Walrasian and Marshallian stability

JOURNAL

OF ECONOMIC

THEORY

Walrasian

34, 371-379

(1984)

and Marshallian LARS E. 0.

Institute

for

Stability*

SVENSSON

International Economic Studies, University S-106 91 Stockholm, Sweden

Received

June 22, 1983; revised

February

of Stockholm,

2, 1984

It is well known that there is no direct relation between stability of the Walrasian titonnement process and stability of the Marshallian quantity adjustment process. It is shown that, if short run Walrasian tdtonnement with a given number of firms is distinguished, there is, under the assumption of no joint production, local stability of the short run Walrasian adjustment process if and only if there is local stability of the long run Marshallian adjustment process with exit and entry of firms according to profit levels. Journal of Economic Literature Classification Numbers: 021, 022. c? 1984 Academic PreSS, h.

1.

INTRODUCTION

In the discussion of stability of competitive markets, two Ldifferent adjustment processes have been considered. One is when prices adjust depending upon excess demand; the other is when quantities supplied adjust, either depending upon differences between prices and marginal costs, or via exit or entry of firms depending upon profit levels. The former adjustment process has been associated with Walras (tatonnement), the latter with Marshall, although both Walras and Marshall were aware of the two process and both used them in the proper context (see Takayama [8] for a general discussion and references to the literature). Partial equilibrium discussions of these processes are standard in intermediate micro textbooks. A standard * The idea for this paper occurred during a seminar at Harvard University given by Hugo Sonnenschein, where results from Sonnenschein [6], Novshek and Sonnenschein [4], and a forthcoming paper by Novshek and Sonnenschein were presented. I thank Hugo Sonnenschein, Andreu Mas-Colell, and Wilfred Ethier for very useful discussions of previous notes, which helped to correct some crucial errors of mine. I have benefited from comments on the present version of the paper by Elhanan Helpman, by participants in the Stockholm Theory Workshop, in particular Harald Lang, and by two anonymous referees. Remaining errors and obscurities are solely my own responsability. I am grateful to NBER for secretarial assistance during the early stage of this work.

371 0022-0531/84

$3.00

Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.

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result is that a market may very well be Walrasian stable and Marshallian unstable, or the other way around. Hence, there is no direct relation between the stability properties of the two processes. The modern discussion of stability in general equilibrium has mostly been within the Walrasian (titonnement) process.’ Very recently, however, the Marshallian adjustment process has been examined in this journal by MasCole11 [3] and Novshek and Sonnenschein [4], both of whom study quantity setting noncooperative equilibria of finite economies approximating a limit economy with infinitesimal firms, the latter being the usual standard competitive equilibrium model (although with infinitesimal firms). In particular, Novshek and Sonnenschein [4, Example 21 emphasize the possibility of instability of the Marshallian adjustment process for very simple economies and they for this reason call into question the significance of the standard Walrasian competitive equilibrium, at least to the extent that the Walrasian equilibrium concept is considered as the limit of noncooperative quantity setting equilibria. Indeed, they give an interesting example of a two-commodity (consumption goods and leisure) economy with a unique long run Walrasian equilibrium and an unstable Marshallian adjustment process. The purpose of this paper is to examine more closely the relation between the Marshallian long run adjustment and the Walrasian short run adjustment. I limit the discussion to competitive economies with infinitesimal firms. In particular, I want to emphasize the relevance of the short run Walrasian adjustment process for a given number of firms, as opposed to the relatively irrelevant “long run” Walrasian tfitonnement process when the number of firms is variable.* Indeed, I will show that-under the assumption of no joint production-the former short run Walrasian process is very closely related to the long run Marshalliam adjustment process. We shall see that local stability, for all speeds of adjustment, of the Walrasian titonnement process in a short run equilibrium with a given number of firms (local Walrasian stability) implies local stability, for all speeds of adjustment, of the long run adjustment where firms enter or exit according to the level of profits (local Marshallian stability). Conversely, local Marshallian stability, for all speeds of adjustment, of the long run adjustment process implies local short run Walrasian stability, for all speeds of adjustment, for the given long run equilibrium number of firms. (However, 1There has also been considerable work on nontitonnement adjustment processes, which, present problem. See Takayama [S] for referenceto earlier

however, are not relevant for the work on nontitonnement.

‘By the long run Walrasian titonnement process we mean the adjustment process where prices move in response to long run excess demands, where long run excess demands involves the long run supply functions with for each price vector the long run equilibrium number of firms.

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local stability, for given nonuniform speeds of adjustment, of the Walrasian process does not imply local stability of the Marshallian process. Neither does local stability, for given nonuniform speeds of adjustment, of the Marshallian process imply local stability of the Walrasian process. Since little is known about different speeds of adjustment, though, it seems at this stage reasonable to restrict the analysis to theorems that can be proved without arbitrary assumptions about particular speeds of adjustment, and hence not discuss stability for given speeds of adjustment.) implying a close relation between Walrasian and These results, Marshallian stability, may at first appear inconsistent with Novshek and Sonnenschein’s two-commodity example of a unique long run Walrasian equilibrium that is Marshallian unstable. For we know that with two commodities a unique Walrasian equilibrium is always stable with a Walrasian tatonnement process. However, it is the relatively irrelevant “long run” tltonnement process with variable number of firms that is stable, not the short run tatonnement process for a given number of firms. Indeed, in Novshek and Sonnenschein’s example, the short run Walrasian process turns out to be unstable and their example is hence consistent with my results. (Svensson [7] discusses Novshek and Sonnenschein’s example in some detail.) For a single-consumer economy, it is well known that the Walrasian adjustment process is globally stable, for all speeds of adjustment. In a forthcoming paper Novshek and Sonnenschein show under great generality that the Marshallian adjustment process is also globally stable for a singleconsumer economy. Below I shall give a simple proof of the latter proposition in the present framework. Section 2 presents the model and discusses local stability. Section 3 discusses global stability. Section 4 presents some conclusions. The Appendix presents some simple technical results on stability.

2.

LOCAL

STABILITY

We consider the following technology: There are n goods/industries, indexed i = l,..., n. Each good i is produced with a homogeneous input, labor. In industry i, all firms are identical and each firm produces one unit of good i, using up a, > 0 units of labor (where a, is a constant). Hence, industry i output yi > 0 is equal to the number of firms in the industry, and industry input of labor is Zi = ai yi. In the long run the number of firms in each industry is variable and there are constant returns to scale. In the short run the number of firms, and hence industry input and output, is fixed. (In Svensson [7] I show that the results also hold for short run variability of output.)

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Let p, denote the price of good i relative to labor (we shall use labor as numeraire throughout). In the short run profits in industry i are ni=

(Pimui)yiv

(1)

which can be positive, zero, or negative. In the long run, profits are zero and the long run equilibrium price fulfills

pi = pi = ai. We assume that the adjustment the differential equation

(2)

of the number of firms in industry

l’i = IdPi - Pi),

i follows

Izi > 0.

(3)

That is, for given goods prices the rate of change of the number of firms depends on profits per firm, with a constant speed of adjustment li. Let there be m consumers, indexed h = l,..., m. Each consumer h has a utility function U,,(c,, I,J where the n-vector c,, = (chi) denotes consumption of goods and I,, is supply of labor. Let the n-vector a,, = (ahi) (with 0 < ahi < 1, C,, ahi = 1) denote the shares in industries’ profits of consumer h. Then the budget constraint for consumer h is (4) where p’c,, denotes the inner product CP~C,,~, etc., 7c(p, y) is the n-vector of profits given by (I), and p and y are the price and output n-vectors. Maximizing the utility function subject to (4) (for given p and y) gives to the goods (n-vector) demand function C,(p, n(p, y)) of consumer h. Then we can define the aggregate demand function as

(5)

C(P, Y) = 1 C,(P, Z(P, Y>). h

We define the (short run) goods excess demand (n-vector)

function

X(P, Y> = C(P, Y) - Y* (By Walras’ Law we can disregard equilibrium for given y we have

the labor market.)

X(P, Y> = 0,

as (6)

In a short

run

(7)

the solution to which gives rise to the (demand) price (n-vector) function P(y), giving equilibrium goods prices as a function of the given industry output vector. The short run adjustment of prices is assumed to follow a tatonnement process hi(t) = k,X,(p(t), y) for each good i, and for given

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industry output y, with a constant positive speed of adjustment. If K is the positive diagonal matrix whose diagonal is the vector of speeds of adjustment, the short run Walrasian process can be written

ii(r) = KX(P(G ~19

(8)

for given y. This process is assumed to be very fast relative to the long run adjustment of the number of firms, resulting in the short run equilibrium price P(y) for any given y. (Indeed, the process occurs in “instantaneous time,” denoted by r, which is distinct from “real time,” denoted by t.) The long run adjustment process (in real time) then follows (3), and we can hence write

i)(t) = A my(t)> - P),

(9)

where A is the diagonal matrix whose diagonal elements are the speeds of adjustment Li. In the long run equilibrium we have J(t) = 0 and P(y(t)) = fi, which results in the long run output vector j.’ We shall now see how local stability of (8) for given y (local Walrasian stability) is related to local stability of (9) (local Marshallian stability). To examine this, we recall that we have local stability of (8) if and only if the (n x n)-matrix KXp = [ki &Yi/api] has negative real parts of all its eigenvalues for p = P(y). (This is a simplification: The condition is a suffkient condition for local stability of the nonlinear system (8), but a necessary condition for local stability of the linearized system derived from (8), only. We here disregard that distinction.) Similarly, (9) is locally stable if and only if the (n x n)-matrix Aip, = [Ai aPi/ayj] has negative real parts of all its eigenvalues for y = J. Now, by differentiating (7) we get P, = 4, where X, = [kKi/ayj].

lx, )

(10)

By (5) and (6) we have xy=c,-I=~cch,ny-I,

(11)

h

but rr,, = D(p - p) where D(p - p) denotes the diagonal matrix

with

the

3 Mas-Cole11 [2] discusses stability of a system like (8) and (9) when the Walrasian and Marshallian processes both occur in the same time. He has n final goods and m + 1 factors, one of which is used as numeraire. Each good is produced with Leontiev technologies. Demand is a function of goods and factor prices and total profits. Goods and factor prices are assumed to adjust according to excess demand in goods and factor markets. Output is assumed to adjust according to the difference between price and unit costs. In Proposition 1 12, p. 2141 he shows that, if the demand function is continuously differentiable and satisfies the Weak Axiom of Revealed Preferences (and if the Jacobian of the dynamic system is nonsingular), then the equilibrium of the system is locally stable.

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vector p -p as its diagonal. For y = 7 we have p =p, hence nY = 0, C, = 0 and X,, = -I. It follows that for y = Y; P, =X$

(12)

Now, local stability of (8) implies that KX,, and hence its inverse has negative real parts of all its eigenvalues, but this does not imply that this is also the case for AP, =AX;‘.4 However, assume that (8) is locally stable for all speeds of adjustment (this concepts of stability, “Dstability,” is further discussed in Quirk and Saposnik [5] and Arrow [ 1 I).’ This is equivalent to Xp being a D-stable matrix, which by (P4) in the Appendix is equivalent to its inverse Xi’ = P, being a D-stable matrix, which is equivalent to AP, having negative real parts of all its eigenvalues, for all speeds of adjustment matrices II. Hence, local stability of (8) for all speeds of adjustment implies local stability of (9) for all speeds of adjustment. Similarly, local stability of (9) for all speeds of adjustment implies local stability of (8) for all speeds of adjustment, at the long run equilibrium y = 7, but not necessarily for other y. To understand this result intuitively, consider the simpliest case with only one good/industry (that is, with two commodities, including labor). In the neighborhood of the long run equilibrium profits are to a first-order approximation zero. Hence, at constant prices an increase in the number of firms a?d hence industry output has no (first-order) effect on profits and on demand for goods, which results in an excess supply for goods. If the Walrasian short run process is stable, the price will fall at this increased output, to restore short run equilibrium. The fall in price leads to negative profits, and by the Marshallian long run adjustment process firms exit and the long run equilibrium is restored. Hence the close relation between stability of the two processes.

X;‘K-’

3. GLOBAL STABILITY

It is well known that the Walrasian process is globally stable for a single consumer economy. To see the analogy with the Marshallian process below, let us quickly repeat the argument. We take V(p, 1’7z(p, y)) to be the utility level for given p and y, where V(p, D) is the indirect utility function and 4 Except for the special case when n equals K ‘. (X; ‘K ’ has the same eigenvalues as ,-‘,;I.) ’ The complications due to nonuniform speeds of adjustment, as well as the crucial distinction between stability for given and for all speeds of adjustment, were pointed out to me by Andreu Mas-Colell.

WALRASIAN

AND

MARSHALLIAN

17 = 1 ‘rr denotes total profits xi rri. Taking derivative, we have, for given y, 3=

v-g + v, l’n,@ = -VnX(&

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STABILITY

the (instantaneous-)

y)‘KX(p,

y) < 0

time

(13)

for X(p, y) # 0, where we have used Roy’s Identity (C(p,ZI) = -VP/V,), n,, = D(y), 1’5 = y’, (6), and (8). Since V has a minimum for p = P(y), global stability follows by the usual Liapunov argument. Intuitively, price is lowered for a good in net supply, which lowers utility (a terms-of-trade deterioration in international trade terminology). For the Marshallian process, as Novshek and Sonnenschein show in a forthcoming paper, the utility function again provides the appropriate Liapunov function. In our framework we can write the utility level in a shortrun equilibrium as U(J, a’y) where total labor input is a’~ = JJ aiyi. Taking the (real-) time derivative, we get

0=&j+

u,a’j=(-u,)(P(y)-p)‘j==(-u,)(P(y)-J7)’n(P(y)-p)>O (14)

for y = 7, where we have used U, < 0, P(y) = -UJ U,, and (9). Since U(y, a’y) has a maximum for y = ~7, global stability follows again by the usual Liapunov argument. Intuitively, goods output and labor input are increased in the proportion 1 : a, for goods with a marginal rate of substitution for leisure l/p, that is less than l/ai, which increases utility.

4.

CONCLUSIONS

It is a standard result that there is no direct relation between stability of the Marshallian adjustment process and stability of the Walrasian (long run) tatonnement process. When we make a distinction between long run Walrasian tatonnement with variable number of firms, which process I find rather irrelevant, and short run Walrasian tltonnement for given number of firms, a process that I find much more relevant, we get very different results. Thus, we have seen demonstrated a close relation between stability of short run Walrasian adjustment and stability of long run Marshallian adjustment. Indeed, with regard to local stability (for all speeds of adjustment) around the long run equilibrium, under the assumption of no joint production we could show a direct equivalence between the two. We also restated results according to which both are globally stable for a one consumer economy. In a way the results are rather striking. It makes a lot of sense to consider

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a long run adjustment process with exit and entry of firms depending upon profit levels. It does not make much sense, though, if the short run adjustment process which establishes those levels of profits is unstable. So it seems reasonable to restrict attention to stable short run adjustment. If, however, that stable short run adjustment is Walrasian tatonnement, it implies, really without any further restriction than no joint production, local stability of the long run Marshallian adjustment, and hence that the long run competitive equilibrium is attained. I would like to emphasize two qualifications, though. First, as mentioned, the above results have only been shown for the case with no joint production. They may or may not hold with joint production (see Svensson [7] for the complications that result from joint production). Second, the stability concept used is stability for all speeds of adjustment, which in the absence of information about speeds of adjustment seems apropriate. With some information about speeds of adjustment of the long run Marshallian process, say, stability of this may be assured for short run Walrasian adjustment process that fulfill less restrictive conditions than stability for all speeds of adjustment. These two qualifications may be suitable for further research.

APPENDIX:

STABILITY AND D-STABILITY OF INVERSES AND TRANSPOSES

An n x n-matrix A is said to be stable if and only if the real parts of all its eigenvalues are negative. It is said to be D-stable (Arrow [ 1 ] and Quirk and Saposnik [5]) if and only if the matrix product DA is stable for all positive diagonal matrices D. The following propositions are easy to show: (Pl) (P2) (P3) (P4)

A is stable if A is stable if A is D-stable A is D-stable

and only if and only if if and only if and only

its inverse A ~’ is stable. its transpose A’ is stable. if A’ is D-stable. if A-’ is D-stable.

(Pl) follows since the eigenvalues of A -I are the reciprocals of the eigenvalues of A. (P2) follows since A and A’ have the same eigenvalues. (P3) follows by the following argument: Let D be a positive diagonal matrix, and let DA be stable. Then A’D is stable by (P2). But then DA’ is stable, since it has the same eigenvalues as A’D. (If A and B are quadratic and of the same order, AB and BA have the same eigenvalues.) Then (P3) follows. Finally, to show (P4), assume DA stable. Then by (Pl) A-‘D-l is stable. But then D -‘A -’ is stable, and (P4) holds.

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REFERENCES 1. K. J. ARROW, Stability independent of adjustment speed, in “Trade, Stability, and Macroeconomics: Essays in Honor of Lloyd A. Metzler,” (G. Horwich and P. A. Samuelson, Eds.), pp. 181-202, Academic Press, New York, 1974. 2. A. MAS-COLELL, Algunas observaciones sobre la teoria de1 titonnement de Walras en economias productivas, An. Economia 21 (1974) 191-224. 3.’ A. MASS-C• LELL, Walrasian equilibria as limits of noncooperative equilibria. I. mixed strategies, J. Econom. Theory 30 (1983), 153-170. 4. W. NOVSHEK AND H. SONNENSCHEIN,Walrasian equilibria as limits of noncooperative equilibria. II. pure strategjes, J. Econom. Theory 30 (1983), 171-187. 5. J. QUIRK AND R. SAPOSNIK, “Introduction to General Equilibrium Theory and Welfare Economics,” McGraw-Hill, New York, 1968. 6. H. SONNENSCHEIN,Price dynamics based on the adjustment of firms, Amer. Econom. Rev. 72 (1980), 1088-1096. 7. L. E. 0. SVENSSON,“Walrasian and Marshallian Stability,” IIES Seminar Paper No. 257, June 1983. 8. A. TAKAYAMA, “Mathematical Economics,” Dryden Press, Hinsdale, Ill., 1974.