The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Working Paper 87-4
THE ECONOMIC EFFECTS OF CORPORATE TAXES
IN A STOCHASTIC GROWTH MODEL
Michael Dotsey
Federal Reserve Bank of Richmond
October 1987
*
The views expressed in this paper are solely those of the author and do not
necessarily reflect the views of the Federal Reserve Bank of Richmond or the
Federal Reserve System.
I.
Introduction
The Economic Recovery Act of 1981 led to the largest postwar decline
in effective tax rates on capital.
The legislation also had its most
significant effect on rates in 1982 due to the rapid decline in inflation.
Although some of the tax cut was rescinded in 1982, effective corporate tax
rates on plant and equipment, measured as the difference between before and
after-tax rates of return to capital as a percentage of before-tax rates of
return, remained at historically low values through 1986.
Accompanying this
tax cut is the current economic recovery which began in November, 1982.
During this recovery we have witnessed relatively large increases in business
fixed investment, a stock market boom, and a large rise in both the ex-post
and ex-ante real interest rate.
It is, therefore, natural to investigate the
linkages between the tax cut and the increase in economic activity.
The effects of this particular tax cut are also consistent with
observed negative correlations between taxes and real interest rates, stock
prices, investment, and output growth associated with other business tax cuts.
For example, using the effective tax rates on plant and equipment reported in
Hulten and Robertson (1982) the correlation coefficients between tax rates and
real gnp, real business fixed investment, and the New York Stock Exchange
price index for the period 1952-1984 are -.81, -.78, and -.87.
correlation coefficients all have a significance level of .0001.
These
Further,
using Livingston data on expected inflation over the period 1960-1984 the
exante real rate of interest and the effective tax rate display a correlation
coefficient of -.38 while the coefficient with respect to the expost real rate
is -.50.
These coefficients have significance levels of .0634 and .0112
respectively.
- 2-
While the general effects of the recent tax cut are consistent with
effects observed in other periods, the relative size of the 1981 tax cut is
quite large.
For instance, Hulten and Robertson calculate that the effective
tax rate on capital in nonresidential business was reduced from roughly 33
percent in 1980 to approximately 1 percent in 1984.
It is not surprising that
a change of this magnitude has generated renewed interest in the interaction
between taxes on capital and their effect on the economy.
Most of the work dealing with the effects of taxes has proceded
within the confines of standard nonstochastic growth models (e.g. Abel and
Blanchard (1982)) or in models in which agents have perfect foresight
regarding the path of tax rates (e.g. Goulder and Summers (1987)).
Little
effort seems to have been given to examining the effects of tax rate changes
when taxes are explicitely depicted as following a particular stochastic
process.
This paper takes the latter approach and investigates the effects of
tax changes in a stochastic growth model in which only tastes, technology, and
the stochastic process for taxes are exogenous.
This procedure takes
seriously the methodology advocated by Lucas (1976) and Sargent (1979) that a
policy should be represented as a given outcome of some stochastic process.
The analysis yields investment, output, and real interest rate behavior that
depend explicitely on tastes and technology parameters as well as the
underlying process generating tax rates.
The qualitative movements in endogenous variables that are generated
by tax rate changes in this model are similar in some instances to results
derived in the nonstochastic or perfect foresight models.
For instance, when
a high tax rate is generated (and the rate is expected to remain high), agents
reduce the capital stock.
This leads to lower real interest rates and lower
- 3 -
real output growth.
The qualitative similarity of the results from the
various models is a natural outcome of the behavior of agents; behavior which
is characterized by analogous intertemporal optimization problems in all three
types of models.
However, an explicit stochastic treatment of taxes allows one to
examine the effects of uncertainty on the decisions of individual agents.
This uncertainty should be an important consideration in determining behavior
since tax law changes are fairly frequent and changes in inflation do result
in significant movements in the effective tax rate on capital.
Also, the
exact nature of the uncertainty is related to the specific process that taxes
are assumed to follow.
For example, the degree of persistance of the process
generating tax rates will have important implications for individual behavior.
Therefore, agents will show quantitatively different behavior for any specific
realizations of tax rates when the inherent randomness of taxes is modelled as
opposed to treating tax rate data as being known with certainty.
If one wants
to derive realistic decision rules, then the stochastic nature of the agents
problem needs to be analyzed explicitly.
The results generated by the stochastic growth model derived in this
paper produce correlations that are consistent with those mentioned above.
The model, therefore, indicates that the 1981 tax change was a potentially
important factor in the current economic recovery.
The paper is structured as follows.
Section II contains a
description of the model, while Section III characterizes the model's
equilibrium.
Of particular interest is the derivation of a closed form
solution to the nonlinear stochastic difference equations that determine the
equilibrium.
Section IV presents the solution when taxes are assumed to
follow either a simple two state Markov process or are white noise.
A
- 4 -
comparison between capital stocks generated by the model for particular
realizations of these processes with those generated by a perfect foresight
model is made in Section V, while a short summary is given in Section VI.
II.
The Model
The model is a one sector stochastic growth model consisting of
three economic entities:
firms, consumers, and the government, that interact
in two markets each period.
First, there is a capital market in which firms
purchase capital from individuals.
Capital is carried over from the previous
period and is therefore supplied inelastically as in Brock (1979).
Next there
is a combined goods and securities market in which individuals allocate their
wealth among goods and securities.
Individuals also decide how much they will
consume and how much capital to carry into the succeeding period.
The
government taxes away some of the firm's revenue and remits the proceeds lump
sum to individuals.
Tax rates are stochastic, and are announced at the
beginning of each period so that there is no uncertainty over the current tax
rate.
a.
Capital Market
At the beginning of period t, the representative individual has
carried over kt units of capital which is sold to the firm for rt units of
output per unit of capital.
One can think of output as seeds which can either
be eaten (consumed) or invested (carried into the next period and sold to
firms).
The firm maximizes its after tax profits
(1) 7rt = (1-Tt)f (kt)
-rtkt
- 5-
where
T
is
the effective tax rate on capital.
This optimization implies that
the price of capital is equated to its after tax marginal product.
Formally,
rt = (1--Tt)f'(kt). Therefore capital is totally depreciated and the tax rate
drives a wedge between the marginal product of capital and its price.
The
distortionary tax rate is announced prior to the capital market so that there
is no uncertainty regarding current taxes.
There is, however, uncertainty
over future tax rates.
b.
Goods and Securities Market
After selling capital to firms and receiving the distribution of
profits and lump sum tax remissions, individuals choose their current
consumption ct. next periods holdings of capital, kt+i, and their share of the
firm st+1 subject to the value of their current wealth, wt, where
(2) wt = (qt + it)st + rtkt+
Tt
f(k d
The first term after the equality represents the current value of the shares
of the firms, qtst, plus dividend payments ntst.
The second term is the
payment for capital and the third term is the lump sum transfer of tax
proceeds.
The bar denotes that this is an average value and that the
individual has no control over its realization. The budget constraint facing
the individual is
(3) ct + kt+i + qt st+1 s wt.
c.
The Individuals Maximization
The individual's problem is to maximize his discounted expected
utility
- 6 -
(4) U = Et
ZE
coT-t
u(cT)
-tT=t
subject to his budget constraint (3).
The problem can be posed in terms of
dynamic programming where the value function is the maximized value of the
right hand side of (5),
(5)
max
V(wt, Tt) =
{u(ct) +
afV(w
,
Tt+ )dF(rt Tt+ ))
Ct kt+l'st+l
for all t, and taxes are assumed for simplicity to follow a first order Markov
process.
(6a)
The first order conditions are given by
u'(c ) = X
t
(6b)
BEt(At+,rt+,)
(6c)
6Et
t+l(qt+1
t
t+l)
tqt
where X is the LaGrange multiplier associated with the time t budget
constraint.
The transversality conditions associated with capital and equity
are
(7a)
limE ~tuI(c+)k +
(7b)
4%mEtaTu (C+)qt+
=+
= 0
Equation (6a) implies that the marginal utility of consumption
equals the marginal utility of wealth.
That is, individuals are indifferent
- 7-
between consuming or holding an extra unit of wealth.
Equation (6b) states
that the discounted value of next periods marginal utility of wealth times the
after tax marginal productivity of capital equals the marginal utility of
wealth.
This means that at an optimum the individual is indifferent between
investing in an extra unit of capital and holding some more wealth today.
Equation (6c) is the difference equation determining the price of equity.
It
implies an indifference at the margin of selling equity today and holding the
equity and selling it next period.
III.
EQUILIBRIUM
a. A Particular Solution
Since the main concern of the analysis is to examine how various
realizations of tax rates affect economic activity, the remainder of the paper
will examine a particular functional form for which a closed form solution
exists.
In particular the utility function u(c) = ln(c) and the production
function f(k) = k , 0 < a < 1, will be investigated. The closed form solution
to this problem is nontrivial and involves the solution of a set of nonlinear
difference equations.
The equilibrium conditions that characterize the model are
(8a)
st = I for all t and
(8b)
ct + kt+i = f(kt) for all t.
Given these condtions, the Euler equations 6(a)-6(c), the firms profit
maximization condition, the budget constraint (3), the transversality
conditions and the assumption that taxes follow a first order Markov process
- 8 -
the equilibrium stochastic processes for {c t+j}
{r
0t
co
c}o , and {q
}i
can be constructed.
t+J J=o
t+j j=0
,
{k t+j+1}
These processes simultaneously
satisfy both profit and utility maximization as well as market clearing in the
capital, output, and securities markets.
Without the remittance of taxes (i.e. if tax proceeds were
destroyed) this problem would be equivalent to the one solved by Brock (1979)
where the production function was subject to multiplicative productivity
shocks.
In that case the fraction of output allocated to investment would
equal a$
and be independent of the stochastic process followed by taxes.
The
assumption that all tax proceeds are remitted lump sum is, to some extent,
like assuming that government spending is valued exactly like consumption.
This allows one to ignore the effects of government spending and to isolate
the effect of taxes.
Also, the remittance of these proceeds implies that it
is the compensated effects of consumption that are being analyzed, and that
even in-the presence of log utility expectations of future taxes will be an
important determinant of current decisions.-/
An intuitive guess regarding the decision rules governing
consumption and investment is that each is a fraction of output and that these
fractions are functions of the current realization of the tax rate (past
realizations would be important for Markov processes of higher order).
particular kt+
(9)
Y(tt )
=
y(Tt)f(kt) and ct = (-y(Tt))fk
a6g(l-Td)
t)
1+cag(1-Tt)
and g(1-Tt) is given by the recursive relationship
where
In
- 9 -
t)
=(1-nt+1)5
gt(
Tt+1)
a
That this is a solution to economy wide equilibrium is given in the appendix.
The closed form solutions for ct and kt+I indicate that the mixture
between consumption and investment is based on the conditional expectation of
the entire path of future taxes.
The expected path of future taxes is
relevant since it affects the value of future capital and the amount of
consumption that can be purchased from the sale of capital to firms.
IV.
Solutions for Specific Stochastic Processes.
Further intuition regarding the economic effects of tax rate change
can be gained by looking at the results obtained when taxes follow a
particular stochastic process.
To highlight the difference between permanent
and transitory tax rate movements both a white noise and simple two state
first order Markov process will be used.
The behavior of investment, equity
prices, and the real rate of interest differ quite markedly under the two
assumed distributions.
These examples, therefore, clearly illustrate the
importance of correctly specifying the stochastic process for taxes if one is
to have confidence in the derived consequences of tax rate changes.
Let the transaction probabilities for the first order Markov process
be given by
TO | Tt
(11a)
prob (Tt+I
(11b)
prob (Tt+= T,
1
where O<T
<T1
<
1.
O
o
Tt = T) =P
Given the discussions in Barro (1979) and Lucas and
Stokey (1983), concerning the use of government debt to smooth tax distortions
-
10 -
over time, one would expect the tax rate to show a good deal of persistance
with both p
and p1 being greater than 1/2 and perhaps close to one.
Further,
the qualitative results generated using the simple process described in (11a)
and (1lb) are not altered by using a Markov process having more than two
states.
Therefore, the qualitative results yielded by this example are of
general interest.
(12)
g(1-
Taking advantage of the recursive nature of g(1-Tt) yields
-
(1-Tj)pj + (1-Ti) (1-pi) - a (J-Ti)(1-T)6
1-ac[(1-T.)p. +
for i,j
=
0,1,igj, and 6 = Pi+ p.
-
(1-T
)p.] + a
2[(l-T )(1-T.)6
1.
Now if taxes are distributed as white noise with mean
g(1-Tt)
a.
=
T,
then
In this case the function g is a constant.
(1-T)/(1-cS(1-T)).
Consumption and Investment
The behavior of consumption and investment are analyzed under the
two different processes for tax rates.
Regarding the Markov process one
observes that
(13)
g(I-
g(
1
)
A[
(T
where A is the denominator in (12).
T )
(p 4pl+
-1) ]
Equation (13) implies that g(1-T )
>
g(1-T1 ) if p + p1 > 1, that is if tax rates are likely to persist over time.
From the definition of y(T) this means y(T )>y(TI).
Hence, a greater fraction
-
11
-
of output will be invested in the low tax state, if the low tax state implies
that taxes are more likely to be low in the future.
Alternatively, if tax rates were white noise with a mean of
the fraction of output devoted to investment would be aa(1-T).
then
T,
This fraction
would be independent of the current realization of taxes.
A comparison between these two process, a first order Markov process
with p
+ P1 > 1 and a white noise process, points out the problems that
arise if one were to simply assume that agents have perfect foresight.
The
behavior of investment for some given realization of taxes depends crucially
on the distribution that tax realizations were drawn from.
As will be shown
in more detail in Section V, arbitrarily assigning expectations to equal
actual realizations may be misleading and certainly affects the quantitative
results of the analysis.
b.
Security Prices
Equation (6c) represents a first order difference equation determing
security prices.
Employing the assumptions regarding the utility and
production functions as well as (1) and (6b) yields
~~~~~~~~~~~~~ciE0B~~~~~~~~.
&E
(14) q
(14
=
t jo
tc
E
= 6(a)c
Z
tjE
&3g(1t)
which is a complicated expression involving expectations of future tax
rates.-/
As in Brock (1979) the asset price represents a return to the
technology k
and is directly related to profits share of output, (1-a), as
well as to the discount rate 6.
For the case where taxes follow a simple first order two state
Markov process described by (Ila) and (lib) equation (14) becomes
-
12 -
n 1-6n
(15) q (--r.
-t) c
=
l
(1- P
n
+1
(1-T)
M -p ) £
I-a
1
618(1
for i,j = 0,1 and icj.
Notice that q(1-T )
q
equals 1-
(1-T
[g(1-T
which is positive if taxes have some persistance.
)
-
g 1-a)cM
Therefore, stock prices are
higher when a low tax state is realized.
If, instead taxes were generated as a white noise process then stock
prices would not vary with tax realizations. Thus in order to forecast stock
prices one must know the process generating tax rates and not merely some
assumed realizations.
c. The Real Rate of Interest
Calculating the equilibrium real rate of interest, p, is
accomplished by considering the price 'p
of consumption next period.
of a bond ,R, that yields one unit
This is done by adding Pt Rt+I to the LHS of (3)
t+
and Rt to the definition of wealth.
The resulting first order condition with
respect to Rt+I is
(16) aE X
t t+1
=
PR
tt
Equation (16) implies that an individual is indifferent at the margin between
sacrificing Pt units of wealth today for one unit of wealth next period.
Using the expressions for consumption and investment and the fact
that 1/(1-Y(T )) = 1+a~g(1-Tt) implies that
1
(17)
1-I-n
=
t
l/ct
pR
BE (1/c
tt+tt1
_
)
l+ag(l-Tt)
SEt(1+cag(l- t+ )
Y(T )a ya-l
t
t
)
- 13 -
For the case where
T
follows the process given by (1la) and (11b) and where
taxes show permanence (p
,pi
>
1/2) it can be shown that the real rate of
interest is higher when a low tax state is realized.-/ This occurs because a
lowering of the tax rate indicates that capital will be more valuable in the
future and that there will be more future output.
Individuals will,
therefore, value a unit of future wealth by less, causing pt to fall and 1+p
~~~~~~t
t
to rise.
Alternatively, individuals will wish to accumulate more capital.
order to induce lower consumption, the real rate of interest must rise.
previous cases, when
T
In
As in
is distributed as white noise the real interest rate
does not respond to movements in tax rates.
V.
A Numerical Example
In this section, a direct comparison is made between capital stock
accumulation under perfect foresight and situations where agents behave under
uncertainty.
To perform the experiment 20 realizations of tax rates were
generated using a random number generator and the actual transition
probabilities associated with the effective tax rate series given in Hulten
and Robertson.-/ In terms of (11a) and (hib),
p1 = .9.
T
= 1/4, T1
=
1/2, and po
The realizations and resulting levels of the capital stock are given
The starting value of the capital stock is the value that would
in table 1.
result if T
=
.375 (the average value of the tax rate) for all time.
In
calculating the values under the assumption of perfect foresight it was
assumed that g(1-T2 1 ) = .60.
This assumption is important regarding the
solution for the capital stock in period 20, but the importance of the end
point constraint quickly vanishes.
From table 1, it is clear that the capital stock under uncertainty
behaves in a quantitatively different manner than it does under perfect
foresight.
The movements in capital tend to be smoothed out by uncertainty
-
14 -
since there is always some positive probability that next periods tax rate
will be different from todays.
Also, under perfect foresight agents respond
one period sooner to tax rate changes than do agents who are unsure of the
value of next periods tax rate.
This difference in behavior would also occur if taxes were assumed
to follow a white noise process.
Under uncertainty the level of the capital
stock would remain at .095 independent of the actual realizations of taxes,
while with perfect foresight the capital stock would respond considerably.
Therefore, if one is to accurately predict how agents will respond to tax rate
changes, one must carefully consider the forecasting problem facing agents.
In order to do this requires an explicit stochastic treatment of the problem.
VI.
Summary
This article analyzes the effects of corporate taxes in a simple
stochastic growth model.
The model is able to produce responses that are
consistent with many of the observed correlations between economic activity
and effective tax rates on capital.
The paper also represents an advance
since it treats tax rates as inherently stochastic. As shown, the actual
process generating taxes is an important determinant in understanding how the
economy will behave with respect to particular realizations of tax rates.
One might also wish to explore the interaction between tax changes
and nominal magnitudes such as inflation and the nominal interest rate.
Extending the model to incorporate money via a cash in advance constraint is
not a problem.
The basic solution governing the real side of the economy is
unchanged if the cash in advance constraint is only on consumption and if one
rules out precautionary demands for cash.
This is easily done so long as
monetary growth is not overly deflationary. The solution is only slightly
changed if the cash in advance constraint includes capital as well.
In these
-
15 -
cases a decrease in taxes causes capital accumulation and an increase in
output.
For given money growth the price level falls and the economy
experiences lower inflation.
Since little is to be gained through these
additions the paper concentrates on real economic variables.
- 16 -
Table 1
time period
tax rate
y(Tt) under
kt+l under
perfect foresight
perfect foresight
0
y(Tt) under
uncertainty
.095
kt+, under
uncertainty
.095
1
.50
.167
.076
.18
.082
2
.50
.167
.071
.18
.078
3
.50
.167
.069
.18
.077
4
.50
.167
.068
.18
.077
S
.50
.167
.068
.18
.076
6
.50
.167
.068
.18
.076
7
.50
.169
.069
.18
.076
8
.50
.180
.074
.18
.076
9
.50
.250
.105
.18
.076
10
.25
.250
.118
.24
.102
11
.25
.250
.123
.24
.112
12
.25
.250
.124
.24
.116
13
.25
.250
.125
.24
.117
14
.25
.250
.125
.24
.117
15
.25
.245
.123
.24
.118
16
.25
.231
.115
.24
.118
17
.50
.167
.081
.18
.089
18
.50
.167
.072
.18
.080
19
.50
.167
.069
.18
.078
20
.50
.167
.068
.18
.077
-
17 -
FOOTNOTES
1/
With a log utility function income and substitution effects can be
shown to be exactly offsetting. The remittance of taxes implies that we are
only interested in analyzing substitution effects.
Since substitution effects
are usually viewed as being of primary importance in most macroeconomic
analysis the examination of the compensated effects of tax changes is more
relevant under the log utility specification.
2/
The second equality in (14) is derived using (1) and (6b).
Using the
equation for profits (1) and the consumption function,
1-Tt
Xr
t+/ct+C
(1-a) 1y-( ~~t+1 )T
t+1 t+1
Equation (6) leads to the difference equation
1-T
t+l
ct J
Tt+1
1-Y(Tt)
t
From the definition of g(1-Tt) the latter expression is also equal to
a~g(1-T ).
-/
The proof of this relies on the fact that if po, p1
>
1/2 then
g(1-t0 ) > g(1-TI) and that
1+acg(I-T0 )
aEg(1-Tt+I Tt=To)
1+acg(I-T).
The proof of the
S(l+ae Eg(l-Tt+l Tt=Tl)
latter inequality involves some cumbersome algebra and is omitted.
- 18 -
-/
The exact procedure was to look at the Hulten and Robertson
effective tax rate series as a realization of a two-state first order Markov
process and use the calculated sample transition probabilities and sample
means.
Then, 20 random numbers between zero and one were generated.
It was
assumed that the initial tax rate was high and that a number between 0-.1
implied a change in the tax rate while a number between .1-1.0 implied that
the tax rate remained at its previous value.
- 19 -
Appendix
In this appendex the solutions for consumption, ct = [l-y(.t)f(kt)
and kt+1 = y(Tt)f(kt), where f(k ) = ka, u(c ) = ln(c ), and T
t
t
t
t
is a first
order Markov process are shown to be an equilibrium solution to the model.
Proof
Equation (6a) implies that Xt = 1/ct for all t.
Substituting for
rt+1 from the firms profit maximization condition,
rt
t+1
= (1-Tt ))a k atI into (6b) yields the difference equation
t+1
t+1
t l-tt+1
t+l
~ ct+1~k +i t+
tOtOE
=
1.
Using the postulated solution for c and k gives
1-X (T[t)
a
t
Y(Tt)
1
Trt+l
1-y(Tt+1)
The solution to this nonlinear first order difference equation is
aag(Ir1t)
Y(Tt
1+a~g(l-t)
Making this substitution and the fact that g(1-Tt) is known at time
t results in
1
E (1-T
I
one notes that g(l-Tt) satisfying g(1-Tt)
E t(1-Tt+1)(1+ag(l-T +))
satisfies this conditions. The function g(1-T ) can also be written
- 20 -
as
g(-
=t
=
(1E
t+[)[11+aaEt+[(1-Tt+2)[
++aE
[
Now given this solution it is easy to see that equilibrium in the goods market
is satisfied. The transversality condtion (7a) is also satisfied since the
conditional expectation of
is finite for all j.
(
g1Tt+j)
The solution for qt given in (14) satisfies (6c) as well as the
transversality condition (7b).
The solution also satisfies the consumers
budget constraint. Therefore, the constructed solution maximizes individual
utility, clears both the goods and asset markets, and maximizes the firm's
profits.
It therefore represents a competitive equilibrium.
- 21 -
REFERENCES
1.
Abel, Andrew B., and Olivier Blanchard.
Saving and Investment."
2.
Barro, Robert J.
"An Intertemporal Model of
Econometrica 51 (May 1983), 675-92.
"On the Determination of Public Debt."
Journal of
Political Economy 87 (October 1979), 940-71.
3.
Brock, William A.
"Asset Prices in a Production Economy," in
The Economics of Information and Uncertainty, edited by John J.
McCall, The University of Chicago Press, Chicago, 1979.
4.
Goulder, Lawrence H., and Lawrence H. Summers.
Prices, and Growth:
"Tax Policy, Asset
A General Equilibrium Analysis." National
Bureau of Economic Research, Working Paper No. 2128, January 1987.
5.
Hall, Robert E.
"The Dynamic Effects of Fiscal Policy in an Economy with
Foresight." Review of Economic Studies 38 (January 1971), 229-44.
6.
Hulten, Charles R., and James W. Robertson.
Economic Growth:
"Corporate Tax Policy and
An Analysis of the 1981 and 1982 Tax Acts."
Discussion Paper, The Urban Institute, December 1982.
7.
Lucas, Robert E., Jr.
"Econometric Policy Evaluation: A Critique."
The Phillips Curve and the Labor Market, edited by K. Brunner and A.
Meltzer, Vol. 1 of Carnegie-Rochester Conference Series in Public
Policy.
8.
Amsterdam: North Holland, 1976.
and Nancy L. Stokey.
"Optimal Fiscal and Monetary Policy in an
Economy without Capital." Journal of Monetary Economics 12 (July
1983), 55-94.
9.
Sargent, Thomas J.
1979.
Macroeconomic Theory.
New York:
Academic Press,
@FedFRASER