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Working Paver 94 10
TAX STRUCTURE, WELFARE, AND THE STABILITY OF EQUILIBRIUM
IN A MODEL OF DYNAMIC OPTIMAL FISCAL POLICY
by Jang-Ting Guo and Kevin J. Lansing
Jang-Ting GUOis a professor of economics at the University
of California, Riverside, and Kevin J. Lansing is an
economist at the Federal Reserve Bank of Cleveland. For
helpful comments, the authors thank David Altig, Costas
Azariadis, Roger Farmer, Gary Hansen, Finn Kydland, and
seminar participants at the 1994 SEDC conference and the
UCLA Tuesday evening Macro Workshop.
Working papers of the Federal Reserve Bank of Cleveland
are preliminary materials circulated to stimulate discussion
and critical comment. The views stated herein are those of
the authors and not necessarily those of the Federal Reserve
Bank of Cleveland or of the Board of Governors of the
Federal Reserve System.
September 1994
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ABSTRACT
This paper shows that the assumed structure of taxation can have dramatic effects on economic
welfare and the stability of the steady state in a dynamic general-equilibrium model of optimal fiscal
policy. Specifically, tax structure refers to the use of separate versus uniform tax rates on labor and
capital income, the level of taxation of firm dividends (single versus double taxation), and the tax
treatment of depreciation. Under each tax structure, the government selects a balanced-budget fiscal
policy (consisting of tax rates and the level of public expenditures) which maximizes the welfare of the
representative household. We find that household welfare is highest under a tax structure that includes
separate tax rates on labor and capital income, double taxation of dividends, and tax deductible
depreciation. Moreover, single taxation of dividends yields an unstable steady state under a structure with
separate tax rates on labor and capital income and tax deductible depreciation. This instability, which
is robust to changes in parameter values, can be removed by implementing various changes to the tax
structure, such as (1) imposing double taxation of dividends, (2) taxing labor and capital income at the
same rate, or (3) eliminating the depreciation allowance. Of these three options, imposing double
taxation of dividends yields the highest welfare.
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1. Introduction
This paper shows that the assumed structure of taxation can have dramatic effects on economic
welfare and the stability of the steady state in a dynamic general-equilibrium model of optimal fiscal
policy. Specifically, tax structure refers to the use of separate versus uniform tax rates on labor and
capital income, the level of taxation of firm dividends (single versus double taxation), and the tax
treatment of depreciation. The framework for the analysis is a deterministic, infinite-horizon
representative household model in which the government solves a dynamic version of the Rarnsey (1927)
optimal tax problem. Public capital serves as a direct input to the firm's neoclassical production function,
which exhibits constant returns to scale in all inputs, public and private. As a result, the firm retains
positive profits equal to public capital's share of output.
Within the constraints defined by each tax structure, we endogenize the choice of fiscal policy
by solving for the optimal tax rates and the optimal levels of public consumption and public investment
that maximize household utility. We consider three distinct aspects of the tax structure. First, we
examine the effects of taxing all types of income at the same rate (a so-called uniform income tax),
versus a structure that allows for separate tax rates on labor and capital income.' Second, we postulate
that profits are initially taxed at the firm level, and study the effects of allowing the government to tax
profits a second time when they are distributed to households in the form of dividends. Third, we
consider the effects of eliminating the tax deductibility of depreciation. The various tax structures are
compared in terms of economic welfare (as measured by steady-state utility), output (as measured by
steady-state GNP), and the local stability of the steady state.
Our primary finding is that household welfare is highest under a tax structure that includes
separate tax rates on labor and capital income, double taxation of dividends, and tax deductible
depreciation. From the perspective of choosing an optimal fiscal policy, a uniform tax structure imposes
an additional constraint on the government's decision problem, namely, that the tax rate on labor income
must be equal to the tax rate on capital income. Because a policy of equal tax rates is available to the
government under a more general tax structure, but is not chosen, we know that the additional constraint
is binding and thus results in a lower level of household utility than in the unconstrained case. From the
'see Guo and Lansing (1994) for a more detailed analysis of this aspect of tax structure in a related model.
1
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standpoint of economic welfare, double taxation of dividends is superior to single taxation because it
allows the government to recapture a larger percentage of public capital's share of output. If a separate
profits tax were available, the government would choose to tax profits at a rate of 100 percent.
Effectively, the tax on profits acts like a user fee for productive services of public capital. Double
taxation improves welfare because it comes closer to the ideal confiscatory rate than does single taxation.
Furthermore, the additional tax revenue is used to finance public consumption and public investment
(both of which provide benefits to households), and permits the use of a lower tax rate on labor income.
Finally, the depreciation allowance improves household welfare because it operates as an implicit
subsidy to capital accumulation, and partially offsets the distortion associated with taxing income from
capital. As we have previously shown in a related paper (see Guo and Lansing [1994]), the welfare cost
of eliminating the depreciation allowance is quite high, over 2 percent of GNP, a fact which highlights
the importance of this tax break in encouraging capital formation.
An additional, rather interesting finding is that single taxation of dividends yields an unstable
long-run equilibrium under a structure which includes separate tax rates on labor and capital income and
tax deductible depreciation. In this case, we find that the Jacobian matrix of the linearized dynamical
system (in the neighborhood of the steady state) displays too many explosive eigenvalues. This indicates
that the long-run equilibrium cannot be characterized by a stable set of stationary policy rules. This
instability, which is robust to changes in parameter values, can be removed by implementing various
changes to the tax structure, such as (1) imposing double taxation of dividends, (2) taxing labor and
\
capital income at the same rate, or (3) eliminating the depreciation allowance. Of these three options,
imposing double taxation of dividends yields the highest level of household welfare.
Our results highlight the importance of taking into account both the structure of the tax system
and the level of tax rates in considering policies designed to improve welfare or stabilize the economy.
Moreover, our results suggest a possible justification for some observed features of the U.S. tax system,
such as the coexistence of the corporate income tax with the practice of subjecting dividends to the
personal income tax. Some previous related research includes Arrow and Kurz (1970), who employ an
optimal growth model with public capital to discuss the theoretical possibility of multiple steady states
exhibiting different stability properties. Cooley and Hansen (1992) evaluate the welfare effects of various
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combinations of exogenous taxes in a neoclassical monetary economy. Finally, Pecorino (1993) and
Stokey and Rebelo (1993) study the effects of tax structure on economic growth in endogenous growth
models with human capital.
The remainder of this paper is organized in the following manner: Sections 2 and 3 describe
the model and the methods we use to compute the steady state and determine the stability of the long-
run stationary equilibrium. The choice of parameter values is discussed in section 4. Section 5 presents
quantitative welfare comparisons based on steady-state analysis. Section 6 concludes.
2. The Model
The model economy consists of many identical, infinitely lived households, identical private
firms, and the government. The government finances expenditures on public consumption goods and
public investment goods by levying distortionary taxes on households and firms. AU goods (public and
private) are produced using a privately owned, Cobb-Douglas technology that exhibits constant returns
to scale in the three productive inputs: labor, private capital, and public capital. The form of the
technology implies that private firms earn an economic profit equal to the difference between the value
of output and payments to private factor inputs. The purpose of introducing profits is to obtain a positive
optimal tax rate on capital, consistent with U.S. observations.' As owners of the firms, households
receive net profits in the form of dividends, but consider them to be outside their control, similar to
wages and interest rates. We assume that the government can distinguish between labor and capital
income, but cannot distinguish between the different categories of capital income, such as profits,
dividends, and capital rental income. Therefore, our model includes only two types of distortionary taxes:
a labor tax and a capital tax.
2.1 The Household's Problem
Households maximize a discounted stream of within-period utility functions over consumption
and leisure, subject to a sequence of budget constraints. The decision problem can be summarized as
2~ones,Manuelli, and Rossi (1993) show that the existence of profits and a restriction on the menu of available tax
instruments (the absence of a separate profits tax) is one method of obtaining a positive optimal tax rate on capital in the steady
state. Without profits, the optimal steady-state tax on capital is zero (see Judd [I9851 and Chamley [1986]).
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-
max
EDf{lnc, - A h ,
+ Blng,)
O<D <1,
A, B 2 0
c,.h,.k,*, t - 0
subject to
In the above equations, J3 is the constant household discount factor and ct represents private
consumption goods. Households are endowed with one unit of time each period and work ht hours during
period t. Household preferences also include a separable term representing the utility provided by public
consumption goods g,. Examples of public consumption goods that might affect household utility are
national defense, police protection, and government provision of food and housing during natural
disasters. Public goods are assumed to be noncongestable and free of specific user charges.
Households maximize the utility function in (1) over c, and h,, but view g, as outside their
control. The form of the within-period utility function has been chosen for tractability. The fact that
utility is linear in hours worked reflects the "indivisible labor" formulation, as described by Rogerson
(1988) and Hansen (1985). The separability in c, and g, implies that public consumption does not affect
the marginal utility of private consumption, a specification supported by parameter estimates in
McGrattan, Rogerson, and Wright (1993). We introduce no uncertainty into this model because our
analysis focuses on steady-state welfare comparisons and the stability properties of a linearized
dynamical system that exhibits the property of certainty equivalence.
Equation (2) represents the period budget constraint of the household. The terms xt and k,
represent gross private investment and private capital, respectively. Households derive income by
supplying labor and capital services to firms at rental rates w, and r, and pay taxes on labor and capital
income at rates 7
, and T,, respectively. An additional source of income is the firms' net profits .ft,,
which are distributed to households as dividends and are taxed at the same rate as capital rental income
rt kt . The term $7, 6kt represents the depreciation allowance, where the parameter $ can be set to either
1 or 0, depending on whether this tax break is included as part of the tax structure. Equation (3) is the
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law of motion for private capital, given a constant rate of depreciation 6. Households view tax rates,
wages, interest rates, and dividends as determined outside their control.
2.2 Household Optimality
The Lagrangian for the household's problem is defined as
The household first-order conditions with respect to the indicated variables and the associated
transversality conditions (TVC) are
TVC:
lim Ptktkt+,= 0.
t +-
The government's problem is solved by finding the set of welfare-maximizing allocations ct , ht
and kt+,,such that the household's first-order conditions ( 5 ) and budget constraint (2) are satisfied. Given
these optimal allocations, the government uses the household equilibrium conditions to recover the
appropriate tax rates z, and z, that will support these allocations in a decentralized e~onomy.~
3 ~ e Chari,
e
Christiano, and Kehoe (1993) for a more complete discussion of this equilibrium concept.
5
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2 3 The Firm's Problem
Output (y,) is produced by identical private firms which seek to maximize after-tax profit,
subject to a technology that exhibits constant returns to scale in the three productive inputs, h,, k, and
k,,, where kGtis the stock of public capital. The f m ' s profits are taxed at the rate
7
.,
The firm's
decision problem can be summarized as follows:
subject to
0
0
0,
Yt = kt ' hz kct
0<Oi<l,
O l + O 2 + e 3 = 1.
In (6), the parameter v controls the level of taxation of firm dividends. When v = 2, dividends
are taxed twice; once at the firm level and again at the household level. When y = 1, dividends are taxed
only once, at the household level. This formulation of double taxation reflects the idea that only a
portion of the total income from capital is taxed twice. Capital rental income (rt kt ) is taxed only once,
at the rate z,
regardless of the value of y.The firm's first-order conditions are4
wt = O,,.
J't
ht
The firm's after-tax profits, distributed to households in the form of dividends, are
2.4 The Government's Problem
The government chooses an optimal program of taxes and public expenditures in order to
maximize the discounted utility of the household. This is a dynamic version of the Ramsey (1927)
optimal tax problem, involving a Stackelberg game between the government and households. To avoid
4
There is no need to distinguish between'variables under the household's control and variables representing "per capita"
quantities here, as must be done when solving directly for a decentralized equilibrium. As pointed out by Lucas and Stokey
(1983), the solution to the government's (centralized) decision problem yields a set of policies which dictate household
equilibrium allocations. These allocations determine the equilibrium prices r, and w,. Thus, prices are not outside the control of
the government, which differs in this respect from an individual household.
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time-consistency problems, we assume that the government can commit to a set of state-contingent,
stationary policy rules announced at time zero. Also, to make the problem interesting, we rule out any
time-zero levies on private-sector assets that might be used to finance all future expenditures. In addition,
we assume that the government adheres to a period-by-period balanced-budget constraint.' With these
assumptions, the government's problem is
subject to
(i)
household first-order conditions and budget constraint,
(ii)
firm profit maximization conditions,
(vi)
z,
-
z,
=
z,
(for the uniform tax structure) .
Constraints (i) and (ii) summarize rational maximizing behavior on the part of private agents and
constitute "implementability" constraints imposed on the government's choice of policy. Constraint (iii)
is the government budget constraint, where the last term on the right-hand side reflects the structure of
dividend taxation. Constraint (iv) is the law of motion for public capital, given a constant rate of
depreciation 6, and gross public investment x,.
Constraint (vi) is a transversality condition on the
accumulation of public capital, where It;,is the marginal utility of public consumption g, . Finally, (vi)
specifies the constraint associated with the uniform income tax. The vector !P,= (x,, , g, , z,,
z,
j
summarizes government policy at time t. The summation of the household budget constraint (2) and the
government budget constraint ( iii) yields the following resource constraint for the economy:
' ~ d d i ngovernment
~
debt to the model introduces complications that we wish to avoid here. Specifically, the perfect
foresight equilibrium for a model with debt and capital imposes an arbitrage condition on the returns from govemment bonds
and private capital. The steady-state level of debt is thus indeterminate (see Chamley [1985]). Furthermore, government debt
introduces another state variable which increases the dimensionality of the dynamical system we intend to analyze.
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Because the resource constraint and the government budget constraint are not independent
equations, equation (1 1) will be used in place of (iii) in solving the government's problem.
3. Solution of the Model
To facilitate computation of the long-run stationary equilibrium, the government's problem
specified in (10) is rewritten as the following infinite-horizon sequence problem:
-
subject to
-
w,h,-L
-,
(for the uniform
s
J=O
tax structure only)
To obtain the formulation in (12), we first substitute the household first-order conditions shown
in (5) into the household budget constraint (2), the resource constraint (1 I), and the household utility
function to eliminate 7, , T, , and c, . The resource constraint (11) is then used to substitute out g, .
Following the solution method of Kydland and Prescott (1980), we define the household lagged-shadow
price h., to be a "pseudo-state variable" for our analysis. Including h,, in the state vector provides a
link to the past by which the policymaker at time t considers the fact that household decisions in earlier
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periods depend on current policy. This is the mechanism by which the commitment assumption is
incorporated into the government's decision problem In (12), the vector of state variables for the
government's problem in period t is st =(kt, kG,, h .,)
and the government's decision variables are 4+,,
k, ,+,
, I , , and h,. The first constraint in (12) is the household budget equation after substituting in the
household first-order conditions. The next constraint imposes the condition of equal tax rates on labor
and capital income for the uniform tax structure. The remaining constraints define the production
technology and the factor prices r, and w,.
The sequence problem in (12) applies for all t > 0. The problem at t = 0 must be considered
separately, as shown by Kydland and Prescott (1980), Lucas and Stokey (1983), and Chamley (1986).
At
t=
0, the stock of private capital is fixed. Optimal fiscal policy thus implies a high initial tax on
capital to take full advantage of this nondistortionary source of revenue. We assume that this form of
lump-sum taxation is insufficient to finance the entire stream of future expenditures. The analysis here
will focus on long-run equilibrium, i.e., when t approaches infinity. Since (12) can be written in the form
of a recursive dynamic programming problem (see Kydland and Prescott [I9801 and Guo and
Lansing [1994]), we confine our attention to stationary equilibria. We do not solve the t = 0 problem or
compute the transition path to the steady state.
,
Given (l2), we obtain first-order conditions with respect to kt+,, kG , I , , h, ,A,, , and, for the
uniform tax structure, h,, where A,, and h, are the Lagrange multipliers associated with the first two
constraints (the remaining constraints having been eliminated by substitution). To compute the steady
state, all time subscripts are dropped from the first-order conditions to form a system of nonlinear
equations in the indicated variables. The system is then solved using a nonlinear equation s01ver.~
Under the more general tax structure (z,#z,),
the first-order conditions from (12) can be
reduced to a system of three second-order, nonlinear difference equations in 4 , k,, , and &-,. In this case,
the first-order condition for h, is used to eliminate A,, from the dynamical system, and the (linearized)
first-order condition for A,, is used to eliminate h, . Similar operations are performed for the uniform tax
structure (z, = z,), except that the dynamics for the state variable &-, are now only first order, and A,,
6 ~ hNLSYS
e
routine in GAUSS version 3.1.4 was used.
9
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is retained as a variable in the dynamical system7 The steady state is then used to construct a log-linear
approximation of the dynamical system under each tax structure. By implementing a transformation of
variables zt = kt+,, ut = k,
,+,, and vt = h, , the system can be rewritten as a set of six first-order linear
difference equations. The difference equations take the following form, where hat (^) variables denote
deviations from steady-state values in logarithms, and J is the 6x6 Jacobian matrix for the linearized
dynamical system:
for the more general tax structure, and
for the uniform tax structure.
Because the government's problem for t > O is recursive, the solutions to the linear systems
shown in (13) and (14) consist of a set of stationary policy rules for k,+, , k,
expressed as log-linear functions of the state variables k,, kct, and h
, and h t which can be
The stability of the stationary
equilibrium is determined by comparing the number of eigenvalues of J located inside the unit circle
to the number of initial conditions (see Farmer [1993]). For this problem, the three predetermined state
variables constitute the three initial conditions. When J has exactly three eigenvalues inside the unit
7 ~ this
n case, the first-order condition for h, is used to eliminate h,,the (linearized) first-order condition for A,, is used to
Thus, A,, remains in the dynamical system.
eliminate h,, and the first-order condition for h, is used to eliminate I,+,.
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circle, the linearized dynarnical system possesses a unique set of stable, stationary policy rules in terms
of k, , k,, , and h
,,.' When J has less than three eigenvalues inside the unit circle, the system explodes.
In this case, the long-run equilibrium cannot be characterized by a stable set of stationary policy rules,
and we label the system as "unstable."
4. Calibration of the Model
To obtain quantitative results from the model, as many parameters as possible are assigned
values in advance on the basis of empirically observed features of postwar U.S. data. Table 1
summarizes the choice of parameter values and is followed by a brief explanation of how they were
selected. The time period in the model is taken to be one year. This is consistent with the frequency of
most government fiscal decisions. Experiments with a quarterly time period (P = 0.99, 6 = 0.02, and
6, = 0.01) produced qualitatively similar results.
Table 1: Parameter Set
Parameters and Values
Agent
Households
= 0.962
Firms
8, = 0.30
8, = 0.60
8, = 0.10
Government
4 = {1,0)
A = 2.60
B = 0.28
6 = 0.08
W = {2, 1)
6, = 0.04
The discount factor P implies an annual rate of time preference of 4 percent. The parameter A
in the household utility function is chosen such that the fraction of time spent working is close to 0.3
at the steady-state. The value of B is chosen to yield a steady-state value of gIGNP near 0.17, the
average ratio for the U.S. economy from 1947 to 1992.9 The selected values of 8, and 8, are in the
range of the estimated shares of GNP received by private capital and labor in the U.S. economy (see
Christian0 [1988]). The output elasticity of public capital, 03, is chosen to yield a steady-state ratio of
' ~ t t= O , the government's problem cannot be represented by
stability of the long-run stationary equilibrium.
(12),so our approach only allows us to characterize the
9 ~ computing
n
this average, public consumption was estimated by subtracting public investment from an annualized series
for government purchases of goods and services (GGEQfrom Citibase). This was done to reduce double counting, since the series
does not distinguish between government consumption and investment goods.
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public investment to GNP (x,/GNP) near 0.05, based on the U.S. average from 1947 to 1992. The
private capital depreciation rate 6 is consistent with values commonly used in the real business-cycle
literature. The public capital depreciation rate, 6, , was estimatedby regressing the linear law of motion
on annual data for k,, and x,, (see Lansing [1994]).
5. Steady-State Welfare and Stability
Based on our choice of parameter values, tables 2a and 2b show the steady-state values of key
model variables for each of the various tax structures. In terms of maximizing steady-state utility and
output, the "optimum" tax structure can be found in the first column of table 2a. This tax structure
includes double taxation of dividends, tax deductible depreciation, and separate tax rates on labor and
capital income. A few general observations about the effects of changes to the tax structure can be made.
First, holding other aspects of the tax structure constant, the imposition of double taxation of dividends
always causes steady-state utility and output to increase. Second, any tax structure that includes a
depreciation allowance yields higher levels of steady-state utility and output than any structure without
this tax break. Third, a uniform income tax does almost as well as a structure with separate tax rates on
labor and capital income. This last observation is consistent with the results of Guo and Lansing (1994),
who employ a model in which profits are introduced by way of monopolistic competition rather than
by way of productive public capital.
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Table 2a: Steady-State Comparison of Tax Structures
Double Taxation of Dividends (y = 2)
With Depreciation Allowance ($ = 1)
Variable
%A
+ Tk
Utility
-2.767
GNP = y
0.464
zh=zk=z
No Depreciation Allowance ($ = 0)
%A
+ Tk
zh=zk=z
Tax Rates
Table 2b: Steady-State Comparison of Tax Structures
Single Taxation of Dividends (y= 1)
With Depreciation Allowance ($ = 1)
Variable
Utility
%A
+ zk
-2.792
No Depreciation Allowance ($ = 0)
'q,=zk=z
Th + Tk
zh=zk=z
-2.793
-2.816
-2.829
0.439
0.423
0.421
GNP = y
0.442
...............................................
-..............................................
-..............................................
-..............................................
-...............................................
c
Tax Rates
0.268
0.264
0.258
0.265
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Table 3 summarizes the welfare and output losses for each of the various tax structures, where
the optimum tax structure noted above is used as the baseline for comparison. These figures show that
tax structure can have dramatic effects on economic welfare and output, even when the government does
its best to maximize the well-being of the household. Starting from the optimum tax structure, a change
from double taxation of dividends to single taxation reduces household welfare by over 1.5 percent of
GNP, which translates to an annual loss of $390 per person in 1993." Double taxation of dividends is
superior to single taxation because it allows the government to recapture a larger percentage of public
capital's share of output. If a separate profits tax were available, the government would choose to tax
profits at a rate of 100 percent (see Jones, Manuelli, and Rossi [1993]). Effectively, the tax on profits
acts like a user fee for the productive services of public capital. Double taxation is welfare improving
because it comes closer to the ideal confiscatory rate than does single taxation. Furthermore, the
additional tax revenue is used to finance public consumption and public investment, and permits the use
of a lower tax rate on labor income.
Table 3 also shows that, starting from the optimum tax structure, eliminating the depreciation
allowance will reduce household welfare by over 2 percent of GNP. (This change corresponds to a
movement across the first row of table 3a to end up in the third column of the table.) Eliminating the
depreciation allowance reduces household welfare because it operates as an implicit subsidy to capital
accumulation and partially offsets the distortion associated with taxing income from capital. This occurs
even though marginal tax rates on capital income are lower in a structure with no depreciation
allowance. Because the govemment wishes to tax profits as much as possible, a high tax rate on capital
combined with a full depreciation allowance is superior to a low tax rate on capital combined with no
depreciation allowance." It is interesting to compare our finding that the depreciation allowance has a
large welfare effect to that of Pecorino (1993), who finds that the tax deductibility of depreciation has
no effect on the maximum steady-state growth rate in an endogenous growth model.
'O~hisnumber is based on a nominal GNP of $6,510 billion and total U.S.population of 258.2 million in 1993.
"when profits are zero (1-8, -8,=0) and the government can tax labor and capital income separately, eliminating the
depreciation allowance has no effect whatsoever on the steady-state allocations. This can be seen from the government's problem
in (12) for the generalized tax structure (z,#z,); note that the depreciation allowance parameter $I only appears in the profit
term of the (transformed) household budget constraint.
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Finally, table 3 summarizes the stability properties of the long-run stationary equilibrium, as
determined by the method described in section 3. In the first column of table 3b, we see that the steady
state is unstable under a structure with single taxation of dividends, tax deductible depreciation, and
separate tax rates on labor and capital income. In this case, the Jacobian matrix of the linearized
dynarnical system displays too many explosive eigenvalues (there are four eigenvalues located outside
the unit circle and only two inside). This indicates that the solution to the (approximate version of) the
government's problem (12) cannot be characterized by a stable set of stationary policy rules. Our
experiments indicate that this instability is robust to changes in the following parameters:
P, A, B, 8,,
8,. 8,. 6, and 6,. However, table 4 shows that the instability can be removed by implementing various
changes to the tax structure, such as (1) imposing double taxation of dividends, (2) imposing a uniform
income tax, or (3) eliminating the depreciation allowance. Of these three options, only one, imposing
double taxation of dividends, causes household welfare to increase. Schematically, this option
corresponds to a movement from the first column of table 3b, directly upward, to the first column of
table 3a, yielding the optimum tax structure noted earlier.
The high dimensionality of the dynamical system makes it difficult to pinpoint the intuition for
the unstable tax structure. However, it appears that the instability is due in some way to the presence
of public capital as an endogenous state variable. With productive public capital, the dynamical system
is characterized by a 6x6 Jacobian matrix. We have experimented with an otherwise similar model
without public capital (see Guo and Lansing [1994]), which is described by a 4x4 Jacobian matrix, and
found no unstable tax structures. We have also experimented with the use of noninteger values for the
tax structure parameters
v and $. Starting from the unstable tax structure, we found that values of
v2 1.19 or $10.11 remove the instability. Thus, a stable tax structure can be obtained by taxing
dividends only slightly higher than single taxation or by allowing only a small fraction of depreciation
expenses to be tax deductible.
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Table 3a: Welfare and Output Losses and Long-Run Stability
Double Taxation of Dividends (y = 2)
With Depreciation Allowance (I$ = 1)
5
f
Tk
No Depreciation Allowance (I$ = 0)
'q,='ck='t
=h # =k
5=2,=2
Welfare Loss
AU /(A yIa
baseline
Output Loss
AYIY
baseline
1.590 %
6.198 %
5.915 %
Long-Run Stability
stable
stable
stable
stable
2, f 2,).
aAU and Ay are normalized using the steady-state values of X and y from the baseline tax structure (y = 2, $I = 1, and
where 1 is the marginal utility of private consumption (to convert AU into consumption units) and y is GNP.
Table 3b: Welfare and Output Losses and Long-Run Stability
Single Taxation of Dividends (W = 1)
With Depreciation Allowance (I$ = 1)
No Depreciation Allowance (I$ = 0)
Welfare Loss
AU l@Y)
Output Loss
AYIY
Long-Run Stability
unstable
stable
stable
stable
Table 4: Policy Options to Remove Long-Run Instability
Policy Change to Tax Structure
Impose Double Tax of
Dividends
Impose Uniform Tax
Structure
Eliminate Depreciation
Allowance
Change in Welfare
AU ~ ( A Y ) ~
2, #
aAU and Ay are normalized using the steady-state values of h and y from the unstable tax structure (v = 1, Q, = 1, and
T,), where X is the marginal utility of private consumption (to convert AU into consumption units) and y is GNP.
16
clevelandfed.org/research/workpaper/index.cfm
6. Concluding Remarks
We have examined the welfare implications of some basic features of the U.S. tax code, namely,
the double taxation of dividends, the tax deductibility of depreciation, and the practice of taxing labor
income differently from capital income. We find that all three of these features are desirable from the
standpoint of maximizing the welfare of the representative household. Moreover, we find that movements
away from this optimum tax structure can not only reduce economic welfare and output, but can also
produce unstable dynamics in our model economy. While our model is admittedly a very abstract and
simplified representation of the vastly complex U.S. tax code, we believe that models of this type can
be useful for examining key questions about the institutional structure of our tax system. In our view,
the strength of this model lies in its ability to capture the general-equilibrium effects of endogenous
fiscal policy. In our future research, we plan to address other features of the U.S. tax structure, such as
the effects of tax progressivity in an economy with heterogeneous agents.
clevelandfed.org/research/workpaper/index.cfm
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