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Working Pauer 9308
DYNAMIC OPTIMAL FISCAL AND MONETARY
POLICY IN A BUSINESS CYCLE MODEL WITH
INCOME REDISTRIBUTION
by Kevin J. Lansing
Kevin J. Lansing is an economist at the Federal
Reserve Bank of Cleveland. The author thanks
David Altig, Costas Azariadis, Anton Braun,
Rajeev Dhawan, Eric Engen, Bruce Fallick, Roger
Farmer, William Gale, Jang Ting Guo, Gary Hansen,
Finn Kydland, Axel Leijonhufvud, and Rodi
Manuelli for helpful discussions and comments.
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 author and
not necessarily those of the Federal Reserve Bank
of Cleveland or of the Board of Governors of the
Federal Reserve System.
November 1993
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ABSTRACT
An optimal program of distortionary taxes, money growth, and borrowing to finance a
stream of expenditures is computed in a monetary real business cycle model for which
distribution issues between the rich and poor play a fundamental role in policy decisions.
Specifically, a simple feedback rule links public spending on goods and services to a
measure of income inequality, and the government is required to provide poor households
with some minimum level of transfers. The stationary equilibrium policy displays positive
capital taxation, progressive labor taxes, and moderate (6 percent) inflation. The capital
tax and the inflation tax fluctuate over time to absorb budget shocks, while the labor tax
remains relatively constant. Model simulations compare favorably in many respects with
postwar U.S. time series on tax rates, money growth, and aggregate business cycle
variables. The solution method employs the recursive algorithm developed by Kydland
and Prescott (1980) to compute optimal policy rules under the assumption of
commitment.
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1. Introduction
Regardless of one's views on whether government should be involved in the business of
redistributing income, it seems clear that the complicated U.S. system of taxes and public spending
programs has been designed, in large measure, with this objective in mind. Policy debates are often
driven by arguments for a more equitable distribution of income, as in this example:
Senate Majority Leader George Mitchell insisted from the beginning that the
wealthiest taxpayers pick up more of the tab... Now, the nearly instant
availability of the distribution tables casts every offer and counter-offer in terms
of fairness between the rich and the poor. Every plan involving a cut in the
capital gains tax invariably showed a windfall for the rich.'
The distribution of wealth and income in the United States is highly skewed, with the top 20
percent of households owning about 80 percent of the wealth and earning about 42 percent of pre-tax
income.' In this environment, policymakers and the public have come to view the capital gains tax as
being paid primarily by the wealthy. This tax and another capital-type tax, the corporate income tax, are
frequently singled out by policymakers as tools for achieving more equity in the U.S. economy. The
government has also developed other redistributive tools, including our system of progressive marginal
tax rates and a myriad of means-tested assistance programs, commonly known as elfar are."^
In this paper, I formulate a model of dynamic optimal fiscal and monetary policy that
incorporates, in a simple way, the government's use of redistributive tools .like the capital tax,
progressive labor taxation, and means-tested transfers. I then subject the model to the same kind of
quantitative comparisons with U.S. data that have been widely used in the real business cycle literature.
As a way of approximating the skewed distribution of U.S. wealth and income, capital ownership in the
'see A. Murray and J. Calrnes, "How the Democrats, with Rare Cunning, Won the Budget War," The Wall Breet Journal,
November 15, 1990.
'see McDermed, Clark, and M e n (1989), figures 13.1 and 13.2, and Rosen (1992), table 8.1. The measure of wealth inequality
cited here is based on net worth from 1983 household survey data. This measure remained approximately constant from 1962 to 1983.
3 ~ hprincipal
e
means-tested transfer programs used to supplement the earnings of the poor are Aid to Families with Dependent
Children (AFDC), Supplemental Security Income (SSI), and the Earned Income Tax Credit (EITC). There are also in-kind transfer
programs such as housing assistance, food stamps, job training, and Medicaid. Social insurance programs may be viewed as implicit
transfer programs. The three major social insurance programs are Social Security, Medicare, and unemployment insurance. For more
details, see Economic Report of the President 1992, chapter 4.
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model is concentrated in the hands of a single group, labeled "rich" households. The govemment solves
a dynamic version of the Ramsey (1927) optimal tax problem, in which a policymaker chooses a
program of distortionary taxes over time to finance a required stream of spending. Monetary policy is
incorporated by viewing inflation as an effective tax on real money balances.
A crucial aspect of the model is the manner in which government outlays are determined. In
particular, I assume that a simple feedback rule links public spending on goods and services to a
measure of income inequality and, further, that the government must provide the poor with some
minimum level of transfers. The transfer payments are a proxy for the various means-tested assistance
programs in the U.S. economy. However, the infinite-horizon framework abstracts from any life-cycle
effects of specific transfer programs, like Social Security. The endogenous policy variables are the tax
rate on capital income, tax rates on labor income (for the rich and poor), and the growth rate of the
nominal money stock. For simplicity, the steady-state level of govemment debt is taken to be exogenous.
The government's problem is solved using a numerical recursive algorithm based on a method developed
by Kydland and Prescott (1980). Specifically, a "pseudo state variable" is defined that permits the use
of dynamic programming to compute optimal policy rules under the assumption of commitment.
A primary finding is that equilibrium policy displays positive capital taxation, progressive labor
taxes (in the sense that the rich are taxed at a higher marginal rate than the poor), and moderate (6
percent) inflation. In simulations, the capital tax and the inflation tax fluctuate over time to absorb
budget shocks, while the labor tax remains relatively constant. As previously identified by Judd (1989)
and Chari, Christiano, and Kehoe (1991)- the fact that household savings in the form of capital or money
balances is inelastic in the short run suggests that state-contingent taxes on these assets can serve as
nondistortionary shock absorbers. Budget shocks in the model are caused by changes in the size of the
tax base (due to business cycle fluctuations) or by changes in exogenous spending requirements.
Predictions for the moments of aggregate economic variables are very close to those of previous
monetary real business cycle models. This result is reassuring because it suggests that these models can
be extended into new areas without sacrificing a reasonable description of the aggregate economy.
Another finding is that some predictions of partial-equilibrium models that have been used in
the past as empirical tests for optimal government behavior are not implied by this general-equilibrium
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model. In simulations, the labor tax is negatively correlated with inflation (or money growth) while the
capital tax is positively correlated with inflation. Partial-equilibrium models generally do not distinguish
between labor and capital taxes and predict a positive correlation between an "income tax" and inflation.
This failure to distinguish between factor incomes may help to explain the inconsistent findings of
previous empirical studies designed to test the partial-equilibrium result (see Mankiw [1987], Roubini
and Sachs [1989], Poterba and Rotemberg [1990], and Roubini [1991]).
Within the infinite-horizon growth framework, models of dynamic optimal fiscal policy have
been applied to the study of heterogeneous-agent economies by Judd (1985), Aiyagari and Peled (199 I),
and Alesina and Rodrik (1991). This paper attempts to go further by bringing in monetary policy and
by directly examining the quantitative implications of the model in comparison to U.S. data." A wellknown result that applies to infinite-horizon growth models is that the optimal steady-state tax on capital
is zero.' Moreover, Judd (1985) has shown that this result holds regardless of the weights placed on
different groups in a social welfare function, even when one group holds the entire stock of physical
capital. This seemingly counterintuitive finding obtains because a zero tax on capital leads to higher
levels of capital accumulation and hence higher wages, thus benefiting all individuals, not just capital
owners. However, variations in the structure of the standard model can overturn the optirnality of a zero
tax rate on capital, for example, when certain kinds of externalities or constraints are present or when
the government faces restrictions on the menu of available policy instruments. Arrow and K u n (1970),
Thompson (1979), Stiglitz (1987), Aiyagari and Peled (1991), and Jones, Manuelli, and Rossi (1992)
all provide examples of such cases.
~
In this paper, I assume that income inequality generates negative externalities that ultimately lead
I
to a drain on productive resources in the form of higher public spending needs. The government's desire
4~taticmodels of optimal redistributive taxation can be found in Fair (1971), Mirlees (1971), and Stiglitz (1987). A dynamic
model with futed saving propensities is developed by Pestieau and Possen (1978). The overlapping-generationsmodels of Atkinson
and Sandmo (1980) and Stiglitz (1987) examine aspects of intergenerationalredistribution. This paper deals with redistribution among
rich and poor households of the same generation. The assumption of an operative bequest motive among finitely lived households
allows the economy to be modeled using an infinite-horizon framework.
he intuition for this result is that the long-run supply elasticity of capital is essentially infmite. Recall that the Ramsey principle
of optimal taxation states that taxes should be set in inverse proporlion to the elasticity of the tax base. The zero tax result is discussed
by Arrow and Kun. (1970), pp. 195-203, and has been further elabomted on by Judd (1985) and Chamley (1986).
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to combat these externalities motivates a positive tax on capital in the steady state. This simple
formulation in which feedback from the economy affects government spending policy introduces a role
for the capital tax in reducing income inequality. Support for this idea can be found in the urban
development literature, where studies of urban decline often refer to the spillover effects of distributional
inequality, such as increased crime, family disintegration, urban population loss, deteriorating
neighborhoods, and low-quality school^.^ In addition to other harmful consequences, these spillovers
contribute to a drop in income and property tax bases in urban areas, forcing cities to rely increasingly
on aid from federal and state governments to provide basic public s e r ~ i c e s .On
~ the empirical side, some
recent cross-country studies suggest that government spending and economic growth are both linked to
inequality. Easterly and Rebelo (1993) report a statistically significant, positive correlation between
income inequality and public spending on education, transportation, and communication, while Persson
and Tabellini (1991), Alesina and Rodrik (1991), and Perotti (1992) find that income inequality has a
detrimental effect on economic growth. All of these studies are consistent with a model of political
equilibrium in which an increase in inequality causes the median voter to support higher levels of
government redistribution expenditures and higher taxes on capital.
In an economy with distortionary taxes, the optimal rate of inflation can depend crucially on the
specific structure of the model. A common result is that the optimal rate of inflation is negative in steady
state, in agreement with the Friedrnan (1969) optimal money rule. Under this policy, the government
reduces the size of the nominal money stock at the rate of time preference, thereby achieving a zero
nominal interest rate. However, the presence of externalities or the use of alternative functional forms
(for household utility or transaction cost functions) can yield optimal rates of inflation that are po~itive.~
Here, I assume that the government is constrained to provide poor households with some
minimum level of transfers. This constraint can be viewed as a way of reflecting that welfare programs
represent a "safety net" for the poor that cannot be reduced below some baseline amount. Alternatively,
'see,
for example, Bateman and Hochman (1971), Bradbury, Downs, and Small (1982), and Mieszkowski and Mills (1993).
7 ~ r o m1950 to 1988, federal and state aid to cities averaged nearly 40 percent of local revenues. See Rosen (1992), table 21.4.
'see
Chari, Christiano, and Kehoe (1991), Braun (1993), and Croushore (1993) for further discussion.
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political factors may render these programs "untouchable" in the eyes of many policymakers. The effect
of the constraint is to introduce an externality into the government's decision problem that can motivate
the use of a positive inflation tax. Since transfer payments are not taxed but must be financed by
distorting taxes, it is efficient for the government to spread the distortionary costs across various tax
bases, including consumption. The inflation tax operates as a consumption tax in this model because a
subset of consumption goods (known as "cash goods") can only be acquired with previously accumulated
cash balances. A steady-state annual inflation rate of 6 percent is obtained when model transfers are
calibrated to match the average level of U.S. means-tested transfers (approximately 2 percent of GNP).
The remainder of the paper is organized as follows. Sections 2 and 3 describe the model and the
recursive solution method. Section 4 describes the choice of parameter values. Section 5 presents
quantitative results from steady-state and dynamic experiments, and section 6 concludes.
2. The Model
The model economy consists of two types of infinitely lived households, identical competitive
firms, and the government. The total number of households is normalized to one so that y and 1-y
represent the fraction of poor and rich households, respectively, where O< y <l. The poor's discount
'factor (pP) is assumed to be less than that of the rich (pR). This implies that the poor are more
impatient than the rich and thus would rather borrow than save at the steady-state real interest rate.9 In
equilibrium, ownership of the capital stock is concentrated in the hands of the rich. Although the poor
would like to borrow to increase their current consumption, it is assumed that they are prevented from
doing so. A rationale for the borrowing constraints is that the rich are unwilling to make loans to the
poor because of difficulties in enforcing repayment. The condition pP<pRensures that the poor will
choose not to save for stochastic equilibria of the type examined here, involving sufficiently small
fluctuations around the steady state.10p11The assumption regarding discount factors is consistent with
he discount factor flis inversely related to the marginal rate of time preference p, according to P=(l+p)-'.
'In a perfect foresight model, Baker (1980) shows that concentrated ownership of capital results when one household type is
more patient than other types. Woodford (1988) applies this result to a stochastic economy. When all households have the same
discount factor, Baker shows that the steady-state distribution of capital across households is indeterminate, i.e.. long-run equilibrium
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some recent empirical evidence suggesting that individuals with lower income and lower education tend
to be more impatient (see Lawrance [1987, 19911). Further distinction between households is made by
assuming that the labor input of the poor is less productive than that of the rich, based on the idea that
the poor have less human capital, perhaps due to less education. Rather than explicitly modeling the
accumulation of human capital, I introduce an exogenous labor efficiency parameter (&)for the poor.
This factor represents the ratio of the poor's productive labor input to their number of hours worked,
where O< & <I. In equilibrium, this factor produces a wage differential between the rich and poor.
2.1 The Household's Problem
Household behavior is described in the context of a "cash-in-advance" economy similar to that
of Cooley and Hansen (1992). Household income is either allocated to consumption, held in the form
of money, or, in the case of rich households, invested in the formation of productive capital. To ensure
that money is held in equilibrium, households are assumed to be subject to a cash-in-advance constraint
of the type described by Lucas and Stokey (1983, 1987). A particular type of consumption goods known
as cash goods can only be acquired with previously accumulated cash balances. Purchases of the other
type, known as credit goods, can be financed out of current income. Household preferences are
summarized by the following utility function:
Superscripts P and R are used throughout the discussion to denote quantities held by poor and
rich households, respectively. In (I), c,,' represents purchases of cash goods and c,'
is purchases of
credit goods. Households are endowed with one unit of time each period and work h,' hours (i=P, R)
is consistent with any wealth distribution. If the discount factor is endogenous (rather than constant), the steady state determines a
unique wealth distribution (see Epstein and Hynes [1983]). AppendixA describes how PP< PR yields concentrated capital ownership.
o or row in^
constraints are necessary to ensure a positive level of steady-state consumption for the poor. Otherwise, poor
households would borrow up to the present value of their income stream to finance current consumption. Here, no borrowing is
allowed. An alternative would be to allow borrowing up to some credit limit that would always be binding.
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during period t. The symbol E, is the expectation operator conditional on information available at time
t. The form of the within-period utility function has been chosen for tractability and for comparability
with previous business cycle literature. The coefficient of relative risk aversion for consumption is
constant and equal to one for this function.
The fact that utility is linear in hours worked reflects "indivisible labor" as described by
Rogerson (1988) and Hansen (1985). This means that all variation in economywide hours worked is due
to variations in the number of employed workers as opposed to variations in hours per worker. In a
decentralized economy, these authors show that the utility function in (1) can be supported by a lottery
that randomly assigns workers to employment or unemployment each period, with the firm providing
full unemployment insurance. Wage contracts call for households to be paid based on their expected,
as opposed to actual, number of hours worked. Real business cycle models with indivisible labor are
better able to match some key characteristics of aggregate labor market data. Specifically, U.S. data
display a large volatility of hours worked relative to labor productivity and a weakly positive or even
slightly negative correlation between hours and productivity.12
The weights a and A in the utility functions are assumed to be equal for both the rich and poor.
Households maximize the utility function in (1) over consumption and leisure, subject to the following
sequence of budget constraints:
Cash-in-advance constraint:.
Budget constraint of poor households:
P
P
cl,
p
+ c,, +
mt+1
5 1 - 7 ,) 8 w
P
h
p
PI
+
mtp
+ TR, ,
p,
'see Hansen (1985), Christiano and Eichenbaum (1992), and Hansen and Wright (1992).
7
0<8<1.
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Budget constraint of rich households:
Equation (2) represents the cash-in-advance constraint faced by all households. A requirement
that this constraint hold with equality is imposed in the computation of government policy.13 Equations
(3) and (4) are the within-period budget constraints of households. Poor households choose not to hold
physical capital due to the assumption pP<pR. In the above constraints, ml,+,represents nominal money
balances carried into the next period, and P, is the price level in period t. Poor households are assumed
to qualify for lump-sum government transfer payments TR, . The timing of transfer payments is such that
they supplement wage earnings and thus do not enter the cash-in-advance constraint. The term TR, is
intended to summarize, in an approximate way, the many forms of means-tested assistance provided by
the government. Although explicit modeling of specific assistance programs is possible (such as the
EITC), the basic nature of the results depends not on the form of the transfers, but rather on the
assumption that some portion is exogenous and cannot be directly taxed by the government.
The terms x, and k, represent the rich household's investment and capital stock, respectively, and
r, is the rental rate on capital. Since the poor have a lower labor efficiency, their equilibrium wage rate
(Jw,) is lower than that of the rich (w,). The tax rate on labor income ( T ~ ' is
) allowed to vary between
household types. Notice that marginal tax rates are the same as average tax rates here, but rich and poor
households face different linear tax schedules. This is an approximation of the nonlinear, progressive
U.S. tax system, where marginal tax rates generally exceed average tax rates. Rich households pay taxes
on capital income at the rate z,
and 7, 6k, is the depreciation allowance built into the U.S. tax code.
In equation (4), b,,, represents real, one-period government bonds carried into period t+l by the rich.
1 3 ~ o the
r constraint to be binding, the condition aUBc,,> aUBc, must hold in expected terms. This implies nonsatiation for real
money balances. With logarithmic utility, this condition will be satisfied when E, [l/(l+p,+,)]> 1/P, where p,+,is next period's money
growth rate. Since P is generally less than one (P=0.99 is a typical value for quarterly data), the constraint will bind whenever the
expected money growth rate is positive, and even for negative expected growth rates that are sufficiently small. The Friedman (1969)
optimal money rule is p=P-1 in steady state, which yields a nominal interest rate of zero.
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These bonds can be viewed as being indexed against inflation so as to pay principal and interest of
b, (l+r,,) in real terms during period t.I4 Equation (5) is the law of motion for capital, given a constant
rate of depreciation 6. Households view tax rates, transfer payments, wages, and interest rates as being
determined outside the& control. The household decision variables in period t are c,,', car, h,', m ',+,,kt+,,
and b,+, , i = P , R.
2.2 The Firm's Problem
Output (Y, ) is produced by identical competitive firms using a constant-return-to-scale
technology. Since profits are zero in equilibrium, there is no need to model a market governing the
ownership of firms. The production technology is subjected to serially correlated exogenous shocks ( 2 , )
that are revealed to agents at the beginning of period t. These shocks produce equilibrium business cycle
fluctuations in the model. The firm's technology can be described as follows:
Z1+,
= PzZI+El+'
9
O<pz<l,
~ , + ~ - i i d ( ~ , z,) ,given.
In (6), H, and Kt are the economywide labor and capital inputs.15 The (per capita) labor input
of the poor is e^ hr. The f i ' s problem is static because it simply rents capital and labor services from
households each period, with the objective of maximizing profits: Y, -r, K, -w,HI. The firm's first-order
1 4 ~ hinflation
e
tax operates only on real money balances in this model. The assumption of indexed debt, though counterfactual,
is needed to pin down an equilibrium policy in a model with money, capital, and government bonds. I will come back to this point
when I describe the government's decision problem.
15~here
is no need to distinguish between variables under the household's control and variables representing per capita quantities,
as is necessary when solving directly for a decentralized, competitive equilibrium. As noted by Lucas and Stokey (1983). solution of
the government's decision problem yields a set of policies that dictate household equilibrium allocations. These allocations determine
the equilibrium prices r , , w , , and PI.Thus, prices are not outside the government's control as they are for households.
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conditions yield the following expressions for the rental rate on capital and the real wage:
To facilitate solving for an equilibrium, a transformation of variables is performed to render the
household problems stationary. Following Cooley and Hansen (1992), the transformation is -defined as
The term MI is the economywide nominal money stock that evolves according to MI+,= (l+h)Ml,
where h is the growth rate observed at time t. The government achieves the desired level of h by
injecting new money into the economy through open-market operations. Since households use identical
currency, h must be the same for all households, unlike tax rates, which may differ between types. In
equilibrium, the economywide money stock is the sum of household money stocks: MI = y m r + (1-y)m;.
In transformed variables, the equilibrium condition is 1= y f i r + (I-y)fi;.
23 Household Optimality
As a preliminary step to obtaining the conditions for household optimality, the variable
transformations in (8) are applied to equations (1)-(5) and the cash-in-advance constraint is imposed with
equality. This procedure yields the following Lagrangians for households:
Poor households:
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Rich households:
The household first-order conditions with respect to the indicated variables and the associated
transversality conditions are
t
2.4 The Government's Problem
The government chooses an optimal program of distortionary taxes, money growth, and
borrowing to finance a stream of expenditures and transfers. The problem is a dynamic version of the
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classic Ramsey case, involving a Stackelberg game between the government and household^.'^ To avoid
time-consistency problems, I assume that the govemment can commit to a set of state-contingent policy
rules announced at time zero. Also, to make the problem interesting, lump-sum taxes are ruled out.
Otherwise, the government would elect to finance all future expenditures with an initial levy on
household assets. With these assumptions, the government's problem can be summarized as follows:
,
.
P
max E ~ ~ ( P ~ ) ~P (~2~~
w hl
( P c ~R ~ ~ ,R 2 hlR))v
1 ~
,
1=O
subject to:
(i)
household first-order conditions and budget constraints
(ii)
firm profit-maximization conditions
(iii) gl +yTRl + (1-y )bl(l +r,,) = y [.r:6w,hlP]
(iv)
M,+, = ( 1 +p,)Ml,
(v)
TRl 2
(vi)
lin~
+
(l-~)[.r:w~h.~+ .r,,(rl-6)kl]
where Ml = y mlP+ (1 -y lmtR
FR > 0
(1 -Y )b,
= 0.
The government employs discount factor
objective functions, W(.). The choice of
PG
in maximizing a sequence of within-period
PG and W(. ) will be discussed shortly. Constraints (i) and
(ii) summarize rational maximizing behavior on the part of private agents and constitute
"implementability" constraints on the government's choice of policy. Constraint (iii) is the government
budget constraint, where the term p&4, lPl ( = p, lp, ) represents seigniorage. Constraint (iv) describes the
1 6 ~ h edynamic Ramsey problem in a representative-agent framework has been studied by numerous authors. A partial list
employing general-equilibrium models with either money or capital is Helpman and Sadka (1979), Kydland and Prescott (1980),
Turnovsky and Brock (1980), Lucas and Stokey (1983), Chamley (1986), Lucas (1990), Jones, Manuelli, and Rossi (1992), Zhu
(1992), Chari, Christiano, and Kehoe (1991),and Braun (1993).Models with both money and capital are analyzed by Drazen (1979)
and Chamley (1985a).
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evolution of the economywide nominal money stock. Constraint (v) is the assumption that the
government must provide poor households with a minimum level of transfer payments,
TR. In
equilibrium, this constraint is always binding such that TR, =TR for all t.17 Finally, (vi) is a
transversality condition that ensures the government budget constraint is satisfied in present-value terms.
Since the model includes heterogeneous households, the question arises as to what form the
government objective function should take. In the words of Arrow and KUR (1970), "no definitive
criterion can be given that will withstand all criticism." I suggest, therefore, an analytically tractable
version reflecting the basic premise that the government cares about household welfare, where welfare
is measured by some concave function of household consumption and leisure. Within this class, the
qualitative behavior of the model is robust to the choice of a specific function. To minimize introduction
of new parameters, the function W(-) is assumed to be an additively separable combination of
'(-)
household within-period utility functions, W(. ) = $17
+ U R(.).
This choice implies that the
government respects household valuation of utility within a given period. The parameter $ > 0 controls
how much the policymaker favors one group over the other. For example, the weight assigned to the
welfare of a given group may exceed the value implied by the group's relative number in the population.
The govemment's discount factor
PG need not coincide with household discount factors. The
case in which public and private discount factors differ is termed fiturity divergence by Arrow and KUR
(1970). To support an equilibrium with public debt, however, I assume that the govemment's discount
factor coincides with that of rich households (pG=pR).I thus attribute the government with having more
patience than the poor, an assumption that seems quite' reasonable considering the existence of
government-mandated savings programs like Social Security, long-term investments in public
infrastructure, and the government's willingness to subsidize activities that build human capital, such
as job training and basic education.''
The choice of
pG=pRand the form of W(.)
yield a government
17~emovingthe constraint essentially provides the government with a nondistortionary tax instrument. In this case, the
equilibrium value of TR, is highly negative, which indicates that the government would like to impose a large lump-sum tax on the
poor, allowing other distortionary taxes to be lowered. Lump-sum taxes have been ruled out to focus on a "second-best" equilibrium.
'*If the government is myopic relative to savers (pG<pR),an equilibrium with government debt can be restored by introducing
an exogenous limit on public borrowing (which would always be binding) or by imposing an exogenous cost of borrowing (which
increases with the level of debt). The assumption pG=pRavoids these additional complications, but is not crucial for the basic results.
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objective that represents a transformed version of a standard utilitarian social welfare function, where
the transformation places more weight on the future utility of the poor.19
A simple feedback rule is assumed to link government spending on goods and services (g,) to
a measure of income inequality in the economy. The law of motion for g, is
In (13), government spending consists of both exogenous and endogenous outlays, where gand
q are positive constants. Exogenous spending follows a stationary stochastic process subject to serially
correlated shocks (v, ) that are revealed to agents at the beginning of period t. Endogenous spending is
assumed to depend linearly on the difference in income between households of each type, where income
is measured by funds available for consumption after taxes are paid and transfer payments are re~eived.~'
The parameter q is the semi-elasticity of government spending with respect to income inequality. The
linear form simplifies computations, but is not crucial for any results. For all parameter values examined,
endogenous spending is positive. As a further simplification, households derive no direct utility from
government spending; it is simply a drain on productive resources.
The vector ?=IT,
z,, K } summarizes government policy implemented at time t.
The interest
rate on government bonds (r,,) is not an independent policy variable in this model. This restriction,
combined with the assumption that debt is indexed against inflation, is necessary to pin down a unique
policy in equilibrium. As Zhu (1992) and Chari, Christiano, and Kehoe (1991) have shown, allowing
Oudiz and Sachs (1985) examine a structural macroeconomic model with a myopic government. Arrow and Kurz (1970), Atkinson
and Sandmo (1980), and Stiglitz (1987) consider the opposite case where government is less myopic than households.
190ne version of a social welfare function is SWF(-)=yx(PP)'U r + (I-y)x(pR)'U:. The government's objective here is
C(pR)' ( o u r + U P ) , which allows for "imperfect altruism." Perfect altruism would imply @=[y/(l-y)](pP/pR)'.
When p<pR, the
value of $ defined in this way approaches zero as r increases. By restricting I$= constant > 0 , the government's objective represents
a transformation of SWF(-).
20~trictlyspeaking, the funds available for consumption in (13) should also include the after-tax interest payments on government
debt held by rich households. For the levels of government debt examined in this model, the additional income is small and is
neglected to avoid complication.
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the government to choose 2 , and r,, independently results in a fundamental indeterminacy between the
equilibrium values of these two variables. To see why, note that the household fmt-order conditions for
kt+,and b,,, in (1 1) represent an ex ante arbitrage condition on expected returns from bonds and capital.
Ex post, after shocks to the economy are revealed, the government is free to alter the combination of
2,
and r,, in many different ways to raise necessary revenue yet still satisfy ex ante arbitrage. A unique
policy can be pinned down by imposing restrictions that reduce the degrees of freedom in setting
effective bond returns. With indexed bonds, the government is prevented from using state-contingent
inflation to manipulate the rate of return. One more degree of freedom can be removed by requiring the
arbitrage condition to hold ex post as well as ex ante. The interest rate on government bonds is thus
determined by rb,=(I-z,)(r,- 8). It must be pointed out, however, that since other restrictions are
possible, the model alone cannot pin down a unique prediction for the time-series behavior of T,.~' Ex
post arbitrage imposes "certainty equivalence" on the government's use of debt. Certainty equivalence
is also exploited in computing a solution to the model, because the method involves a linear-quadratic
approximation of the government's decision problem.
The summation of the household budget constraints and the government budget constraint yields
the following resource constraint for the economy. Note that the resource constraint and the government
budget constraint are not independent equations. To simphfy the formulation, the resource constraint is
used in place of the government budget constraint in the recursive version of the problem.
Y [c1:
+
c,:]
+
(l-Y)[c1:
+
c,:
+
x,]
+
g, = Y , .
2.5 Recursive Formulation of the Problem
The government's problem under commitment can be solved using the unique recursive
algorithm developed by Kydland and Prescott (1980). Standard recursive methods cannot be used
210therrestrictions might be a period-by-period balanced budget constraint (so that government debt is not a state variable) or
the assumption that either r,, or z, is not state contingent (the equilibrium value is not a function of state variables).
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because the problem does not satisfy Bellman's principle of optimality." Specifically, households'
optimum decision rules, which must be incorporated into the return function W(.), depend not only on
current policy, but also on the anticipated sequence of future policies. The value of the return function
W(.) at time t is thus dependent on future policy variables,,,z
,,,,z
, and p,,, for j>0. This influence
of future policy on current returns destroys the recursivity of the problem. However, the method of
Kydland and Prescott allows the problem to be redefined in a way that recovers a recursive structure.
The crucial aspect involves defining the lagged shadow prices hP,, and hR,, to be pseudo state variables.
Including these prices in the state vector provides a link to the past by which the policymaker at time
t takes into account the fact that household decisions in earlier periods depend on current policy by
means of expectations. This link to the past is crucial in order to solve the commitment problem using
dynamic programming. In a no-commitment regime, the policymaker at time t ignores the effect of
current policy on household decisions in earlier periods.23
To reformulate (12), we first substitute the household first-order conditions in (1 1) into the
transformed household budget constraints, as seen in the Lagrangians. The substitution eliminates ,z,
,z, p,, cDP,and cDRand yields the following set of equations?"
22~ellman(1957), pp. 81-83. Bellman defines a recursive problem as one in which the optimal decision rules depend only on
current-period state variables.
23~ngame theoretic terms, the government's optimal strategy under commitment is "memory based," where hP,, and hR,.,
summarize the history of the game. Under no commitment, there are potentially many equilibrium strategies, including "memoryless"
strategies that are functions only of current-period state variables z , , v,, k,, and b,. Oudiz and Sachs (1985) provide an excellent.
summary of these equilibrium concepts in the context of a structural macroeconomic model.
% ~ u to
e the presence of the expectation operator in the first-order conditions for kt+, and b,,, the substitution has been
accomplished using the expression E f , (-)=f,(.) - u ,, where f , (.) is a function of random variables and u , is the forecast error.
The assumption of rational expectations implies E ,.,u ,=O.
,
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The household first-order conditions can also be used to eliminate 7, and 7, from the expression
for income inequality in the feedback rule for government spending. The result is
The first-order condition form,,, can be used to obtain a useful relation between rji; and m
:
Finally, c,,' and c', ( i = P , R ) are eliminated from both the return function W(.)and the resource
constraint (14) using the cash-in-advance constraints and the first-order condition for c,.' The vector of
state variables for the government's problem is s , = { z , , v , , k t , b , , Apt-,, hR,,).In the transformed
, h,' , h', , kt+,, b,+,, i =P , R . Using primes (') to
problem, the govemment's decision variables are fii1+,
denote next-period quantities, the recursive version of.the government's problem is shown in (19).
The Bellman equation in (19) summarizes the recursive nature of the problem. The first line of
constraints lists the cash-in-advance constraints and the first-order conditions for credit goods. The
second line is the relationship between household money stocks from (18); and the transfer payment
constraint. The next three lines are the household budget constraints and the resource constraint. The
remaining constraints define the production technology and the laws of motion for g, k, z, and v.
The dynamic programming problem defined in (19) applies for all t > 0. The problem at t=O must
be considered separately, as shown by Kydland and Prescott (1980), Lucas and Stokey (1983), and
Charnley (1986). At t=O, the stocks of capital, bonds, and money are fixed. Optimal policy thus implies
very high values for the initial tax rate on capital and the initial money growth rate, to take full
advantage of nondistortionary sources of revenue. I assume that this form of lump-sum taxation is
insufficient to finance the entire stream of future expenditures.
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E-, {W (cl , c2 , h P , c ~ , c ~ , h R ) + p c [ ~ ( s ' ) ~ s ] } ,
~ ( s =) max
':. k'.
b'
hi.)t, i-P,R
d
where:
s = {
Z,
v, k, b , h!l,
hfl 1
W(.) = @ U P ( cP 1 , Pc 2 , h P+UR(c;,c;,hR)
)
subject to:
The analysis here will focus on policy in stationary stochastic equilibrium, i.e., when t is very
large. The linear-quadratic approximation method used to solve (19) is accurate only in the neighborhood
of the deterministic steady state. Consequently, I do not solve the t=O problem or compute the transition
path to the stationary equilibrium. One complication that arises with this approach is that the steady-state
level of government debt cannot be determined solely on the basis of steady-state analysis. Rather,
steady-state debt is a function of both the initial level of debt, b,, and the entire transition path of taxes
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and spending from t=O until the steady state is reached. As an alternative to performing this difficult
computation, I simply choose the level of steady-state debt to reflect a debt-to-GNP ratio consistent with
U.S. data. I assume that b, and the transition path are set such that the government budget constraint is
satisfied in present-value terms.25
,'
Kydland and Prescott (1980) prove the existence of a stationary equilibrium in a representative
household version of the Rarnsey problem. Proving existence and uniqueness of a stationary equilibrium
in this model is difficult due to the borrowing constraints imposed on poor households. Instead, I simply
assume that some (unspecified) institutional mechanismprohibits the use of time-varying policy rules.
Equilibrium is defined as a value function V(s) and an associated set of stationary decision rules that
satisfy (19). The decision rules dictate a set of household allocations and prices at time t that can be
implemented by means of the government's chosen policy. The government's explicit policy rules for
tax rates and money growth can be recovered by substituting the implementable allocations and prices
into the household first-order conditions and budget constraints and by imposing r,, = (1-7, )(r, - 6).
3. Computation Procedure
The dynamic programming problem in (19) is solved numerically using a variant of the linearquadratic approximation technique first used by Kydland and Prescott (1982). An approximate version
of (19) is obtained by first substituting all nonlinear constraints into the government's objective function
W ( . ) and then forming a qua.dratic approximation of the resulting expression in terms of the logarithms
of all variables.16 The solution algorithm exploits the certainty equivalence property of linear-quadratic
control problems. The optimal decision rules for the approximated economy can be obtained by solving
the deterministic version of the model." An initial guess V,, is made for the optimal value function V(s)
2 5 ~ hindeterminacy
e
of steady-state debt is discussed by Chamley (1985b). Auerbach and Kotlikoff (1987) show how the steadystate level of debt can be computed by explicitly modeling the transition path in a life-cycle model with no uncertainty.
he log-linear version of the Kydland-Prescott method is described in Christian0 (1988). A step-by-step guide to linearquadratic solution algorithms can be found in Hansen and Prescott (1991).
Sargent (1987), p. 36. Specifically, the stochastic terms E,, 4 , u:, and u; are set equal to their unconditional means
(zero) in the numerical algorithm. With a quadratic objective, the first-order conditions are linear in all variables. This allows the
expectation operator in (19) to be passed through the expressions, dropping out any stochastic terms.
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in the quadratic version of (19). Sequential candidate value functions Vi are then computed by
successively iterating on the Bellman equation until the value function has converged, i.e., until Vi is
sufficiently close to V,,. Once the process has converged, log-linear decision rules that dictate household
equilibrium allocatioxis are computed. Log-linear policy rules for T,, T,, and R can then be computed
using the household first-order conditions in (1 1) and the household budget constraints, log-linearized
around the steady state. To improve accuracy during the simulations, the nonlinear versions of the policy
rules are used in computing period-by-period values for the policy variables.
4. Calibration of the Model
To explore the quantitative predictions of the model, as many parameters as possible are assigned
values in advance based on empirically observed features of postwar U.S. data. Parameter choices are
also guided by the desire to obtain steady-state values for key model variables that are consistent with
postwar averages in the U.S. economy. For parameters that are difficult to pin down, such as q, @, and
TR,a range of values is examined. Table 1 summarizes the baseline parameter values and is followed
by a brief explanation of how they were selected.
Table 1: Baseline Parameter Set
Agent
Parameters and Values
Households
Finns
Government
y~~~ = 0.02
g/Y = 0.22
(1- y) b f l = 0.25
p, = 0.95
0,
= 0.02
The relative number of poor households ( y ) determines the distribution of wealth and income
in the model. Using data from the 1983 Survey of Consumer Finance, McDermed, Clark, and M e n
(1989) estimate a Lorenz curve that summarizes the highly skewed distribution of wealth in the U.S.
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economy. Their results indicate that the richest 20 percent of households own 80 percent of U.S. wealth
(based on net worth). Here, poor households have no wealth. The choice w . 8 0 implies that the richest
20 percent of households in the model own 100 percent of the wealth, a distribution that roughly
approximates the empirical Lorenz curve. For the model, it can be shown that the analytical Gini
coefficient (based on wealth) is equal to y. The value w . 8 0 is very close to empirical estimates of
wealth-based Gini coefficients for the U.S. economy. Using data on household net worth, Wolff and
Marley (1989) report Gini coefficients of 0.772 for 1962 and 0.788 for 1983.28
The parameter a determines the relative importance of cash versus credit goods in the household
utility function. Empirical estimates of this parameter vary, depending on the choice of monetary
aggregate and the sample period. Cooley and Hansen (1992) estimate a value of @.84.
Using a
somewhat different utility function, Chari, Christiano, and Kehoe (1991) estimate the relative weight on
cash goods to be 0.43. The chosen value of 0.6 lies about midway between the two estimates. It turns
out that higher values of a induce the government to choose higher rates of inflation in equilibrium.
The parameter A is picked to yield an economywide average number of hours worked close to
0.3. This is consistent with time-use studies, such as Juster and Stafford (1991), which indicate that
households spend approximately one-third of their discretionary time in market work.29
The time period in the model is taken to be one quarter. With quarterly time periods, the
common discount factor for rich households and the government is set at pR=pG=0.99.This value
implies an annual rate of time preference equal to 4 percent. The discount factor for poor households
is set at PP=0.985, which implies an annual time preference rate of 6.2 percent. Engen (1992) estimates
annual rates of time preference in the range of 4 to 7.9 percent, while Lawrance (1991) estimates rates
in the range of 0 to 19 percent. In Lawrance's study (tables 3 and 5), time preference rates of households
with below-median incomes are 2 to 5 percentage points higher than those with above-median incomes.
2 8 ~ hGini
e coefficient is a measure of inequality that ranges between zero and one. A value of zero implies no inequality among
households. A value of one implies all wealth (or income) accrues to a single household. The coefficient can be computed by taking
twice the area between the Lorenz curve and the 45 degree diagonal.
2 9 ~ nthe model, poor households spend a larger fraction of their time working than do rich households, hP> hR.This is because
the rich derive a substantial portion of their income from capital and earn a higher hourly wage.
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Households without college educations have time preference rates about 2 percentage points higher than
those with college educations. The chosen values for PP and PR imply time preference rates for the poor
that are about 2 percentage points higher than rates for the rich.
The share of output that represents payments to capital (8) is set at 0.36, about midway in the
range of 0.25 to 0.43 estimated by Christian0 (1988). The quarterly depreciation rate of &O.025 is
commonly used and, together with PR and 8,yields a realistic steady-state ratio of economywide capital
to output of 8.5 and a ratio of total investment to output of 0.21. The process governing technology
shocks has been estimated by Prescott (1986). The parameters governing the shocks, pz=0.95 and
0,=0.007, represent values commonly used in real business cycle models.
Empirical estimates of the labor efficiency parameter (e") as a function of savings behavior or
wealth are not available. Estimates are available, however, as a function of age and education. Using
panel data on labor earnings, Engen (1992) estimates e" as a quadratic function of age over an
individual's lifetime for various education levels. Three-fourths of the sample consumers have no college
education. The ratio of the average lifetime e" for individuals with no college education to those with a
college education is about 0.75. If the non-saving, poor households in this model. are viewed as
representing individuals with no college education, then the empirical evidence would suggest a value
of e"=0.75. The values of e" and y affect the skewness of the income distribution in the model. As an
additional calibration source, the distribution of income in the model can be compared t o the U.S.
economy. Rich households in the model earn 42 percent of total income (before taxes and transfers).
This figure coincides with the average share earned by the top fifth of U.S. households from 1947 to
1989. The model's income-based Gini coefficient is 0.22, a value somewhat lower than the average
value of 0.37 for the postwar U.S. economy.30
The semi-elasticity parameter q controls the degree to which government spending responds to
income inequality. It turns out that the value of q (together with
level of
2,.
Given the values for
PR and y) determines the steady-state
PR and y described above, q is set to yield 2,=0.41.
This tax rate is
about midway in the range of estimates for the average marginal tax rate on capital in the U.S. economy.
3 0 ~ eRosen
e
(1992), table 8.1, and Economic Report of the President, 1992, chapter 4, chart 4-4.
22
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With 11=0.04,the steady-state level of endogenous government spending is equal to 4.5 percent of GNP.
To put this number in perspective, federal and state aid to local governments averaged 3.6 percent of
GNP from 1950 to 1988 (see Rosen [19921, table 21.4). A range of values for q is also investigated.
The law of motion for exogenous spending, gexp (v,), is designed to mimic the quarterly time
series of government purchases of goods and services in the U.S. economy. Data on total government
purchases were used in the estimation because it is not possible to isolate and exclude that portion driven
by income inequality. Exogenous spending accounts for about 80 percent of g, in the model, however.
The value of
gis set to yield a steady-state ratio of total government purchases to GNP of 0.22, the
postwar U.S. average. The parameters p, and o5 govern the behavior of the exogenous spending
The value of TRis set to approximate the average ratio of transfer payments to GNP in the U.S.
economy. This ratio varies, depending on the type of payments included in the definition. Means-tested
transfer payments (including in-kind transfers) increased from 1.2 percent of GNP in 1965 to 3.6 percent
in 1988. If social insurance programs (Social Security, Medicare, and unemployment insurance) are
included in the definition, the average level of transfer payments from 1950 to 1990 increases to more
than 6 percent of GNP. It turns out that the value of
TR significantly affects the
government's
equilibrium choice of money growth. Therefore, the steady-state ratio of transfer payments to GNP
is set at 0.02, and a range of values is investigated. The steady-state ratio of government
debt to GNP is set at 0.25. This value is at the lower end of the range of net federal debt as a share of
GNP since 1950. The basic results are not significantly affected by the level of steady-state debt.32
The parameter 0 controls how much the government favors one group relative to the other and
thus significantly affects the progressivity of equilibrium labor taxes. I choose 0 such that the revenueweighted average of 7, across all households is close to estimates for the U.S. economy. The baseline
3 1 ~ h elaw of motion for exogenous government spending is equivalent to the following AR(1) specification:
ln g , = (I-p)ln g+pln g ,.,+ 5,. Using this form, Christian0 and Eichenbaum (1992) estimate p=0.96 and 0\=0.02.
3 2 ~ a ton
a government purchases and total transfer payments are from Citibase. Data on net federal debt to GNP are from Federal
Deb1 and Interest Costs, Congressional Budget Office (1993). Data on means-tested transfers are from Rosen (1992) and Economic
Report of the President, 1992, chapter 4.
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value is set at $=2.7, and a range of values is investigated.33
5. Quantitative Properties of the Model
5.1 Steady-State Experiments
Figures 1-4 show the effect on steady-state policy of varying four key parameters in the model,
namely, q , $,
TR,and
In each figure, only a single parameter is varied, with remaining parameters
set at the baseline values in table 1.
Figure 1 shows the effect of varying q , which controls the sensitivity of government spending
to income inequality. An analytical expression for the optimal steady-state tax on capital as a function
of q can be derived and is shown below.
2,=1-
P
P +T-P + 6 '
1-Y
where
1
p =--I
(PR=pG).
PR
Equation (20) is derived by combining the government's first-order condition for k,+,with the
corresponding household first-order condition in (1 1) and making use of the assumption pR=pG.When
q=0, the result is 7, =O. Notice that the steady-state tax on capital is not affected by $, the weight placed
on the poor's welfare in the government objective function. These results agree with those proved in
Judd (1985) and Chamley (1986) in models with no externalities. From (201, we see that
&,/a>O and
&,/a>O. Higher levels of capital accumulation accentuate income inequality. This effect imposes a
negative externality on the economy (as determined by q ) in the form of higher public spending because
the additional spending must be financed by distortionary taxation. Positive values of
7,
force rich
households to help pay for this externality. An increase in q also tends to reinforce the progressivity of
labor taxes. As the number of poor households ( y ) increases, the income distribution becomes more
skewed. This increase in inequality causes more spending, calling for higher levels of
7,.
Figure 1
shows that the amount of endogenous spending necessary to induce high levels of 7, is relatively small,
3 3 ~ i t@=2.7,
h
the government places less weight on the within-period utility of the poor thin is implied by their relative number.
With ~ 0 . 8 and
0 1 - ~ 0 . 2 0there
,
are four times as many poor households as rich. Here, the government places only 2.7 times as
much weight on the poor's within-period utility. This behavior might be justified either as a way of compensating for pG>pP or as
a reflection of lower voting rates among the poor. The model abstracts from an explicit description of political equilibrium, however.
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about 4 to 5 percent of GNP.
Figure 2 shows the effect of varying the political weighting factor
0.
When Q=0, the
government's optimal policy calls for highly regressive labor taxes. This is because the government
views the poor's labor supply as completely inelastic and thus it imposes a very high tax on this activity,
in accordance with Rarnsey's principle of optimal taxation. As
$I
increases, labor taxes become more
progressive. Due to the diminishing marginal utility property of U'(.), the government perceives more
benefits from a dollar in the hands of the poor than a dollar in the hands of the rich.
Figure 3 shows that the quarterly money growth rate (which equals the quarterly inflation rate
in steady state) increases rapidly with the level of required transfer payments FR.In a standard cash-inadvance model with utility functions of the form used here and no externalities, optimal money growth
adheres to the Friedman rule.34 In this model, transfer payments represent a negative externality for the
government because they are not taxed but must be financed by distortionary taxation. This drives a
wedge between the government's marginal utility of consumption and that of households. Moreover,
transfer payments induce the poor to work less and cause their labor supply to become more elastic, thus
increasing the distortionary costs of labor taxation. To spread out distortionary costs across tax bases,
the government levies a tax on consumption in the form of inflation. In a representative household
version of the model, with
Y= 1
and pG=pP=p,it is possible to derive the following steady-state
expression for optimal money growth:
From (21), when TR=O
the result is p=p- 1 (the Friedman rule). The term A, > 0 is the Lagrange
multiplier on the household budget constraint in the government's first-order conditions. This represents
the perceived benefit to the government of increasing private consumption by one unit. When
TR> 0,
optimal monetary policy calls for a positive nominal interest rate in steady state. The government's
inability to tax transfers directly motivates the imposition of a tax through the back door, by raising the
34~ecallthat the Friedman rule in steady state is p=P-1. For further discussion, see the references cited in footnote 8.
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nominal price of consumpti~n.~~
Equation (21) illustrates the well-known fact that standard optimality
results may not go through in the presence of externalities. At the baseline level of transfers, quarterly
money growth is p=0.014, implying an optimal inflation rate of about 6 percent per year.
Figure 4 shows the effect of increasing the steady-state ratio of government purchases to GNP.
The ratio is varied by increasing
which controls the level of exogenous purchases. As g/Y increases,
tax rates on labor increase in a linear fashion. It is efficient for the government to finance long-run
(steady-state) increases in g with labor taxes because the long-run elasticity of labor supply is less than
the long-run elasticities of capital or money balances. As labor tax rates approach 0.60, the money
growth rate accelerates dramatically. At this point, maximum revenue is being collected from labor taxes.
As required spending continues to go up, the government is forced to rely more heavily on seigniorage.
Revenues from seigniorage are limited by households' willingness to hold money balances, as measured
by the parameter a. From (21), higher values of a result in higher money growth rates.
As a final steady-state experiment, table 2 compares revenues collected from various sources in
the model and in the postwar U.S. economy. Model results are for the baseline parameters, and all
revenues are normalized by GNP. The labor tax is the largest source of revenue. The capital tax provides
significantly less revenue than the labor tax, even though the tax rate on capital is higher in the model.
This is due to the depreciation allowance. Finally, seigniorage is the smallest source of revenue. The
relative sizes of revenue compare remarkably well with the U.S. averages. However, revenue sources
in the data do not always fit neatly into one of the three categories.
Table 2: RevenuelGNP from Different Sources
Source of Revenue
Model
U.S. Economy"
Labor Income Tax
0.177
0.159
Capital Income Tax
0.062
0.067
Seigniorage
0.0046
0.0035
a ~ a revenues
x
are average values from various issues of Revenue Statistics of OECD Member Countries, 1965-1990,table
61. Labor tax revenue is defined to include federal and state individual income taxes and Social Security taxes. Capital tax revenue
is defined to include federal and state corporate taxes, capital gains taxes, and property taxes. Seigniorage is from Neumann (1992)
for 1951 -90, defined as (M,-M,, )/P,, where MI is the monetary base.
3 s ~ h iinterpretation
s
is basedon a discussion of transfers in Jones, Manuelli, and Rossi (1992), p. 36.
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5.2 Dynamic Experiments
5.2.1 Optimal Policy Rules
The solution to the approximate version of (19) yields the following set of log-linear optimal
policy rules, which are valid in the neighborhood of the deterministic steady state.
Table 3: Optimal Policy Rules
Constant
Z I
" I
In ( k ,
In (b I )
In (KI)
11, (KI)
Shocks to the government budget are caused by unexpected changes in the size of the tax base
or by unexpected increases in exogenous spending requirements. The government's optimal response to
these shocks can be seen by examining the coefficients on state variables z , and v, . For example, a
positive technology shock causes large decreases in 7, and p, (in proportion to their steady-state values)
relative to 7,. A positive z, causes GNP and household incomes (the tax base) to rise, allowing revenue
requirements to be met with lower taxes. In contrast, a positive expenditure shock (v,) calls for an
increase in 2, and p, to collect additional required revenue. Absorbing shocks in this way is efficient
because capital and money balances are completely inelastic within a given period. Judd (1989) and
Chari, Christiano, and Kehoe (1991) also obtain shock-absorbing behavior in related models. Notice that
the policy rule for & reflects the notion of countercyclical monetary policy, in that money growth moves
opposite to output fluctuations. However, the neoclassical framework precludes any role for
"stabilization" in the sense of preventing large swings in unemployment over the business cycle.
The shock-absorbing features of 7, and U. allow the government to maintain relatively stable
tax rates on labor, reminiscent of the tax-rate-smoothing hypothesis of Barro (1979, 1986). This
hypothesis has been the subject of numerous empirical studies designed to test whether tax rates or
inflation follows a random walk (or martingale).36In this model, however, the optimal policy rules show
36~ee,
for example, Sahasakul (1986), Mankiw (1987), and Bizer and Durlauf (1990).
27
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that there is no theoretical prediction that policy variables follow a random walk.37
As a direct test of the model, it would be desirable to compare the policy rules in table 3 with
empirical versions estimated with U.S. data. An estimation problem exists, however, because shadow
could-,/I
prices hP,,and hRclare unobservable. Kydland and Prescott (1980) point out that the ratio &
conceivably be estimated from the household's first-order conditions, but this still allows the shadow
prices to be scaled in an arbitrary way. Empirical testing of key characteristics of the optimal policy
rules is an area for future research.38
5.2.2 Policy Simulations
Figures 5-10 plot simulated policy from the model together with U.S. data on marginal tax rates
and money growth. Tables 4 and 5 provide a quantitative comparison of the series. In these tables, the
= (P,- P,-,)/PI, has also been included. Although
inflation tax rate, defined as ~c,
is the instrument of
monetary policy directly under the government's control, n, has the advantage of lying between zero
and one, analogous to the other tax rates 2, and 2,. The two measures of monetary policy are related
by q = l
- p,, 1[PI (l+p,,)], where p, is defined in (8).
The model does reasonably well in capturing the standard deviations and serial correlations of
the policy variables (table 4), but is less successful regarding the contemporaneous correlations (tables
5a and 5b). A basic prediction is that the capital tax and the inflation tax should both be more volatile
than the labor tax, a feature generally confirmed by the data. The capital tax series estimated by
Jorgenson and Sullivan (1981), shown in column 3, has a much higher standard deviation than the series
estimated by Joines (1981), shown in column 4. The values are 16.38 percent and 5.09 percent,
respectively. The Jorgenson and Sullivan series is an estimate of the effective corporate tax rate, while
the Joines series also includes property taxes and taxes paid by individuals on capital gains and
dividends. Neither series takes into account the imputed subsidy on investment in residential housing.
-
-
---
3 7 ~ h ipoint
s
was originally made by Chari, Christiano, and Kehoe (1991).
3 8 ~ hcoefficients
e
on lipI.,are equal to zero in table 2 because poor households do not save. In fact, since lip,.,is directly related
to hRl.,by (18), hP,.,could have been eliminated as a state variable.
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Other estimates of U.S. tax rates on capital can be found in Auerbach and Poterba (1988), Fullerton and
Karayannis (1987), King and Fullerton (1984), Jorgenson and Yun (1989), and Judd (1989).
Table 4: Simulated Policy versus U.S. Economy (All Variables Detrended)
Modela
7h:
=kt
Mean
0.278
Std. Dev. (%)
0.99
u.s
U.S.
Economy'
Economyb
(-1)
0.49
corr (-2)
0.01
C O (-3)
~
-0.22
-0.67
-0.59
Mean
0.412
0.299
0.540
Std. Dev. (%)
8.41
16.38
5.09
-0.21
-0.24
-0.37
0.06 1
0.048
0.050
COIT
COIT
(-1)
corr (-2)
C O (-3)
~
Mean
Std. Dev. (%)
Pt
50.50
29.78
(-1)
0.48
corr (-2)
-0.01
(-3)
-0.22
-0.18
-0.24
Mean
0.056
0.040
0.041
Std. Dev. (%)
28.0 1
COIT
COIT
'=,
22.50
49.75
36.28
corr (-1)
corr (-2)
'Model statistics are means over 100 simulations, each 124 quarters long. During each simulation, annualized series were
constructed using revenueweighted averages to compute tax rates and end-of-year money stocks and prices to compute p, and n:.
The annualized series were then detrended using the Hodrick-Prescott filter with a smoothing parameter of 100.
bHere,z, is from Barro and Sahasakul(1986) for 1947-83,z, is from Jorgenson and Sullivan (1981, table 11) for 1947-80,
pf is based on the MI series constructed by Rasche (1987) for 1947-89, and n,is based on the CPI (all items) from Citibase for 194789. Data for pI and q were annualized as in the model, and all variables were detrended using the Hodrick-Prescott filter.
'Here, 2, and z, are from Joines (1981, tables 2 and 10) for 1947-75, where z, is "MTRL4 and z, is "MTRK." Data
for p, are from the monetary base series in Citibase for 1947-89, and q is based on the GNP deflator for 1947-89, also from Citibase.
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Table 5a: Contemporaneous Correlation in Modela
' ~ o d e lstatistics are means over 100 simulations. where all series have been annualized and detrended as in table 4.
Table 5b: Contemporaneous Correlation in U.S. Economva
* ~ asterisk
n
indicates that the correlation coefficient has the same sign as in the model. The top and bottom numbers in
each cell represent correlations using the U.S. variables described in footnotes a and b, respectively, of table 4. The U.S. series were
each annualized and detrended over periods for which a full set of variables was available. For the top numbers, this period was 194780. For the bottom numbers, the period was 1947-75.
Also from table 4, we see that the tax rate on labor in the model has a much lower standard
deviation than either U.S. series (0.99 percent versus 5.65 or 4.44 percent). Money growth and the
inflation tax both display very high standard deviations (more than 20 percent). Comparisons with the
data are slightly more favorable for the monetary base series (as opposed to M1) and the GNP deflator
series (as opposed to the CPI index). In a related model, Chari, Christiano, and Kehoe (1991) report a
much higher standard deviation for simulated money growth than the value shown here. However, their
model includes nominal government debt, and the inflation tax is the only available shock absorber.
The correlation coefficients in the model match the signs in U.S. data for about half the cases
in table 5. The model generally predicts strong correlations among the variables, while many U.S.
correlations (which are based on only 29 to 34 observations) are quite weak and can even vary in sign,
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depending on the source. All variables have been detrended in table 5 because the U.S. labor tax and
money growth rate both display upward trends, while the U.S. capital tax displays a slight downward
trend (see figures 5, 7, and 9). These trends have no counterpart in the model because the ratio of
government outlays to GNP is stationary. In U.S. data, the ratio of outlays to GNP has risen over time,
mainly due to the rapid growth in transfer payments. From figures 3 and 4, the model predicts that
higher steady-state outlays relative to GNP should be accompanied by increases in the labor tax and the
money growth rate.39
Another basic prediction of the model is that the labor tax should be negatively correlated with
inflation (and money growth), while the correlation between the capital tax and inflation should be
positive. Partial-equilibrium models generally do not distinguish between labor and capital taxes and
predict a positive correlation between a single "income tax" and inflation. This failure to distinguish
between factor incomes may help to explain the conflicting findings of previous U.S. and cross-country
empirical studies designed to test for the partial-equilibrium result (see Mankiw [1987], Roubini and
Sachs [1989], Poterba and Rotemberg [I9901 and Roubini [1991]).
As a final check of the model's dynamic behavior, tables 6 and 7 summarize predictions for key
business cycle statistics. Table 6 shows the corresponding statistics from Cooley and Hansen (1989),
who study a cash-in-advance model with no distortionary taxes and exogenous stochastic money growth.
The model statistics are virtually identical to the Cooley-Hansen results. Table 7 displays the model
predictions for two labor market statistics that have received particular attention in recent real business
cycle literature, namely 1) the volatility of hours worked relative to labor productivity, o,/o,,
, and 2)
the contemporaneous correlation between hours and productivity, corr(H,YIH ). The model results are
comparable to those obtained by Christian0 and Eichenbaum (1992) and Hansen and Wright (1992) in
models without money or distorting taxes. These results are encouraging because they suggest that
monetary real business cycle models can be extended into new areas, such as policy analysis or perhaps
even forecasting, without sacrificing a reasonable description of the aggregate economy.
391n the United States, the upward trend in <, is possibly linked to the trend in p, by the phenomenon of "bracket creep," which
existed before tax schedules were indexed for inflation in 1985. Regarding the capital tax T~ Auerbach and Poterba (1988) argue
that the downward trend is due to increasingly generous investment tax credits and accelerated depreciation schedules.
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Table 6: Business Cycle Statistics for Models and U.S. Economy
Series
U.S. Economy"
Standard Deviation in Percent
Modelb
Cooley-Hansenc
Output
1.74
1.74
1.73
Consumption
0.81
0.61
0.62
Investment
8.45
5.79
5.69
Capital Stock
0.38
0.49
0.48
Hours Worked
1.41
1.24
1.33
Productivity
0.89
0.64
0.50
Price Level
(CPI) 1.59
(GNP) 0.98
U.S. Economy
Series
Contemporaneous Correlation with Output
Model
Cooley-Hansen
--
Consumption
Investment
0.91
Capital Stock
0.28
Hours Worked
0.86
Productivity
0.59
Price Level
(CPI) -0.48
(GNP) -0.53
a ~ h U.S.
e statistics are from table 1 of Cooley and Hansen (1989) for the period 1955:IlIQ to 1984.IQ (115 quarters).
b ~ o d estatistics
l
are mean values over 100 simulations, each 115 quarters in length. All variables were logged and detrended
using :he Hodrick-Presc~ttf&cr with a smoothing parameter of 1600. The value 0,=0.0077 was used for the technologj shock to
achieve a standard deviation of output equal to 1.74. Productivity is defined as output/hours.
'statistics are from Cooley and Hansen (1989), table 1, with quarterly money growth of 0.015 and 0,=0.00721.
Table 7: Comparison of Labor Market Statistics
Statistic
U.S. Economya
Modelb
Christiano-Eichenbaumc
Hansen-Wright"
?he U.S. statistics are from Hansen and Wright (1992), table 2, for the period 1947:IQ to 1991:mQ (179 quarters). The
top and bottom numbers refer to the household and establishment surveys, respectively.
b ~ o d estatistics
l
are means over 100 simulations, each 179 quarters in length, with 0,=0.007.
'~hristiano and Eichenbaum (1992), table 4, with government consumption, indivisible labor, and 0,=0.012.
d~ansenand Wright (1992), table 3, with home production and 0,=0.007.
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6 . Concluding Remarks
The goal of any quantitative model of the economy should be to capture the basic incentives and
interactions among agents that govern the process of interest. In the case of government policy, it is clear
that real-world policymakers are fundamentally concerned with distribution issues. Monetary real
business cycle models have been reasonably successful in describing the behavior of aggregate
fluctuations. This paper uses such a framework as the starting point for endogenizing the choice of fiscal
and monetary policy over time in a model with the following characteristics: 1) the distribution of wealth
and income among households is highly skewed, 2) income inequality affects tax and spending policies,
and 3) the government must provide transfers to the poor.
I subjected the model to comparisons with postwar U.S. data on tax rates, money growth, and
inflation, and obtained varying degrees of success in capturing observed behavior of the various time
series. Comparisons with the data are difficult, however, because estimates of average marginal tax rates
are available only at annual frequency and consist of a small number of observations. A noteworthy
result is that the model predicts distinctly different behavior for the labor tax and the capital tax
regarding the optimal interaction with inflation, thereby pointing out the importance of distinguishing
between these taxes in empirical tests for optimal government behavior. Finally, the model was shown
to deliver business cycle statistics very dose to models in which government policy is treated as an
exogenous state variable.
The methodology of this paper can be used to perform quantitative studies in other important
policy areas, such as characterizing the optimal behavior of public investment over the business cycle
or quantifying the effects of international policy coordination on aggregate fluctuations. Regarding
monetary policy, a more complete description of the banking sector (which captures the liquidity effect
of a money shock) would be desirable. It would also be interesting to perform the policy simulations
done here in the context of an overlapping-generations framework (see Rios-Rull [1992]) to allow for
age heterogeneity as well as for rich and poor households. For example, such a model would allow
~mrohoroglu,and Joines [1992]).
consideration of optimal Social Security policy (see ~mrohoro~lu,
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APPENDIX A
Equilibrium with Different Discount Factors
This appendix briefly explains how the assumption of different discount factors, pP<pR,results
in concentrated ownership of capital in the hands of the more patient, rich households. See Becker
(1980) for a formal treatment of equilibrium when households have different discount factors. The
argument here is based on Woodford (1988). If we assume that all households face the same after-tax
interest rate, the following condition must hold if poor households would rather borrow than save:
The condition for rich households to save is
In steady state, these two conditions become
pPcpRfor concentrated ownership of
and A.2 can be satisfied even if pP=pR.
Combining the expressions in A.3 yields the condition
capital. It should be pointed out, however, that equations A.l
An example is when the poor face a lower after-tax interest rate than the rich. This situation might arise
if the poor incur some type of transacti'on cost for investing small amounts that effectively reduces their
rate of return. Judd (1985) uses an argument similar to this to justify concentrated ownership of capital
when all households have the same discount factor.
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FIG 1:
FIG 2:
TAX RATES vs FEEDBACK EXTERNALITY
TAX RATES v s WEIGHT ON U(P)
1 .o
0.8
-
0.6
+p (quarterly rote)
Endog Spending/GNP
0.4
0.2
a
0.0
1
t- -0.2
f -0.4
-0.6
-
-0
-0.8
+7, (Rich
Households)
7
, (Poor Households)
+p
(quorlerly rate)
-1.4
1.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Government Weight on Utility of Poor (
Feedback Externality f r o m Income Inequality ( 7 )
FIG 4:
-
+)
TAX RATES vs GOVT SPENDING/GNP
1 .o
T~ (Rich Households)
s. (Poor Houaahalds)
+p (quarterly
-
o
0.00
.
0.02
+p (quorlerly rote)
rote)
l
0.04
I
0.06
1
l
l
I
0.08
0.10
0.12
Transfer Payments t o Poor ( yTR/Y )
Source: Author'scalculations.
-0.1
0.04
0.10 0.16
0.22 0.28
0.34 0.40 0.46
Total Govt Spending/GNP
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FIG 5: U.S. TlME SERIES FOR LABOR TAX
0.40
FIG 6: SIMULATED TIME SERIES FOR LABOR TAX
(Borro & Sohoro*rl-lD88)
I
1
0.40
-r,-,rkl.nLI"""
0.35
'
0.30
'
.
0
B
0
-I
5
-
0.25
.
a
B
0.20
'
I
i
0.15
0.l9947
i9s2
1057
ion
1082
1072
ion
Yeor
FIG 7: U.S. TlME SERIES FOR CAPlTK TAX (Jorpenron & Sullivan-1981)
FIG 9:
U.5. TIME SERIES FOR MONEY GROWTH (MI doto)
I
101I
'
'
0.1Oo
10
20
15
30
FIG 8: SIMULATED TIME SERIES FOR WlTAL TAX
.FK: 10: SIMULATED TIME SERIES FOR MONEY GROWTH
* - Ynq Clo.th hi.
Year
Source: Author's calculations.
25
Ymr
35
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