, (iii) is the model presented
in this paper.
6
Table 3: A Better Amplification Mechanism
Parameters
Indivisible Labori
High Substitutionii
Cho and Cooleyiii
Home Productioniv
Our Modelv
q
1:00
3:00
1:00
1:00
1.00
®
0:36
0:36
0:36
0:29
0.36
¯
0:984
0:984
0:990
0:989
0.984
±
0:025
0:025
0:025
0:023
0.025
½
0:979
0:989
0:950
0:950
0.900
¾
0:0072
0:0012
0:0102
0:0070
0.0032
¾shock
0.002469
0.000131
0.002081
0.000980
0.000102
Table 3. Parameter q denotes relative risk aversion parameter, ® capital share in a Cobb-Douglas
production function, ¯ the subjective discount factor, ½ and ¾ the autocorrelation coe¢cient and
the standard deviation of innovation process, respectively. Finally, ¾shock denotes variance of the
stochastic shocks, and it is computed as ¾ shock = ¾=(1 ¡ ½). References (i): Hansen (1985), (ii):
Burnside, Eichenbaum and Rebelo (1995). (iii): Cho and Cooley (1994). (iv): Greenwood, Rogerson
and Wright (1995). (v) Model presented in this paper. Note that, even if we have more than one
stochastic shock, ¾ are the same.
7