Inflation, wealth and interest rates in an intertemporal optimizing model

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Transcript of Inflation, wealth and interest rates in an intertemporal optimizing model
Journal of Monetary Economics 16 (1985) 7385. NorthHolland
INFLATION, WEALTH AND INTEREST RATES IN AN INTERTEMPORAL OPTIMIZING MODEL*
Daniel COHEN CEPREMA P, 75013 Paris, France
This paper analyzes the relationship between the growth of the money supply and capital accumulation in a monetary optimizing model. Under certain conditions we show that a large intertemporal elasticity of substitution makes the nominal rates of interest undershoot its longterm value, that a low elasticity gives rise to overshooting, while a unitary elasticity is shown to make the nominal rate of interest a constant. These considerations plus explicit attention paid to the income and substitution etIects induced by the changes in the nominal rates of interest provide us with an understanding of the correlation between growth of money and capital accumulation.
1. Introduction
The link between capital accumulation and monetary policy has been recognized in various papers, mostly written during the sixties, such as Mundell (1963) Tobin (1965) and Sidrauski (1967b). By pushing up inflation, expan sionary monetary policy would raise the nominal rate of interest, and reduce the demand for money. How the story would continue was modelspecific. For example in Tobin’s analysis, the reduced demand for money implied an increased demand for other financial assets and triggered capital accumulation. This shift, however, was derived from postulated demand equations and not from explicit optimizing behavior. By contrast the intertemporal optimizing model of Sidrauski (1967a) implies that the longterm capital accumulation is determined by the consumers’ rate of time preference and thus is immune to changes in monetary policy.
More recently, Fischer (1979) has examined the transition path of the Sidrauski model and has shown that expansionary monetary policy does have an expansionary impact on capital accumulation, except in one case: when agents have a unit intertemporal elasticity of substitution. This latter case however was a puzzle since, again, it raised the question of the general validity of the link between capital accumulation and monetary policy.
*Initial impetus for this work came from stimulating talks with Andrew Abel, Olivier Blanchard and JeRrey Sachs whom I gratefully thank. I am particularly indebted to Michael Jerison, Daniel Laskar, Philippe Michel, Jeffrey Sachs and Charles Wyplosz for their helpful comments on a previous draft of this paper and to an editor of this journal for his many suggestions. None of the above are responsible for remaining mistakes.
03043923/85/$3.3001985, Elsevier Science Publishers B.V. (NorthHolland)
74 D. Cohen, Inflation, wealth and interest rates
The purpose of this paper is to provide the intuition behind Fischer’s results. Returning to the insight of the early papers on the topic, we focus on the behavior of the nominal rate of interest and on its impact on the saving decision. In the model examined by Fischer, we show that nominal rates vary along the transition path in response to two conflicting influences: they tend to decrease along with real rates, and tend to increase along with the inflation rate. The two effects cancel out when the intertemporal elasticity of substitu tion is one. Otherwise, the nominal rates overshoot (resp. undershoot) their longterm value when this coefficient is below (resp. above) unity. Monetary policy can change the extent of the overshooting (resp. undershooting). It is through this channel that it modifies capital accumulation, by inducing a change in intertemporal relative prices; it is not through its influence on the level of the nominal rates nor through the wealth effect it induces. In the (puzzling) case when the agents have a unit intertemporal elasticity of substitu tion, nominal rates are constant so that monetary policy can induce no changes in the intertemporal relative prices: capital accumulation stays unchanged.
Section 2 sets up the model. Section 3 examines the behavior of nominal interest rates and shows that they are constant when agents have a unit intertemporal elasticity of substitution. Section 4 shows why an increased growth of the money supply reduces initial consumption. Section 5 shows its impact on capital accumulation. A conclusion summarizes the paper.
2. The model
We follow Fischer’s model, but we keep the economy decentralized (we show the equivalence in our last section). There is no uncertainty in the model and a.lI values are perfectly foreseen.
Finns. All producers have access to a technology of production with only one input, capital:
Call {rc)rTO the equilibrium path of real rates of interest, and let
(2)
The firms maxim&e the discounted value of their cash flow. Assuming no installation cost or depreciation, the firms solve
R(‘)[f(k,)k,] dt,
D. Cohen, Injarion, wealth and itrferest rates 75
subject to initial capital, k,, given. Implicit in this formulation is the fact that firms do not use money. Their decisions only depend on the real rates of interest and the solution to (3) is characterized by
f’(k) = r,* (4)
Government. The government’s only activity is to print money and to distrib ute the proceeds to the households in a lumpsum way. Let {H,}, 20 be the flow of money creation each period and {M,}, LO the stock of nominal money. Then
tit= H,, MO given. (5)
We shall only consider changes in monetary policy such that
stays a constant.
Monetary wealth. We let { p,}, ~ 0 be the equilibrium path of the price level. Then
W,,O = J
* e R& dt + M, 0 PI PO
(6)
is the wealth which is created by the government and distributed to the households.
Households. Private agents maximize an intertemporal utility function of the form (we assume away population growth)
U=]meg’U(c,,m,)dt, 0
with
u(c,m)= [ 1 pmla 1s 1s if SrO, sz 1,
= log cam1 (I if S=l,
where c, is the flow of consumption at time t and m, is the stock of real cash balance held by the households. Let 0 = l/S. We shall refer to t9 as to the intertemporal elasticity of substitution.’
‘In stochastic models, S would be interpreted as the coefficient of relative risk aversion.
76 D. Cohen, Inflation, wealth and interest rates
The private opportunity cost of holding real cash balances (nominal interest rate) is
i,=r,+P,/P,. (7)
The total wealth of the households is composed of the ownership of the firms and of monetary transfers from the government. The intertemporal budget constraint faced by the households can then be written
/
m
0 evR(‘)[c,+ i,m,] = Wk, + W,,O = W,.
With the utility function under study, this problem leads to
Cf a ci ml llx ”
g+[r,6](la)(&lg. , t
63)
00)
When c, and m, are aggregated into a composite good Q, = cprn:“, whose price index is ijlol), eqs. (9) and (10) can be equivalently written as (9) and (10’):
=rt(l )EL6 1 1 dQ, 8 Q, dt
a i,dt ’
(10’) is the standard equation stating that the marginal rate of substitution between future and present consumption of the composite good, modified by the discount factor 6, is equal to the composite good rate of interest [see, e.g., Arrow and Kurz (1969)]. We return to this interpretation in section 4. From (lo), we see that in the stationary state lim,,,r, = S. This is the origin of Sidrauski’s superneutrality result; i.e., longrun capital accumulation is de termined by f’( k,) = 6.
Fischer linearizes the model around the steady state in a threedimensional system whose variables are {k,, c,, m,} and show:
Proposition 1. An increase in the rate of growth of the money supply speeds up capital accumulation on the transition path to the steady state, in all cases but when the intertempbral elasticity of substitution is one. In this latter case, money is superneutral on the transition path.
The case when 8 = 1 is puzzling since all general statements on the relation ship between the money supply and capital accumulation will prove invalid in that case.
D. Cohen, Injation, wealth and interest rates 17
In order to investigate the origin of Proposition 1, we shall separate the consumers’ and the producers’ decisions. To do so, we shall take as given the equilibrium values of the real rate of interest: This will define the producers’ strategies and keep productive wealth Wk, constant. Associated to this se quence, we shall derive from the money demand equation the equilibrium values of the nominal rates (in section 3), and the consumers’ decision (in section 4). Only in section 5, shall we calculate the equilibrium value of the real rates by imposing that the good market clears.2
3. The behavior of nominal rates of interest
In this section and in the following one we shall assume that the path of present and future real rates of interest { r, }, 2 c is given and perfectly foreseen. We shall assume that { r, )I L O declines toward. its longterm limit S. (This property will be satisfied in the neighborhood of the steady state, see section 5.)
Associated to this sequence { r,}, r ,,, we now calculate the equilibrium value of the nominal rates of interest. In all that follows we shall restrict our attention to the solution {i,}, ~ O which converges in the stationary state toward a finite value [see Obstfeld and Rogoff (1983) for a discussion of this assump tion]. This value will necessarily be lim,,,i, = i, = 6 + u (a is the rate of growth of money supply).
We now show:
Proposition 2. Associated with a path of real rates of interest {r,}, r O, there exists a unique converging solution for the sequence of nominal rates of interest. The solution { i, }, $ O is an increasing (resp. a decreasing) sequence if 19 > 1 (resp. 8 c i). It is a constant when 8 = 1. The rate of growth of { i,},20 is a decreasing (resp. an increasing) function of the rate of growth of the money supply when 13 > 1 (resp. t3 < 1).
The second part of the proposition can be written
with ah/a030 when e21.
Since
~=ev[fu(~,{rS})du, +=explmfU(o,{rs})du, m I
‘A related (and independent) work by Jones (1982) also illuminates Fischer’s results out of the analysis of the marginal utility of wealth. I thank Professor Fischer for having pointed out to me this reference.
18 D. Cohen, lnjlution, wealth and interest rates
this will enable us in the following section to derive that i,/i, and i,/ii are reduced (resp. increased) by an increase in u when 0 > 1 (resp. 8 < 1).
The proof of Proposition 2 is given in appendix A, where we show that i, can be written
~=hjm(exphj’[i,(8l)(rU6)]du)ds, , I r
with 1
“=l+(la)(&q’ i,=a+6*
In order to give some intuition about Proposition 2, we now deal with the case of static expectations on nominal interest rates. This case will say that the consumers believe at each point in time that the nominal rates will not vary. A priori, we already know [from eq. (lo)] that this hypothesis will yield the correct solution when 8 = 1. (In this case, expected nominal interest rate changes do not enter into the consumption decision; see below in section 4 for an interpretation in terms of income and substitution effects.) When agents’ expectations are static, we can write consumption’s changes and expected inflation to be [from eqs. (10) and (9)]
so that the value of nominal rates becomes
i,=u+6(0l)(r,6). 03)
Eq. (13) shows that the perfect foresight equilibrium described in (11) is simply an harmonic average of the static expectation case.
The intuition behind this latter case comes easily from eq. (12). Inflation is a decreasing function of the real rates. When the intertemporal elasticity of substitution is large, the influence of inflation offsets the influence of the real rates so that nominal rates increase along with inflation. When 0 is small, the reverse occurs: The nominal rates decrease along with the real rates. When t9 = 1, the two effects cancel out and the nominal rates are constant. From (13) the rate of growth,of the nominal rates can be written
1 di L1= 1 d i, dt o+S(BI)(r,6)‘e1’~‘r,6’.
In absolute terms, one sees that the rate of growth of i, is always reduced, while the sign depends on B >< 1 for the reasons explained above.
D. Cohen, Injution. wealth und interest rates 79
4. The behavior of consumption
We have described the equilibrium sequence {i,),;, 0 as a function of {‘rl,zo We can now derive the optimum consumption path selected by the households. From Fischer’s results we expect that an increased rate of growth of the money supply will initially reduce consumption, ldaving room for more capital accumulation. We show:
Proposition 3. An increased rate of growth of rhe money suppry tilts forward the consumption stream and reduces initial consumplion.
This proposition can be established in the following two steps. First, from (lo), we can find that the value of consumption is
(1@(Bl)
expt9 Ca(r,G)dr. / I 04
From Proposition 2, we see that the increase in the growth of the money supply will increase i/i, with 0 > 1 and decrease i/i, when 0 < 1. In both cases this will reduce c/c,. Hence, the first part of the proposition.
We now show that this will reduce initial consumption. In the next section, the superneutrality in the stationary state will impose
c, =f(k,) with k, = (f’)‘(6),
so that initial consumption will decrease out of the reduction of co/c,. We need not rely on longrun supemeutrality, however, to prove this result, as we now show.
Monetary wealth is defined by (6), but can also be &own to be
Wm, E M, J PO 0
meR(~,!S dt =/meR(‘)i,m,dt. PI 0
(15)
[See Sachs (1983) and appendix B.]
This equality shows that the value of monetary wealth will always equal (for all values of the real rates of interest) the discounted cost that the consumers will pay to hold their money balances. From (8), we then see that
/
00
0 ebR(‘)c, dt = Wk,. (16)
In order to understand eq. (14), we shall return to the interpretatiog given in eq. (10’) and use the fact that the instantaneous utility function allows us to
80 D. Cohen, Isolation, wealrh atld itlteresr tcues
aggregate c, and m, into a composite good, which we call Q,, whose true price index is ijrmn). Given the value of Q,, c, and m, are simply determined by
c, = aij’a) Qv i,m, = (1  a)ij*a)Q,.
The optimal choice of Q, comes from the program
subject to
/
co eR(Oijia)Q,dt = W,,
0
From (9) and (16), the equilibrium value of W, is simply
w, = i Wk,, (18)
and is independent of monetary policy.
We can now split the influence of an expansionary monetary policy into two parts: (i) It raises all nominal rates. The wealth is unchanged when measured in consumption units [from (18)], but is reduced when measured in terms of the composite goods. (ii) It distorts the equilibrium sequence of the relative price of the composite good from one period to another.
To separate (i) and (ii) we shall consider them from the following two points of view:
First, let us assume all nominal rates are increased in a way which keeps the relative prices (i,/i,)(‘“) unchanged. Since the utility function U is homo thetic, and since W, is unchanged, QJQo is unchanged and Q. is decreased in proportion to iA1  =). This keeps co( = aif @Q,) unchanged. We then see that neither the rise in the general level of the nominal rates, nor their implication for wealth, will create a reduction of consumption.
Second, let us now examine the influence of the distortion in relative prices (i/i,)““) on the decision to consume, by assuming that i, stays unchanged. The usual competing income and substitution effects are at work. When 0 > 1, we know [see, e.g., Abel and Blanchard (1983)] that substitution effects dominate. The reduction in (i,/i,)(‘“) will therefore reduce initial spending Q,, hence initial consumption [by (17) and our assumption that i, is un changed]. On the other hand, when 13 < 1 income effects dominate, but now an expansion& monetary policy increases (i,/io)“a’ so that, again, co is reduced. Only when 4 = 1 do income and substitution effects exactly cancel out. The changes in the nominal rates do not enter into the consumption
D. Cohen, Inflation, wealth and interest rates 81
decision (and this is why our static expectation case yielded the perfect foresight equilibrium in the previous section).
5. Changes in capital accumulation
We now calculate the changes in capital accumulation associated with the change of monetary policy. From the previous section, we already know that an increased growth of the money supply will initially reduce consumption, and will therefore tend to increase capital accumulation. This, in turn, tends to increase productive wealth and might annihilate the initial impact. We know from Fischer that this will not be the case and that the speed of capital accumulation will be increased all along the transition path. We now prove this result again but in a way which differs from Fischer’s. To his threedimensional system (k,, M,, c,}, we substitute (T,, i,, c,} and solve the model in a way which respects the distinction between consumers and producers.
The clearing of the goods market will be written
f( k,) = ic, + c,. (19)
In order to linearize the system around its steady state, we let
x,=r,8, y,= i, i co* z,=c,cc,.
Linearization of eq. (19), (10) and (12) gives
i, = Sx, + az,, (204
i,= ex,+(lCY)(el);?f coo, m 1 (20'$ Y,=hi,[y,+(elb,], (204
where
a= f”(k,) (a>O).
One way to find the characteristic equation of this system is to recognize that there is only one predetermined variable, x,, so that, if one calls /A the negative root of the system, the saddlepath will be described by
x, = x0 eept, Y, =yoelr’, z, = zoeCc
Here, the higher /.L, the faster x, goes to zero and the faster is capital accumulation.
Eq. (20a) allows us to write
C,Cc,= S+lJ
X a I? (21)
which describes the supply of consumption good: An increased capital accu mulation lowers the supply of consumption good. It does not depend on monetary policy.
Eqs. (20b) and (20~) allow us to write
c,cc,=  1 i(1 or)(8*)‘&#.x,.
m (22)
which is the household’s demand for consumption good. Eq. (22) is a linearized version of the eq. (14) which we used to prove Proposition 2.
Equating (21) and (22) yields the value of ~1 which is compatible with the clearing of the goods market, and yields the characteristic equation found by Fischer. Thanks to our decomposition, we can now give an intuitive interpreta tion of this equation. The supply is independent of monetary policy [eq. (21)] while the demand is related to monetary policy in a way which we described in Proposition 2: An increased rate of growth of the money supply shifts downward the demand for consumption unless 0 = 1. Except in this latter case, this leads to an increase in CL, the speed of capital accumulation, due to an increased growth of the money supply: This shows Proposition 1.
6. Conclusion
We have derived the relationship between the rate of growth of the money supply and the speed of capital accumulation in an intertemporal optimizing framework. In our model, consumption and real cash balances could be aggregated into a composite good whose price was an increasing function of the nominal rate of interest. We have shown that the link between monetary policy and capital accumulation arises from the variation over time of this price index, and not from the wealth effect induced by its absolute change. In the case when capital accumulation was invariant to monetary policy, we have shown that the nominal rate of interest (hence the price of the composite good) was constant over time, so that monetary policy induced no dynamic effect on the intertemporal pattern of consumption.
Appendix A: Proof of Proposition 2
From (9) and (lo), one sees that i, solves
k%=h[i,i,+(8l)(r,a)].
D. Cohen, Infklrion. wealth and interest rates 83
By setting z, = l/i,, the equation becomes a linear one and is solved as written in eq. (11). This equation also implies
2 ‘= =,
X i,(8l)(r,6) 1
[X~~exph[[~~(tJl)(~Ud)du]dr]~l), (A.l)
so that Z/Z >< 0 according to 8 5 1, which proves the first part of the proposition (recall that r, is a declining sequence).
Now, let us prove that Z/Z( =  l/i,(diJdt)) will be an increasing (resp. a decreasing) function of i, when 0 > 1 (resp. 8 < 1). One has
(e1)(&)6) 1 hIme “‘f(t) dt ’
0 1 f(t)=exp(8l)(r,a), (A.2)
so that
00
d i [()I [ [ / te“m’f(r)dt
di, z ,a =x1 O
/ Q)
eeXi‘j(l)dl * ’ 0 1
(A.3)
Let us prove the last part of the proposition in the case when 0 > 1. We want to prove that the above derivative is positive, that is, we want
/
m texim’f(t)dtl
0 ~meA’‘j(r)dr]2. (A4)
To prove this inequality, it is enough to integrate by parts the LHS to find
LHS=~mdrJme“‘“/(u)d~=~we~i~‘(~we’i~u~(~+~)du)dr. I 0 0
Because f is a decreasing function f(t + u) <f(u), and because it is above unity f(u) <f (t)f (u), so that (A.4) obviously holds. Q.E.D.
a4 D. Cohen, @Tation, wealth and interest rutes
Appendix B: Proof and interpretation of Eq. (15)
(a) Proof. Integrate by parts Wn,O to find
exp/mrUdu$ + 0 I J to 0
*expR(I)% I
(B.1)
(b) To understand eq. (B.l), one may turn it into an arbitrage condition. To see that, let us show [recalling i, = r, + (l/p,)(dp,/dt)] that
Proof. By definition of i, and integration by parts, the LHS is
LHS=/ om[exp(~,,d~)][~~+~]d~
= [(e+cdu])[;I];=;.
Eq. (B.2) (which always holds) is a particular case of (B.l) when written for a constant money supply. Eq. (B.2) can now be easily understood as an arbitrage condition: In order to accomplish their transactions, the agents may as well hold one unit of money or borrow this unit every period. The two costs must be equal: This is what is stated in eq. (B.2). Eq. (B.l) is simply a forwardly repeated version of (B.2) [see Boyer and Hodrick (1982)].
References
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Boyer, R. and R. Hodrick, 1982, The dynamic adjustment path for perfectly foreseen changes in monetary policy, Journal of Monetary Economics 9,185201.
Fischer, S.. 1979, Capital accumulation on the transition path in a monetary optimizing model, Econometrica 47,14331439.
Jones, E. P., 1982, Neutrality of monetary policy, Working paper HBS 8307. June (Harvard Business School, Cambridge, MA).
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D. Cohen, Inflation, wealth and interest rates 85
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Sidrauski, M., 1967b, Inflation and economic growth, Journal of Political Economy, Dec., 796810.
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