Bursting bubbles: Further results

Journal of Monetary Economics 17 (1986) 425-431. North-Holland

BURSTING

BUBBLES: FURTHER

RESULTS

Alan G. ISAAC* Northern

Illinois

University,

DeKalb,

IL 60115,

USA

It is widely recognized that simple perfect foresight m aximizing models often allow a continuum of hyperinflationary equilibria in the presence of a fixed money supply. Various attempts have been made to restrict the number of equilibria in such models. This paper illustrates some new aspects of a recent proposal.

1. Introduction

Perfect foresight maximizing models are often restricted in order to rule out non-stationary equilibria. In overlapping generations (OLG) models, a popular approach has been to assume logarithmic utility functions for identical individuals who have no endowment for their old age. Recently, Farmer (1984) offered a proposal with more economic intuition and somewhat less restriction on model structure: consumption can be traded only in units of size A > 0. In models with a fixed money supply, this clearly precludes hyperinhationary equilibria. In a hyperinflationary speculative bubble, money would become worthless in a finite period of time and hence would never be held. This paper first demonstrates the existence of stationary equilibria under Farmer’s proposal. Then it is shown that, although Farmer’s proposal succeeds in ruling out speculative bubbles, a number of difllculties remain. For example, model restrictions which previously guaranteed a unique stationary monetary equilibrium no longer do so. Recasting Farmer’s proposal as a (small) cost to trade avoids this dikulty. 2. Individual

optimization

The model is a standard OLG model deriving from Samuelson (1958). Each period there is one young agent with 6xed endowment of the consumption good e,, and one old agent with fixed endowment e, of the consumption good and M of money. In a ‘monetary equilibrium’, the young exchange part of their endowment of the single perishable good for money, the single durable asset. This allows them to consume more than their endowment when they are *An anonymous referee’s comments are gratefully acknowledged. 0304-3923/86/$3.50Q1986,

Elsevier Science Publishers B.V. (North-Holland)

426

A.G.

Isaac,

Bursting

bubbles:

Further

results

old. If we let p(t) be the commodity price of money in period t and assume perfect foresight, the young will choose nominal money balances M,, to

where monotonically increasing, differentiable, strictly quasi-concave U describes ordinal utility rankings of two-period consumption. Assume Vi/U, has a limit of zero (infinity) as [ey -p(t)ikl,]/[e, +p(t + l)M,,] approaches infinity (zero). (Here Ui is the partial derivative of U[. , . ] with respect to the ith argument.) Then for positive prices and in the absence of any restrictions on the choice set, this implies a demand for money MJ p( t), ~(t + l)] which is homogeneous of degree minus one in prices. The old simply supply all their money, M, for as much of the consumption good as they can get. 3. Macroeconomic

equilibrium

In the model as presented, macroeconomic equilibrium

~,b(~)~~(~+l)l

=M.

obtains if’ (2)

This non-linear difference equation in prices describes all possible perfect foresight monetary equilibria (PFME), i.e., non-zero price sequences compatible with macroeconomic equilibrium under perfect foresight. Unless the cost of not trading is infinite [e.g., U,( ., e,) = co] there may be a continuum of PFME. Consider economies characterized by such a continuum. As Farmer notes, almost all of these PFME are hyperinflationary despite the constant money stock.’ However, there will also be a unique stationary PFME characterized by market clearing at a constant price, p*. That is, there is a unique price p* such that

My(p*, p*> = hf.

(3)

Many economists view the possibility of hyperinflation given a constant money stock as anomalous. Farmer addresses this anomaly by proposing to restrict trades to multiples of A, a positive indivisible unit of real consumption. As he argues, this plausible economic restriction rules out price paths such that p(r)M < A at any finite r. Such paths would render money worthless ‘Inflation drives p(f) to zero since this paper deals, for notational convenience, with the commodity price of money rather than the money price of the commodity. Deflationary paths, p(r) + 00, are ruled out by the finite endowment of the young (i.e., negative consumption is impossible).

A.G. Isaac, Bursting bubbles: Further results

42-l

at time t, implying that under perfect foresight money would never be held. Thus, Farmer’s proposal rules out hyperinflationary paths under perfect foresight: his plausible economic assumption yields a plausible economic result. However, there is a cost to this approach. In restricting consumption to discrete increments, Farmer restricts the set of feasible equilibrium commodity prices of money, FECP. Farmer recognizes this, but does not explore its implications. In fact, he asserts that the model under his proposal ‘has the same stationary solutions which were discussed for the case of a continuous state space’ [Farmer (1984, p. 33)]. In agreement with Farmer’s characterization of the possible equilibrium money prices of the commodity, characterize the possible equilibrium commodity prices of money by FECP=

{kA/M]k=O,l,...,

e/A}.

(4)

Note that p*, the stationary PFME found in the continuous state space formulation of the model, is a function of the endowments and need not belong to FECP. If it does, it is clearly a stationary equilibrium in the discrete state space formulation as well. Yet if p* @ FECP, the existence of a stationary equilibrium under Farmer’s proposal remains to be proven. The following proof applies in either case. The consumer choice problem (1) can be represented under stationary prices by mpU[e,-kA,e,+kA],

(5)

subject to kA/M

E FECP.

Let k* be a solution to this problem and define p; = k*A/M. Clearly a stationary sequence of prices pk* is a perfect foresight equilibrium under Farmer’s proposal. If it were not, the representative young consumer would be better off consuming at least A more or A less than k*A (=p,*M) given the price sequence. This, however, contradicts the assumption that k* is a maximum. Similarly, if pk = kA/M, where k is not a solution to the consumer choice problem, then pk cannot form a stationary perfect foresight equilibrium since desired holdings of real balances will differ from pkM by at least A (otherwise k would be a maximum, contradicting its assumed non-optimality). Note that U[ .; ] increases (decreases) monotonically in k for k < ( > )p*M/A. Therefore, the work of the preceeding paragraph implies that, if

428

A.G.

Isaac,

Bursting

bubbles:

Further

results

we choose pi, pi+, E FECP such that p* E [pi, pi+ J, these are the only two possible stationary equilibria prices. Since piM = id and P~+~M = (i + l)A, (1) pi is an equilibrium

stationary price, if

U[e,- (piM+A)>e,+piM+A] C2)

Pi+1

is an equilibrium

5 U[e,,-piM,e,+piM],

(6)

stationary price, if

4 ey- (Pi+lM-A),e~+Pt+lM-A] (7)

If these conditions are met with equality, then pi and pi+l are both equilibrium stationary prices. Equivalently, (5) has two solutions: k = p,M/A and k =P,+~M/A.

To summarize, the existence of a stationary PFME in the continuous state space representation of an economy guarantees at least one and at most two stationary perfect foresight equilibria under Farmer’s proposal. Unfortunately, it is not possible to show that the economy is monetized in such an equilibrium, i.e., that the stationary equilibrium commodity price of money is greater than zero. For example, let U[c,, CJ = u[ci] + /lu[cJ where u[ -1 is any strictly concave function on [0, ey] and 0 < B < 1. This popular representation of consumer preferences allows a stationary PFME in the continuous state space representation when eY= 24 and e, = A, but under Farmer’s proposal the solution to (5) is easily found to be k* = 0. Correspondingly, pZ = 0 is the only stationary perfect foresight equilibrium in the latter case. This anomaly can be ruled out only by assumption. Intuitively, A must be assumed to be small enough not to rule out gains from trade in a stationary PFME. 4. Further problems

Gale (1973) first noted that some of the non-stationary equilibria in the continuous state space OLG model may be cyclical, and examples can be found in Azariadis (1980) and Cass, Okuno and Zilcha (1980). Brock and Scheinkman (1980) suggest that as a result non-stationary PFME may exist even in OLG models incorporating any of the usual devices marshalled to preclude asymptotic demonetization. Here Farmer’s proposal cuts two ways. It clearly can eliminate cyclical PFME in some models which would otherwise allow them, simply by restricting the price space. For example, A = e,, implies FECP = (0, A/M} which rules out cyclicity. On the other hand, the same proposal may introduce cyclical PFME in some models which otherwise preclude them.

A.G.

Isaac, Bursting

bubbles:

Furiher

429

results

Consider a popular version of the OLG model: gross substitution in each period’s consumption is assumed in order to eliminate the possibility of cyclicity by insuring a monotonic response of the demand for real balances to the rate of inflation. In the discrete state space formulation it may still be possible to choose pi, pi+l E FECP as in the last section and 8nd that

and

U[e,- (Pi+lM-A),eo+PiM-AP,/pi+,] If so, an oscillation between pi and pi+l is a PFME under Farmer’s proposal despite the assumption of gross substitutability in consumption.* Intuitively, the deflation in (8) is not sufficient to induce the young to accumulate A more real balances once they are holding ~$4. Similarly, despite the inflation in (9) it doesn’t pay the young to decumulate their real balances by A below pi+ ,M. As a result, a perfectly foreseen cycle of prices between pi and pi+l leads to a cyclical PFME where the demand for nominal balances is M for each period. 5. A simplification

Numerous other proposals have been made for the elimination of hyperinflationary equilibria in the basic continuous state space OLG model with a constant nominal money supply. One hoary proposal invokes a minimum real tax upon the young which must be paid in currency. Brock (1980) explores restrictions on tastes which preclude demonetization of the economy. Such approaches make money in some sense ‘essential’ to the young. Although such essentiality is arbitrarily imposed, these approaches do eliminate hyperintlationary PFME without discarding the useful traditional tools of the differential calculus. In contrast, Farmer neither restricts tastes nor invokes the deus ex machina of a minimum real tax. Instead, he makes the simple, extremely realistic assumption that there exists some degree of indivisibility in consumption. Unfortunately, this intuitively attractive approach to the elimination of ‘bootstrap’ equilibria requires relatively cumbersome argument in a discrete ‘Proof in appendix. The simplest example of such an equilibrium log-additive, e,, = 0, and e,/A is any odd number greater than one.

arises when utility is

430

A.G.

Isaac,

Bursting

bubbles:

Further

results

state space. However, the essential goals and insights of Farmer’s proposal can be preserved without this encumbrance. Rather than stress the indivisibility of the consumption good, now assumed infinitely divisible, it suffices to assume a minimum real cost to trade. The cost of trade will be represented as positive, finite, and non-decreasing in the level of trade. As the simplest example, consider the cost of trade function

C(PW = c, defined on ph4 E (0, ey]. If the old consumer bears the cost of trade, the choice problem in the presence of trade is now m,axu[e,-p(t)M,,e,+p(r+l)M,-c], Y which may be subjected to the same macro-level analysis as before. Of course, c may be large enough to prohibit trade, just as A could be under Farmer’s proposal. If it is not, however, hyperinflationary paths are ruled out as before. A rising inflation drives p to zero and in a finite time insures p (t + l)M < c, at which point it no longer pays the old to engage in trade or the young to accumulate currency. Since money would be worthless in a finite time along such a path, it would never be held. Deflationary paths are still ruled out by finite eY, therefore, only stationary and cyclical paths can be rationally expected in such an economy. 6. Conclusion It is generally accepted and empirically observed that the long-run stability of an economy’s price level corresponds to the stability of its money supply process. For this reason many economists have viewed as anomalous the hyperinflationary equilibria occurring in OLG models with no money growth. Farmer’s (1984) ingenious proposal offers an economically appealing justification for discarding these paths, but the cost in lost analytical tools and anomalous results is high. The key insight that these paths are incompatible with certain economic indivisibilities can, however, be incorporated without completely discarding the use of the differential calculus. For example, a positive minimum real cost to trade excludes hyperinflationary paths without introducing the anomalous economic behavior found on a discrete state space. For those who prefer Farmer’s proposal, it is shown that as long as indivisibilities are not too large, existence of a stationary perfect foresight monetary equilibrium (PFME) in the underlying continuous state space model guarantees the existence of a stationary PFME under Farmer’s proposal as well.

A.G. Isaac,

Bursting

bubbles:

Further

results

431

Some new notation is convenient. Let m* be the optimal real balances in the continuous state space problem: maximize U[e, - m, e, +p(t f l)m/zdtN Consider the deflationary case, recalling pi p,iV. If U[e, - (mi + A), e, + (mi + A)pi+,/pi] s U[ey - mi, e, + mi+J, then mi -!-A > m* since U[ . , * ] above is monotonically increasing (decreasing) in m for m E (0, m*] (for m E [m*, e,]). For the same reason, further accumulation of real balances or reduction of real balances below mi will only decrease utility. In other words, m,(M) is an optimal level of real (nominal) balances for the young when pi and P,.+~ are chosen as in section 3, p(t) =pi, p(t + 1) =P~+~, and real savings must be an integer multiple of A. This justifies eq. (8) as a suflicient condition for equilibrium when the young face the deflationary case. Eq. (9) is justified for the inflationary case in a parallel fashion. Of course these arguments have ignored the utility reducing consequences of non-integer p( t + 1)/p(r): in period t + 1 the old will be unable to spend a fraction of A in useless ‘real balances’ they will be holding. Thus eqs. (8) and (9) are suflkient but not necessary for cyclicity.

Azariadis, Costas, 1980, Self-fuIhlling prophecies, Journal of Economic Theory 25,280-396. Brock, W.A. and J.A. S&&&man, 1980, Some remarks on monetary policy in an overlapping generations model, in: J.H. Kareken and N. Wallace, eds., Models of monetary economics (Federal Reserve Bank of Minneapolis, MN) 211-232. Cass, David, Masahiro Okuno and Ikhak Zilcha, 1979, The role of money in supporting tbe pareto optimahty of competitive equilibrium in consumption loan type models, Journal of Economic Theory 20,41-80. Farmer, Roger, 1984, Bursting bubbles: On the rationality of hyperinflations in optimizing models, Journal of Monetary Economics 14,29-35. Gale, D., 1973, Pure exchange equiIibrium of dynamic economic models, Journ of Economic Theory 6,12-36. Samuelson, Paul, 1958, An exact consumption loan model of interest, with or without the social contrivance of money, Journal of Political Economy 66,467-482. &heir&man, J.A., 1980, Discussion, in: J.H. Kareken and N. Wallace, eds., Models of monetary economies (Federal Reserve Bank of Minneapolis, MN) 91-96.