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Article Excerpt
Summary. We argue that real uncertainty itself causes long-run nominal inflation. Consider an infinite horizon cash-in-advance economy with a representative agent and real uncertainty, modeled by independent, identically distributed endowments. Suppose the central bank fixes the nominal rate of interest. We show that the equilibrium long-run rate of inflation is strictly higher, on almost every path of endowment realizations, than it would be if the endowments were constant.
Indeed, we present an explicit formula for the long-run rate of inflation, based on the famous Fisher equation. The Fisher equation says the short-run rate of inflation should equal the nominal rate of interest less the real rate of interest. The long-run Fisher equation for our stochastic economy is similar, but with the rate of inflation replaced by the harmonic mean of the growth rate of money.
Keywords and Phrases: Inflation, Equilibrium, Control, Interest rate, Central bank, Harmonic Fisher equation.
JEL Classification Numbers: C7, C73, D81, E41, E58.
1 Introduction
Our goal is to understand the behavior of prices and money in a simple, stationary economy with exogenous production subject to independent and identically distributed shocks. We show that there is a unique, neutral stationary equilibrium, both for the case when the economy has no loan market and also when there is a central bank that sets an interest rate [rho], the same for both savers and borrowers. If the economy has no loan market, then the money supply remains constant and, in equilibrium, prices are independent and identically distributed.
For the economy with a central bank, inflation (or deflation) is possible. We calculate exactly the long-run rate of inflation as a function of the interest rate [rho] and the distribution of the random shocks. Surprisingly, we find that larger productivity shocks lead to higher long-run inflation. We couch our analysis in terms of a representative agent with an arbitrary concave utility function u(dot), a single good, and independent, identically distributed endowments. We prove that there is a unique neutral, stationary equilibrium, and derive explicit formulae for it.
According to the famous Fisher equation, the rate of inflation should depend on the monetary rate of interest and on the time-preference of the agents, and on nothing else. In a nonstochastic, stationary economy, this is precisely the case:
[p.sub.n+1]/[p.sub.n] = [m.sub.n+1]/[m.sub.n] = [beta](1 + [rho]). (1.1)
Here [beta] denotes the discount rate of the agents (its reciprocal is the rate of time-preference), 1 + [rho] denotes the gross monetary rate of interest, [p.sub.n] is the price on day n, and [m.sub.n] is the money supply on day n. If the central bank sets the rate of interest (the same for borrowers and for depositors) equal to the rate of time-preference for agents, the equilibrium rate of inflation and of monetary growth will be zero.
In a stochastic stationary economy with well-informed agents, who know the value of their endowment before deciding on expenditures, no simple formula like (1.1) can hold at each moment in time. Indeed, denoting consumption at time n by [Y.sub.n], we must have that
u'([Y.sub.n])/[p.sub.n] = (1+[rho])[beta] x E [u'([Y.sub.n+1])/[p.sub.n+1]],
and since [Y.sub.n+1] and [p.sub.n+1] are not independent, one cannot hope for a clean expression for E[[p.sub.n+1]/[p.sub.n]]. However, in our stochastic economy we do derive a precise formula for the long-run rate of inflation [lim.sub.r[right arrow][infinity]][[rth root of ([p.sub.n+r]/[p.sub.n])]]. A Fisher-like equation still holds, but with the one-period rate of inflation replaced by the harmonic mean of the one-period money growth rate, giving an inflationary bias:
[lim.[r[right arrow][infinity]]] [rth root of ([p.sub.n+r]/[p.sub.n])] > {E [[m.sub.n]/[m.sub.n+1]]}[.sup.-1] = [beta](1 + [rho]) almost surely. (1.2)
In case [beta](1+[rho]) = 1, the harmonic mean of [m.sub.n+1]/[m.sub.n], the gross rate of monetary growth, is one. As long as there is any random variation in the rate of monetary growth, its geometric mean must therefore be greater than one. With independent draws, the law of large numbers guarantees that the long-run (geometric) growth rate of the money supply is greater than one, on almost every path. Since the long-run rate of inflation must be equal to the long-run rate of growth of the money supply, this shows that when [beta](1+[rho]) = 1, the slightest bit of monetary fluctuation typically (5) creates an inflationary bias, irrespective of the utility function of the representative agent.
There is some historical evidence that periods of stable output are associated with low inflation. Our analysis linking inflation with real uncertainty provides a modest reason for that empirical association. On the other hand, it can easily be shown in our model that money cannot grow faster than the nominal interest rate, and hence the long-run inflation is bounded from above by 1+[rho]. The inflationary bias is therefore at most (1-[beta])(1+[rho]), which should be small, since [beta] is ordinarily close to 1. We give a precise formula in Section 4. A bigger inflationary bias from real uncertainty might be generated in a model in which the central bank adjusts the interest rate, but we do not pursue that question here.
Suppose that physical endowments [Y.sub.n] are independent and identically distributed (iid) and that Y u'(Y) is not constant, where u'(Y) is the marginal utility of consumption. We show that no matter what fixed interest rate [rho] the central bank maintains, monetary stocks [m.sub.n] and prices [p.sub.n] must fluctuate unboundedly. Even if Y takes on just two values, prices [p.sub.n] will eventually become unboundedly large or small, or both. Only by active management, making [rho] a function of the physical endowment Y, can the central bank ensure that prices will stay bounded. Even such an active bank, however, cannot maintain absolutely fixed price levels [p.sub.n] = p for all n.
We show that if agents do not know their endowment before they are called upon to commit themselves to expenditures, then the original Fisher equation is restored irrespectively of the agents' utility function, and setting the rate of interest equal to the rate of time preference will result in an expected rate of inflation equal to zero.
Our model is in the spirit of the representative agent approach of Lucas (1978); we use dynamic programming methods in a microeconomic model of money, in the tradition of Shubik (1972), Shubik and Whitt (1973), Lucas (1980, 1990), Lucas and Stokey (1983), Stokey and Lucas (1989), Woodford (1994), and Karatzas, Shubik and Sudderth (1994). The microeconomic tradition of analyzing policy and money in a market-clearing model is vast; see, for example, Phelps (1967, 1970, 1973), Kydland and Prescott (1977), Barro (1990), Chari et al. (1991), Mankiw (1992), Sargent (1987, 1999), Alvarez, Lucas and Weber (2001), and Dubey and Geanakoplos (1992, 2003). To the best of our knowledge, however, the connection between real uncertainty and long-run inflation addressed in this paper seem to be treated here for the first time. The model of Lucas (1990), for instance, is extremely close to ours, but analyzes the case where the central bank interest rate is random and output is fixed. Models of Mehra and Prescott (1985), Weil (1992) and others examine the real interest rate, that is, the interest rate on bonds that pay one unit of good in each period. They find that real uncertainty might increase or decrease the real rate of interest, depending on the third derivative of utility.
There are precedents for our conclusion that prices must wander to either infinity or to zero no matter what fixed nominal interest rate [rho] the central bank fixes. For example, Matsuyama (1991) showed that even for a nonstochastic economy, cyclical or chaotically fluctuating prices are possible. (But that is in a nonstationary equilibrium.)
An alternative generalization of the Fisher equation to stochastic economies is derived for models with exogenous prices by Benninga and Protopapadakis (1983) and Sarte (1998). These authors express the one-period deviation for the classic equation in terms of the covariance between the ratio of marginal utilities u'([Y.sub.n+1])/u'([Y.sub.n]), and [p.sub.n+1]/[p.sub.n].
We derive an explicit formula for the long-run rate of inflation, for arbitrary utility u, iid endowments {[Y.sub.n]}, and nominal interest rate [rho], without any approximation. Over the last decade it has become fashionable to investigate the properties of monetary economies by using log-linear approximations around the riskless steady state economy (see, for example, Woodford, 2004). We have been able to avoid the need for such shortcuts by confining our attention to a representative agent economy. We do not know how to compute explicit formulas for heterogeneous agent economies. We also confine our attention to (what we prove) is the unique neutral stationary equilibrium, ignoring sunspot equilibria and nonstationary equilibria. These latter are studied in Woodford (1994) in an economy without real uncertainty.
Lastly, we note that our interest rate pertains to the trading period; an agent who wishes to sell in order to raise the revenue to make a simultaneous purchase must borrow the money at rate [rho]. This embodies a genuine cash-in-advance constraint.
1.1 Preview
The derivation of the harmonic Fisher equation (1.2) will be undertaken in a completely specified general equilibrium model with a representative agent. It may be instructive to see briefly how to derive the harmonic Fisher equation in a reduced-form model based on two premises, stationarity and money-neutrality. First, we suppose that prices at time n are proportional to the supply of money at time n, namely
[p.sub.n] = p([Y.sub.n])[m.sub.n] (1.3)
for an appropriate function p(dot) to be determined (see equation (4.6')), where [Y.sub.n] is the random endowment of the perishable good at time n. Secondly, we suppose that the money supply at time n + 1 is proportional to the supply at time n, with a proportionality constant that depends only on the random endowment at time n, namely
[m.sub.n+1] = [tau]([Y.sub.n])[m.sub.n] (1.4)
for an appropriate function [tau](dot), to be determined (see equation (4.8)). In equilibrium, the agent is indifferent between spending a dollar on consumption and depositing it in the bank with interest to consume during the next period:
u'(y)/[p.sub.n] = [beta](1 + [rho]) x [E.sub.n] [u'([Y.sub.n+1])/[p.sub.n+1]], on{[Y.sub.n] = y}.
Here and in the sequel, [E.sub.n][dot] = E[[dot]|[F.sub.n]] denotes conditional expectation with respect to the information [F.sub.n] available to agents at time n; this information includes [Y.sub.n] and [m.sub.n]. Substituting for [p.sub.n], [p.sub.n+1] and [m.sub.n+1], gives
u'(y)/p(y)[m.sub.n] = [beta](1+[rho]) x [E.sub.n] [u'([Y.sub.n+1])/[p([Y.sub.n+1])[m.sub.n+1]]] = [beta](1+[rho]) x [E.sub.n] [u'([Y.sub.n+1])/[p([Y.sub.n+1])[tau](y)[m.sub.n]]]
= [[beta](1 + [rho])/[tau](y)[m.sub.n]] x [E.sub.n] [u'([Y.sub.n+1])/p([Y.sub.n+1])], on{[Y.sub.n] = y}.
Let z(y) [??] u'(y)/p(y). Cancelling [m.sub.n] from both sides, bringing [tau](y) to the left, and then inverting both sides, gives
1/[tau](y) = [1/[beta](1 + [rho])][z(y)/[E.sub.n][z([Y.sub.n+1])]], on{[Y.sub.n] = y}.
Assuming that the [Y.sub.n+1] is independent of [F.sub.n] for all n [greater than or equal to] 1, and taking expectations, we obtain
E [[m.sub.n]/[m.sub.n+1]] = E [1/[tau]([Y.sub.n])] = [1/[[beta](1 + [rho])]][[E[z([Y.sub.n])]]/[E[z([Y.sub.n+1])]]] = 1/[[beta](1 + [rho])], (1.5)
because [Y.sub.n] and [Y.sub.n+1] have the same distribution. This is the harmonic Fisher equation of (1.2).
2 Equilibrium
2.1 The model
We consider a representative agent model extending over days or time-periods n = 1, 2,.... On each day the agent receives a random endowment [Y.sub.n]([omega]) of a single perishable commodity, where [Y.sub.n] is a random variable on a given probability space ([OMEGA], F, P) and [omega] is an element of [OMEGA]. The random variables [Y.sub.1], [Y.sub.2],..., corresponding to the successive random endowments of the agent, are assumed to be independent with a common...
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