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CESifo Area Conference on
Macro, Money and International
Finance
25 - 26 February 2005
CESifo Conference Centre, Munich
Distributional Effects of Monetary Policies in
a New Neoclassical Model with Progressive
Income Taxation
Burkhard Heer and Alfred Maußner
CESifo
Poschingerstr. 5, 81679 Munich, Germany
Phone: +49 (89) 9224-1410 - Fax: +49 (89) 9224-1409
E-mail: [email protected]
Internet: http://www.cesifo.de
Distributional Effects of Monetary Policies
in a New Neoclassical Model with Progressive
Income Taxation∗
By Burkhard Heera,b and Alfred Maußnerc
a
Free University of Bolzano-Bozen, School of Economics and Management, Via Sernesi 1,
39100 Bolzano-Bozen, Italy, [email protected]
b
CESifo
c
University of Augsburg, Department of Economics, Universitätsstraße 16, 86159 Augsburg, Germany, [email protected]
JEL classification: E31, E32, E52, D31, D58
Keywords: unanticipated inflation, wealth distribution, income distribution, progressive
income taxation, Calvo price staggering
Abstract:
In our dynamic optimizing sticky price model, agents are heterogenous with regard to their age,
their assets, and their income. Unanticipated inflation redistributes income and wealth. In order
to model the business cycle dynamics of the wealth distribution, we study a 240-period OLG
model with aggregate uncertainty. A positive technology shock increases the concentration of
wealth as measured by the Gini coefficient considerably. In particular, a one percent increase of
the technology level results in a one percent increase of the Gini coefficient over the medium run.
An unexpected expansionary monetary policy is found to increase the inequality of income, but
to only have a negligible effect on the wealth inequality. In addition, we find that the business
cycle dynamics in the OLG model in response to both a technology shock and a monetary shock
are similar, but not completely identical to those found in the corresponding representative-agent
model.
∗
We would like to thank Victor Rı́os-Rull for his comments. All remaining errors are ours. Burkhard
Heer kindly acknowledges support from the German Science Foundation (Deutsche Forschungsgemeinschaft
DFG) during his stays at Georgetown University, University of Pennsylvania, and Stanford University.
1
Introduction
The effect of inflation on the distribution of income and wealth is a matter of ongoing
concern for economists. Quite recently, the relationship between inflation and income
distribution has received renewed interest. Easterly and Fischer (2001) as well as Romer
and Romer (1998) point out in their empirical analysis that inflation hurts the very poor.
Galli and van der Hoeven (2001) provide a survey of the empirical literature. Most of
the studies find a negative effect of inflation on the equality of the income distribution;
however, in many cases, the effect is not significant once some basic control variables are
included in the regression analysis. Moreover, there is no evidence on the empirical effects
of inflation on the distribution of wealth, basically for the reason that there is hardly any
high-frequency micro data for wealth. Given that the correlation of wealth and income is
rather low empirically due to the life-cycle effect of savings,1 we are also uncertain with
regard to the empirical effects of inflation on the distribution of wealth.
In recent years, due to the progress in the computer technology, there are some, even
though very few quantitative studies of the distributional effects of inflation that are based
on general equilibrium models. These include studies by Erosa and Ventura (2002) and
Heer and Süssmuth (2003).2 These two studies focus on the effects of a change in the
long-run inflation rate and, therefore, their studies amount to a comparative steady-state
analysis of the endogenous wealth distributions. In both papers, a rise in the anticipated
long-run inflation rate results in a rise of the wealth inequality. Erosa and Ventura (2002)
emphasize the inflation’s effect on the composition of the consumption good bundle. Higher
inflation results in an increase of the consumption of the credit good at the expense of
the consumption of the cash good, and richer agents have lower credit costs. Heer and
1
The reason is simple. In general, the wealth-rich people, e.g. the retired people, are not necessarily
the high-income people. Labor earnings peak in the early 50s of one’s life-time (see, e.g. Hansen, 1993).
Dı́az-Giménez et al. (1997) report a correlation of US earnings (income) and wealth equal to 0.230 (0.321)
using data from the 1992 Survey of Consumer Finances.
2
Relatedly, Bhattarchya (2001) studies the effect of higher inflation on the distribution on income. In
his work, inflation increases the external finance premium.
1
Süssmuth (2003) model the observation that not all agents have access to the stock market
and, therefore, that poorer agents are less likely to hold assets whose real return is not
reduced through higher inflation. All these studies, however, refrain from modelling the
short-run effects of inflation.
In order to model the distribution of wealth, we study an Overlapping Generations (OLG)
model. The life-cycle motive of savings has been described as a prominent factor in the
explanation of the wealth heterogeneity, among others by Huggett (1996). In order to
model the short-run effects of monetary policy, we assume that prices are sticky and adjust
as in Calvo (1983). Following an unexpected rise of the money growth rate, we observe
unexpected inflation. Prices and markups adjust endogenously in our economy. As a consequence, both factor prices and the distribution of income change. In addition, income is
taxed progressively in our model. If there is unexpected inflation, the real tax burden of the
income-rich agents increases. We also consider the effect of inflation on pensions. In particular, unexpected inflation results in a reduction of real pensions as the government adjusts
the pensions for higher-than-average inflation only with a lag. Consequently, following an
expansionary monetary policy with an unexpected rise of inflation, the non-interest income
distribution becomes more equal, while the effect on the total income distribution depends
on the distribution of interest income (and, hence, the distribution of wealth).
The paper is of equal interest for the researcher in the field of business cycles. Common
business cycle research focuses on the behavior of the representative agent model and
neglects the effects that are caused by the heterogeneity of agents. As one of the very few
exceptions, Rı́os-Rull (1996) has analyzed a stochastic OLG model rather than a stochastic
Ramsey model for the study of the effects of a technology shock. He has shown that
the business cycle dynamics of the life-cycle model is basically the same as those of the
infinitely-lived representative-agent model. In his analysis, he only considers a Walrasian
economy and uses a period equal to one year. Different from him, we introduce money
and sticky prices in a life-cycle model, consider a non-Walrasian economy, and calibrate the
model period equal to one quarter rather than one year. We consider a model period of one
2
quarter to be more appropriate for the study of business cycle dynamics, even though the
computational problem becomes much more demanding. For this reason, we have used an
algorithm that, to the best of our knowledge, has not yet been applied to such large-scale
stochastic OLG models.
The paper is organized as follows. In section 2, we describe the OLG model. The model
is calibrated in section 3, where we also present the algorithm for our computation in
brief. The results are presented in section 4. We will present the effect of a monetary
shock and a productivity shock on the distribution of wealth. Furthermore, we compare
the heterogeneous-agent with the corresponding representative-agent economy and make
the interesting observation that the business cycle dynamics in these two models are not
completely identical. Section 5 concludes. The appendix describes the corresponding
representative-agent Ramsey model and the solution method for the dynamics of the OLG
model.
2
The Model
The model is based on the stochastic Overlapping Generations (OLG) model with elastic
labor supply and aggregate productivity risk, augmented by a government sector and the
monetary authority. The model is an extension of Rı́os-Rull (1996).
Four different sectors are depicted: households, firms, the government, and the monetary
authority. Households maximize discounted life-time utility with regard to their intertemporal consumption, capital and money demand, and labor supply. Firms in the production
sector are competitive, while firms in the retail sector are monopolistically competitive and
set prices in a staggered way a la Calvo (1983). The intermediate good firms produce using
labor input and capital. The government taxes income progressively and spends the revenues on government pensions and transfers. Both aggregate productivity and monetary
policy are stochastic.
3
2.1
Households
Households live T + T R = 240 periods (corresponding to 60 years). Each generation is
of measure 1/(T + T R ). The first T =160 (=40 years) periods, they are working, the
last T R = 80 periods (=20 years), they are retired and receive pensions. The s-year old
household holds real money holdings Mts /Pt and capital kts in period t. He maximizes
expected life-time utility at age 1 in period t with regard to consumption cst , labor supply
s+1
s+1
nst , and next-period money balances Mt+1
, and real capital kt+1
:
Et
T
+T R
β
s−1
u
cst+s−1 ,
s=1
s
Mt+s−1
s
, 1 − nt+s−1 ,
Pt+s−1
(1)
where β is a discount factor and expectationsare conditioned
on the information set of the
t
household at time t. Instantaneous utility u ct , M
, 1 − nt is assumed to be:
Pt
⎧
⎨ γ ln c + (1 − γ) ln M + η (1−n)1−η
0
M
P
1−η
1−σ
u c, , 1 − n =
M 1−γ
γ
1−η
⎩ c (P )
P
+ η (1−n)
0
1−σ
1−η
if σ = 1
if σ = 1
(2)
where σ > 0 denotes the coefficient of relative risk aversion.3 The agent is born without
capital kt1 = 0.4
The s-year old working agent faces the following nominal budget constraint in period t:
s+1
s+1
− (1 − δ)kts + Mt+1
− Mts + Pt cst = Pt rt kts + Pt wt s nst + Pt trt + Pt Ωt
Pt kt+1
Pt yts
, s = 1, . . . , T.
− Pt τt
Pt−1 π
3
(3)
We follow Castañeda et al. (2004) in our choice of the functional form for the utility from leisure.
In particular, this additive functional form implies a relatively low variability of working hours across
individuals that is in good accordance with empirical evidence.
4
In order to avoid that utility is not defined at age 1, we do not choose m1t ≡ 0, but rather endow
the household at age 1 with a small constant amount of money, m1t = ψ. In our computation, we pick
ψ = 0.001. However, the choice of ψ does not have any quantitative effect on our results.
4
Individual productivity s depends on age s. The working agent receives income from
effective labor s nst and capital kts as well as government transfers trt and profits Ωt which
s+1
s+1
are spent on consumption cst and next-period capital kt+1
and money Mt+1
. He pays taxes
on his nominal income Pt yts :
Pt yts = Pt rt kts + Pt wt s nst .
The government adjust the tax income schedule at the beginning of each period for the
average rate of inflation in the economy which is equal to the non-stochastic steady state
rate π. Therefore, nominal income is divided by the price level, Pt−1 π, and the tax schedule
τ (.) is a time-invariant function of (deflated) income with τ > 0. Notice that when we
have unanticipated inflation, πt =
Pt
Pt−1
> π, the real tax burden increases and we have cold
progression.
The nominal budget constraint of the retired worker is given by
s+1
Pt kt+1
− (1 −
δ)kts
+
s+1
Mt+1
−
Mts
+
Pt cst
=
Pt rt kts
+ P ent + Pt trt + Pt Ωt − Pt τt
Pt yts
Pt−1 π
,
s = T + 1, . . . , T + T R
with the capital stock and money balances at the end of the life at age s = T + T R being
equal to zero, ktT +T
R +1
= mtT +T
R +1
T +T R
and nTt +1 = nTt +2 = . . . = nt
≡ 0, because the household does not leave bequests,
≡ 0. P ent are nominal pensions and are distributed
lump-sum. They are not taxed. Again, the government adjusts pensions each period for
expected inflation according to P ent = pen Pt−1 π, where pen is constant through time. If
inflation is higher then expected, πt > π, the real value of pensions declines.
The real budget constraint of the s-year old household is given by
s+1
kt+1
+ms+1
t+1 =
⎧
⎨ (1 + rt − δ)kts +
⎩ (1 + rt − δ)kts +
mst
πt
mst
πt
s y π
+ wt s nst + trt + Ωt − τt tπ t − cst ,
s yt π t
π
+ pen
+
tr
+
Ω
−
τ
− cst ,
t
t
t
πt
π
5
s = 1, . . . , T,
s = T + 1, . . . , T + T R .
(4)
where we define mt ≡
Mt
.
Pt−1
The necessary conditions for the households with respect to consumption cst , s = 1, . . . , T +
s+1
R
T R , next-period capital kt+1
, and next-period money ms+1
t+1 at age s = 1, . . . , T + T − 1
in period t are as follows:
λst
λst
λst
s (1−γ)(1−σ)
Mts
mt
s
s γ(1−σ)−1
= uc
, 1 − nt = γ (ct )
,
Pt
πt
π t+1
s+1 πt+1
1
−
δ
+
r
1
−
τ
y
,
= βEt λs+1
t+1
t+1
t+1
π ⎤ π
⎡
s+1
s+1 Mt+1
s+1
s+1
u
c
,
,
1
−
n
M/P
t+1 Pt+1
t+1 ⎥
⎢λ
= βEt ⎣ t+1 +
⎦
πt+1
πt+1
cst ,
= βEt
s+1 γ(1−σ) s+1 (1−γ)(1−σ)−1 mt+1
(1
−
γ)
ct+1
λs+1
t+1
+
.
πt+1
πt+1
The optimal labor supply of the workers at age s = 1, . . . , T is given by:
π π Mts
t
s
s
s t
s
λt wt 1 − τ yt
= u n ct ,
, 1 − nt = η0 (1 − nst )−η .
π π
Pt
2.2
(5)
(6)
(7)
(8)
Production
The description of the production sector is similar to Bernanke et al. (1999). A continuum
of perfectly competitive firms produce the final output using differentiated intermediate
goods distributed on [0,1]. These goods are manufactured by monopolistically competitive
firms. Firms in the intermediate goods’ sector set prices according to Calvo (1983).
2.2.1
Final Goods Firms
The firms in the final goods sector produce the final good with a constant returns to scale
technology using the intermediate goods Yt (j), j ∈ [0, 1] as an input:
6
1
Yt =
−1
Yt (j)
−1
dj
.
(9)
0
Profit maximization implies the demand functions:
Yt (j) =
Pt (j)
Pt
−
Yt ,
(10)
with the zero-profit condition
Pt =
1
1−
Pt (j)
1
1−
dj
.
(11)
0
2.2.2
Intermediate Goods Firms
The intermediate good j ∈ [0, 1] is produced with capital Kt (j) and effective labor Nt (j)
according to:
Yt (j) = zt Kt (j)α Nt (j)1−α .
(12)
All intermediate producers are subject to an aggregate technology shock zt being governed
by the following AR(1) process:
ln zt = ρz ln zt−1 + εzt ,
(13)
where εzt is i.i.d., εzt ∼ N (0, σz2 ).
The firms choose Kt (j) and Nt (j) in order to maximize profits. In a symmetric equilibrium
profit maximization of the intermediate goods’ producers implies:
1
αzt Ktα−1 Nt1−α ,
xt
1
(1 − α)zt Ktα Nt−α ,
=
xt
rt =
(14)
wt
(15)
7
where xt denotes the mark-up of the goods prices over marginal costs.
Calvo Price Setting
Let φ denote the fraction of producers that keep their prices unchanged. Profit maximization of symmetric firms leads to a condition that can be expressed as a dynamic equation
for the aggregate inflation rate:
π̂t = −κx̂t + βEt {π̂t+1 } .
(16)
with κ ≡ (1 − φ)(1 − βφ)/φ > 0 and π̂t is the percent deviation of the gross inflation rate
from its non-stochastic steady state level π.5
2.3
Monetary Authority
Nominal money grows at the exogenous rate θ:
Mt+1
= θt .
Mt
(17)
The seignorage is transferred lump-sum to the government:
Seignt = Mt+1 − Mt .
(18)
The growth rate θt follows the process:
θ̂t = ρθ θ̂t−1 + εθt ,
where εθt is assumed to be i.i.d., ηt ∼ N (0, σθ2 ).
5
A detailed derivation of this relation can be found in Herr and Maußner (2005), Section A.4.
8
(19)
2.4
Government
Nominal government expenditures consists of pensions P ent , and government lump-sum
transfers Pt T rt to households. Government expenditures are financed by an income tax
T axt and seignorage:
Pt T rt + P ent = T axt + Seignt .
(20)
The income tax structure is chosen to match the current income tax structure in the US
most closely. Gouveira and Strauss (1994) have characterized the US effective income tax
function in the year 1989 with the following function:
− 1 τ (y) = a0 y − y −a1 + a2 a1 ,
(21)
and estimate the parameters with a0 = 0.258, a1 = 0.768 and a2 = 0.031. We use the
same functional form for our benchmark tax schedule. The average nominal income in
1989 amounts to approximately $50,000.6
2.5
Equilibrium conditions
1. Aggregate and individual behavior are consistent, i.e. the sum of the individual
consumption, effective labor supply, wealth, and money is equal to the aggregate
6
We follow Castañeda et al. (2004).
9
level of consumption, effective labor supply, wealth, and money, respectively:
T
+T R
Ct =
s=1
T
Nt =
s=1
nst s
,
T + TR
T
+T R
Kt =
s=1
T
+T R
mt =
s=1
cst
,
T + TR
kts
,
T + TR
mst
.
T + TR
2. Households maximize life-time utility (1).
3. Firms maximize profits.
4. The goods market clears:
zt Ktα Nt1−α = Ct + Kt+1 + (1 − δ)Kt .
5. The government budget (20) balances.
6. Monetary growth (17) is stochastic and seignorage is transferred to the government.
7. Technology is subject to a shock (13).
The non-stochastic steady state and the log-linearization of the model at the non-stochastic
steady state are described in more detail in the appendix. In addition, we will compare our
OLG model with the corresponding representative-agent model which is also presented in
the appendix.
10
3
Calibration and Computation
The model is calibrated with regard to the characteristics of the US postwar economy. We
use standard values for the parameters of the model. Periods correspond to quarters. The
first T = 160 periods, agents are working, the remaining T R = 80 periods they are retired.
3.1
Preferences
β is set equal to 0.997 implying a non-stochastic steady state annual real rate of return
equal to r − δ = 3.40% and a capital-output ratio equal to K/Y = 2.73. The relative
risk aversion coefficient σ is set equal to 2.0. η0 is set so that the average labor supply
is approximately equal to 1/3, n̄ ≈ 1/3. Furthermore, we choose η = 7.0 which implies
a conservative value of 0.3 for the Frisch labor supply elasticity.7 γ is chosen so that
the (annualized) average velocity of money P Y /M is equal to the velocity of M1 during
1960-2004, which is equal to approximately 6.0. For the values η0 = 0.24 and γ = 0.9845,
average labor is equal to 0.330 and the velocity of money amounts to 6.0.
3.2
Government
Pensions are constant. We choose a non-stochastic replacement ratio of pensions relative
to net wage earnings equal to 30%, ζ =
pent
,
(1−τ̄ )wt n̄t
where n̄t and τ̄ are the average labor
supply and the income tax rate of the average income in the non-stochastic steady state
of the economy. The calibration of the tax schedule follows Goveira and Strauss (1994).
7
The estimates of the Frisch intertemporal labor supply elasticity ηn,w implied by microeconometric
studies and the implied values of γ vary considerably. MaCurdy (1981) and Altonji (1986) both use PSID
data in order to estimate values of 0.23 and 0.28, respectively, while Killingsworth (1983) finds an US labor
supply elasticity equal to ηn,w = 0.4. Domeij and Floden (2001), however, argue that these estimates are
biased downward due to the omission of borrowing constraints.
11
3.3
Monetary authority
In accordance with Cooley and Hansen (1995), the quarterly inflation factor is set equal to
π̄ = 1.013. Money growth follows an AR(1)-process. For the postwar US economy, Cooley
and Hansen estimate ρθ = 0.49 and σθ2 = 0.0089.
3.4
Production
The production elasticity of capital α = 0.36 and the quarterly depreciation rate δ = 0.194
are taken from Prescott (1986) and Cooley and Hansen (1995), respectively. The annual
fraction of producers that do not adjust their prices in any quarter is set equal to 1 − φ =
0.75 as in Bernanke et al. (1999). Following empirical evidence presented by Basu and
Fernard (1997), we set the average mark-up at the amount of x̄ = 1.2 implying a constant
elasticity of substitution between any two intermediate goods equal to 6.0. The parameters
of the AR(1) for the technology are set equal to ρz = 0.95 and σz = 0.007 as in Cooley and
Hansen (1995).
3.5
Computation
In order to compute business cycle dynamics of the model, we first need to compute the
non-stochastic steady state of the model. Secondly, we log-linearize the model around the
non-stochastic steady state. The following algorithm describes the computation of the
non-stochastic steady state:
1. Make initial guesses of the aggregate capital stock K, aggregate effective labor N ,
transfers tr, and aggregate real money M/P .
2. Compute the values of w and r that solve the firm’s Euler equations. Compute the
pensions.
12
3. Compute the household’s decision functions.
4. Compute the steady-state distribution of the state variable {k s , ms } .
5. Update K, N , tr, and M/P , and return to step 1 until convergence.
The non-stochastic steady state can be computed with very high accuracy. In essence, we
computed the solution to a system of some hundred non-linear equations that consists of the
first-order conditions of the households. Accuracy of the solution, therefore, is determined
by the accuracy of our non-linear equations solver routine which amounts to less than 10−8 .
The problem is to find a good initial value for the algorithm. The method is explained
in more detail in Chapter 7.2.2 of Heer and Maußner (2005). Thereafter, we log-linearize
the model around the non-stochastic steady. This linear rational expectations model can
be solved by, e.g., applying the method of Blanchard and Kahn (1980), (see King, Plosser,
and Rebelo, 1988) or of King and Watson (2002).8
4
Results
In this section, we present the results on the distribution effects of inflation. We will first
describe the benchmark model and then look at the effects of a productivity shock and a
monetary shock on the distribution. In addition, we compare the heterogeneous-agent to
the representative-agent case and will find out that the two economies display similar, but
not identical behavior, a result that is in good accordance with those in the non-monetary
models of Krussell and Smith (1998) and Rı́os-Rull (1996).
8
Again, these methods are described in more detail in Chapter 2.3 in Heer and Maußner (2005) and
applied to large scale dynamic systems of several hundred state variables and controls in Chapter 7.2.2.
13
4.1
The non-stochastic steady state
Our OLG model displays the behavior that is typical for this kind of model. The wealthage profile is hump-shaped as displayed in the upper left graph in figure 1. Notice that
due to the hump-shaped age-productivity profile (not displayed) households dissave during
the first 44 quarters (=11 years). Only at real lifetime age 31 do they start to build
up positive savings. Furthermore, labor supply (lower left graph) attains a maximum at
around age 30 because the age-specific productivity is rather low at young ages. Labor
supply also attains its maximum prior to the maximum in the hourly wages because older
agents have higher wealth and work fewer hours. In addition, consumption as displayed
in the lower right graph in figure 1 is increasing over the life-time as the discount rate is
smaller than the interest rate.9 Notice that the household behavior changes abruptly as
they enter retirement. This kind of behavior is absent from most standard OLG models.
Consumption growth increases at retirement, while there is a downward jump in the real
money stock. The reason is the presence of progressive income taxation in our model. In
the first period of retirement at age 60.25, taxable income falls and the tax rate on capital
income is much smaller than during working life. For this reason, the after-tax rate of
return on real capital income increases. As a consequence, consumption growth is higher,
and the household readjusts its portfolio allocation. The premium on the return on capital
relative to the one on money has increased, and the real money stock is reduced as can be
seen from the upper right picture in figure 1.
In our economy, income and wealth are distributed unequally. The heterogeneity of income,
however, is a little lower than observed empirically. In particular, the Gini coefficient of
total gross income amounts to 0.296. For the US economy, Dı́az-Giménez et al. (1997)
find a value of 0.51 for households aged 36-50, while Henle and Ryscavage (1980) estimate
9
In order to imply a more realistic consumption-age profile, we may have introduced stochastic survival
probabilities; in this case, consumption declines at old age. However, our quantitative results are not
sensitive to this modelling choice and, therefore, we kept the model as simple as possible.
14
Figure 1: Non-stochastic steady state
an average US earnings Gini coefficient for men of 0.42 in the period 1958-77.10 The
main reason for our low Gini values is the neglect of productivity heterogeneity within
generations.11 As a consequence, the distribution of wealth is also much more equal than
observed empirically. In our model, the Gini coefficient of wealth amounts to 0.444, whereas
Greenwood (1983), Wolff (1987), Kessler and Wolff (1992), and Dı́az-Giménez et al. (1997)
estimate Gini coefficients of the wealth distribution for the US economy in the range of
0.72 (single, without dependents, female household head) to 0.81 (nonworking household
head).12 In summary, our model is able to replicate a large portion, but not the total
10
11
Income transfers are excluded in the respective definition of earnings.
In future work, we are planning to introduce this kind of heterogeneity. At this moment, however, we
do not know of any numerical method that is able to handle the large state-space that results from this
more realistic modelling.
12
Huggett (1996) shows that we are able to replicate the empircally observable heterogeneity of wealth
in a computable general equilibrium model if we introduce both life-cycle savings and individual earnings heterogeneity. Only the wealth concentration among the very rich agents is not well described in
15
heterogeneity of the income and wealth distribution that is observed empirically. Given
the current state of numerical methods, however, this is the best we can achieve.13
4.2
Productivity shock and distribution
In figure 2, the impulse response functions of aggregate variables to a technology shock
εz,2 = 1 in period 2 (and zero thereafter) are presented. In the first row, the percentage
deviations of the variables technology level zt , output Yt , consumption Ct , and investment
It are graphed, in the second row, we illustrate the percentage deviations of capital Kt ,
employment Nt , real money mt , and the inflation factor πt , while in the third row, you
find the behavior of marginal costs gt (the inverse of the mark-up), profits Ωt , the real
interest rt , and the wage rate wt . Following an unexpected increase of the technology level
by 1%, output and employment instantaneously increase by 1.04% and 0.04%, respectively.
Inflation declines and most of the price adjustment takes place in the first period. As the
productivity increases, both factor prices (wages and interest rate) increase. Profits also
increase as the marginal costs decline while prices are sticky.
The behavior of the wealth and income distribution in response to a technology shock is
displayed in figure 3. The most profound effect of the technology shock is on the distribution
of income. The Gini coefficient of income increases by 4% following a 1% increase in
productivity (lower right graph in figure 3). Both the increase of the interest rate and
the wage rate result in a more unequal distribution of total income. Agents with highproductivity have a more elastic labor supply than low-productivity (young) agents. As
wages go up, they work more hours relative to the young agents. Similarly, as the interest
rates go up, agents substitute consumption intertemporally and increase savings. Similarly,
the rich agents benefit from lower tax rates as the deflation reduces the real tax burden. In
such a model. In order to replicate the wealth distribution of the top quintile, Quadrini (2000) models
entrepreneurship endogenously.
13
Please see Krueger and Kuebler (2005) and Heer and Maußner (2004) for a discussion of the current
state of computational methods and the curse of dimensionality.
16
Figure 2: Technology shock in the OLG model
addition, the real values of pensions increases. As the maximum of the wealth holdings is in
the periods 180 and 181 (corresponding to the model periods T and T+1), this effect also
increase the inequality of the capital distribution. As a consequence, inequality increases
in the economy, and the Gini coefficient of capital increases by more than one percent.
However, the effect on the distribution of total wealth (capital plus money), surprisingly,
is rather small and the Gini coefficient only changes by less than 0.01%.
Furthermore, we find that the behavior of our heterogeneous-agent economy in response to a
technology shock is similar to the one in the corresponding representative-agent economy,
that is described in more detail in the appendix. In figure 4, we display the impulse
responses of the aggregate variables in the representative agent model. The ordering of
the variables is exactly as in figure 2. Qualitatively, the response is the same for all
variables. Notice, however, that some small difference prevail. For instance, the response
of employment is positive for 12 periods (=3 years) in the Ramsey model, while it is
negative after 2 years in the OLG model. In addition, the increase in employment in the
17
Figure 3: Technology shock in the OLG model and distribution
Ramsey model displayed in figure 3 is twice as high as those in the OLG model. In essence,
however, we confirm the result of Rı́os-Rull (1996) that the life-cycle economy displays a
similar behavior in response to a technology shock as the representative-agent economy.
4.3
Monetary shock and distribution
An expansionary monetary shock increases demand. As prices are sticky and firms are monopolistic competitors in the intermediate goods sector, output and employment increases.
The impulse response functions of aggregate variables to a monetary growth shock εθ,2 = 1
in period 2 (and zero thereafter) are presented in figure 5. Again, the ordering is as in
figure 2 except for the upper left graph where the response of the money growth rate is
displayed rather than the response of the technology shock. The percentage changes of output, employment and investment are small and only amount to 0.04%, 0.07% and 0.2%,
respectively. Inflation, the real interest rate, and wages all increase. Notice that in this
18
Figure 4: Productivity Shock in the Representative Agent Economy
sticky price model we are unable to model the liquidity effect that nominal interest rates
decrease following an expansionary monetary policy. In addition, profits decline. This is
one of the major shortcoming of the sticky-price model that has been documented in the
literature.
The impulses responses of the Gini coefficients of capital, money, wealth, and income
are displayed in figure 5. Following a monetary expansion, the distribution of income is
becoming more unequal, while the distribution of wealth is becoming more equal. The
Gini coefficient of income even increases by 4%. This large effect is basically driven by the
decline in real pensions. Even though the wage rate and the real return on capital increase,
this effect on inequality is rather small as taxes also increase. An increase of the money
growth rate by 1% also results in an instantaneous decline of the Gini coefficient of capital
by 0.16%. Again, this effect is basically explained by a decline in real pensions. Retired
agents temporarily increase their rate of dissaving. However, the wealth distribution effect
of unexpected inflation is small in our economy, especially if one also considers the effect
19
Figure 5: Monetary shock in the OLG model
of higher inflation on the distribution of money.
The impulse response functions of the aggregate variables in the Ramsey model with Calvo
price staggering are graphed in figure 7. Again the ordering of the variables is identical
to the one in figure 5. Notice that the qualitative behavior of the variables in response
to a monetary expansion is the same in the two economies except for consumption. Consumption declines in our OLG model because real pension payments decline. Inflation is
bad for retired people. They have to reduce consumption. In addition, the quantitative
response of output and employment in the Ramsey model is almost twice as much as in the
OLG model. The increase in demand in the Ramsey model is much higher as consumption
demand also increases.
We conclude this section by a comparison of the time series properties of the OLG and
the Ramsey model. For this reason, we simulate the two economies over 1,000 periods.
The technology shock and the growth shock are generated by the processes (13) and (17),
20
Figure 6: Monetary shock in the OLG model and distribution
respectively. We use the same sequence of shocks for the OLG and the Ramsey model.
The time series moments of the two models are displayed in the table 1 for the OLG and
the Ramsey model, respectively. In the second and fifth column, the standard deviation
of the HP-filtered variables are displayed. Column 3 and 6 display the correlation with
output, while column 4 and 7 present the autocorrelation of the variables in the two models,
respectively. Notice that the differences are negligible. Again, we conclude that life-cycle
economies behave similar to the infinitely-lived representative-agent economies over the
business cycle. Notice further that the wealth distribution is not very cyclical, but the
inequality of the income distribution seems to be pro-cyclical, an effect that has also been
documented by Castañeda et al. (1998).
21
Figure 7: Monetary Shock in the Representative Agent Economy
5
Conclusion
So far, only the long-run distribution effects of monetary policy have been analyzed in
computable general equilibrium models, as e.g. in Erosa and Ventura (2002) or Heer
and Süssmuth (2003). The short-run effect of unexpected inflation on the distribution of
wealth, which many economists believe to be more important than the change of the longrun inflation rate (as long as higher inflation rates are not subject to higher volatility), have
not received any attention yet, to the best of our knowledge. In this paper, we presented a
model framework for the analysis of the distribution effects of unanticipated inflation. Our
framework can only be regarded as a first step to a fully-fledged analysis of the short-run
22
Table 1:
OLG model
Ramsey model
Variable
sx
rxy
rx
sx
rxy
rx
Output
0.90
1.00
0.67 0.92 1.00
0.67
Investment
3.24
1.00
0.67 2.88 1.00
0.67
Consumption 0.30
0.98
0.69 0.27 0.98
0.69
Hours
0.08
0.58
0.30 0.12 0.61
0.21
Real Wage
1.07
0.77
0.38 1.06 0.79
0.38
Inflation
1.66 -0.04 -0.08 1.67 0.00
-0.08
Gini Wealth
0.00
0.08
0.85
Gini Income
5.10
0.76
0.36
Notes: sx :=standard deviation of HP-filtered simulated series of variable x,
rxy :=cross correlation of variable x with output, rx :=first order autocorrelation of variable x.
distribution effects of monetary policy. Nevertheless, our model can serve as a benchmark
case for future work that may include a more sophisticated modelling of the idiosyncratic
earnings process and may even allow for a third asset besides money and capital, namely
housing. In particular, we suggested a framework that replicates the following important
channels of monetary policy on the distribution of income and wealth: 1) cold progression,
2) inflation-dependent pensions, and 3) the response of prices, and hence the change in the
mark-ups, interest rates, wages and, ultimately, the incomes of the individuals.
Our results can be summarized as follows. A positive technology shock increases income
and wealth inequality because factor prices increase and the real tax burden falls. A
monetary expansion is associated with a rise in inflation and higher income heterogeneity
23
basically because retired agents have lower real pensions. As a consequence, wealth is also
distributed more equally, even though the quantitative effect is negligible. For this reason,
our results do not give support to the idea that monetary policy should be very concerned
with its distribution effects. Again, we emphasize that we have neglected housing from
our analysis.14 In addition, we also find another very important result with relevance
for the literature on computable general equilibrium. The business cycle dynamics of the
heterogeneous-agent economy with sticky prices are very similar to those found in the
corresponding representative-agent economy. As a consequence, our studies gives support
to the focus of the business cycle literature on the Ramsey model. Our result for the nonWalrasian monetary economy with quarterly periods is therefore in very good accordance
with those of Rı́os-Rull (1996) for a Walrasian real economy with annual periods.
14
The effect will most likely be dominant in countries such as Italy where mortgage payments are flexible
and tied to the short-run interest rate.
24
6
Appendix
6.1
Non-stochastic steady state
In the stationary state state (constant money growth θ̄ and zt ≡ 1), the following equilibrium conditions hold:
1. π = 1 + θ̄
2. x̄ =
.
−1
3. r = x1 αK α−1 N 1−α − δ
4. w = x1 (1 − α)K α N −α .
5. Ω = 1 − x1 K α N 1−α .
6. seign =
6.2
Seignt
Pt
= θ̄m.
Log-linearization around the non-stochastic steady state
Log-linearization of first-order conditions around the non-stochastic steady state results in:
s
s
s
λt = (γ(1 − σ) − 1) ct + (1 − γ)(1 − σ) mt − πt , s = 1, . . . , T + T R .
λst
r
s+1
·
= Et λ
t+1 +
1 − δ + r (1 − τ (y s+1 ))
s+1
s+1
1 − τ y s+1 r̂t+1 − τ (y s+1 )y s+1 y
+
π̂
(y
)π̂
−
τ
,
t+1
t+1
t+1
s = 1, . . . , T + T R − 1,
(22)
25
where yts = rt kts + wt nst es for s = 1, . . . , T :
wns es rk s s
s
s
yt = s rˆt + kt +
ŵt + nt
y
ys
(23)
and yts = rt kts for s = T + 1, . . . , T + T R :
yts = rt + kts
(24)
s
s+1
s+1
·
(2
πλs λst + πλs Et {π̂t+1 } = βλs+1 λ
t+1 + πλ − βλ
s+1
s+1
γ(1 − σ)c
, s = 1, . . . , T + T R −
t+1
t+1 + ((1 − γ)(1 − σ) − 1) mt+1 − π
1
ns s
s s s
s
y
n , s = 1, . . . , T.
τ
(y
)y
+
π̂
(y
)
π̂
+
τ
=
η
t
t
t
1 − τ (y s )
1 − ns t
λst + ŵt −
(26)
Furthermore, we need to log-linearize the working household’s budget constraint around
the steady state for the one-year old with k 1 ≡ 0:
2
k 2 k
t+1
+
2
m2 m
t+1
1 1
= wn 1 1 1
1
ŵt + nt + trtrt + ΩΩ̂t − τ (y )y yt + π̂t − c1 c1t
and for s = 2, . . . , T :
s+1
k s+1 k
t+1
+
s+1
ms+1 m
t+1
ms s
s s
s
mt − π̂t + wn wt + nt
= (1 + r −
+ rk rt +
π
t − τ (y s )y s yts + πt − cs cst
+trt
rt + Ω Ω
(27)
δ)k s kts
s
Log-linearization of the retired agent’s budget constraint around the non-stochastic steady
state results in:
s+1
k s+1 k
t+1
+
s+1
ms+1 m
t+1
ms s
mt − π̂t − pen π̂t
= (1 + r −
+ rk rt +
π
t − τ (y s )y s yts + πt − cs cst
+tr t
rt + Ω Ω
δ)k s kts
26
s
(28)
Finally, consumption at age s = T + T R is given by:
R T +T R
cT +T c
t
R mT +T
T +T R
mt
= (1 + r −
+ rk
rt +
− π̂t − pen π̂t
π
T +T R
T +T R T +T R
yt
(29)
+tr trt + Ω Ωt − τ (y
)y
+ πt
R T +T R
δ)k T +T k
t
T +T R
Therefore, we have the following T + T R + T + T R + T + 5 = 665 controls: cst , s =
1, . . . , T + T R , nst , s = 1, . . . , T , yts , s = 1, . . . , T + T R , wt , rt , Kt , Nt , mt ,
T + T R − 1 + 1 = 240 costates λst , s = 1, . . . , T + T R − 1, πt , and xt ,
and T + T R − 1 + T + T R − 1 = 478 predetermined variables kts , s = 2, . . . , T + T R , mst ,
s = 2, . . . , T + T R .
We also need to compute the aggregate capital stock, employment, and money balances:
kts
T +T R
, we get:
From Kt =
R
s=2
T +T
K̂t =
T
+T R
s=2
Similarly, log-linearization of Nt =
nst s
T
s=1 T +T R
N̂t =
T
s=1
Aggregate money mt =
ks
1
ks .
K T + TR t
T +T R mst
s=2
T +T R
m̂t =
implies:
ns s 1 s
n
T + TR N t
can be decomposed as follows:
T
+T R
s=2
ms
1
s .
m
m T + TR t
The wage rate is given by the marginal product of labor:
wt = ẑt − x̂t + αK̂t − αN̂t
27
(30)
Similarly, the real interest rate can be log-linearized as follows:
rt = ẑt − x̂t + (1 − α)N̂t − (1 − α)K̂t
(31)
Calvo Price Staggering implies:
π̂t = −κx̂t + βEt {π̂t+1 } .
Profits Ωt = 1 −
1
xt
(32)
zt Ktα Nt1−α are given by:
Ω̂t =
1
x̂t + ẑt + αK̂t + (1 − α)N̂t .
x−1
(33)
The government budget (20) is given by:
t =
tr tr
T
+T R
s=1
m
τ (y s ) y s s
(ŷ
+
π̂
)
+
(θ
−
1)
t
T + TR t
π
TR
θ
θ̂t + m̂t − π̂t +
pen π̂t
θ−1
T + TR
Money growth is defined as follows:
m̂t+1 + π̂t − m̂t = θ̂t
Finally, we have the law of motion for the exogenous state variables zt :
ẑt = ρz ẑt−1 + εzt
(34)
θ̂t = ρθ θ̂t−1 + εθt ,
(35)
and θt :
28
6.3
Local stability of the non-stochastic steady state
In order to conduct a local stability analysis, it is convenient to express our system of
stochastic difference equations as follows:
Cu ut = Csλ
Dsλ Et
st+1
λt+1
+ Fsλ
st
λt
st
+ Cz zt ,
(36)
= Du Et ut+1 + Fu ut + Dz Et zt+1 + Fz zt .
(37)
Therefore, we define:
29
λt
⎛
c1t
c2t
..
.
R
cT +T
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜ t
⎜
⎜ n1
⎜
t
⎜ 2
⎜ nt
⎜
⎜
..
⎜
.
⎜
⎜ T
⎜ nt
⎜
⎜ ŷ 1
t
ut = ⎜
⎜
..
⎜
.
⎜
⎜ T +T R
⎜ ŷt
⎜
⎜ r
t
⎜
⎜
⎜ wt
⎜
⎜
⎜ K̂t
⎜
⎜ N̂t
⎜
⎜
⎜ m̂t
⎜
⎜ tr
⎝ t
Ω̂t
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
st = ⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞
kt2
kt3
..
.
T +T R
k
t
m̂2t
..
.
m̂Tt +T
R
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
λ̂t = ⎜
⎜
⎜
⎜
⎜
⎝
λ̂1t
⎞
⎟
⎟
⎟
⎟
⎟
⎟,
⎟
λ̂59
t ⎟
⎟
π̂t ⎟
⎠
x̂t
λ̂2t
..
.
zt =
ẑt
θ̂t
The dynamic system has exactly ns = 478 eigenvalues within the unit circle for both
flexible (x̂t = 0) and sticky prices (32). The equilibrium is saddlepoint stable.
6.4
6.4.1
The representative agent model
First-order-conditions and parameter choice
We compare the behavior of the heterogeneous-agent OLG model with the behavior of the
corresponding representative-agent Ramsey model. Of course, in this model we are unable
to model pensions. Everything else is unchanged.
30
The representative household maximizes his infinite life-time utility
β t u(ct , Mt /Pt , 1 − nt )
subject to
kt+1 +
Mt+1
= (1 − δ + rt (1 − τ [(πt /π)(wt nt + rt kt )]))kt
Pt
.
Mt
+
+ (1 − τ [(πt /π)(wt nt + rt kt )])wt nt + trt + Ωt − ct .
Pt
His decision variables in period t = 0 are M1 , k1 , c0 , and n0 .
In this model, there are two predetermined state variables, the stock of capital kt and
beginning-of-period real money balances
mt := Mt /Pt−1 ⇒
mt
Pt
Mt
=
, πt :=
.
Pt
πt
Pt−1
Using these definitions, we can write the first-order conditions as follows:
Mt
λ t = uc c t ,
, 1 − nt = γ (ct )γ(1−σ)−1 (mt /πt )(1−γ)(1−σ) ,
Pt
λt = βEt λt+1 (1 − δ + rt+1 (1 − τ [(πt /π)(wt nt + rt kt )](πt /π))) ,
⎤
⎡
Mt+1
u
,
,
1
−
n
c
t+1
t+1
M/P
Pt+1
λt+1
⎦
+
λt = βE ⎣
πt+1
πt+1
λt+1 (1 − γ) (ct+1 )γ(1−σ) (mt+1 /πt+1 )(1−γ)(1−σ)−1
= βE
+
,
πt+1
πt+1
Mt
, 1 − nt = η0 (1 − nt )−η = λt wt (1 − τ [(πt /π)(wt nt + rt kt )](πt /π)) .
un c t ,
Pt
(38)
(39a)
(39b)
(39c)
(39d)
(39e)
In this model we calibrate the tax function so that the average tax rate paid by the
representative household equals the average income tax rate of the US-economy. The
government’s tax revenues are transferred lump-sum to the representative agent. Capital’s
share is α = 0.36 and δ equals 0.019. The parameters that determine the properties of
31
the productivity shock and the money supply shock are the same as those used in the
simulations of the OLG model. The remaining parameters are set as follows: β, γ and η0
are chosen so that
• the pre-tax gross real interest rate,
• the velocity of money,
• and working hours
equal the respective magnitudes in the OLG model (namely, 11.4 percent p.a., 1.5, and
0.33, respectively).
6.4.2
Stationary State
The stationary solution of the representative agent model is characterized by the following
set of equations. Since real money balances are constant, the inflation factor π equals the
money growth factor µ := 1 + θ:
π = µ.
(40a)
Calvo price staggering implies
x=
.
−1
(40b)
The stationary version of the Euler equation for capital,
1 − β(1 − δ)
= n1−α k α−1 τ [(1/x)n1−α k α ]
αβ(1/x)
can be solved for k given our predetermined value of n = 0.33. Given the solution for k we
can determine y. The stationary version of the economy’s resource constraint,
y = c + δk
32
(40c)
allows us, then, to compute c. Finally, the Euler equation (39c) implies the stationary
solution for the ratio between consumption and real money balances:
C
γ
µ
=
−1 .
M/P
1−γ β
(40d)
We use this equation and (40c) to determine the value of γ. This is all we need to compute
the policy function of the log-linearized model.
6.4.3
Linearization
The log-linear version of (39a) is given by
[γ(1 − σ) − 1]ĉt = −(1 − γ)(1 − σ)m̂t + λ̂t + (1 − γ)(1 − σ)π̂t .
(41a)
Log-linearizing (39e) delivers:
η
τ
n
n̂t − ŵt + ∆0 ŷt = λ̂t −
π̂t + ∆0 x̂t ,
1−n
1 − τ
−1
a0 − τ −a1
−a1
(1
+
a
)
1
−
(gy)
+
a
(gy)
∆0 =
,
1
2
1 − τ
(41b)
(41c)
where τ is the marginal tax rate computed at the steady state solution of wn + rk. The
cost-minimizing conditions (15) and (14) provide two additional equations:
αn̂t + ŵt = αk̂t − x̂t + ẑt ,
(41d)
(α − 1)n̂t + r̂t = (α − 1)k̂t − x̂t + ẑt .
(41e)
The log-linear version of the aggregate production function is given by:
(α − 1)n̂t + ŷt = αk̂t + ẑt .
(41f)
The definition of gross investment it = yt − ct implies
[(y/i) − 1]ĉt − (y/i)ŷt + ît = 0.
33
(41g)
Finally, the profit equation Ωt = yt (1 − (1/xt )) provides the following log-linear equation:
−ŷt + Ω̂t = ( − 1)x̂t .
(41h)
The five equations that determine the dynamics of the log-linear model are derived from
the economy’s resource constraint kt+1 = (1 − δ)kt + yt − ct , the Euler equations for capital
and money balances, (39b) and (39c), the definition of beginning-of-period money balances
(38), and from the Calvo price staggering model. They are:
k̂t+1 + (δ − 1)k̂t = (y/k)ŷt − (c/k)ĉt ,
−Et λ̂t+1 + λ̂t +
(42a)
τ (1 − β(1 − δ))
Et π̂t+1
1 − τ
−(1 − β(1 − δ))∆0 Et x̂t+1
= (1 − β(1 − δ))Et r̂t+1 − ∆0 Et ŷt+1 , (42b)
−(β/π)Et λ̂t+1 + λ̂t + ∆1 Et π̂t+1 + ∆2 Et m̂t+1 = ∆3 Et ĉt+1 ,
−βEt π̂t+1 + π̂t +
(42c)
Et m̂t+1 − m̂t + π̂t = µ̂t ,
(42d)
(1 − φ)(1 − βφ)
x̂t = 0,
φ
(42e)
−(1 − (β/π))[(1 − γ)(1 − σ) − 1] =: ∆1 ,
β/π + (1 − (β/π))(1 − γ)(1 − σ) =: ∆2 ,
(1 − (β/π))γ(1 − σ) =: ∆3 .
34
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