Price Distortionand Shortage Deformation,or What Happened to the Soap? By MARTIN L. WEITZMAN* The model of this paper generalizes the classical theory of consumer behavior to the more general case of prices that are not necessarily market-clearing.Suppose that, in addition to the money cost, some sort of search, waiting, or other quasi-fixed "effort-cost" is needed to obtain goods. The presence of this quasi- fixed cost element will triggeran inventorypolicy. A shortage equilibriumoccurs when effort costs are such that, in the correspondinginventory policy, the flow of desired consumption does not exceed the available supplyflow. Stock hoarding, a critical phenomenon in the economics of shortage, emerges as a natural component of this model. A complete characterization of a stationary shortage equilibrium is given. Comparative statics and welfare analysis are performed. The dynamic transition between steady states is analyzed to give insight into the mechanics of how shortages develop. (JEL D50) It is known, in a general way, that of consumer behavior to a situation where price distortions lead to shortages, queues, prices are not necessarily market-clearing. searching, hoarding, and so forth. Yet it The model essentially consists of an equilib- seems fair to say that the exact mechanism rium approach that combines inventory the- integrating each main element of a "shor- ory with demand theory. tage syndrome," especially the stockpiling A particularly vivid illustration of the phenomenon, has not been clearly articu- phenomenon I have in mind is illustrated by lated.1 The main aim of this paper is to recent Soviet experience in the consumer- provide a usable model of shortages by ap- goods market. Consider Soviet soap as a propriately generalizing the classical theory metaphorical example. The example is metaphorical because, while the shortage phenomenon in the consumer-goods market *Department of Economics, Harvard University, I seek to describe is quite general, the par- Cambridge, MA 02138. This research was supported by ticular commodity most illustrative of the a grant from the National Science Foundation. general phenomenon can vary. 'Some models dealing with somewhat different as- pects of this phenomenon are described in Janos Throughout most of 1989 (at the time this Kornai and Jorgen Weibull (1978), Victor Polterovich paper was written), there was virtually no (1983), Dale Stahl and Michael Alexeev (1985), Kent soap available on the shelves of Soviet Osband (1989), and the references cited in these works. stores. When officials in charge of planning The case typically treated has the waiting-effort cost proportional to the amount of the good bought. This is were asked about this problem, they acted equivalent to the assumption that people are limited to embarrassed and annoyed. In newspaper ar- buying one small unit at a time and must wait in line ticles and television talk shows, they ex- anew for each small purchase that is made. (It is then a plained repeatedly that production this year straightforward exercise to generalize to an "as if" was actually up 10 percent over last year, market equilibrium where the "as if " price is the money price plus the appropriately normalized disutil- which was itself 4 percent higher than the ity of waiting-effort price.) I much prefer the opposite previous year. Furthermore, not only was assumption: after waiting in line for a sufficiently long production accelerated somewhat as this time, or happening upon the good, the customer can embarrassing shortage became evident, but effectively buy as much as he wants. I think this is a more realistic assumption of the two extremes; addi- more than $8 million of valuable foreign tionally it leads naturally to an analysis of the key exchange was spent on buying soap abroad. inventory-stock aspect of the problem. Finally, they pointed out that statistics of 401 402 THE AMERICAN ECONOMIC REVIEW JUNE 1991 per capita soap consumption show the So- rational economic interpretation and can be viet Union to be not very far behind the coherently analyzed within the appropri- advanced Western capitalist countries. All ately extended framework of standard eco- of this sounds quite believable; and on the nomic theory. whole Soviet citizens do not seem to be going unwashed. I. The Model The above story can be repeated for any number of commodities. Soviet leaders fre- Suppose there are n goods in the econ- quently characterize the economy as be- omy, denoted i = 1, 2,..., n. For the sake of ing in a "crisis situation" ("krizisnoye argument, all consumers are assumed to be polozheniye"); but what, exactly, is the cri- identical. (Allowing consumers to be dif- sis? What is the appropriate model of ferent would not affect the existence or gen- causality leading from budget deficits and a eral form of an equilibrium, but it would monetary overhang to disorders in the con- render less sharp the characterization of its sumer-goods markets? What is happening? properties.) What should be done? Each consumer has the same utility func- To such questions no clear answers tion of the form emerge. Some cite breakdowns in the distri- bution system. (Railroads seem particularly (1) U(d)-V(e)-W(s). to be accused.) Others blame a "hoarding psychology" that causes panic buying and is In the above formula, d = (di) is the usual somehow related to the deficit and mone- consumption n-vector. The variable ec tary overhang. Theft by workers, sabotage, stands for the amount of "effort" required and speculation by cooperatives are also to obtain good i each time that it is ob- candidates. The officials seem unified only tained. Conceptually it is perhaps easiest to on promising increased production to meet think of ei as the time wait in line needed the shortage and on calling for formation to buy good i. (There is a separate line for of committees to investigate formally the each good, and after waiting the required problem. time,2 the consumer can buy as much as Some informal investigation reveals an desired.) However, ei could more generally interesting, if perhaps not unexpected, fact. represent search effort of any sort expended Although few official figures are available, to obtain the commodity. On the most ab- observations, conversations, and anecdotes stract level, ei is interpreted as the degree suggest strongly that Soviet people are of difficulty in obtaining good i. If good i is hoarding soap and other commodities in obtained at frequency fi, then the total ef- massive amounts. Significant parts of bath- fort (per unit time) expended on obtaining rooms, closets, hallways, and other areas all goods3 is have been given over to storage. My aim is to model carefully the general (2) e-,fiei. process of "hoarding psychology," which is a fairly widespread occurrence in shortage While it is conceptually easiest to think of situations, even if it is not typically so ex- effort el as deterministic, there is no prob- treme as the above case. I believe the model has potential applications to a wide variety of situations where prices are stuck at 2Note that et represents the time wait in line, not "wrong" values for whatever reason. Thus, the length of the line. If m people, each of whom some conclusions may have relevance for stocks up amount s, are waiting in line and the total malfunctioning markets in capitalist flow of goods into the store is d, then the time wait in economies and may even help to under- line is e = ms/d. stand certain features of 3By writing disutility as a function of total effort, "fixed-price" V(e), I am implicitly assuming that the sum of a large macroeconomics. It will be shown that number of fixed costs incurred at different times can, in hoarding psychology can be given a quite effect, be smoothed. VOL. 81 NO. 3 WEITZMAN:PRICE DISTORTIONAND SHORTAGES 403 lem with an interpretation that makes each capita are q = (q1). Any feasible consumer e, an independently distributed random demand must satisfy the additional con- variable so long as it is small relative to n. straints Then by the law of large numbers, e itself will be (almost) deterministic, equal to a (5) d < q. weighted sum of the form (2), where el is now interpreted as the expected effort that The supplementary constraints (5) distin- must be spent on obtaining good i. guish the present model from classical con- The variable s1 stands for the stock of sumer theory. In the classical case, in effect good i that is purchased and must be stored q = oo for all i, or else p represents equilib- when the good is obtained. The coefficient rium prices that just exactly make demands hi represents the opportunity cost per unit d equal to supplies q. For the classical case, of good i carried (per unit time). The mag- in effect there are no explicit constraints on nitude of h, would reflect such things as the consumer purchases other than the overall opportunity cost of storage space i takes up budget constraint (4). Here, the interesting (on shelves, in refrigerators, in warehouses, case is when constraint (5) "bites" for some or whenever applicable), shrinkage, the cost goods, representing inadequate supply at of guarding, interest forgone, the inconve- the fixed prices, presumably arising ulti- nience of hoarding, and so forth. Total stor- mately from production limitations. age cost is then The present formulation is sufficiently rich to cover a number of special situations of (3) s- EhSi. interest. For example, the price might be artificially repressed on only one or a few The representative consumer's utility goods, which then become "deficit" com- function is assumed to be of the additively pared to the bulk of commodities, which are separable form (1).