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On the effects of macroeconomic policy in a maximizing model of a small open economy
ROBERT J. HODRICK Carnegie-Me/h University International Monetary Fund
On the Effects of Macroeconomic Policy in a Maximizing Model of a Small Open Economy This paper investigates the effects of government policies in a small open economy. Determination of the exchange rate and the effects of macroeconomic policies on the balance-of-payments accounts are examined in a model of infinitely lived, utility-maximizing agents who have perfect foresight. Residents of the country and the government borrow and lend in a world capital market which constrains them to live within their means. The dependence of consumption, leisure, and real-balance holdings on the present discounted value of government expenditure and real debt is demonstrated. Government policies must be dynamically consistent.
1. Introduction This paper investigates the effects of government policies in a small open economy. The macroeconomic issues of the determination of the exchange rate and the effects of policy on the balanceof-payments accounts are examined in a model of infinitely lived, utility-maximizing agents who have perfect foresight. The model is similar to those of Kouri (1976), Flood (1979), Boyer and Hodrick (1982), and Obstfeld (1981) in that agents consume a single consumption good whose foreign price is given and have access to a world capital market in which they can borrow or lend in real terms. As in Obstfeld’s model, the agents are placed in a maximizing framework although this analysis postulates a constant rate of time preference. The development of the model in Section 2 follows straightforwardly from Mussa (1976). The dependence of agents’ current decisions about consumption and leisure and their current demands for real balances are demonstrated to depend on the present discounted value of government spending. It is also demonstrated that the present discounted value of government spending plus the current real value of the government debt must be financed by the present discounted value of government revenue from wage income taxation and the inflation tax. Government policies must be dynamjournul Copyright
of Macroeconomics, 0
1982
by Wayne
Spring State
1982, Vol. University
4, NO. Press.
2, pp.
195-213
195
Robert J. Hodrick ically consistent in the sense that only two of the three present values discussed above are independent. The specification of a time path for government spending and a rate of growth for the money supply determines a present value of wage taxation. This result generalizes the discussions in Barr-o (1979) and Kydland and Prescott (1980) to allow for the inflation tax. Section 3 analyzes the equilibrium of the model with constant government policies, and Section 4 considers the response of the economy to a change in policy. Tax-financed increases in government spending depreciate the exchange rate and cause a decrease in consumption and an increase in leisure. The effects of an increase in the rate of monetary growth are shown to depend on the elasticity of demand for real balances with respect to the rate of inflation since wage taxation must go up, remain the same, or go down as the demand for real balances is elastic, unitary elastic, or inelastic. Section 5 examines anticipated increases in policy and the results are shown to depend critically on when taxes are changed to balance the budget. That depreciation of the exchange rate in excess of the rate of monetary growth and a current account surplus are not inconsistent in a perfect foresight model is demonstrated in this section. Section 6 contains some conclusions.
2. The Model Consider a small open economy that takes as given the foreign currency prices of all goods. For the purpose of this analysis, these commodities can be aggregated into a single composite good whose foreign currency price can be set equal to one. Under free trade the domestic currency price of the commodity, P, is equal to the exchange rate: P=S
(1)
where
S is the domestic currency price of foreign exchange. Output of the good in the country is produced by competitive firms which use only the labor input of households. The production function is Y
=fb)
(2)
Effects
of Macroeconomic
Policy
where y is the value of output and f is nonnegative and homogeneous of degree one. Firms take nominal wages, W, and prices as given and maximize profits. Consequently, the marginal product of labor equals the real wage,
f’(n) = W/P = w , and the value of output
equals the return
(3) to labor,
y=wn.
(4)
Households are assumed to maximize the present discounted value of an instantaneous utility function. The representative households instantaneous utility is modeled as a separable function of consumption, c, leisure, e, and real balances, rn:l
U(c,&m)
= u(c) + h(l) + v(m)
(5)
where real balances are defined by deflating the households nominal money balances, M, by the price level, m = M/P. The functions u, h, and u are taken to be nonnegative, strictly concave, and twice continuously differentiable, with the properties that lim u’(c) = lim h’(8) = lim u’(m) = O”. C-W e-m m-m
(6)
In addition it is assumed that there is nonsatiation in consumption, leisure, and real balances implying that the derivatives u’, h’, and 8’ are strictly positive. It is assumed that the only assets available to residents of the country are domestic money which is not held by foreigners and an internationally traded bond which has constant purchasing power and a rate of return equal to r *. Households are allowed to borrow and lend at this constant foreign interest rate, and net bond holdings are denoted b.
‘The convention of placing real balances in the utility function in order to capture the nonpecuniary services of real balances in a tractable way as in Sidrauski (1967) or Brock (1974) has been criticized by Wallace (1980) and others. See Kareken and Wallace (1980) and especially the paper by Lucas (1980) for a discussion of these issues and alternative modeling strategies. 197
Robert J. Hodrick At each moment in time households must allocate their real market wealth, a, between real balances and real bond holdings:
a, = m, + b, .
(7)
Additionally, the household must decide its labor supply and the allocation of its income between consumption and the accumulation of market wealth. Labor input, n, plus leisure, 4, equals total time available to households which is normalized to one. Household’s disposable income is the sum of its labor income and the expected returns on holding market assets minus the taxes it pays to the government. The returns on assets are r*b for net foreign assets and -rm for real balances where n is the expected rate of inflation. The government is assumed to tax labor income at the proportional rate 7. Planned real saving at time t is therefore d, = ton, (1 - rt) + r*b, - ntmt - c, .
03)
The final constraint which binds the households choice problem is imposed by the world capital market. As Arrow and Kurz (1969) and Mussa (1976) note, the natural feasibility condition on the households plan is that its market indebtedness at any point in time, -a,, must not exceed the households ability to repay its debts out of its future wage income. Formally, this requires that cc
-a, I
[zon,(l - 7,)]e-r*(s-f) a!s I
(9)
t
for all t, where the right hand side of (9) is nonasset wealth at time t. Since r* is taken to be a constant for all time, the discount factor e-rr(s-t) measures the market price of a unit of real consumption at time s in terms of real goods at time t. The household’s choice problem is m
e -B U(c,, C,, m,)dt
subject to the constraints 198
(7), (B), and (9) where
the discount
rate
Effects
of Macroeconomic
Policy
6 is taken to be a constant.2 In solving (10) households take the paths of the price level and the tax rate as given. It is clear from (9) that the world capital market must also assess the paths of taxation and labor supply in order to constrain the household not to live beyond its means. Without an explicit consideration of uncertainty the only sensible assumption regarding the formation of expectations of the future by households and by the world capital market is to assume that agents in the model have perfect foresight. Under this assumption households and the capital market are assumed to know how government policy is conducted and to understand that the market-clearing process will result in prices that are expected to occur given that the demands for consumption goods and real balances and the supplies of labor are derived from (10) and are based on the same expectations. Maximization of (10) subject to (7), (8), and (9) requires that households choose, c, 4, m, b, and the shadow price, 5, at each instant to maximize the current value Hamiltonian:
H = u(c) + h(t) + u(m) + t(a - m - b) + X(w(1 - e)(l - ~)+r*b--m-c) where all variables are evaluated ditions are
at time s 2 Oa3The first-order
(11) con-
u’(c) = A,
(12)
h’(e) = Xw(1 - T),
(13)
u’(m) = 5 + XIT,
(14)
r-*X = 5,
(15)
-b=o.
(16)
and a-m
‘Obstfeld (1981) considers a similar problem, but he employs the variable-discount-rate model of Uzawa (1968). See Section 4 for a discussion of the differences in the results arising from the alternative assumptions. The solution of the household’s intertemporal allocation problem follows the presentation of Mussa (1976) relatively closely. 199
Robert J. Hodrick Equations (12) through (16) determine the currently determined variables, c, e, m, b, and 5 as functions of the state variable, a, and the costate, A. The next step in the solution of the households intertemporal allocation problem is to determine the time path of the state and the costate variables up to the initial value of the costate, A,. As Mussa (1976) demonstrates, a useful alternative representation of the transition law for the state, (8), is obtained by adding the implicit value of the nonpecuniary services of real cash balances to both income and consumption. The implicit value of these services is the level of real balances multiplied by the opportunity cost of holding cash rather than real foreign assets which is the difference in the two rates of return, r* + T. Defining total consumption as the sum of consumption of goods and the services of real balances, 2, = c, + (r* + 7r,)m,, the transition
law for the state variable
(17)
can be written
as
ci, = r*a, + ~(1 - e,)(l - 7,) - 2, .
(18)
The solution of (18) is
t { I
a, = e’*t a0 +
[w(l
0
- e,)(l - 7,) - z,]e-r*s a!s I
(19)
where a, is the historically given level of real market wealth at time 0. The costate variable must satisfy
i, = 6X, - g
= (6 - ‘*)A, ,
(20)
which has the solution, A
t
=
Aoe@-‘*‘)’
where A, remains to be determined.
>
Clearly,
(21) since the household
take w, r*, T,, and T, as given, and since c,, e,, and m, are functions of a, and A,, the paths of a, and A, are fully determined by
Effects of Macroeconomic
Pohcy
(19) and (21) given the initial values of a,, and A,,. The initial value of a, is given and the value of A, can be determined from the transversality conditions which are necessary conditions for an optimal path. These conditions require that lim e+ t--r-
A, 2 0 ,
(259
and lim ea8” Atat = 0 . t-m
(23)
From (21), (22), and the first-order conditions, A, > 0 is required since otherwise consumers would attempt to consume an infinite amount without working in violation of the lifetime budget constraint. Substituting (19) and (21) into (23) gives
’ [w (1 - e,)( 1 - 7,) - zS]e-r*s o!.s = 0 , which
implies
that m
cc I0
(24)
=,e -I-*’ & = a, +
w(1 - C,)(l - rS)e-‘*’ ds . I0
(25)
Equation (25) demonstrates that the initial value of A, will be chosen such that the present discounted value of lifetime total consumption is equal to the sum of initial market assets and initial nonasset wealth. This is an equivalent expression to (9) and demonstrates that (9) will hold with equality along an optimal path. From the first-order conditions we know that the left-hand side of (25) is monotonically decreasing in A, while the right-hand side is monotonically increasing in A,. Consequently, if an equilibrium exists, it is intuitively clear that it will be unique.4
4As Arrow
and
Kurz
(1970)
demonstrate
in Proposition
11.8,
when
the
optimized
Hamiltonian is concave in the state variable as it is here, any policy that satisfies the first-order conditions and the transversality conditions is an optimal policy. Existence of an optimal policy depends on convergence of the integrals in (25) which in turn depends on the specification of government policy. If government policies are constrained such that the integrals converge, the policies possess “process consistency” in the terminology of Flood and Garber (1980). 201
Robert J. Hodrick From (21) and consideration of the first-order conditions, if 6 > r* and if an equilibrium exists with constant government policies, the equilibrium would be characterized by monotonically shrinking consumption and increasing labor supply. For 6 < r*, consumption and leisure would increase over time as the country became wealthier. Only if 6 = r* is A, ever a constant, and in this case, it is unchanging at A, = A,. In what follows, this latter case will be the only one considered.’ Determination of the equilibrium time path for the economy requires a specification of the government sector. The government is assumed to buy goods in amount g, to collect taxes, and to issue money. 6 The government also is allowed to borrow and lend at the world real interest rate. Its flow budget constraint is g, + r*bf
= wntTt + 6: + ptm,
(26)
where bf is the level of real government indebtedness and pt = A&/M, is the rate of growth of the nominal money supply. The world capital market will also constrain the government to keep its indebtedness to be less than or equal to its ability to repay which implies that
which must hold for all t. From ment debt evolves according to
(26) we know that actual govern-
%ince there is no capital in the model, a perpetual divergence between the discount rate and the real interest rate is possible. See Bazdarich (1978) and Fischer and Frenkel (1972) for models of trade in debt with capital accumulation. ‘In what follows no consideration will be given to the optimal@ of government policy or to how government spending is determined. Since government spending does not feed back into the model, it is implicitly assumed that the government sector is too large relative to some optima level because agents would be better off by a reduction in spending and taxation. It is also assumed that the government does not have access to distortionless head taxes. 202
Effects
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Policy
where b$ is an initial level of indebtedness. Multiplying (27) and (28) into (27), rearranging terms and let(28) by e-‘*‘, substituting ting t + 00 yields
bf, +
g,e -‘*’ d.s zs
(wn,T, + p8mS)e+g a!s ,
(29)
which indicates that the present discounted value of government expenditure plus the initial government indebtedness must be less than or equal to the present discounted value of the government’s tax collections plus its revenue from money creation.’ Equation (29) will hold as an equality when the government does not plan to accumulate wealth. Consideration of (26) and (29) indicates that at a point in time the government has three degrees of freedom in determining its four actions of spending, taxation, money creation, and debt creation. Nevertheless, the existence of a rational capital market implies that when these four actions are viewed as policies which will persist into the indefinite future there are really only two degrees of freedom. The prohibition against perpetual debt finance imposed by the capital market in (27) combined with the necessity of specifying government policies or actions at each point in time if agents are forward looking implies a policy constraint which binds government policy makers.’ Substitution of the government’s flow budget constraint (26) into the households budget constraint (8) gives bt - 6: = writ + r*(b,
- bf) - g, - C, ,
(39)
which demonstrates the equality of the capital account deficit and the current account surplus under flexible exchange rates. Since
‘Equation (29) including revenue
generalizes from money
Barro’s (1979) constraint on public creation. While the existence of
indebtedness government
by debt
implies that some future taxes will be used to finance it, which specific taxes are used is a choice variable of the government. *It is obvious that actual governments have not sufficiently specified their policies that agents know the future with certainty. The introduction of uncertainty into future policy as well as the existence of nominal government debt would increase the relevance of this paper without altering the necessity that policy must be specified in a dynamically consistent manner. 203
Robert J. Hodrick (30) must hold at all times, multiplication tegration with respect to time gives
of (30) by e -‘*’ and in-
co
cc --T*s
bo
+
bn,
which, when combined
-
&
=
bg
+
-‘*’
&
,
(31)
with (29) and (25), yields
m
m
(r* + 7F,)mse-‘*” 0% = m. + I
g,e
de
0
o (psms)e-r*s d-s .
(32)
I
These equations demonstrate an interesting separation of the real and the monetary sectors of this model. While these constraints will be satisfied along a perfect foresight path, as Mussa (1976) notes, households must still be thought of as maximizing the problem in (10) since otherwise total wealth would not be the constraint on their behavior. We can use (31) and (32) to characterize the solution since along a perfect foresight path such a division will occur.
3. System Equilibrium In this section we study the properties of the model for particularly simple government policies. Given initial values of private foreign assets, b,, government indebtedness, b& and nominal money balances, and under the assumption that real government expenditures are constant at g and that the rate of monetary growth is constant at a; (12), (13), (14), (29), (31), and (32) can be solved simultaneously for values of C, 2, tii, ti, X, and t which are both the instantaneous and the long-run equilibrium values of these variables. The equilibrium relationships after solving for i are the following:
r*(b,
h’(Z) = U’@)W(l - ‘i) )
(33)
o’(fi)
(34)
= u’(E)(r*
- b$) + ~(1 - 2) - E = g,
r*bg+g=pfi+yw(l-Z), 204
+ fr) ,
(35) (36)
Effects
of Macroeconomic
Policy
and
(r* + 5) $ = ci + p vii/?-*.9
(37)
The equilibrium relationships have straightforward interpretations. From (37) we see that the actual and the expected rates of inflation are equal to the growth rate of the money supply. From (33) and (34) we find that the marginal rate of substitution of leisure for consumption is equal to the net of tax real wage and that the marginal rate of substitution of real balances for consumption is equal to the opportunity cost of holding money. Equations (35) and (36) indicate that the current account is in equilibrium and that the government budget is balanced in the sense that no new debt is being issued. The two sources of government revenue, taxation of labor income and inflation tax, just offset government spending and interest on the outstanding debt.
4. Alternative Government Policies In this section we consider the effects of changes in government policies. Two interpretations can be given to the analysis. First, one can think of the exercise as determining how a second economy would differ from the first economy studied in the previous section if the second economy had larger values of government spending or a faster rate of monetary growth. Alternatively, one can think of the analysis as determining how the economy would react to surprises in government policy which are expected to be permanent. While the latter interpretation will be used in this study, it is clear that specification of a stochastic environment is necessary before one can adequately discuss unanticipated changes in policy. lo
‘Equation (35) demonstrates that government debt is subtracted from private wealth in this model. An increase in the debt must be offset by future taxes, but either the labor income tax or the inflation tax can be used to finance it. “It is common in the literature to confront agents who have perfect foresight with an “unanticipated” change in policy and to examine the movement of endogenous variables along the new perfect foresight path. With postulated behavioral functions such experiments can be defended by asserting that behavior depends only on the expected value of a variable. This result will also be produced by assuming that agents maximize the expected value of a quadratic utility function.
205
Robert J. Hodrick Now consider the effects of a change in government spending policy that is financed by a change in the tax rate. The effects on c, e, m, and T are the following: (33)
de -= & dm dg=
-uy+
(39)
p)w(h”(l
- 4) - u’w(l
- $})
(49)
and
d7 -= 4s
1
i {UnuRw2(l - 7)’ + h”u”(r*
p)p - B”}
(41)
where
and 2 = w(1 - e)(l - 7) . For small values of T and t.~, the fundamental determinant, A, will be positive. Consequently, when the government increases spending, consumption falls, real balances fall (that is, the exchange rate depreciates) and the tax rate increases. For a utility function such as the exponential, leisure will increase since the term in brackets
10 continued from p. 205) Since explicit introduction of uncertainty and maximization of expected utility would have complicated many of the points in the previous section, I avoided those problems at the cost of straining the interpretation of the model in this section. Nevertheless, it remains my conjecture that the thought experiments provide sensible answers regarding the direction of response to questions of how agents would respond to permanent changes in government policies in a stochastic environment. Obviously, a superior approach to the subject would be to tackle the stochastic specification directly. (Footnote
206
Effects
of Macroeconomic
Policy
in (39) will be negative. The substitution effect of the increase in taxes outweighs the wealth effect which decreases consumption and would tend to decrease leisure. Since the instantaneous values are the long-run values, the economy moves immediately to its new long-run equilibrium. These results are similar in spirit to those of Kouri (1976) and Boyer and Hodrick (1982) w h o h ave investigated the issue in a nonmaximizing framework. In those models a tax-financed increase in government spending creates a current account deficit, a loss of wealth, and lower (long-run) consumption. The results differ markedly from Obstfeld’s (1981) maximizing model which employs the variable rate of time preference model of Uzawa (1968). In Obstfeld’s model an increase in government spending does not change the steady-state level of consumption but causes consumption to fall in the short run by an amount sufficient to generate a current account surplus which allows private foreign assets to increase to finance the increased taxes. While the variable time preference model provides more interesting dynamics than this model, whether the specification is more appropriate than assuming a constant rate of time preference cannot be determined by a priori reasoning since the appropriateness of the specification depends on the nature of the time preference in the intertemporal utility function. Turning now to consideration of an increase in the growth rate of the money supply, the changes in the equilibrium values of the variables are the following: 0% -= 6
-ul)u~w2 (42)
A
(43)
’ dm -= dp
zur’ufw2 A
1 + h”(1 - 4)
(44
U”Wlj
and I d7 - -i) (u”w2(1 &A
- 7) + ,“)
From (42), (43), and (45), if m = %,
neither
consumption,
lei207
Robert J. Hodrick sure, nor taxation changes. Substituting this value of m into (44) dm ’ implies = u from which it is easily seen that in this case the dp u” elasticity of demand for real balances with respect to the rate of inflation is equal to minus one. When this is the case, a change in p changes m equiproportionately leaving the government revenue from inflation,
pm, unchanged.
If m > $$,
the demand
for
real balances is inelastic with respect to the rate of inflation, and an increase in l.~ increases government revenue from the inflation tax causing a fall in the tax rate on labor income, an increase in consumption, and a decrease in leisure. When the demand is elas-u’ p tic, m < an increase in lo decreases government revenue v” ’ from inflation causing the tax on wage income to rise in order to balance the government budget which increases leisure and decreases consumption. Regardless of the magnitude of the elasticity, an increase in lo, causes a fall in real balances which corresponds to an increase in the exchange rate. Since there are no further dynamics in this model, an increase in lo does not cause overshooting of the exchange rate as in the models of Kouri (1976), Flood (1979), Boyer and Hodrick (1982), or Obstfeld (1981). In those models real balances decline by more in the short run than in the long run and the economy accumulates foreign assets as it moves to the new steady state. The rate of depreciation of the exchange rate is less than the new rate of growth of the money supply. Here, after the initial jump in the exchange rate, it depreciates at the new F.
5. Anticipated Changes in Policy In the previous section changes in government policy were unanticipated and permanent. There were no current account reactions to changes in policy since the economy merely chooses new long-run equilibrium values instantaneously. In this section we consider the behavior of the economy in response to anticipated changes in policy. Suppose that at some point in time TO the economy expected the government to increase its spending at some time T, in the future. How do the consumption, leisure, and real balance deci-
Effects
of Macroeconomic
Policy
sions of the economy change and how does this affect the exchange rate and the accounts of the balance of payments? From consideration of the results of the dynamic maximization problem we know that the agents will choose new constant lower levels of consumption and of real balances implying an instantaneous depreciation of the exchange rate. The time path of leisure depends on when the government raises taxes to finance the increased government spending. If they are raised at T,,, a new higher level of leisure will be chosen immediately. If taxes are raised at T,, people will initially choose to work harder, prior to the tax increase, then after T,, agents will increase leisure. It is clear from (SO), which must be zero after T,, that the current account will be in surplus prior to T,, since after T,, g will be increased and n will either remain constant or be decreased to a lower constant level. Consider now an expected increase in the rate of growth of the money supply that will occur at T,. Since the exchange rate cannot be anticipated to jump, it must depreciate immediately at T,, and move continuously thereafter. If there are no changes in government policy after T,, the economy must be in its steady state at that time which means that real balances will be constant at I’, and thereafter. Between T,, and T, the government revenue from inflation is declining since the exchange rate is depreciating faster than the rate of growth of the money supply. After T,, the inflation tax increases since lo. increases and m is fixed. Whether a tax policy needs to be changed depends upon whether the increase in the inflation tax at T, is sufficient to offset the increased debt service from the budget deficit between T,, and T,. If it is not, taxes will have to be raised sometime, and the time path of consumption will fall immediately at T, reflecting the loss of wealth. If taxes are not raised until T,, leisure will fall between T,, and T, which will offset the budget deficit to some extent, and after the increase in taxes, leisure will increase. If taxes are raised to a new permanent level level, and the at To, leisure will increase to its new permanent budget will be in deficit until T,. At that time the increase in the inflation tax will balance the budget. In this latter case there will be no effect on the current account since n, g, and c are all at new equilibrium values. The public merely acquires the debt of the government. When taxes are increased at T,, which decreases labor supply, the current account must be in balance. This implies that the current account was in surplus prior to T,.
209
Robert J. Hodrick 6. Conclusions This paper has examined the effects of government policy in a small open economy that has perfect foresight. With access to an international market in borrowing and lending, infinitely lived economic agents determine their consumption, leisure and portfolio allocations subject to constraints that reflect the present discounted value of government expenditures and the method of financing those expenditures. Current consumption, leisure, the demand for real balances and, consequently, the exchange rate all depend on expectations of future policies and not just on the current policy. The interdependence of government policies was also examined. It was demonstrated that the present discounted value of government spending plus the current outstanding government debt must be financed by the present value of taxation on wage income and by the present value of the inflation tax. Specifying a time path of government spending and a rate of growth of the money supply implies a present value of taxation which must be raised.” It was demonstrated that permanent, unanticipated changes in government policy have no dynamic effects on the economy, a result which differs from the literature. Anticipated increases in government spending or in the rate of growth of the money supply do have significant dynamic effects on the model. In the case in which an anticipated increase in the rate of growth of the money supply must be accompanied by an increase in taxation to balance the government budget, it was demonstrated that the rate of depreciation of the exchange rate would be faster than the current rate of growth of the money supply while the current account was in surplus. While the predictions of this model depend in an important way on the specification of a constant rate of time preference for the utility function, extensions of the model to explicitly incorporate uncertainty in the formulation of government policy may be more important first steps than the relaxation of the constancy of the rate of time preference. In this regard, understanding the dynamic constraints on government policy under certainty helps us to formulate the expectations problem which agents face under certainty. Although constant government policies produced uninteresting dynamics in the certainty model, introducing uncertainty about “Kydland and Prescott (1980) examine these environment and discuss problems arising when termined through application of optimal control. 210
issues in a rational expectations the government policies are de-
Effects
of Macroeconomic
Policy
the level of government expenditure, the rate of monetary growth, or tax policy can simultaneously vary the current and expected future policies in which case the model does produce nontrivial dynamic results. A final caveat concerns the specification of the utility function and the opportunities available to agents for holding assets. The model abstracts from a direct specification of the role of money in the economy. Capturing the medium-of-exchange properties of money in a tractable analytical model is quite difficult. One possible procedure is to follow Lucas (1980) who has employed the cash-in-advance constraint advocated by Clower (1967). Stockman (1980) has worked with this type of model in a two-country framework. Nevertheless, in multicountry models, the Clower constraint clearly begs the question of why a particular money is chosen by residents of a country. An alternative approach is the overlapping-generations model such as that of Kareken and Wallace (1980) who have demonstrated that without government intervention the exchange rate must be either constant or indeterminate. Nickelsburg (1979) has examined threats of intervention as a mechanism which resolves the indeterminacy. The point of this discussion is that the results of the model presented here may be sensitive to the possibility that currencies may be substitutes.‘2 The overlapping-generations models with laissez-faire allow currencies to be perfect substitutes while they have been treated here as distinct and nonsubstitutable. Surely, the reality is somewhere between the two extremes since even in hyperinflations, domestic currencies are not immediately abandoned for foreign currencies. I would argue that the reason domestic currencies are not abandoned is the very strong medium-ofexchange property of money. The Clower-constraint models attempt to capture this phenomenon while the overlapping-generations models consider only the store-of-value role for money. Neither formal model of money is adequate for the open economy which is why I have chosen to capture the services of real balances by placing them into the utility function.13 Received: December, 1980 Final version received: September,
1981
‘*Calve and Rodriguez (1977, Girton and Roper (1981). and Miles (1978) have discussed currency substitution. 13Boyer and Kingston (1980) have examined issues of currency substitution in a maximizing framework with two different real balances in the utility function. If the real balances are perfect substitutes, the exchange rate is demonstrated to be indeterminate. 211
Robert J. Hodrick References Arrow, Kenneth J. and Mordecai Kurz. “Optimal Consumer Allocation over an Infinite Horizon. ” Journal of Economic Theory 1 (June 1969): 68-91. and -. Public Investment, the Rate of Return, and Optimal Fiscal Policy. Baltimore: Johns Hopkins Press, 1970. Barro, Robert J. “On the Determination of the Public Debt. ” Journal of Political Economy 87 (October 1979): 940-71. Bazdarich, Michael J. “Optimal Growth and Stages of the Balance of Payments. ” Journal of lnternational Economics 8 (August 1978): 425-44. Boyer, Russell S. and Robert J. Hodrick. “Perfect Foresight, Financial Policies, and Exchange Rate Dynamics. ” Canadian Journal of Economics, Forthcoming May 1982. Boyer, Russell S. and Geoffrey H. Kingston. “Currency Substitution: A Perfect Foresight Optimizing Analysis. ” University of Western Ontario Working Paper, November 1980. Brock, William. “Money and Growth: The Case of Long-Run Perfect Foresight. ” international Economic Rev& 15 (October 1974): 750-77. Calvo, Guillermo A. and Carlos A. Rodriguez. “A Model of Exchange Rate Determination under Currency Substitution and Rational Expectations. ” Journal of Political Economy 85 (June 1977): 617-27. Clower, Robert N. “A Reconsideration of the Microfoundations of Money. ” Western Economic Journal 6 (1967): l-9. Fischer, Stanley and Jacob A. Frenkel. “Investment, the Two-Sector Model and Trade in Debt and Capital Goods.” Journal of International Economics 2 (1972): 211-33. Flood, Robert P. “An Example of Exchange Rate Overshooting. ” Southern Economic Journal (July 1979): 168-77. and Peter M. Garber. “An Economic Theory of Monetary Reform. ” Journal of Political Economy 88 (February 1980): 24-58. Girton, Lance and Don Roper. “Theory and Implications of Currency Substitution. ” Journal of Money, Credit and Banking 13 (February 1981): 12-30. of EquiKareken, John and Neil Wallace. “On the Indeterminacy librium Exchange Rates.” Quarterly Journal of Economics 96 (May 1981): 207-22. eds. Models of Monetary Economies. Federal and Reserve Bank of Minneapolis, 1980. Kouri, Pentti J.K. “The Exchange Rate and the Balance of Pay212
Effects
of
Macroeconomic
Policy
ments in the Short Run and the Long Run: A Monetary Approach.” Scandinavian Journal of Economics 7812 (1976): 280-304. Kydland, Finn E. and Edward C. Prescott. “Dynamic ‘Optimal Taxation, Rational Expectations and Optimal Control. ” Journal of Economic Dynamics and Control 2 (February 1980): 79-92. Lucas, Robert E., Jr. “Equilibrium in a Pure Currency Economy. ” In Models of Monetary Economics. J. Kareken and N. Wallace, eds. Federal Reserve Bank of Minneapolis, 1980. Miles, Marc A. “Currency Substitution, Flexible Exchange Rates, and Monetary Independence. ” American Economic Review 68 (June 1978): 428-36. Mussa, Michael. A Study in Macroeconomics. Amsterdam: NorthHolland, 1976. Nickelsburg, Gerald. “Flexible Exchange Rates and Uncertain Government Policy: A Theoretical and Empirical Analysis.” Ph.D. Dissertation, University of Minnesota, 1979. Obstfeld, Maurice. “Macroeconomic Policy, Exchange Rate Dynamics and Optimal Asset Accumulation. ” Journal of Political Economy 89 (December 1981): 1142-61. Sidrauski, Miguel. “Rational Choice and Patterns of Growth in a Monetary Economy.” American Economic Review 57 (May 1967): 534-44. Stockman, Alan C. “A Theory of Exchange Rate Determination. ” Journal of Political Economy 88 (August 1980): 673-99. Uzawa, H. “Time Preference, the Consumption Function, and Optimum Asset Holdings. ” In Value, Capital, and Growth: Papers in Honour of Sir John Hicks. J.N. Wolfe, ed. Chicago: Aldine, 1968. Wallace, Neil. “The Overlapping Generations Model of Fiat Money.” In Models of Monetary Economies. J. Kareken and N. Wallace, eds. Federal Reserve Bank of Minneapolis, 1980.
213
On the Effects of Macroeconomic Policy in a Maximizing Model of a Small Open Economy This paper investigates the effects of government policies in a small open economy. Determination of the exchange rate and the effects of macroeconomic policies on the balance-of-payments accounts are examined in a model of infinitely lived, utility-maximizing agents who have perfect foresight. Residents of the country and the government borrow and lend in a world capital market which constrains them to live within their means. The dependence of consumption, leisure, and real-balance holdings on the present discounted value of government expenditure and real debt is demonstrated. Government policies must be dynamically consistent.
1. Introduction This paper investigates the effects of government policies in a small open economy. The macroeconomic issues of the determination of the exchange rate and the effects of policy on the balanceof-payments accounts are examined in a model of infinitely lived, utility-maximizing agents who have perfect foresight. The model is similar to those of Kouri (1976), Flood (1979), Boyer and Hodrick (1982), and Obstfeld (1981) in that agents consume a single consumption good whose foreign price is given and have access to a world capital market in which they can borrow or lend in real terms. As in Obstfeld’s model, the agents are placed in a maximizing framework although this analysis postulates a constant rate of time preference. The development of the model in Section 2 follows straightforwardly from Mussa (1976). The dependence of agents’ current decisions about consumption and leisure and their current demands for real balances are demonstrated to depend on the present discounted value of government spending. It is also demonstrated that the present discounted value of government spending plus the current real value of the government debt must be financed by the present discounted value of government revenue from wage income taxation and the inflation tax. Government policies must be dynamjournul Copyright
of Macroeconomics, 0
1982
by Wayne
Spring State
1982, Vol. University
4, NO. Press.
2, pp.
195-213
195
Robert J. Hodrick ically consistent in the sense that only two of the three present values discussed above are independent. The specification of a time path for government spending and a rate of growth for the money supply determines a present value of wage taxation. This result generalizes the discussions in Barr-o (1979) and Kydland and Prescott (1980) to allow for the inflation tax. Section 3 analyzes the equilibrium of the model with constant government policies, and Section 4 considers the response of the economy to a change in policy. Tax-financed increases in government spending depreciate the exchange rate and cause a decrease in consumption and an increase in leisure. The effects of an increase in the rate of monetary growth are shown to depend on the elasticity of demand for real balances with respect to the rate of inflation since wage taxation must go up, remain the same, or go down as the demand for real balances is elastic, unitary elastic, or inelastic. Section 5 examines anticipated increases in policy and the results are shown to depend critically on when taxes are changed to balance the budget. That depreciation of the exchange rate in excess of the rate of monetary growth and a current account surplus are not inconsistent in a perfect foresight model is demonstrated in this section. Section 6 contains some conclusions.
2. The Model Consider a small open economy that takes as given the foreign currency prices of all goods. For the purpose of this analysis, these commodities can be aggregated into a single composite good whose foreign currency price can be set equal to one. Under free trade the domestic currency price of the commodity, P, is equal to the exchange rate: P=S
(1)
where
S is the domestic currency price of foreign exchange. Output of the good in the country is produced by competitive firms which use only the labor input of households. The production function is Y
=fb)
(2)
Effects
of Macroeconomic
Policy
where y is the value of output and f is nonnegative and homogeneous of degree one. Firms take nominal wages, W, and prices as given and maximize profits. Consequently, the marginal product of labor equals the real wage,
f’(n) = W/P = w , and the value of output
equals the return
(3) to labor,
y=wn.
(4)
Households are assumed to maximize the present discounted value of an instantaneous utility function. The representative households instantaneous utility is modeled as a separable function of consumption, c, leisure, e, and real balances, rn:l
U(c,&m)
= u(c) + h(l) + v(m)
(5)
where real balances are defined by deflating the households nominal money balances, M, by the price level, m = M/P. The functions u, h, and u are taken to be nonnegative, strictly concave, and twice continuously differentiable, with the properties that lim u’(c) = lim h’(8) = lim u’(m) = O”. C-W e-m m-m
(6)
In addition it is assumed that there is nonsatiation in consumption, leisure, and real balances implying that the derivatives u’, h’, and 8’ are strictly positive. It is assumed that the only assets available to residents of the country are domestic money which is not held by foreigners and an internationally traded bond which has constant purchasing power and a rate of return equal to r *. Households are allowed to borrow and lend at this constant foreign interest rate, and net bond holdings are denoted b.
‘The convention of placing real balances in the utility function in order to capture the nonpecuniary services of real balances in a tractable way as in Sidrauski (1967) or Brock (1974) has been criticized by Wallace (1980) and others. See Kareken and Wallace (1980) and especially the paper by Lucas (1980) for a discussion of these issues and alternative modeling strategies. 197
Robert J. Hodrick At each moment in time households must allocate their real market wealth, a, between real balances and real bond holdings:
a, = m, + b, .
(7)
Additionally, the household must decide its labor supply and the allocation of its income between consumption and the accumulation of market wealth. Labor input, n, plus leisure, 4, equals total time available to households which is normalized to one. Household’s disposable income is the sum of its labor income and the expected returns on holding market assets minus the taxes it pays to the government. The returns on assets are r*b for net foreign assets and -rm for real balances where n is the expected rate of inflation. The government is assumed to tax labor income at the proportional rate 7. Planned real saving at time t is therefore d, = ton, (1 - rt) + r*b, - ntmt - c, .
03)
The final constraint which binds the households choice problem is imposed by the world capital market. As Arrow and Kurz (1969) and Mussa (1976) note, the natural feasibility condition on the households plan is that its market indebtedness at any point in time, -a,, must not exceed the households ability to repay its debts out of its future wage income. Formally, this requires that cc
-a, I
[zon,(l - 7,)]e-r*(s-f) a!s I
(9)
t
for all t, where the right hand side of (9) is nonasset wealth at time t. Since r* is taken to be a constant for all time, the discount factor e-rr(s-t) measures the market price of a unit of real consumption at time s in terms of real goods at time t. The household’s choice problem is m
e -B U(c,, C,, m,)dt
subject to the constraints 198
(7), (B), and (9) where
the discount
rate
Effects
of Macroeconomic
Policy
6 is taken to be a constant.2 In solving (10) households take the paths of the price level and the tax rate as given. It is clear from (9) that the world capital market must also assess the paths of taxation and labor supply in order to constrain the household not to live beyond its means. Without an explicit consideration of uncertainty the only sensible assumption regarding the formation of expectations of the future by households and by the world capital market is to assume that agents in the model have perfect foresight. Under this assumption households and the capital market are assumed to know how government policy is conducted and to understand that the market-clearing process will result in prices that are expected to occur given that the demands for consumption goods and real balances and the supplies of labor are derived from (10) and are based on the same expectations. Maximization of (10) subject to (7), (8), and (9) requires that households choose, c, 4, m, b, and the shadow price, 5, at each instant to maximize the current value Hamiltonian:
H = u(c) + h(t) + u(m) + t(a - m - b) + X(w(1 - e)(l - ~)+r*b--m-c) where all variables are evaluated ditions are
at time s 2 Oa3The first-order
(11) con-
u’(c) = A,
(12)
h’(e) = Xw(1 - T),
(13)
u’(m) = 5 + XIT,
(14)
r-*X = 5,
(15)
-b=o.
(16)
and a-m
‘Obstfeld (1981) considers a similar problem, but he employs the variable-discount-rate model of Uzawa (1968). See Section 4 for a discussion of the differences in the results arising from the alternative assumptions. The solution of the household’s intertemporal allocation problem follows the presentation of Mussa (1976) relatively closely. 199
Robert J. Hodrick Equations (12) through (16) determine the currently determined variables, c, e, m, b, and 5 as functions of the state variable, a, and the costate, A. The next step in the solution of the households intertemporal allocation problem is to determine the time path of the state and the costate variables up to the initial value of the costate, A,. As Mussa (1976) demonstrates, a useful alternative representation of the transition law for the state, (8), is obtained by adding the implicit value of the nonpecuniary services of real cash balances to both income and consumption. The implicit value of these services is the level of real balances multiplied by the opportunity cost of holding cash rather than real foreign assets which is the difference in the two rates of return, r* + T. Defining total consumption as the sum of consumption of goods and the services of real balances, 2, = c, + (r* + 7r,)m,, the transition
law for the state variable
(17)
can be written
as
ci, = r*a, + ~(1 - e,)(l - 7,) - 2, .
(18)
The solution of (18) is
t { I
a, = e’*t a0 +
[w(l
0
- e,)(l - 7,) - z,]e-r*s a!s I
(19)
where a, is the historically given level of real market wealth at time 0. The costate variable must satisfy
i, = 6X, - g
= (6 - ‘*)A, ,
(20)
which has the solution, A
t
=
Aoe@-‘*‘)’
where A, remains to be determined.
>
Clearly,
(21) since the household
take w, r*, T,, and T, as given, and since c,, e,, and m, are functions of a, and A,, the paths of a, and A, are fully determined by
Effects of Macroeconomic
Pohcy
(19) and (21) given the initial values of a,, and A,,. The initial value of a, is given and the value of A, can be determined from the transversality conditions which are necessary conditions for an optimal path. These conditions require that lim e+ t--r-
A, 2 0 ,
(259
and lim ea8” Atat = 0 . t-m
(23)
From (21), (22), and the first-order conditions, A, > 0 is required since otherwise consumers would attempt to consume an infinite amount without working in violation of the lifetime budget constraint. Substituting (19) and (21) into (23) gives
’ [w (1 - e,)( 1 - 7,) - zS]e-r*s o!.s = 0 , which
implies
that m
cc I0
(24)
=,e -I-*’ & = a, +
w(1 - C,)(l - rS)e-‘*’ ds . I0
(25)
Equation (25) demonstrates that the initial value of A, will be chosen such that the present discounted value of lifetime total consumption is equal to the sum of initial market assets and initial nonasset wealth. This is an equivalent expression to (9) and demonstrates that (9) will hold with equality along an optimal path. From the first-order conditions we know that the left-hand side of (25) is monotonically decreasing in A, while the right-hand side is monotonically increasing in A,. Consequently, if an equilibrium exists, it is intuitively clear that it will be unique.4
4As Arrow
and
Kurz
(1970)
demonstrate
in Proposition
11.8,
when
the
optimized
Hamiltonian is concave in the state variable as it is here, any policy that satisfies the first-order conditions and the transversality conditions is an optimal policy. Existence of an optimal policy depends on convergence of the integrals in (25) which in turn depends on the specification of government policy. If government policies are constrained such that the integrals converge, the policies possess “process consistency” in the terminology of Flood and Garber (1980). 201
Robert J. Hodrick From (21) and consideration of the first-order conditions, if 6 > r* and if an equilibrium exists with constant government policies, the equilibrium would be characterized by monotonically shrinking consumption and increasing labor supply. For 6 < r*, consumption and leisure would increase over time as the country became wealthier. Only if 6 = r* is A, ever a constant, and in this case, it is unchanging at A, = A,. In what follows, this latter case will be the only one considered.’ Determination of the equilibrium time path for the economy requires a specification of the government sector. The government is assumed to buy goods in amount g, to collect taxes, and to issue money. 6 The government also is allowed to borrow and lend at the world real interest rate. Its flow budget constraint is g, + r*bf
= wntTt + 6: + ptm,
(26)
where bf is the level of real government indebtedness and pt = A&/M, is the rate of growth of the nominal money supply. The world capital market will also constrain the government to keep its indebtedness to be less than or equal to its ability to repay which implies that
which must hold for all t. From ment debt evolves according to
(26) we know that actual govern-
%ince there is no capital in the model, a perpetual divergence between the discount rate and the real interest rate is possible. See Bazdarich (1978) and Fischer and Frenkel (1972) for models of trade in debt with capital accumulation. ‘In what follows no consideration will be given to the optimal@ of government policy or to how government spending is determined. Since government spending does not feed back into the model, it is implicitly assumed that the government sector is too large relative to some optima level because agents would be better off by a reduction in spending and taxation. It is also assumed that the government does not have access to distortionless head taxes. 202
Effects
of Macroeconomic
Policy
where b$ is an initial level of indebtedness. Multiplying (27) and (28) into (27), rearranging terms and let(28) by e-‘*‘, substituting ting t + 00 yields
bf, +
g,e -‘*’ d.s zs
(wn,T, + p8mS)e+g a!s ,
(29)
which indicates that the present discounted value of government expenditure plus the initial government indebtedness must be less than or equal to the present discounted value of the government’s tax collections plus its revenue from money creation.’ Equation (29) will hold as an equality when the government does not plan to accumulate wealth. Consideration of (26) and (29) indicates that at a point in time the government has three degrees of freedom in determining its four actions of spending, taxation, money creation, and debt creation. Nevertheless, the existence of a rational capital market implies that when these four actions are viewed as policies which will persist into the indefinite future there are really only two degrees of freedom. The prohibition against perpetual debt finance imposed by the capital market in (27) combined with the necessity of specifying government policies or actions at each point in time if agents are forward looking implies a policy constraint which binds government policy makers.’ Substitution of the government’s flow budget constraint (26) into the households budget constraint (8) gives bt - 6: = writ + r*(b,
- bf) - g, - C, ,
(39)
which demonstrates the equality of the capital account deficit and the current account surplus under flexible exchange rates. Since
‘Equation (29) including revenue
generalizes from money
Barro’s (1979) constraint on public creation. While the existence of
indebtedness government
by debt
implies that some future taxes will be used to finance it, which specific taxes are used is a choice variable of the government. *It is obvious that actual governments have not sufficiently specified their policies that agents know the future with certainty. The introduction of uncertainty into future policy as well as the existence of nominal government debt would increase the relevance of this paper without altering the necessity that policy must be specified in a dynamically consistent manner. 203
Robert J. Hodrick (30) must hold at all times, multiplication tegration with respect to time gives
of (30) by e -‘*’ and in-
co
cc --T*s
bo
+
bn,
which, when combined
-
&
=
bg
+
-‘*’
&
,
(31)
with (29) and (25), yields
m
m
(r* + 7F,)mse-‘*” 0% = m. + I
g,e
de
0
o (psms)e-r*s d-s .
(32)
I
These equations demonstrate an interesting separation of the real and the monetary sectors of this model. While these constraints will be satisfied along a perfect foresight path, as Mussa (1976) notes, households must still be thought of as maximizing the problem in (10) since otherwise total wealth would not be the constraint on their behavior. We can use (31) and (32) to characterize the solution since along a perfect foresight path such a division will occur.
3. System Equilibrium In this section we study the properties of the model for particularly simple government policies. Given initial values of private foreign assets, b,, government indebtedness, b& and nominal money balances, and under the assumption that real government expenditures are constant at g and that the rate of monetary growth is constant at a; (12), (13), (14), (29), (31), and (32) can be solved simultaneously for values of C, 2, tii, ti, X, and t which are both the instantaneous and the long-run equilibrium values of these variables. The equilibrium relationships after solving for i are the following:
r*(b,
h’(Z) = U’@)W(l - ‘i) )
(33)
o’(fi)
(34)
= u’(E)(r*
- b$) + ~(1 - 2) - E = g,
r*bg+g=pfi+yw(l-Z), 204
+ fr) ,
(35) (36)
Effects
of Macroeconomic
Policy
and
(r* + 5) $ = ci + p vii/?-*.9
(37)
The equilibrium relationships have straightforward interpretations. From (37) we see that the actual and the expected rates of inflation are equal to the growth rate of the money supply. From (33) and (34) we find that the marginal rate of substitution of leisure for consumption is equal to the net of tax real wage and that the marginal rate of substitution of real balances for consumption is equal to the opportunity cost of holding money. Equations (35) and (36) indicate that the current account is in equilibrium and that the government budget is balanced in the sense that no new debt is being issued. The two sources of government revenue, taxation of labor income and inflation tax, just offset government spending and interest on the outstanding debt.
4. Alternative Government Policies In this section we consider the effects of changes in government policies. Two interpretations can be given to the analysis. First, one can think of the exercise as determining how a second economy would differ from the first economy studied in the previous section if the second economy had larger values of government spending or a faster rate of monetary growth. Alternatively, one can think of the analysis as determining how the economy would react to surprises in government policy which are expected to be permanent. While the latter interpretation will be used in this study, it is clear that specification of a stochastic environment is necessary before one can adequately discuss unanticipated changes in policy. lo
‘Equation (35) demonstrates that government debt is subtracted from private wealth in this model. An increase in the debt must be offset by future taxes, but either the labor income tax or the inflation tax can be used to finance it. “It is common in the literature to confront agents who have perfect foresight with an “unanticipated” change in policy and to examine the movement of endogenous variables along the new perfect foresight path. With postulated behavioral functions such experiments can be defended by asserting that behavior depends only on the expected value of a variable. This result will also be produced by assuming that agents maximize the expected value of a quadratic utility function.
205
Robert J. Hodrick Now consider the effects of a change in government spending policy that is financed by a change in the tax rate. The effects on c, e, m, and T are the following: (33)
de -= & dm dg=
-uy+
(39)
p)w(h”(l
- 4) - u’w(l
- $})
(49)
and
d7 -= 4s
1
i {UnuRw2(l - 7)’ + h”u”(r*
p)p - B”}
(41)
where
and 2 = w(1 - e)(l - 7) . For small values of T and t.~, the fundamental determinant, A, will be positive. Consequently, when the government increases spending, consumption falls, real balances fall (that is, the exchange rate depreciates) and the tax rate increases. For a utility function such as the exponential, leisure will increase since the term in brackets
10 continued from p. 205) Since explicit introduction of uncertainty and maximization of expected utility would have complicated many of the points in the previous section, I avoided those problems at the cost of straining the interpretation of the model in this section. Nevertheless, it remains my conjecture that the thought experiments provide sensible answers regarding the direction of response to questions of how agents would respond to permanent changes in government policies in a stochastic environment. Obviously, a superior approach to the subject would be to tackle the stochastic specification directly. (Footnote
206
Effects
of Macroeconomic
Policy
in (39) will be negative. The substitution effect of the increase in taxes outweighs the wealth effect which decreases consumption and would tend to decrease leisure. Since the instantaneous values are the long-run values, the economy moves immediately to its new long-run equilibrium. These results are similar in spirit to those of Kouri (1976) and Boyer and Hodrick (1982) w h o h ave investigated the issue in a nonmaximizing framework. In those models a tax-financed increase in government spending creates a current account deficit, a loss of wealth, and lower (long-run) consumption. The results differ markedly from Obstfeld’s (1981) maximizing model which employs the variable rate of time preference model of Uzawa (1968). In Obstfeld’s model an increase in government spending does not change the steady-state level of consumption but causes consumption to fall in the short run by an amount sufficient to generate a current account surplus which allows private foreign assets to increase to finance the increased taxes. While the variable time preference model provides more interesting dynamics than this model, whether the specification is more appropriate than assuming a constant rate of time preference cannot be determined by a priori reasoning since the appropriateness of the specification depends on the nature of the time preference in the intertemporal utility function. Turning now to consideration of an increase in the growth rate of the money supply, the changes in the equilibrium values of the variables are the following: 0% -= 6
-ul)u~w2 (42)
A
(43)
’ dm -= dp
zur’ufw2 A
1 + h”(1 - 4)
(44
U”Wlj
and I d7 - -i) (u”w2(1 &A
- 7) + ,“)
From (42), (43), and (45), if m = %,
neither
consumption,
lei207
Robert J. Hodrick sure, nor taxation changes. Substituting this value of m into (44) dm ’ implies = u from which it is easily seen that in this case the dp u” elasticity of demand for real balances with respect to the rate of inflation is equal to minus one. When this is the case, a change in p changes m equiproportionately leaving the government revenue from inflation,
pm, unchanged.
If m > $$,
the demand
for
real balances is inelastic with respect to the rate of inflation, and an increase in l.~ increases government revenue from the inflation tax causing a fall in the tax rate on labor income, an increase in consumption, and a decrease in leisure. When the demand is elas-u’ p tic, m < an increase in lo decreases government revenue v” ’ from inflation causing the tax on wage income to rise in order to balance the government budget which increases leisure and decreases consumption. Regardless of the magnitude of the elasticity, an increase in lo, causes a fall in real balances which corresponds to an increase in the exchange rate. Since there are no further dynamics in this model, an increase in lo does not cause overshooting of the exchange rate as in the models of Kouri (1976), Flood (1979), Boyer and Hodrick (1982), or Obstfeld (1981). In those models real balances decline by more in the short run than in the long run and the economy accumulates foreign assets as it moves to the new steady state. The rate of depreciation of the exchange rate is less than the new rate of growth of the money supply. Here, after the initial jump in the exchange rate, it depreciates at the new F.
5. Anticipated Changes in Policy In the previous section changes in government policy were unanticipated and permanent. There were no current account reactions to changes in policy since the economy merely chooses new long-run equilibrium values instantaneously. In this section we consider the behavior of the economy in response to anticipated changes in policy. Suppose that at some point in time TO the economy expected the government to increase its spending at some time T, in the future. How do the consumption, leisure, and real balance deci-
Effects
of Macroeconomic
Policy
sions of the economy change and how does this affect the exchange rate and the accounts of the balance of payments? From consideration of the results of the dynamic maximization problem we know that the agents will choose new constant lower levels of consumption and of real balances implying an instantaneous depreciation of the exchange rate. The time path of leisure depends on when the government raises taxes to finance the increased government spending. If they are raised at T,,, a new higher level of leisure will be chosen immediately. If taxes are raised at T,, people will initially choose to work harder, prior to the tax increase, then after T,, agents will increase leisure. It is clear from (SO), which must be zero after T,, that the current account will be in surplus prior to T,, since after T,, g will be increased and n will either remain constant or be decreased to a lower constant level. Consider now an expected increase in the rate of growth of the money supply that will occur at T,. Since the exchange rate cannot be anticipated to jump, it must depreciate immediately at T,, and move continuously thereafter. If there are no changes in government policy after T,, the economy must be in its steady state at that time which means that real balances will be constant at I’, and thereafter. Between T,, and T, the government revenue from inflation is declining since the exchange rate is depreciating faster than the rate of growth of the money supply. After T,, the inflation tax increases since lo. increases and m is fixed. Whether a tax policy needs to be changed depends upon whether the increase in the inflation tax at T, is sufficient to offset the increased debt service from the budget deficit between T,, and T,. If it is not, taxes will have to be raised sometime, and the time path of consumption will fall immediately at T, reflecting the loss of wealth. If taxes are not raised until T,, leisure will fall between T,, and T, which will offset the budget deficit to some extent, and after the increase in taxes, leisure will increase. If taxes are raised to a new permanent level level, and the at To, leisure will increase to its new permanent budget will be in deficit until T,. At that time the increase in the inflation tax will balance the budget. In this latter case there will be no effect on the current account since n, g, and c are all at new equilibrium values. The public merely acquires the debt of the government. When taxes are increased at T,, which decreases labor supply, the current account must be in balance. This implies that the current account was in surplus prior to T,.
209
Robert J. Hodrick 6. Conclusions This paper has examined the effects of government policy in a small open economy that has perfect foresight. With access to an international market in borrowing and lending, infinitely lived economic agents determine their consumption, leisure and portfolio allocations subject to constraints that reflect the present discounted value of government expenditures and the method of financing those expenditures. Current consumption, leisure, the demand for real balances and, consequently, the exchange rate all depend on expectations of future policies and not just on the current policy. The interdependence of government policies was also examined. It was demonstrated that the present discounted value of government spending plus the current outstanding government debt must be financed by the present value of taxation on wage income and by the present value of the inflation tax. Specifying a time path of government spending and a rate of growth of the money supply implies a present value of taxation which must be raised.” It was demonstrated that permanent, unanticipated changes in government policy have no dynamic effects on the economy, a result which differs from the literature. Anticipated increases in government spending or in the rate of growth of the money supply do have significant dynamic effects on the model. In the case in which an anticipated increase in the rate of growth of the money supply must be accompanied by an increase in taxation to balance the government budget, it was demonstrated that the rate of depreciation of the exchange rate would be faster than the current rate of growth of the money supply while the current account was in surplus. While the predictions of this model depend in an important way on the specification of a constant rate of time preference for the utility function, extensions of the model to explicitly incorporate uncertainty in the formulation of government policy may be more important first steps than the relaxation of the constancy of the rate of time preference. In this regard, understanding the dynamic constraints on government policy under certainty helps us to formulate the expectations problem which agents face under certainty. Although constant government policies produced uninteresting dynamics in the certainty model, introducing uncertainty about “Kydland and Prescott (1980) examine these environment and discuss problems arising when termined through application of optimal control. 210
issues in a rational expectations the government policies are de-
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of Macroeconomic
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the level of government expenditure, the rate of monetary growth, or tax policy can simultaneously vary the current and expected future policies in which case the model does produce nontrivial dynamic results. A final caveat concerns the specification of the utility function and the opportunities available to agents for holding assets. The model abstracts from a direct specification of the role of money in the economy. Capturing the medium-of-exchange properties of money in a tractable analytical model is quite difficult. One possible procedure is to follow Lucas (1980) who has employed the cash-in-advance constraint advocated by Clower (1967). Stockman (1980) has worked with this type of model in a two-country framework. Nevertheless, in multicountry models, the Clower constraint clearly begs the question of why a particular money is chosen by residents of a country. An alternative approach is the overlapping-generations model such as that of Kareken and Wallace (1980) who have demonstrated that without government intervention the exchange rate must be either constant or indeterminate. Nickelsburg (1979) has examined threats of intervention as a mechanism which resolves the indeterminacy. The point of this discussion is that the results of the model presented here may be sensitive to the possibility that currencies may be substitutes.‘2 The overlapping-generations models with laissez-faire allow currencies to be perfect substitutes while they have been treated here as distinct and nonsubstitutable. Surely, the reality is somewhere between the two extremes since even in hyperinflations, domestic currencies are not immediately abandoned for foreign currencies. I would argue that the reason domestic currencies are not abandoned is the very strong medium-ofexchange property of money. The Clower-constraint models attempt to capture this phenomenon while the overlapping-generations models consider only the store-of-value role for money. Neither formal model of money is adequate for the open economy which is why I have chosen to capture the services of real balances by placing them into the utility function.13 Received: December, 1980 Final version received: September,
1981
‘*Calve and Rodriguez (1977, Girton and Roper (1981). and Miles (1978) have discussed currency substitution. 13Boyer and Kingston (1980) have examined issues of currency substitution in a maximizing framework with two different real balances in the utility function. If the real balances are perfect substitutes, the exchange rate is demonstrated to be indeterminate. 211
Robert J. Hodrick References Arrow, Kenneth J. and Mordecai Kurz. “Optimal Consumer Allocation over an Infinite Horizon. ” Journal of Economic Theory 1 (June 1969): 68-91. and -. Public Investment, the Rate of Return, and Optimal Fiscal Policy. Baltimore: Johns Hopkins Press, 1970. Barro, Robert J. “On the Determination of the Public Debt. ” Journal of Political Economy 87 (October 1979): 940-71. Bazdarich, Michael J. “Optimal Growth and Stages of the Balance of Payments. ” Journal of lnternational Economics 8 (August 1978): 425-44. Boyer, Russell S. and Robert J. Hodrick. “Perfect Foresight, Financial Policies, and Exchange Rate Dynamics. ” Canadian Journal of Economics, Forthcoming May 1982. Boyer, Russell S. and Geoffrey H. Kingston. “Currency Substitution: A Perfect Foresight Optimizing Analysis. ” University of Western Ontario Working Paper, November 1980. Brock, William. “Money and Growth: The Case of Long-Run Perfect Foresight. ” international Economic Rev& 15 (October 1974): 750-77. Calvo, Guillermo A. and Carlos A. Rodriguez. “A Model of Exchange Rate Determination under Currency Substitution and Rational Expectations. ” Journal of Political Economy 85 (June 1977): 617-27. Clower, Robert N. “A Reconsideration of the Microfoundations of Money. ” Western Economic Journal 6 (1967): l-9. Fischer, Stanley and Jacob A. Frenkel. “Investment, the Two-Sector Model and Trade in Debt and Capital Goods.” Journal of International Economics 2 (1972): 211-33. Flood, Robert P. “An Example of Exchange Rate Overshooting. ” Southern Economic Journal (July 1979): 168-77. and Peter M. Garber. “An Economic Theory of Monetary Reform. ” Journal of Political Economy 88 (February 1980): 24-58. Girton, Lance and Don Roper. “Theory and Implications of Currency Substitution. ” Journal of Money, Credit and Banking 13 (February 1981): 12-30. of EquiKareken, John and Neil Wallace. “On the Indeterminacy librium Exchange Rates.” Quarterly Journal of Economics 96 (May 1981): 207-22. eds. Models of Monetary Economies. Federal and Reserve Bank of Minneapolis, 1980. Kouri, Pentti J.K. “The Exchange Rate and the Balance of Pay212
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