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CESifo Area Conference
on
PUBLIC SECTOR ECONOMICS
21 – 23 April 2006
CESifo Conference Centre, Munich
Taxes and Stabilization in Contemporary
Macroeconomic Models
Kenneth Kletzer
CESifo
Poschingerstr. 5, 81679 Munich, Germany
Phone: +49 (0) 89 9224-1410 - Fax: +49 (0) 89 9224-1409
[email protected]
www.cesifo.de
Taxes and Stabilization in Contemporary Macroeconomic
Models
Kenneth Kletzer
Department of Economics
University of California
Santa Cruz, CA 95064
July 2005
revised, November 2005
Abstract
The role of proportional and procyclic labor income taxes for automatic stabilization
with stochastic productivity is analyzed in a contemporary macroeconomic model based
on imperfect competition. The importance of short-run nominal wage rigidity for the
effectiveness of progressive taxes on labor income for stabilizing output and raising
household welfare is examined in a model that yields complete analytical solutions with
stochastic output shocks. Increasing the procyclicity of labor income tax rates raises welfare
with and without rigid nominal wages in the model economy. With fully flexible prices and
wages, a positive covariance between the distortionary tax rate and productivity reduces
the volatility of production and employment. This effect disappears under nominal wage
rigidity, although progressive taxation can still raise welfare by reducing the distortion
caused by a proportional labor tax. With rigid nominal wages and flexible consumer goods
prices, payroll taxes levied at rates that rise with output can serve as automatic stabilizers.
1
1.
Introduction
In the traditional Keynesian approach, proportionate taxes on income act as automatic
stabilizers by reducing the volatility of aggregate demand in response to stochastic disturbances
to supplies or demands. Countercyclical public sector budget deficits work to stabilize output
and consumption against exogenous shocks in naive models with demand-determined output
and sticky nominal prices and wages. In competitive equilibrium models with fully flexible
prices, countercyclical fiscal policies can be welfare improving if the tax instruments available
to the government distort private sector decisions at the margin. The approach of tax-smoothing
models has the advantage of deriving aggregate demands and output from optimizing behavior by
households and firms. The extension of these models to allow for nominal price or wage rigidities
and imperfect competition can allow the analysis of the Keynesian role for fiscal stabilization
along with the welfare-improving smoothing of the cost of tax distortions. Contemporary analyses
of monetary policy now rely on general equilibrium models with optimizing households who face
stochastic shocks with known distributions.
A natural question for fiscal policy in contemporary models with optimizing behavior is how
nominal price or wage rigidity influences the role of taxation for stabilization against stochastic
disturbances to productivity or aggregate demand. Only recently have a few authors used such
models for fiscal policy analysis, in contrast to the large literature on monetary policy with
nominal rigidities. An important feature of contemporary models of demand-determined short-run
output based on Blanchard and Kiyotaki [1987] is that firms are monopolistically competitive
producers of heterogeneous goods. Imperfect competition allows expansion of output with fixed
prices because firms meet demand to maximize short-run output within the constraint that the
2
markup of price over marginal cost is nonnegative. In this paper, the impact of nominal wage
rigidity on the stabilization role of factor income taxes is considered in a basic general equilibrium
model with imperfect competition and stochastic productivity. The model is set up to allow for
a complete solution of the impact of stochastic disturbances to productivity with and without
nominal wage rigidity allowing for analytical welfare comparisons following Obstfeld and Rogoff
[2000]. To do this, an intertemporal version of the model does not have capital and will not
generate net national savings in the open economy in equilibrium. These drawbacks compensate
for the benefit of allowing a solution for how progressive labor income taxes affect the volatility
of output and expected utility under the alternative assumptions about nominal wage stickiness.
The model economy also has the advantage that it can be analyzed as a static model because the
static stochastic equilibrium gives all the properties of equilibrium for a dynamic version.
The model demonstrates that a proportionate labor income tax levied at a rate rising with
productivity stabilizes output and raises household welfare in the absence of nominal wage rigidity.
Introducing rigid nominal wages eliminate capacity of progressive tax rates on labor earnings to
reduce output and employment volatility. In the monopolistically competitive economy, increasing
the progressivity of taxes on labor income raises the expected household utility with and without
rigid nominal wages. The analysis of how welfare responds to an increase in the progressivity of
taxes requires some restriction on changes in tax revenues. In this case, the expectation of tax
revenues is held constant motivated by the notion that expected tax revenues would be determined
from tax-smoothing considerations in an intertemporal model with stochastic productivity and
exogenous government expenditures. With productivity shocks, a progressive tax rate lowers the
expected tax rate required to raise fixed expected tax revenues. This reduction in the average tax
3
rate raises welfare in the presence of the uninsurable output risk.
Therefore, progressive proportionate taxes on household labor earnings act as automatic
stabilizers with flexible wages, but not with rigid nominal wages. Before adding nominal
rigidities, there are two distortions in the economy. These are the absence of lump-sum taxes
and given tax revenue constraint and the presence of monopolistic competition to motivate
demand-determined production when nominal wages are fixed. A conclusion of the analysis is
that adding rigid nominal wages changes the welfare effects of progressive taxation by increasing
the impact of productivity shocks (because the labor market does not clear) and eliminating the
direct effect of procyclic tax rates on output volatility.
With flexible wages and prices, a proportionate payroll tax is equivalent to a proportionate
labor income tax. In the economy presented, nominal wages are fixed but the nominal prices of
final goods are not. Nominal price rigidity is consequent to nominal wage rigidity under aggregate
demand shocks because the markup of prices over wages is constant due to constant elasticities of
substitution in the model economy. The price-wage gap fluctuates with productivity and a payroll
tax rate with fixed nominal wages in this economy. Therefore, progressive payroll tax rates can
act as automatic stabilizers reducing the volatility of output and raising expected utility.
The literature on the role of nominal price and wage rigidities for the efficacy of automatic
tax stabilizers in contemporary macroeconomic models based on individual household and firm
optimization is limited. This paper concentrates on how labor income taxes affect the level and
volatility of output and household welfare in a model that yields complete analytical results.
Disturbances to aggregate demand can be represented by shocks to the money supply in the model
economy with sticky nominal wages. Agell and Dillen [1994] study the role of taxes and subsidies
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as automatic stabilizers against aggregate demand shocks with nominal rigidities but allow
lump-sum financing of subsidies. Kleven and Kreiner [2003] eliminate the implicit lump-sum
taxes in Agell and Dillen and show that the utility deviations due to a monetary disturbance are
exacerbated by proportional taxes on labor income, represented as either taxes on household labor
earnings or as payroll taxes.
Demand management can also be achieved through the deficit financing of government
expenditures. The model economy can be extended to an intertemporal economy introducing
public debt and deficits so that countercyclical fiscal policies can offset aggregate demand
insufficiencies and promote allocative improvements with productivity shocks under nominal
wage rigidity. A dynamic model with capital accumulation, imperfect competition and nominal
price stickiness is used by Schmidt-Grohè and Uribe [2005] to analyze capital income subsidies,
labor income taxes and inflation. They show that the case for capital income subsidies is
strengthened by nominal price stickiness and that optimal subsidies are volatile. Their model
must be approximated and calibrated, so that expressions for expected output and utility cannot
be derived. In the calibrated dynamic model, Schmidt-Grohé and Uribe also find that the labor
income tax rate is quite stable. This may be consistent with the analytical result that progressive
labor income taxes cannot reduce the volatility of output with productivity disturbances. However,
the result found below that progressive taxes can raise welfare suggests that the optimal variability
of the tax rate with output should be sensitive to underlying parameters of the economy.
The analysis of distortionary taxes could be extended to allow tax smoothing over time and
borrowing or lending across borders. In general, either extension can eliminate closed-form
solutions and require approximations around stationary states. Two particularly interesting
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possible extensions are proposed. One is to extend the model to an intertemporal economy with
government debt to consider how tax smoothing might increase or reduce the static case for
progressive labor income taxes with and without nominal wage rigidities. The other is to analyze
the insurance role of proportional income taxation with idiosyncratic regional productivity shocks
in fiscal federations.
The remainder of the paper is organized as follows. The next section presents the model
economy with productivity shocks and nominal wage rigidity. The third section reports the
analysis of labor income taxes with fully flexible wages and prices, and the fourth section tax
shows the effects of labor income taxes with nominal wage rigidity. Section 5 compares the
two cases and summarizes the implications of nominal wage rigidity for welfare-improving
progressive labor income taxation. Section 6 proposes extensions of the model to consider tax
smoothing and implicit fiscal transfers in a fiscal union. The last section gives a brief conclusion.
2.
Productivity Shocks and Taxes in Simple Macroeconomic Model
The efficacy of fiscal policy for short-run stabilization in contemporary stochastic general
equilibrium models with nominal wage or price rigidities can be illustrated using a simple
closed-economy version. The model used here extends immediately to a model of an open
economy or of a two-country world economy. Following Obstfeld and Rogoff [2000], a stochastic
version of the Blanchard and Kiyotaki [1987] model of an economy with heterogeneous goods
produced by monopolistically competitive firms is used to consider how nominal wage rigidity
affects fiscal stabilization. Because contemporary models with imperfect competition and
wage-price rigidities are based on optimization by households and firms, these allow welfare
analyses of policy in static or dynamic versions. The purpose of this section is to illustrate how
6
taxation can be studied in these models and highlight some implications of nominal wage rigidity
for fiscal stabilization using distortionary taxes.
The economy consists of a continuum of households, each supplying a unique type of labor to
a continuum of producers of final goods. Households are indexed along the unit interval, [0, 1], as
are producers. Both final goods and labor are imperfect substitutes. This allows each firm to act as
a monopolist in the market for its particular product and each household to behave as a monopolist
in the supply of its particular type of labor. Firms are perfectly competitive as purchasers of labor
services, and households are perfectly competitive as purchasers of final goods.
A single firm produces each particular good j in the interval [0, 1] using a technology given by
the production function,
]
Yj = A
1
0
31
Lij
31
di
,
(1)
for > 1, where Lij is the input of type i labor by firm j. The coefficient A represents a
multiplicative productivity shock.
The typical household i has preferences represented by the period utility function,
Ci134
Ui =
+
14 10
Mi
P
130
K D
L ,
D i
(2)
for D 1, 4 > 0 and 0 > 0 (for 4 = 1 or 0 = 0, the appropriate substitution of log Ci and log MPi
should be made). The consumption index over consumption of heterogeneous goods is given by
]
Ci =
1
0
w31
w
w
w31
cij dj
,
for w > 1, where cij is consumption of good j by household i. The price index, P , is given by
]
P =
0
1
p13w
j dj
1
13w
.
7
The household seeks to maximized its utility in the static economy over a budget set given by
]
P Ci + Mi (1 tw ) Wi Li +
0
1
(1 tp ) ij dj + P i ,
(3)
where Wi is the nominal wage rate for labor of type i, ij equals nominal profit from ownership
by household i of shares of firm j, tw is the labor income tax rate, tp equals the profit tax rate and
P i are lump-sum transfers from the government. The government balances its budget for now
leading to the constraint,
]
0
1
]
Mi di +
0
]
1
P i di 0
1
]
(tw + tpr ) Wi Li di +
0
1
(1 tp ) j dj P G,
(4)
for real government expenditures, G, and payroll tax levied at the rate tpr . In the static economy,
government expenditures will be set equal to zero for simplicity of analysis. In equilibrium, the
labor demanded by firms equals that supplied by households,
]
Li =
1
Lij dj.
0
This condition will determine labor supply with nominal wage stickiness under monopolistic
competition.
The demand function for final goods is derived first and is given by
cij =
p 3w
j
P
Ci ,
so that demand for the output of firm j is
Yj =
where aggregate consumption is C =
U1
0
p 3w
j
P
C,
Ci di. The derived demand for labor is found by
8
maximizing after-tax firm profit as
1
Lij =
A
Wj
W
3
Yj ,
where the wage index, W , is
]
W=
0
1
1
13
Wi13 di
.
Profit maximization also yields the markup pricing rule,
pj =
1 W.
A1
(5)
With a payroll tax imposed at the proportionate rate tpr , the markup pricing rule becomes
pj =
1 W (1 + tpr ) .
A1
Returning to household optimization provides the equilibrium supply of labor and final goods
consumption for flexible nominal wages and prices,
KLD31
=
i
w1
Wi 34 w 1 1
(1 tw )
C =
(1 tw ) ACi34 .
w
P i
w
(6)
The demand for real balances satisfies
Ci34
Mi
=
P
30
(7)
because money balances only have current value in the static economy. This model can be
extended readily to a dynamic model because there is no capital accumulation in the economy
and savings is zero when the static economy is repeated if labor productivity and money growth
are independently and identically distributed (see, for example, Obstfeld and Rogoff [1996],
[1998] and Corsetti and Pesenti [1998]). Obstfeld and Rogoff [2000] show how this static model
extends to a two-country global economy introducing the exchange rate and terms of trade with a
9
continuously balanced current account in equilibrium.
With nominal wage rigidity, the household needs to maximize its utility with respect to a preset
nominal wage rate given the possible disturbances to productivity or the money supply. The
resulting wage setting equation is given by
Wi =
E (KLDi )
.
1 E (1 tw ) LPi Ci34
(8)
As in Blanchard and Kiyotaki [1987] and successive models with imperfect competition, final
goods output and employment are determined by demand in equilibrium with nominal wage-price
rigidity. In this model, nominal wages are sticky and nominal prices can be set ex post. Only in
the case of a multiplicative productivity disturbance will nominal prices respond under markup
pricing.
This model is set up to allow a closed form solution with stochastic disturbances to A, K and
M that are distributed lognormally. Since the details are given elsewhere (see Obstfeld and Rogoff
[1998]), the derivation of the solutions for the closed form is skipped. The variables expressed
in lower case represent natural logarithms of the level variables in the model (with k = log A,
V = log K and = log L). Symmetry leads to dropping household and firm subscripts to obtain
economy-wide equilibrium conditions.
3.
Productivity Shocks and Stabilization with Flexible Wages and Prices
The analysis of distortionary labor income taxes as automatic stabilizers begins with the case
of perfectly flexible prices and wages. The government imposes taxes labor income taxes on
wage recipients or payroll taxes on employers at a proportional rate that can vary with output.
The motivation for labor income taxes is an implicit assumption that the government can only
10
impose such distortionary taxes and uses the revenue to finance public sector expenditures on final
goods. For expositional thrift, tax revenues are redistributed to households as lump-sum transfers.
This assumption does not affect the conclusions. With price and wage flexibility there are two
distortions in the model economy. These are the presence of monopolistic competition and tax
distortions.
The flexible price and wage case is approached by first taking logarithms of the equilibrium
conditions with government expenditures set equal to zero. The equilibrium condition for final
goods output is given by
Y = AL = C.
Taking logarithms of this equilibrium condition and equations (5),(7) and (6) leads to the four
relationships,
y = + k = c,
w = p k + log
(9)
1
,
(10)
4c = 0 (m p)
(11)
and
(D 1) + V = (1 4) k 4 + log (1 tw ) + log
w11
.
w
(12)
Productivity disturbances enter the model economy in two possible ways. The first is as a
multiplicative shock to the output produced from the employment of a given bundle of labor inputs
expressed by the parameters k (or A in levels). The second is as a shock to the disutility of labor
expressed by the parameter V (or K in levels). These shocks affect equilibrium for the economy
11
differently and are analyzed separately.
The multiplicative productivity shock, A, is assumed to be lognormally distributed. The mean
and standard deviation of k are given by k and j 2k , respectively, and for now V is constant. Taking
the expectation of the labor market clearing condition leads to the solutions for the expectation of
the logarithm of output and its standard deviation given by
(D + 4 1) y = Dk + E log (1 tw ) + log
w11
w
(13)
and
j 2y
=
1
D +41
2
2 2
D j k + j 2# + 2Dj k# .
(14)
Here # denotes the portion of labor income received post tax, 1 tw , and is assumed to be
lognormally distributed as well. For constant rate taxes, j 2# and j k# are both zero. For progressive
income tax rates, j k# will be negative. The assumption of lognormality simplifies expressions but
allows positive probabilities of labor income subsidies, 1 tw > 1. This is ignored by assuming
that tax rates become negative only for very improbable productivity shocks.
The flexible-wage solution for output is
Y D+431 = (1 tw ) AD
1 w11
,
K w
(15)
so that expected output is given by
Y =
w11
wK
1
D+431
Dk + #
1 2
exp
+ j ,
D +41 2 y
where # E log (1 tw ). Output is positively correlated with productivity. An increase in the
variance of the productivity shock raises the expectation of output in this economy because the
12
marginal utility of leisure is non-increasing with leisure consumption (expressed by D 1).1
An increase in a constant proportional labor income tax, represented by a decrease in #, reduces
expected output as it should. Tax rates that vary with productivity, for example with labor income,
raise expected output through the combination of the last two terms in equation (14). Progressive
income tax rates can reduce the correlation between productivity shocks and output in this
economy. For example, the standard deviation of output is lower when # is perfectly negatively
correlated with k than when the labor income tax is levied at a constant rate.
A primary advantage of contemporary Keynesian models based on optimizing household
behavior is that they allow welfare analysis. Expected utility can be calculated by ignoring
the additive term in real balances which serves to generate a demand for money but cannot be
interpreted satisfactorily in terms of household welfare. The remaining expression,
Ci134
K D
EU = E
Li ,
14
D
is used to calculate the effect of labor income taxes on expected utility. Using equation (6), we
have that
KLDi
w11
=
(1 tw ) Ci134 ,
D
Dw
(16)
after multiplying by Li = L = Y /k = C. Substitution leads to the expression for expected utility,
EU = E
1
w11
134
(1 tw ) Ci
,
14
Dw
(17)
for 4 9= 1. The calculations for the unitary elasticity of substitution case can be easily derived and
are not included for the remainder of the paper.
1
The quantity, + 1, which appears repeatedly is positive for 1.
13
For a labor income tax levied at a constant rate, expected household utility is given by
EU =
1
w11
(1 tw ) E Ci134 .
14
Dw
Calculation of the expectation in this expression for multiplicative productivity disturbances under
flexible prices leads to
&
%
134 (1 4)2 2
E Ci
jc
= exp (1 4) c +
2
%
134
2 2 &
D+431
(1 4) D
w11
jk
(1 4) D
(1 tw )
.
exp
k+
=
wK
D +41
D +41
2
These expressions imply that the expected utility of the representative household rises with the
mean of the multiplicative productivity shock,
j 2k
E (A) = exp k +
,
2
and decreases with a mean-preserving increase in the variance of the productivity shock. An
increase in the labor income tax rate raises expected output increasing both consumption and labor
supply. The net effect of an increase in the income tax rate on expected utility is negative if
3D+431
w11
< (1 tw ) D+431 ,
w
which holds for all 4 if 1 tw >
w31 31
.
w
Progressive labor income taxes do raise household welfare with flexible wages in this economy.
The welfare effects of labor income tax rates that covary with output is calculated by substituting
the two expressions,
E Ci134 =
w11
wK
134
D+431
%
(1 4) (Dk + #)
+
exp
D +41
14
D +41
&
2 2
j 2#
2 jk
D
+
+ Dj k#
2
2
(18)
14
and
$
#
2 2
2
2
j
(1
4)
j
D
(1
4)
D
k#
#
+
,
E (1 tw ) Ci134 = E Ci134 exp # +
(D + 4 1)
(D + 4 1)2 2
(19)
into the expression for expected utility given by equation (17). The covariance term in expected
utility is given by
&
&
%
%
2
2
w11
1
14
D
Dj k# (1 4) j k# .
exp
exp
14
D+41
Dw
D+41
This is decreasing in j k# , the covariance of the logarithms of productivity and 1 tw , for any
positive coefficient of relative risk aversion. Therefore, mean-preserving progressive income
taxation (that is, a positive covariance between productivity and the tax rate) raises welfare under
flexible wages. Progressive taxation can serve as an automatic stabilizer in the flexible price
economy in terms of welfare.
Disturbances to the disutility of labor through the parameter, K, also represent shocks to
efficiency labor and have been used in the analysis of sticky price models with monopolistic
competition. The solution for output under lognormally distributed shocks to K using equation
(15) is given by
Y =
w11
w
1
D+431
1
V + #
+ j 2y
exp
D +41 2
where
j 2y
=
1
D +41
2
2
j V + j 2# + 2j V# ,
for A = 1. The rest of the analysis for productivity shocks represented by shocks to the disutility
of labor is also analogous to that for multiplicative productivity shocks in the flexible wage case.
The conclusion that countercyclical progressive labor income taxation can act as an automatic
15
stabilizer raising welfare holds under perfectly flexible wages and prices for either type of
productivity shock.
4.
Productivity Shocks and Stabilization with Nominal Wage Rigidity
The impact of productivity disturbances on output and utility can be quite different with preset
nominal wages. The equilibrium condition for labor market clearing in the short run plays a
key role in the analysis of proportional and progressive labor income taxes in the flexible wage
economy. When wages are set in advance of production, employment is demand determined
in short-run equilibrium. The response of household demand for output to productivity shocks
depends on real money balances with short-run nominal rigidities. As a consequence, the
covariance of output with productivity shocks will depend on the money market equilibrium
condition rather than the labor market equilibrium condition. Proportionate labor income taxes
will affect labor supply, expected output and welfare through ex ante wage setting. Further, the
qualitative effects under multiplicative productivity shocks to output or to the disutility of labor
are quite different with nominal wage rigidity. Payroll taxes are also considered below because
these have a different effect on the distribution of output with preset nominal wages.
A starting point for the analysis of how taxes affect output and welfare with productivity
disturbances begins with the determination of fluctuations in output demand. Again, the example
of stochastic multiplicative productivity in the production function is considered first. With fixed
nominal wages, the money market equilibrium condition (7) determines the change in output and
the price level under markup pricing, P =
W .
A 13
Taking the logarithm of this equation, ex post
16
consumption demand is given by
4c = 0 (m p)
,
0
y=
m + k log
w .
4
1
Keeping the money supply constant, the mean and standard deviation of the logarithm of output
are given by
0
y=
m + k log
w
4
1
and
j 2y
2
0
=
j 2k .
4
Fluctuations in output growth caused by productivity shocks are not affected by labor income
taxes. Taxes will only affect the expectation of the logarithm of output with rigid nominal wages.
Output is determined using these moments and the optimal wage setting equation (8). Because
of markup pricing, the real wage is substituted into equation (8) before taking expectations
for lognormal disturbances. Taking logarithms after evaluating expectations, the wage setting
equation leads to
0 = V # + (D + 4 1) y +
1 2
1
1
D (1 4)2 j 2y Dk + D 2 j 2k (1 4) j y# j 2# .
2
2
2
The expectation of output is now given by the expression
Y
1 2
= exp y + j y
2
(20)
3
2 4
1
0
#
$
2
j 2k F
E V + Dk 2 D + (D (D 1) + 4 (1 4)) 4
# + (1 4) j y# + 12 j 2#
F.
= exp
exp E
C
D
D +41
D +41
The first term in equation (20) depends on labor income taxes, while the second term is exogenous.
An increase in a constant rate proportional labor income tax (an decrease in #) lowers expected
output. Because the labor market is monopolistically competitive, such an increase lowers welfare
and the optimal labor tax is negative. A progressive tax on labor income raises expected output if
17
the coefficient of relative risk aversion is greater than one.
The welfare effects of labor income taxes are demonstrated by calculating expected utility
under nominal wage rigidity with multiplicative productivity shocks. Welfare is again evaluated
using the expression
Ci134
K D
EU = E
Li ,
14
D
where market equilibrium implies that Ci = Y and ALi = Y . Using the wage-setting equation
(8) and markup pricing equation for final goods (5), the disutility of labor term in expected utility
can be written as
E
K D
L
D i
=
1 E (1 tw ) Ci134
D
in equilibrium with preset nominal wages so that expected utility becomes
EU = E
1
1
134
(1 tw ) Ci
.
14 D
The first expectation in this expression is given by
#
$
2
1 2
134 (1
4)
#
+
(1
4)
j
+
(1
4)
j
y#
#
2
E Ci
,
= B 134 exp
(D + 4 1)
and the second is given by
E (1 tw ) Ci134 = B 134 exp
where
#
D# + D (1 4) j y# + D 12 j 2#
(D + 4 1)
$
,
2 4
2 0
2
D + D (D 1) + 4 (1 4)
j 2k F
4
E V + Dk F
B = exp E
C
D
D +41
3
1
2
is exogenous to the labor income tax. Direct substitution and differentiation verifies that for any
value of 4 greater than zero, expected utility rises or falls with the sum, # + (1 4) j y# + 12 j 2# , as
18
this expression is negative or positive. To sign this, note that the mean tax rate on labor income
given by
1 2
tw = 1 exp # + j #
2
is positive only if # + 12 j 2# is negative. Therefore, increasing the correlation between the labor
income tax rate and output starting from a constant rate tax and keeping the expected tax rate
constant raises welfare if the coefficient of relative risk aversion is greater than one and reduces
welfare if relative risk aversion is less than one. In the unitary risk aversion case, the covariance
term drops out and only the expectation of the tax rate affects welfare. The restriction,
1
# + (1 4) j y# + j 2# < 0,
2
(21)
will be assumed.
When comparing tax schedules that offer differing degrees of automatic stabilization against
productivity disturbances, it makes sense to hold expected tax revenues constant. Expected real
tax revenues are given by
W
1
E tw L =
E (tw Y ) .
P
Since it is the net of tax share of labor income, 1 tw , that is lognormally distributed, expected
tax revenues are calculated as
1
W
E tw L
=
(EY E (1 tw ) Y )
P
1
1 2
1 2
2
=
exp y + j y exp # + y + j y# +
j + jy
.
2
2 #
19
Substituting the solution for expected output (equation (20)) into this expression leads to
#
#
$
#
$$
# + 12 j 2# + (1 4) j y#
(D + 4) # + 12 j 2# + Dj y#
1
W
E tw L =
B exp
exp
,
P
D +41
D +41
(22)
which is positive given assumption (21) unless the covariance, j y# , is a large positive number.
Consider an increase in the progressivity of the tax rate. This increases tax revenues if the mean
tax rate remains constant for any positive coefficient of relative risk aversion. Differentiation of
equation (22) also reveals that an increase in tax progressivity that keeps expected tax revenues
constant requires an increase in the left-hand side of the inequality (21). This raises welfare.
However, this increase in welfare is due to a rise in expected output and is a consequence of
monopolistic competition. Progressive taxes do not affect the standard deviation of output.
With rigid nominal wages and flexible output prices, payroll taxes are not functionally identical
to labor income taxes as they are with full price and wage flexibility. Adding a proportionate
payroll tax changes the markup pricing equation (5) to
pj =
1 W (1 + tpr ) ,
A1
where tpr is the possible varying payroll tax rate. The money demand function becomes
0
y=
m + k log
log (1 + tpr ) w ,
4
1
which implies that the standard deviation of output depends on the standard deviation of
µ log (1 + tpr ) and its covariance with the productivity shock as
j 2y
2
0 2
=
j k 2j µk + j 2µ .
4
The calculations for expected output and household welfare with rigid nominal wages can be
20
repeated, but the additional terms increase the length of the expressions. The additional terms all
come from the effect of the payroll tax and its covariance with the productivity disturbance on
the volatility of output. The direct effect of a progressive payroll tax is to reduce the volatility of
output given constant payroll tax revenues. The effects on welfare through expected output due to
the effect of the moments of the payroll tax on the preset nominal wage are analogous to those just
demonstrated for the labor income tax. The additional effect of the payroll tax arises because it
changes the markup between marginal cost and final goods prices when it varies with output. The
possibility of imposing a tax on firm profits is included in the model (in equation (3)). The optimal
profit tax rate is one-hundred percent if expenditures are at least as great as the revenue generated.
When productivity disturbances are represented by shocks to the marginal utility of leisure, K,
the markup pricing condition implies that the price level does not respond to a productivity shock.
Therefore, demand-determined output satisfies the conditions
0
y = (m p)
4
and
j 2y
2
0
=
j 2m ,
4
so that these productivity shocks have no effect on output under nominal wage rigidity (as shown
by Obstfeld and Rogoff [1996]). In this case, the wage setting equation (8) in logarithms becomes
1
1
(D + 4 1) y = V j 2V + # + j 2# + Dk.
2
2
(23)
Productivity shocks affect the utility from the consumption of leisure by households but not the
supply of labor and output when nominal wages are set in advance of employment. The wage
setting condition implies that an increase in the volatility of productivity expressed by the disutility
of labor lowers mean output and that labor income taxes cannot serve as automatic stabilizers for
21
output. An increase in the expected labor income tax rate,
1 2
tw = 1 exp # + j # ,
2
lowers expected output and expected utility. The correlation of taxes with productivity have no
effect on employment, production or welfare with preset nominal wages. However, payroll taxes
with rigid nominal wages and flexible final goods prices can affect the distribution of output and
of expected utility if the tax rate varies procyclicly with productivity since these still change the
markup of final goods prices over preset wages.
5.
Comparison of Labor Income Taxation with and without Sticky Wages
The analysis of this model economy demonstrates how the progressivity of a labor income
tax affects representative household welfare with and without nominal wage rigidities. However,
the role of progressive taxation as automatic stabilizers against productivity shocks is quite
different with sticky wages than with flexible wages and prices. Given that tax revenue must be
raised through distortionary labor income taxes, a progressive tax on labor earnings can reduce
the impact of the uninsurable productivity shocks on output raising welfare for a sufficiently
risk averse household with perfectly flexible wages. However, with preset nominal wages, the
progressivity of the income tax has no effect on the variance of output and employment. This
means for automatic stabilization is precluded by setting wage in advance of realization of the
productivity shock and production. In the case of nominal rigid wages, the progressivity of the
labor income tax affects welfare because it allows a reduction in the average rate of labor income
taxation which raises welfare in the distorted economy.
The effect of progressive income taxes can be compared by repeating the calculation of how
22
the labor income tax affects welfare keeping expected tax revenues constant for flexible nominal
wages. This is done in terms of the exogenous productivity shock instead of output which is
endogenous to the joint variation of the tax rate and productivity shock when prices and wages are
flexible.
The effect of the progressivity of the labor income tax rate on welfare in the flexible price case
is given by combining equations (17), (18) and (19) to form the sum
%
&
2 2
j#
1
(1 4) #
14
EU =
(24)
D exp
+
+ Dj k#
14
D +41
D +41
2
&
%
2 2 2
j#
D#
D
D
w11
D exp
+
+
(1 4) j k# ,
Dw
D +41
D +41
2
D +41
where the common term is given by
D=
w11
wK
134
D+431
#
exp
(1 4) Dk
+
D +41
(1 4) D
D +41
2
j 2k
2
$
.
It is helpful to recognize that
#
E (1 tw )
134
D+431
= exp
(1 4) #
+
D +41
14
D+41
2
j 2#
2
$
and that
D=E
w11 D
A
wK
134
D+431
.
With flexible prices, expected tax revenues are given by
D+4
1
W
Dj k#
(D + 4) Dj k#
1 D+431
D+431
E tw L =
E (1 tw )
E (1 tw )
,
exp
D exp
P
(D + 4 1)2
(D + 4 1)2
where
D =E
w11 D
A
wK
1
D+431
.
Differentiation of this expression shows that a sufficient condition for an increase in the
23
progressivity of the tax rate (decrease in j k# ) to raise expected tax revenues for a given expected
tax rate is that tax revenue from labor income be less than half of labor income.2 Assuming this
implies that an expected revenue neutral increase in the progressivity of the labor income tax with
output allows a reduction of the expected tax rate. Differentiation of the expression for expected
utility, equation (24), verifies that a decrease in the covariance of the post-tax labor income and
productivity raises expected utility for any coefficient of relative risk aversion. That is, a more
progressive labor income taxes raise welfare for given expected revenues.
The comparison between the case with fully flexible wages and preset nominal wages can be
made using the expressions here and in the previous sections. This is tedious, but the elements for
such an analysis are given. Two important conclusions are reached. The first is that progressive
labor income taxes act as automatic stabilizers of output against multiplicative productivity shocks
with fully flexible nominal wages and prices but not with rigid nominal wages. The pre-existing
distortions of a proportional labor income tax and monopolistic competition also lead to potential
welfare gains from progressive tax rates. The impact of nominal wage rigidity on the welfare
effect due to these distortions depends on the parameters. The second conclusion is that while a
payroll tax has identical effects to a labor earnings tax with flexible wages and prices, the two
taxes are different with rigid nominal wages. In contrast to the earnings tax, a payroll tax affects
the distribution of output in the presence of productivity shocks with nominal wage rigidity in this
model economy because nominal final goods prices are not rigid. A progressive payroll tax does
act as an automatic stabilizer with rigid nominal wages, but its impact is lessened.
2
To verify this, recall that the expected tax rate is less than one and 1.
24
6.
Possible Extensions
Whether automatic stabilizers reduce the impact of productivity shocks in an economy with
imperfect competition and nominal wage rigidity is analyzed here in a static and closed economy.
In either case, proportional labor income taxes generate higher government revenues with positive
productivity disturbances. A natural and first extension of the analytic solution would be the
calculation and numerical comparison of optimal progressive taxes that generate fixed expected
tax revenues with and without rigid nominal wages. However, the optimal progressivity of income
taxes in this economy may depend upon the static assumption.
In the static economy, larger revenues implicitly finance either larger lump-sum transfers or
higher government expenditures. In particular, this approach could be extended to a multi-period
or infinite-horizon economy to analyze tax smoothing over time with exogenous government
expenditures. Basic tax-smoothing models following Barro [1979] allow output to depend
negatively on tax revenues or on a distortionary tax rate. The monopolistic competition model
with perfectly flexible wages and prices can be used to model the single-period economy in a
tax-smoothing model. Introducing government debt as a single asset along with the single-period
public sector budget identity and conventional solvency constraint to the model with constant
government expenditures is sufficient to allow an analysis of tax smoothing with productivity
shocks if tax instruments are restricted to labor income taxes. Optimal taxation could also be
studied analytically in the closed-economy model presented here iterated over an infinite horizon.
It is an open question whether a simple tax smoothing exercise can be done analytically in this
model, although it may be possible with the single non-productive financial asset.
Extending the model to investigate optimal tax-smoothing could allow an analytical solution for
25
optimal labor income taxes in the monopolistic competition economy with stochastic productivity.
Schmidt-Grohé and Uribe [2005] investigate taxation in a dynamic economy with monopolistic
competition, nominal price stickiness and capital accumulation that requires a second-order
approximation to simulate capital income taxes, labor income taxes and inflation. Their main
findings are that the labor income tax varies very little with shocks while the optimal capital
subsidy is very volatile and exceeds the markup of price over marginal cost. In contrast, the
optimal capital subsidy under imperfect competition with flexible prices just compensates for the
markup of prices over marginal costs as shown by Judd [2002] following the original insight of
Joan Robinson [1933]. These results, however, are demonstrated in a particular calibration of an
approximated economy. The model presented here may allow a general solution in the absence
of capital accumulation that could highlight the separate roles of nominal rigidities and of capital
accumulation.
In the static economy, optimal labor income tax rates should be procyclical with output with
either flexible or preset nominal wages. Tax smoothing in a repeated economy with constant
government expenditures and stochastic productivity would lead to an Euler condition relating
the current ex post labor income tax rate to ex ante distribution of tax rates for the next period.
The linking of realized tax revenues to expected tax revenues for the subsequent period may
mitigate tax progressivity. A proportional labor income tax yields more revenue in a given period,
lowering the required expected revenues for the next period. The degree of tax progressivity
will need to address the trade off between raising single-period expected utility and promoting
the intertemporal smoothing of marginal utilities. Whether tax smoothing raises or lowers the
desirable progressivity of labor earnings or payroll taxes with or without wage rigidity is an open
26
question with an answer that likely depends on the parameters.
Thus, the implication that nominal rigidities reduce the role for tax progressivity by eliminating
its automatic stabilizing effect in the model could be strengthened or weakened by allowing
consumption and production over time with government debt. The proposal that the analysis can
be extended to a dynamic economy with government debt as the single financial asset comes from
the intuition that adding the Euler condition linking taxation across periods should suffice for
analyzing the covariance of the tax rate with productivity without approximation around steady
states.
Another direction for extending the analysis of automatic tax stabilizers is to consider a
two-country economy with international financial capital mobility. Access to an integrated
international financial market allows the government to smooth taxes as the private sector smooths
consumption through international borrowing in the presence of stochastic productivity. Opening
the economy with active international borrowing and lending comes at an analytic cost. The
models of Corsetti and Pesenti [1998] and Obstfeld and Rogoff [2003] generate closed-form
solutions for the open economy only under restrictions that eliminate current account imbalances.
A closed-form solution with commodity trade requires that demands display unitary elasticities
of substitution between composites of goods produced in different countries and that preferences
over home and foreign tradable goods be identical across borders.
A related issue concerns the usefulness of fiscal policy as a substitute for autonomous monetary
policies for stabilizing output or employment in a monetary union. The role of fiscal federalism
in a monetary union is studied by Kletzer and von Hagen [2001] who show that lump-sum
state-contingent transfers between governments in a monetary union can be used to stabilize output,
27
employment or utility. The cost of monetary unification arises because monetary autonomy allows
aggregate demand management against idiosyncratic national or regional productivity shocks
given nominal wage or price rigidity. The motivation for fiscal federalism in a monetary union
then concerns the question of whether fiscal stabilizers, including inter-governmental transfers,
can partially replace independent monetary policies and nominal exchange rate flexibility as
stabilizers of regional output against regional productivity shocks.
Kletzer and von Hagen present a model that allows operative savings through the current
account balance but requires approximation. In that model, a representative home country
household consumes two types of goods, foreign and domestic, and has preferences that depend
on the consumption bundle given by
Cih
)
)31
)31
h )31
)
f
= Ci ) + (1 ) Ci
,
for an elasticity of substitution between home and foreign goods, ) > 1, where
]
Cih =
0
1
h w31
cij w dj
w
w31
and
w
] 1 w31 w31
w
Cif =
dk
,
cfik
0
chij is consumption of home product j and cfij is consumption of foreign product k by the home
household i. Foreign preferences are similar but with consumption shares switched to display
home bias in consumption in either country. The terms of trade between foreign and domestic
goods are given by the relative price indices, P/P W and there is a single money supply for both
countries. Production in each country is subject to multiplicative productivity shocks.
This model allows a comparison of self-insurance through international borrowing and
mutual insurance through lump-sum intergovernmental transfers. Introducing distortionary labor
income taxes and ruling out lump-sum taxes and transfers can enrichen the analysis of the role
28
of automatic stabilization against idiosyncratic productivity disturbances in fiscal federations.
Indeed, proportional and progressive income taxes are a prime vehicle for effecting inter-regional
transfers under fiscal federalism. The mutual insurance role of progressive labor income taxes is
a natural application of the model presented in this paper. Considering it in the static economy
would allow an analysis of the insurance gains from automatic tax stabilizers in a heterogeneous
economy while exploiting the log-linearity of the model for closed-form solutions. The further
extension to a dynamic economy with idiosyncratic regional productivity shocks would allow the
comparison of self-insurance by households through financial savings with insurance through
implicit transfers with proportional labor income taxes.
7.
Conclusion
Automatic stabilization using distortionary labor income taxes is influenced by nominal wage
rigidity is analyzed in an economy in which production and consumption behavior are based on
firm and household optimization. A closed-form solution for a special case of economies with
sticky prices and imperfect competition provides one starting point for investigating the efficacy
of tax stabilizers against productivity shocks. The model assumes nominal wage rigidity but
allows nominal final goods prices to fluctuate with productivity disturbances. Progressive labor
income tax rates reduce output volatility raising expected utility in equilibrium with flexible prices
and wages. Nominal wage rigidity has two effects on automatic stabilization using taxes on labor
earnings. Progressive rates of taxation of labor earnings cannot affect output and employment
volatility with rigid nominal wages. However, increasing the progressivity of labor income taxes
still raises welfare with preset nominal wages in the monopolistically competitive economy with
stochastic productivity. With rigid nominal wages, payroll taxes are not equivalent to taxes on
29
labor earnings because the sellers of labor services preset nominal wages. Procyclic payroll
tax rates can reduce output volatility and increase welfare with rigid nominal wages because
final goods prices are not sticky in this model economy. The model presented is simpler than
dynamic models with and without capital so that it can allow a analysis of automatic tax stabilizers
in closed-form solutions. Allowing deficit financing of public expenditures, adding capital or
introducing an interesting role for international interdependence would come at an analytical price.
30
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