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Government purchases and real interest rates with endogenous time preference
Economics Letters North-Holland
131
35 (1991) 131-136
Government purchases and real interest rates with endogenous time preference Michael
B. Devereux
Queen’s University, Kingston, Received Accepted
Ont. K7L 3N, Canada
29 May 1990 22 June 1990
‘The observation of low real interest rates during periods of high temporary government purchases, such as wars, can be reconciled with the neoclassical growth model extended to allow for endogenous time preference. Temporary increases in government purchases lead to a fall in the instantaneous rate of time preference. This may result in rising investment and falling real interest rates along an adjustment path.
1. Introduction
A standard prediction of the one sector neo-classical growth model is that periods of temporarily high government purchases, such as wars, should be associated with high real interest rates and low rates of private investment spending (e.g. Barro, 1981, 1987). High real interest rates are required to dissuade people from consuming, to increase labour supply, and to reduce private investment during the period of high government purchases. The empirical evidence on real interest rates during wars gives very little support to this prediction. Barro (1987), finds that if anything, U.S. real interest rates have fallen during wars. The standard neoclassical growth model assumes time-additive preferences with a constant subjective rate of time preference. We extend the model to recursive but non time-additive preferences, with endogenous time preference, based closely on Epstein (1987). The dynamic effects of temporary and permanent shocks to government purchases are examined. Extended in this way, the model no longer predicts that real interest rates should be higher during periods of high government spending. Relative to the steady state, real interest rates may be higher or lower following the spending shock. It is possible for interest rates to be lower for the fuiI duration of the shock. In this model the instantaneous rate of time preference is a positive function of current welfare. A temporary rise in government purchases reduces welfare, causing a discrete fall in the rate of time preference. As a consequence, savings and investment may rise along the adjustment path. 1 ’ Mankiw (1987) provides consumption goods. 0165-1765/91/$03.50
an alternative
explanation
for the observations
0 1991 - Elsevier Science Publishers
noted
B.V. (North-Holland)
above,
based
on the presence
of durable
M.B. Devereux
132
/ Governmenl purchases and real interest rates
2. The model
We follow Epstein U(C)
=
(1987) closely. Define
the utility
functional
as
jomu e-'hU(C)dT dt.
C is defined as a strictly positive consumption path, and u and u are real valued, twice differentiable functions of instantaneous consumption. Define rC as the consumption path from time T onwards, and U(,C) as the utility from this. Labour supply is assumed inelastic. The functional (1) is recursive 2 but not time-additive. It is similar to the more familiar specification of Uzawa (1968), and none of our results hinge on this difference. The implicit rate of time preference is both endogenous and time-varying outside of steady states. The local rate of time preference p is defined as the negative of the proportional rate of change of marginal utility over time, evaluated at a constant consumption path.
p(c, U(J))
= -
dlogdy)1
=
C(T)=0
[
M4%W - 4cbcl U(A%W - %W]’
(2)
where U,(C) is the relevant marginal utility of consumption at time T. 3 Along a constant consumption path, this reduces to p(c, U) = u(c). Epstein imposes a number of restrictions on the form of (1). These include (i) p( c, U( &)) > 0, positive time preference (ii) 24, > a p2 > 0, time preference increasing in utility (iii) U,(C) > 0, positive marginal utility (iv) - d log Ut(C)/dT > p(c, U(,C)), for 6(T) > 0 (Related to diminishing marginal utility). With the additional assumption of quasi-concavity, Epstein proves existence, uniqueness, and local stability of both social planning and competitive allocations. Assumption (ii) discussed at length in Epstein (1987) and Epstein and Hynes (1983), is the key factor behind our results. It is Epstein (1987) shows that it is a necessary often referred to as ‘increasing marginal impatience’. condition for stability of the steady state in dynamic models with recursive utility. Finally note that for the special case u, = 0, (1) collapses to the constant time preference model. We may express (1) in terms of the differential equation
m =-f(c(t)?
+(t)L
where f= u(c) - @u(c), and G(t) = U(J). We make use of the equivalence between The fictitious planner solves (PI).
Maximize
(1) subject
w =W(t))
(3) 4 The assumptions above imply that f, > 0, f,, < 0. a social planning problem and a competitive equilibrium.
to:
- c(t) -g(t),
k(0)
= k,.
* Equation (1) can be written as U$C) = loTo e-j6udT dt +e-jTudT U(TC). ’ Epstein shows that U,(C) = e-10 “dr [v, - u,U(Y)]. 4 For @p(t) to solve (3), it is necessary to specify also a terminal condition;
(4
lim, _ m @p(t) e-jAudc
dt = 0.
M.B. Devereux
/ Government purchases and real interest rates
133
F(k) is output net of depreciation, where k(t) is the per capita capital stock. Assume (i) F’(k) > 0, (ii) E”(k) < 0, (iii) F(0) = 0. The variable g(r) resembles a negative, additive, time varying technology shock, but is to be understood in the analysis to capture the effect of an increase in exogenous government spending financed by lump-sum taxes. The time path of g(t) is specified below. The equivalence between these two interpretations is discussed in Blanchard and Fischer (1989). 5 The solution to problem (Pl) is obtained in Epstein by defining a new state variable z(t) where i(r) = u(c), and using the Hamiltonian;
+p[u(c)].
H(c, Z, k)=u(c)e-““‘+X[F(k)-c(t)-g(f)] Using (5) we may derive the following
+) - F’W)~
E =f,/f&+ i=
-f(c,
+),
k(O)=k,,
necessary
conditions
k=F(k) lim G(t) r+cc
(5)
for (Pl)
-c-g, e-‘6UdT=0.
(6)
Epstein (lemma 2) demonstrates that, given the restrictions placed on (l), conditions (6) are also sufficient for a solution of (Pl). The real interest rate is just the instantaneous rate of return on capital F’(k). The first equation in (6) describes consumption growth along an optimal path as being proportional to the difference between the rate of return on capital and the instantaneous rate of time preference. We now examine the impact of government spending shocks. First focus on the steady state. 6 This is particularly easy in this model. Take g(t) as a constant g. In the steady state the rate of time preference equals the return on capital, and net investment is zero. Thus,
u(C) = F’(k),
(7)
F’(k)
(8)
=?+jj.
Figure 1 illustrates these. The uu locus describes eq. (7), while the kk locus describes the steady state market clearing condition, (9). A permanent rise in government spending shifts down the market clearing curve and causes a fall in steady state consumption, but also a rise in the steady state capital stock and a full in the real interest rate. This contrasts with the neoclassical growth model, where, with u = 6, the uu locus is vertical, and a rise in the permanent level of government spending reduces consumption one for one, leaving the capital stock unchanged. The key difference is that in the present model, lower steady state consumption leads to a fall in the steady state level of impatience. This reduces the steady state real interest rate, and the consequent higher capital stock partially offsets the effects of government spending on private consumption. To examine the effect of a temporary increase in government spending, we follow Judd (1985). Let the government spending path be given by g(t) = S + y A(t), where A(t) is assumed to converge to a constant number as 1+ co. (6) may be written as a three dimensional dynamic system in c, + and k, ’ The analysis contains the implicit assumption that government spending provides no direct utility benefits. A similar assumption is made by Wynne (1988) in the calibration of the neo-classical growth model to U.S. wartime economy. Ah the results would still go through were government spending to provide consumption services, as for instance, in Barro (1981). but the algebra of the model would be somewhat messier. ’ Saddle-path stability is established below.
134
M.B. Devereux
/ Government purchases and real interest rates
Fig. 1
where the solutions are implicit functions of t and yA(t). That is, the solutions are functions c(t, y A(t)), +(t, yA(t)) and k2(t, y A(t)). In addition, there is one initial condition; k(O) = k,, and two terminal conditions; one given in footnote 3, and the other by limit, em k(t) = k. The method is to differentiate the system with respect to y around an initial steady state, with y = 0. This leads to the linearized system
(9)
The coefficient matrix has its trace positive and determinant negative, which with the initial and boundary conditions ensures saddlepoint convergence. One of the positive roots is r. Let the other roots be X,, X, with h, < 0 -C X,. The roots satisfy the conditions X, +X2 = r, and X,A, = (~~/YSv,~ - .“). Now take the Laplace
transform
of (9), to get
00) where L,(c,) = lam ePS’c,(t, 0) dt, s > 0, etc. The transformation uses the fact that k,(O, 0) Equation (10) holds for all s > 0, and in particular for s = r, h,. Use these values in (lo), terminal conditions requiring k2(t, 0) and &(t, 0) to be finite [see Judd (1985)]. This gives conditions that may be solved for the initial response of consumption and utility, c,(O, 0) ~~(0, 0). To simplify the exposition, let A(t) follow the path A(t) =A, t E [0, T], A(t) t E (T, co). The solutions for c,(O, 0) and (~~(0, 0) may then be written
+*(O,
0) = -L+(l
- ePrT).
= 0. and two and = 0,
(12)
M. B. Devereux
/ Government purchases and real interest rates
135
The first inequality follows from the properties of the roots of (9). The initial response of investment can then be written as
f&(0,0)=
-1
A
+
(1 _ eehlT)
fcucr
1 _
fcc(r-
i
; fycr
b)h
i
-_‘^;’
‘;
cc
r
(13)
1 r
From (11) consumption always falls in response to a temporary government spending shock. This fall is greater the higher is T. For U, # 0, it is possible that consumption falls by more than one hundred percent of the increased government spending. Investment may then rise or fall. If U, = 0, recovering the neoclassical model, investment always falls, giving the aforementioned response of real interest rates. ’ For the general specification of recursive preferences however, this result no longer obtains. The higher is T, the more likely that investment rises in response to the increased government spending, since for a permanent A(t) shock, investment will always rise. This will lead to .falfing real interest rates in the aftermath of the shock. Finally, we may solve for the inverse Laplace transforms to obtain the solution for the time path of the response of capital, k2(f, 0), in the interval [0, T]. After some lengthy algebra, this gives
k2(t70) =
(Al_\,)[e”l’-
A
x
-
1
(1 - eW +
[
rX2
lh
(x
e”2t] ePXIT+
e_rT
1 (r-4)
(eX2,
1
) 2
_
err)
_
e-*1’(e’2’
h,
- eX1’)
r
e-‘r ‘(‘-A,)
( ert - e? ,] ’
where
Q-y.
iI
cc
For U, = 0, this is always negative. Real interest rates are above their steady state level for the full duration of the increase in government purchases. When U, > 0 the sign of (14) is ambiguous. Real interest rates may be higher or lower than steady state. As T + co, k2(t, 0) > 0 for all t. By continuity, there must be a T-c co such that k2(t, 0) > 0 for t E [0, t] i.e., the capital stock will be above its steady state value for the full duration of the increased government purchases, converging downwards to its original steady state level. Thus real interest rates can be lower during periods of temporarily high government purchases. The observation of low real interest rates during wars is consistent with this model. With a constant rate of time preference, an increase in real interest rates is required to dampen private consumption, with any degree of concavity in preferences. This reduces investment expenditures. Thus in this case the (temporary) rise in government purchases leads to rising real interest rates on impact. With an endogenous rate of time preference, however, the negative welfare effects of the increase in government purchases causes an immediate fall in the implied rate of time preference. This allows for a large fall in consumption, perhaps exceeding the initial increase in government purchases. If the latter occurs, there is an immediate investment boom and falling interest rates along the adjustment path. ’ Note also that for u, = 0 and T = 00, investment government
purchases.
This is another
well-known
is unchanged, and consumption is crowded result of neoclassical theory (Barr0 1987).
out by 100% of the rise in
136
M.B. Devereux / Government purchases and real interest rates
References Barro, R.J., 1981, Output effects of government purchases, Journal of Political Economy 89, 1086-1121. Barro, R.J., 1987, Macroeconomics, 2nd. ed. (Wiley, New York). Blanchard and Fischer, 1989, Macroeconomics (MIT Press, Cambridge, MA). Epstein, L.G., 1987, A simple dynamic general equilibrium model, Journal of Economic Theory 41, 68-95. Epstein, L.G. and J.A. Hynes, 1983, The rate of time preference and dynamic economic analysis, Journal of Political Economy 91, 611-625. Judd, K.L., 1985, Short-run analysis of fiscal policy in a simple perfect foresight model, Journal of Political Economy 93, 298-319. Mankiw, G., 1987, Government purchases and real interest rates, Journal of Political Economy 95, 407-419. Uzawa, H., 1968, Time preference, the consumption function, and optimum asset holdings, in: J.N. Wolfe, ed., Value, Capital and Growth: Papers in Honour of Sir John Hicks (Aldine, Chicago, IL). Wymte, M., 1988, The effects of government purchases in a perfect foresight model, Mimeo. (University of Rochester, New York).
131
35 (1991) 131-136
Government purchases and real interest rates with endogenous time preference Michael
B. Devereux
Queen’s University, Kingston, Received Accepted
Ont. K7L 3N, Canada
29 May 1990 22 June 1990
‘The observation of low real interest rates during periods of high temporary government purchases, such as wars, can be reconciled with the neoclassical growth model extended to allow for endogenous time preference. Temporary increases in government purchases lead to a fall in the instantaneous rate of time preference. This may result in rising investment and falling real interest rates along an adjustment path.
1. Introduction
A standard prediction of the one sector neo-classical growth model is that periods of temporarily high government purchases, such as wars, should be associated with high real interest rates and low rates of private investment spending (e.g. Barro, 1981, 1987). High real interest rates are required to dissuade people from consuming, to increase labour supply, and to reduce private investment during the period of high government purchases. The empirical evidence on real interest rates during wars gives very little support to this prediction. Barro (1987), finds that if anything, U.S. real interest rates have fallen during wars. The standard neoclassical growth model assumes time-additive preferences with a constant subjective rate of time preference. We extend the model to recursive but non time-additive preferences, with endogenous time preference, based closely on Epstein (1987). The dynamic effects of temporary and permanent shocks to government purchases are examined. Extended in this way, the model no longer predicts that real interest rates should be higher during periods of high government spending. Relative to the steady state, real interest rates may be higher or lower following the spending shock. It is possible for interest rates to be lower for the fuiI duration of the shock. In this model the instantaneous rate of time preference is a positive function of current welfare. A temporary rise in government purchases reduces welfare, causing a discrete fall in the rate of time preference. As a consequence, savings and investment may rise along the adjustment path. 1 ’ Mankiw (1987) provides consumption goods. 0165-1765/91/$03.50
an alternative
explanation
for the observations
0 1991 - Elsevier Science Publishers
noted
B.V. (North-Holland)
above,
based
on the presence
of durable
M.B. Devereux
132
/ Governmenl purchases and real interest rates
2. The model
We follow Epstein U(C)
=
(1987) closely. Define
the utility
functional
as
jomu e-'hU(C)dT dt.
C is defined as a strictly positive consumption path, and u and u are real valued, twice differentiable functions of instantaneous consumption. Define rC as the consumption path from time T onwards, and U(,C) as the utility from this. Labour supply is assumed inelastic. The functional (1) is recursive 2 but not time-additive. It is similar to the more familiar specification of Uzawa (1968), and none of our results hinge on this difference. The implicit rate of time preference is both endogenous and time-varying outside of steady states. The local rate of time preference p is defined as the negative of the proportional rate of change of marginal utility over time, evaluated at a constant consumption path.
p(c, U(J))
= -
dlogdy)1
=
C(T)=0
[
M4%W - 4cbcl U(A%W - %W]’
(2)
where U,(C) is the relevant marginal utility of consumption at time T. 3 Along a constant consumption path, this reduces to p(c, U) = u(c). Epstein imposes a number of restrictions on the form of (1). These include (i) p( c, U( &)) > 0, positive time preference (ii) 24, > a p2 > 0, time preference increasing in utility (iii) U,(C) > 0, positive marginal utility (iv) - d log Ut(C)/dT > p(c, U(,C)), for 6(T) > 0 (Related to diminishing marginal utility). With the additional assumption of quasi-concavity, Epstein proves existence, uniqueness, and local stability of both social planning and competitive allocations. Assumption (ii) discussed at length in Epstein (1987) and Epstein and Hynes (1983), is the key factor behind our results. It is Epstein (1987) shows that it is a necessary often referred to as ‘increasing marginal impatience’. condition for stability of the steady state in dynamic models with recursive utility. Finally note that for the special case u, = 0, (1) collapses to the constant time preference model. We may express (1) in terms of the differential equation
m =-f(c(t)?
+(t)L
where f= u(c) - @u(c), and G(t) = U(J). We make use of the equivalence between The fictitious planner solves (PI).
Maximize
(1) subject
w =W(t))
(3) 4 The assumptions above imply that f, > 0, f,, < 0. a social planning problem and a competitive equilibrium.
to:
- c(t) -g(t),
k(0)
= k,.
* Equation (1) can be written as U$C) = loTo e-j6udT dt +e-jTudT U(TC). ’ Epstein shows that U,(C) = e-10 “dr [v, - u,U(Y)]. 4 For @p(t) to solve (3), it is necessary to specify also a terminal condition;
(4
lim, _ m @p(t) e-jAudc
dt = 0.
M.B. Devereux
/ Government purchases and real interest rates
133
F(k) is output net of depreciation, where k(t) is the per capita capital stock. Assume (i) F’(k) > 0, (ii) E”(k) < 0, (iii) F(0) = 0. The variable g(r) resembles a negative, additive, time varying technology shock, but is to be understood in the analysis to capture the effect of an increase in exogenous government spending financed by lump-sum taxes. The time path of g(t) is specified below. The equivalence between these two interpretations is discussed in Blanchard and Fischer (1989). 5 The solution to problem (Pl) is obtained in Epstein by defining a new state variable z(t) where i(r) = u(c), and using the Hamiltonian;
+p[u(c)].
H(c, Z, k)=u(c)e-““‘+X[F(k)-c(t)-g(f)] Using (5) we may derive the following
+) - F’W)~
E =f,/f&+ i=
-f(c,
+),
k(O)=k,,
necessary
conditions
k=F(k) lim G(t) r+cc
(5)
for (Pl)
-c-g, e-‘6UdT=0.
(6)
Epstein (lemma 2) demonstrates that, given the restrictions placed on (l), conditions (6) are also sufficient for a solution of (Pl). The real interest rate is just the instantaneous rate of return on capital F’(k). The first equation in (6) describes consumption growth along an optimal path as being proportional to the difference between the rate of return on capital and the instantaneous rate of time preference. We now examine the impact of government spending shocks. First focus on the steady state. 6 This is particularly easy in this model. Take g(t) as a constant g. In the steady state the rate of time preference equals the return on capital, and net investment is zero. Thus,
u(C) = F’(k),
(7)
F’(k)
(8)
=?+jj.
Figure 1 illustrates these. The uu locus describes eq. (7), while the kk locus describes the steady state market clearing condition, (9). A permanent rise in government spending shifts down the market clearing curve and causes a fall in steady state consumption, but also a rise in the steady state capital stock and a full in the real interest rate. This contrasts with the neoclassical growth model, where, with u = 6, the uu locus is vertical, and a rise in the permanent level of government spending reduces consumption one for one, leaving the capital stock unchanged. The key difference is that in the present model, lower steady state consumption leads to a fall in the steady state level of impatience. This reduces the steady state real interest rate, and the consequent higher capital stock partially offsets the effects of government spending on private consumption. To examine the effect of a temporary increase in government spending, we follow Judd (1985). Let the government spending path be given by g(t) = S + y A(t), where A(t) is assumed to converge to a constant number as 1+ co. (6) may be written as a three dimensional dynamic system in c, + and k, ’ The analysis contains the implicit assumption that government spending provides no direct utility benefits. A similar assumption is made by Wynne (1988) in the calibration of the neo-classical growth model to U.S. wartime economy. Ah the results would still go through were government spending to provide consumption services, as for instance, in Barro (1981). but the algebra of the model would be somewhat messier. ’ Saddle-path stability is established below.
134
M.B. Devereux
/ Government purchases and real interest rates
Fig. 1
where the solutions are implicit functions of t and yA(t). That is, the solutions are functions c(t, y A(t)), +(t, yA(t)) and k2(t, y A(t)). In addition, there is one initial condition; k(O) = k,, and two terminal conditions; one given in footnote 3, and the other by limit, em k(t) = k. The method is to differentiate the system with respect to y around an initial steady state, with y = 0. This leads to the linearized system
(9)
The coefficient matrix has its trace positive and determinant negative, which with the initial and boundary conditions ensures saddlepoint convergence. One of the positive roots is r. Let the other roots be X,, X, with h, < 0 -C X,. The roots satisfy the conditions X, +X2 = r, and X,A, = (~~/YSv,~ - .“). Now take the Laplace
transform
of (9), to get
00) where L,(c,) = lam ePS’c,(t, 0) dt, s > 0, etc. The transformation uses the fact that k,(O, 0) Equation (10) holds for all s > 0, and in particular for s = r, h,. Use these values in (lo), terminal conditions requiring k2(t, 0) and &(t, 0) to be finite [see Judd (1985)]. This gives conditions that may be solved for the initial response of consumption and utility, c,(O, 0) ~~(0, 0). To simplify the exposition, let A(t) follow the path A(t) =A, t E [0, T], A(t) t E (T, co). The solutions for c,(O, 0) and (~~(0, 0) may then be written
+*(O,
0) = -L+(l
- ePrT).
= 0. and two and = 0,
(12)
M. B. Devereux
/ Government purchases and real interest rates
135
The first inequality follows from the properties of the roots of (9). The initial response of investment can then be written as
f&(0,0)=
-1
A
+
(1 _ eehlT)
fcucr
1 _
fcc(r-
i
; fycr
b)h
i
-_‘^;’
‘;
cc
r
(13)
1 r
From (11) consumption always falls in response to a temporary government spending shock. This fall is greater the higher is T. For U, # 0, it is possible that consumption falls by more than one hundred percent of the increased government spending. Investment may then rise or fall. If U, = 0, recovering the neoclassical model, investment always falls, giving the aforementioned response of real interest rates. ’ For the general specification of recursive preferences however, this result no longer obtains. The higher is T, the more likely that investment rises in response to the increased government spending, since for a permanent A(t) shock, investment will always rise. This will lead to .falfing real interest rates in the aftermath of the shock. Finally, we may solve for the inverse Laplace transforms to obtain the solution for the time path of the response of capital, k2(f, 0), in the interval [0, T]. After some lengthy algebra, this gives
k2(t70) =
(Al_\,)[e”l’-
A
x
-
1
(1 - eW +
[
rX2
lh
(x
e”2t] ePXIT+
e_rT
1 (r-4)
(eX2,
1
) 2
_
err)
_
e-*1’(e’2’
h,
- eX1’)
r
e-‘r ‘(‘-A,)
( ert - e? ,] ’
where
Q-y.
iI
cc
For U, = 0, this is always negative. Real interest rates are above their steady state level for the full duration of the increase in government purchases. When U, > 0 the sign of (14) is ambiguous. Real interest rates may be higher or lower than steady state. As T + co, k2(t, 0) > 0 for all t. By continuity, there must be a T-c co such that k2(t, 0) > 0 for t E [0, t] i.e., the capital stock will be above its steady state value for the full duration of the increased government purchases, converging downwards to its original steady state level. Thus real interest rates can be lower during periods of temporarily high government purchases. The observation of low real interest rates during wars is consistent with this model. With a constant rate of time preference, an increase in real interest rates is required to dampen private consumption, with any degree of concavity in preferences. This reduces investment expenditures. Thus in this case the (temporary) rise in government purchases leads to rising real interest rates on impact. With an endogenous rate of time preference, however, the negative welfare effects of the increase in government purchases causes an immediate fall in the implied rate of time preference. This allows for a large fall in consumption, perhaps exceeding the initial increase in government purchases. If the latter occurs, there is an immediate investment boom and falling interest rates along the adjustment path. ’ Note also that for u, = 0 and T = 00, investment government
purchases.
This is another
well-known
is unchanged, and consumption is crowded result of neoclassical theory (Barr0 1987).
out by 100% of the rise in
136
M.B. Devereux / Government purchases and real interest rates
References Barro, R.J., 1981, Output effects of government purchases, Journal of Political Economy 89, 1086-1121. Barro, R.J., 1987, Macroeconomics, 2nd. ed. (Wiley, New York). Blanchard and Fischer, 1989, Macroeconomics (MIT Press, Cambridge, MA). Epstein, L.G., 1987, A simple dynamic general equilibrium model, Journal of Economic Theory 41, 68-95. Epstein, L.G. and J.A. Hynes, 1983, The rate of time preference and dynamic economic analysis, Journal of Political Economy 91, 611-625. Judd, K.L., 1985, Short-run analysis of fiscal policy in a simple perfect foresight model, Journal of Political Economy 93, 298-319. Mankiw, G., 1987, Government purchases and real interest rates, Journal of Political Economy 95, 407-419. Uzawa, H., 1968, Time preference, the consumption function, and optimum asset holdings, in: J.N. Wolfe, ed., Value, Capital and Growth: Papers in Honour of Sir John Hicks (Aldine, Chicago, IL). Wymte, M., 1988, The effects of government purchases in a perfect foresight model, Mimeo. (University of Rochester, New York).