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Home , Market clearing, Shortage economy

Distortionand 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 that are not necessarily -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 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 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 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 . 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). The first element U(d) is market-clearing; or there could be a general just the traditional utility function of classi- deficit of commodities, meaning the prices cal consumer theory, having all the usual of most goods are artificially low relative to properties. The function V(e) represents the incomes and availability. Also covered is the disutility of effort, while W(s) is the disutil- case in which the same good is available ity of storage. The underlying assumption in cheaply in limited amounts at state stores (1) of independence would appear not to be and simultaneously at market-clearing prices terribly restrictive, and perhaps even rea- in private stores. Yet another situation cov- sonable, in the present context. More gen- ered with only slight modification of the eral formulations could be treated but with present framework is that in which a given some loss in crispness of results. It is as- fixed vector of goods is to be allocated, so sumed that U(d) is smoothly concave, while that consumers end up with the same final V(e) and W(s) are smoothly convex. allocation of goods in any case, but for some The representative consumer faces fixed given subset of (deficit) goods prices are nominal prices p = (pl) and is endowed with frozen at below market-clearing levels, while nominal money income I. Thus, the usual for the remaining (available) goods, prices budget constraint (4) pd Ap, - di* = ql.

II. Coefficient of Price Distortion While Api represents the nominal price of an extra unit of good i (normalized so In what follows, it will be useful to have a that the marginal utility of income is 1), Ui quantitative measure of the degree to which measures the actual value of an additional values and prices are distorted in the econ- unit of good i, or what people would actu- omy under consideration. To that end, con- ally be willing to pay. In the present context, sider the following mathematical program- it is then natural to define the coefficient of ming problem: value distortion or price distortion of good i (weighted by the amount of the good) as (6) max[U(d)] (13) bi (Ul -Api)d*i I subject to The coefficient bi measures the differ- (7) d

THEOREM 1: The unique stationary short- [Equation (29), rearranged, is the famous age equilibrium is the solution of the follow- square-root law of inventory theory.] ing equations.8 The rest of the proof is essentially by (21) d* inspection. Assignments (21)-(27) and the di= supplementary assignment

(22) s=wh (30) ,u = A

s2 are proposed as the solution of (18)-(20). (23) ei vwhd* Using conditions (4), (5), (12), and (13), it is straightforward to verify that the proposed where solution, (21)-(27) and (30), does indeed satisfy (28), (29), (19), and (20). (24) w- W'(s) is the marginal disutility of storage evaluated With the shortage equilibrium expressed in at s (= Zhisi) satisfying the simple closed form (21)-(27), it is easy to perform comparative-statics exercises. (25) sW'(s) = 8 Note from (22) that s, is proportional to 81, while from (23) ei is proportional to 2. while Thus, small shortages show themselves pri- marily in increased stock hoarding, with just (26) v--V'(e) very small increases in search activity. On the other hand, large shortages result in is the marginal disutility of effort evaluated at large inventories and very large waiting lines e (= Efiei) satisfying or search times. These observations suggest how shortages might evolve or devolve. (27) eV'(e) = S. While from (23) effort per purchase is proportional to the square of the coefficient PROOF: of price distortion, total effort is not. This is In the optimization problem (16)-(17) because, as waiting lines increase, the con- that defines condition (18), let ut2 0 be the sumer reacts by buying bigger bundles less shadow price multiplier for inequality (17). frequently. The total effort per unit time The corresponding necessary and, by as- spent on obtaining good i is, from (15) and sumption, sufficient first-order conditions for (21)-(23), any d, s 2 0 satisfying (17) to be the unique maximizer of (16) are, for all i, (31) f1el =, 1/ v.

Substituting (31) and (22) into (13) yields, (28) U-V e =_P respectively,

fe, (29) (ed )W'(h). (32) U,=Ap1+ d

(33) w hd', 8Conditions (21)-(27) are presented in the given U,=ApAP sequence and form to facilitate their economic inter- pretation. From the strictly mathematical standpoint of presenting an algorithm that uniquely solves (18)-(20), Thus, the difference between the intrinsic it is preferable to think of the following order: first (21) value of a good and its nominal price is defines {d,}, then (25) defines s and (27) defines e, then made up by the effort expended per unit of (24) defines w and (26) defines L', then (22) defines {s,) the good to obtain it [equation and and (23) defines {e,}. Equations (25) and (27) have (32)] unique solutions in s and e, from the convexity of V(e) also by the cost expended per unit of the and W(s). good to store it [equation (33)]. In shortage VOL. 81 NO. 3 WEITZMAN:PRICE DISTORTIONAND SHORTAGES 407 equilibrium, the consumer ultimately gets economy-wide price distortion. This is be- what goods [equation (21)] he would have cause, unlike the standard second-order gotten under the optimal rationing scheme deadweight loss (from tax theory) of dis- (6)-(8). However, from (32) and (33), it is torted prices in markets that clear, here the cost of nonproductive search and stor- malformed prices cause real shortage defor- age activity that raises the "as if" equilib- mations akin to rent-seeking activity.9 Fur- rium price from Api to Ui, the hypothetical thermore, the social cost of these shortage price that would have to be paid to clear the deformations enters twice; that is, distorted "as if" competitive market. prices here do double damage. First, they In the simplified world of this model, it is create nonproductive search activity at a easy to analyze the welfare loss of a dis- social cost of 8 / a, which has no other torted price system. As a reference point, I function than to allocate the artificially un- will take the constrained second-best opti- derpriced goods. Second, price distortions mal-rationing solution (9) of problem result in goods being tied up by buyers at (6)-(8). Such an allocation maximizes utility social cost b/b in socially unnecessary in- subject to budget constraint (4) and goods- ventories held throughout the system.10 availability constraint (5). In some sense, Somewhat paradoxically, the higher the the utility difference between it and the elasticity of disutility of effort or storage, shortage equilibrium, to be denoted L, rep- the less damage is done by a faulty price resents the waste of search and storage ac- system. This is because search or storage tivity. There is a simple expression linking activities do not play a directly productive L with the coefficient of price distortion 8. role here. Their only purpose is to allocate artificially underpriced goods. Since buyers THEOREM 2: The welfare loss of a station- can only end up consuming on average what ary shortage equilibrium compared with an is being made available on average, the optimal rationing scheme is goods will end up getting distributed the same way no matter what are the V(e) or 1 1 (34) L= 5 - b W(s) functions. When people suffer greater incremental pain from additional search or storage activity, they will actually end up where a is the elasticity of disutility of effort with less total disutility, because they will simply avoid waiting in lines or stocking up V their closets. Hence, the paradoxical conclu- (35) av v sion is that greater potential pain is less actual pain. while b is the elasticity of disutility of storage w 9For an analysis of rent-seeking activity, see James (36) b -s. Buchanan et al. (1980) and the references cited therein. w 10The reader may wonder why distorted prices do double damage in this model. For concreteness, take PROOF: the linear case a = b = 1. Then, (34) becomes L = 23. Yet in a standard model where the waiting-effort cost From (21), the allocation of goods is the is proportional to the amount of the good bought, the same in (18)-(20) as in (6)-(8). Therefore, total loss of consumer surplus would be equal to the from (1) the difference in utility attained is price subsidy times the quantity purchased, or 3. Where, then, does the extra term of 3 come from? In a L = standard model, there is no need to hold inventories. (37) V(e) + W(s). In the present square-root inventory model, the marginal cost of carrying inventory is always equal to Making use of definitions (25), (27), (35), half the average cost. Since equilibrium essentially and (36), expression (37) can be rewritten as requires that the marginal cost of holding inventory be (34). equal to the amount of price subsidy, it follows that the average cost of holding inventory must equal twice the price subsidy. I am indebted to Paul R. Milgrom for Note that shortage losses are first-order providing this intuitive explanation of why L = 23 when in the appropriately normalized measure of a = b = 1. 408 THE AMERICAN ECONOMIC REVIEW JUNE 1991

There is another seeming paradox here, tion of goods, lower fixed prices or higher which makes an important point. In an envi- nominal income decreases consumer wel- ronment of widespread shortage due to a fare. malfunctioning price system, ordinary no- Thus, standard measures of "real income" tions of "real income" lose their meaning. may be a quite unreliable indicator of wel- Actually, if there is sufficient price distor- fare in situations of economic shortage. It is tion, higher income can, other things being then not difficult to understand the tempta- equal, mean lower welfare. If enough goods tion to eliminate search and storage costs by are in shortage, more "real income" can imposing rationing, even though such mea- translate primarily into longer lines and sures introduce problems of their own.'1 greater hoarding rather than increased con- In the model as presented, the produc- sumption per se. tion or supply side is taken as more or less Other things (including prices) being held given. If the supply side were adversely af- equal, when nominal income is raised there fected by shortages, either because people are two effects. More money is available to are so busy finding and keeping goods that spend on nondeficit goods (iIUi = Api), they have less time to work or because it is which increases welfare. This is the usual difficult to maintain morale on the job when sense in which higher real income is better. pay buys so little, the ingredients are pre- However, in a shortage situation there is sent for a vicious circle.'2 An increased also a detrimental side, which might be money overhang decreases welfare, which called the "money-overhang effect." As in- then feeds back to lower production, which come is raised, the marginal utility of in- depresses welfare further. In such a world, come (A) declines, which increases the dif- income-increasing tendencies that exacer- ference between value and price (Ui - Ap) bate a money overhang can be very danger- for deficit commodities, which in turn leads ous indeed. to greater search (ei) and storage (s) effort This point can be illustrated rather simply and raises the welfare loss L. This money- as follows. Suppose that a unit of labor-time overhang effect is more pronounced as the input produces a unit of homogeneous out- deficit commodities constitute a greater put. Each person is endowed with one fixed fraction of all goods. In the extreme case, unit of labor-time, which can be divided the money-overhang effect can be so severe continuously between producing output or that more income can actually result in lower engaging in non-directly-productive search welfare. and storage activity. The delivery price of There is a slightly different way of con- the single good (or, in an alternative inter- structing and interpreting the model that pretation, the wage) is fixed by the govern- makes the same basic point. Take as given a ment at w, whereas the sales price of the fixed vector of goods q to be allocated. good is fixed by the government at p < w. Consumers end up with the same final allo- Thus, the subsidy of cost over price per unit cation of goods d = q no matter what. For a of the good is y w - p. Suppose each given subset of (deficit) goods, prices are person must pay a fixed per capita tax of frozen at below market-clearing levels. For 0 > 0 to the government. Let y be the the remaining (available) goods, prices move amount of the good produced per capita. freely to their competitive levels, which just (The remaining time 1 - y is spent on search clear the fixed supplies, taking account of and storage activity.) Then per capita dis- consumer income. As money income in- creases in this setup, the consumers will be made unambiguously worse off. Price distor- tion on the available goods will always be 11Throughout this paper, black-market activities are zero. However, for the deficit commodities, ignored. It could be argued that pressure toward black the markets might increase with the degree of price distor- coefficient of price distortion will in- tion. crease with income as the marginal utility of 12Indeed, the Soviet economy is showing signs of income declines. For the same final alloca- this vicious circle. VOL. 81 NO. 3 WEITZMAN:PRICE DISTORTIONAND SHORTAGES 409 posable income is wy - 0, which in equilib- rium, reinforces it. As has been shown, the rium must equal the value of goods pur- analysis of steady-state shortage equilibrium chased py. Rewriting this latter relation is eminently tractable. Some suggestive in- gives y = 0/y. Other things being equal, sights emerge about changes in behavior the greater the price subsidy y, the greater across steady states, but there has been no is the incentive to engage in nonproductive dynamic analysis as such. search and storage activity, and the lower is In a model such as this, where inventories per capita production (and consumption) of are playing a central role, dynamic transi- actual goods and services. Note too that tions display an important property not visi- other things being equal, a higher per capita ble from examining steady-state behavior tax (within the relevant range) actually raises alone. With a change between steady states productive output, because it lessens the comes a change in average inventory stocks, monetary overhang caused by price subsi- which in turn forces a corresponding alter- dization. It should be readily apparent that ation of consumption flows during the in- if this kind of economy gets caught in any terim. Adjustments toward higher stocks of kind of spiral between lower standards of equilibrium inventories, without a change in living and increased money-wage subsidiza- supply, must involve an accumulation at the tion, production can implode. expense of consumption, a kind of "inven- For the sake of a sharp characterization, tory squeeze" of consumption. This in turn the analysis has proceeded as if all con- leads to an overshooting property in the sumers are identical. A shortage equilib- waiting time of queues, so that short-run rium will typically exist in the more general welfare temporarily falls below its long-run case when consumers are different, but it equilibrium level. The purpose of the pre- will be more complicated to analyze. Con- sent section is to explore this theme in some sumers with less income or a lower value of detail. It turns out that a formal analysis of time will end up waiting in line and storing dynamics for the general case is an ex- the subsidized products, while people with tremely formidable task. All that I am able higher incomes or a greater value of time to do here is to provide a suggestive exam- will tend to seek out the higher-priced goods ple of the dynamics of one simple case. I with shorter lines.13 believe this example captures well the main The model of shortage equilibrium pre- elements that are involved in the dynamic sented here presupposes a regular, sta- properties of an "inventory squeeze." I also tionary, well-behaved process. Actually, believe that the basic features of the exam- shortage phenomena frequently display an ple must generalize. However, disequilib- erratic, nonstationary aspect. To that ex- rium dynamics is a notoriously difficult area tent, the analysis presented here probably about which rather little is known except understates the social loss of price distor- that a lot of different things typically can tions. happen. I do not want to claim that all adjustment processes will be as smooth as IV. Dynamics the example I will describe in detail. Suppose, then, that the economy starts The analysis so far has been purely static. out in some steady-state shortage equilib- A stationary shortage equilibrium is postu- rium. This old steady-state equilibrium had lated. Everyone expects this steady state to been expected to continue indefinitely. persist forever. Consumer behavior, which Then, suddenly and without warning, some is contingent upon this stationary equilib- underlying parameters change, which in- crease the degree of shortage in the econ- omy. To be specific, suppose S1increases to but the underlying supply flow of goods 13It is even theoretically possible that low-income, a', low-value-of-time people could be made better off by does not change. The easiest thing to envis- price subsidies, although this possibility is excluded in a age is an increase of nominal income that representative consumer world. exacerbates the degree of monetary over- 410 THE AMERICAN ECONOMIC REVIEW JUNE 1991 hang in the economy. Suppose the new situ- make his purchases and restock his invento- ation is naively expected to last forever. ries at time t > 0, will attempt to adjust. Using (21)-(23), it is easy to analyze the Projecting the same effort level to obtain differences between the two steady states. the good but at a higher value relative to With 8/' > 8i, the new stationary equilibrium price, the consumer will initially attempt to will feature more search effort and larger buy and to consume more of the good. inventories. Yet this cannot be the entire Since the flow of desired purchases would story of the transition. When the underlying then exceed the flow of available goods production structure has not changed, con- and since price is fixed (by assumption), sumption will be the same (equal to produc- the length of the queue must increase to tion) in both steady states, or d = d. If throttle back desired purchases to available consumption in the new stationary equilib- supply. Even should waiting lines have in- rium is to be the same as the old, because creased enough to trigger the new steady- what has changed is not the underlying pro- state inventory policy, in which desired duction structure but the price of good i consumption flow is equal to available relative to income, the economy must go supply flow, that is not enough. In the new through a wrenching dislocation in making a equilibrium, each consumer wants to hold transition between steady states. There is more inventories and make purchases less no way that inventory stocks can be built up often because the ratio of search effort to from an average per capita level of s, /2 to nominal price is higher than before. That a higher average per capita level of sif/2 means that every buyer coming to market in and have consumption flows remain unin- the time immediately after t = 0 wants to terrupted throughout. With consumption buy more than before in order to stock up levels the same in beginning and final steady to the new inventory level. Then, waiting states, but inventory levels higher in the times must be even higher than the new final state, there must be a transition period equilibrium level, to keep potential pur- during which consumption is curtailed to chasers away. The effort level to obtain the allow stocks to accumulate. In turn, the only good in the transition period must be so allocation mechanism available to force high that potential buyers are forced to lower consumption is increased waiting time delay purchases by cutting back on con- or search costs. sumption and waiting it out until the lines A change from a lower to a higher state come down to the new equilibrium level. of equilibrium shortage involves a change In Figure 1 are depicted typical "before," from a lower to a higher level of search "during," and "after" patterns of effort, in- time. The greater required effort of the new ventory, and consumption. At least this is steady state is bad news enough. The effort the pattern of the example I wish to pre- level during the transition period is even sent. Before the shock at t = 0, inventory more awful news because, to induce con- stocks for a representative consumer display sumers to curtail consumption and permit the usual sawtooth pattern with periodicity inventory stocks to accumulate, effort must s "overshoot" its new long-run equilibrium (38) T and then only gradually decline to the new d steady-state level, which is merely worse than the old. (For notational convenience, the subscript i Suppose the change occurred at time t= will henceforth be dropped whenever its use 0. At time t < 0, everyone was expecting the is superfluous from the context.) After ad- old shortage equilibrium to last forever. justment, the new steady-state inventory Then, at time t =0, 8i unexpectedly in- sawtooth has periodicity creased to 68/. This new condition, too, is expected to be permanent. The typical con- St (39) T sumer, when he now goes to wait in line to d VOL. 81 NO. 3 WEITZMAN.PRICE DISTORTIONAND SHORTAGES 411

d d~~~~~~~~----dt

td-T t 0 t T' t

FIGURE 1. TYPICAL TRAJECTORIES OF EFFORT, CONSUMPTION, AND INVENTORY

I will exhibit one relatively simple dy- Also shown is a typical consumer's corre- namic transition path that lasts exactly T' sponding inventory stock and consumption time units. At time t < 0, the economy is in flow trajectories. the old steady-state equilibrium. During When the shock hits at t = 0, the con- time 0 < t < T', the economy is in a state of sumer is unexpectedly caught somewhere in transition. For times t > T', the economy is the previously optimal inventory cycle. For in its new steady-state equilibrium. The all t < 0, inventories had been picked up transition itself is a dynamic equilibrium and stored in batches of size s, to be run because everyone believes it will happen down at consumption rate d. Suppose at this way, and their subsequent optimizing t = 0 a consumer has on hand stock o- (O < behavior makes it actually come about. af < s). In other words, this consumer made There may be other consistent transition his last purchase at time paths, but I believe they must demonstrate the same basic "inventory squeeze" aspect s ar ( of the path I will describe, only in more (40) to= -d 1 complicated form, because average inven- tory stocks cannot be built up without some- From t = 0 on, the consumer faces the how squeezing consumption flows in the declining effort profile depicted in Figure 1. process. I am not bothered by limiting my- There is then an incentive to delay the next self to the relatively simple example of a purchase time by slowing down current con- transition described here because: a) it is sumption, because the further off a pur- already complicated enough, b) it illustrates chase can be put into the future, the lower well the basic principles involved, and c) will be the effort cost of the purchase. there is little purpose to misplaced general- In the proposed adjustment, every con- ity here when the model itself and the other sumer is induced to lower consumption from aspects of dynamic behavior are purposely rate d to rate being simplified as much as possible. At time 0, it comes as a complete surprise (41) d d() when 8 changes to 8'. Suppose that after t = 0 everyone believes that the effort re- quired to obtain good i has jumped up until stocks run down to zero at time discontinuously to level e (> e') and is de- clining linearly back down to level e' during (42)t'- the transition period [0, T'], thereafter to d' ds remain forever at the new steady-state level e'. This effort profile is depicted in Figure 1. From time t' on to infinity, this typical con- 412 THE AMERICAN ECONOMIC REVIEW JUNE 1991 sumer repeats the new steady-state cycle: Equation (44) is exactly the ratio of the inventories of size s' are picked up and stock each buyer purchases during the tran- stored with periodicity T' [equation (39)], sition interval to the stock each buyer pur- being run down by consuming at rate d. chases in the old equilibrium. Thus, the Thus, the typical consumer with inventory total stock demand per unit time is the oa at t = 0 suffers a decline in consumption same, equal to supply, and the proposed by the fraction s / s' over his adjustment trajectory is feasible. period [0, t']. By time t = T', all consumers Next it must be shown that, given the are synchronized in the new steady-state. effort profile e(t) depicted in Figure 1, each To prove that the proposed time profiles consumer desires to follow the prescribed e(t), s(t), and d(t) depicted in Figure 1 consumption and stock trajectories d(t) and represent a consistent dynamic transition s(t). For the steady states of t < 0 or t > T' path, two conditions must be shown to hold. this has already been shown by (21)-(23). First, the feasibility of these inventory and To show that the effort profile e(t) of Fig- consumption patterns must be demon- ure 1 induces the depicted pattern of d(t) strated. Then it must be proved that each and s(t) for t E [0, T'], I must make some consumer wants to follow this prescribed assumptions about consumption payments pattern. and inventory costs in the transition period. Feasibility means that at any instant the I do not think that these assumptions are flow of supply is equal to the amount of critical in the sense that they are being stock being demanded per unit time. The made primarily to serve as sufficient condi- proposed trajectory is feasible for t < 0 and tions for the relatively neat patterns of Fig- for t > T' because these are, respectively, ure 1 to emerge. I believe that more general old and new steady-state policies satisfying assumptions of a reasonable sort would give (18)-(20). For t E [0, T'], a "spreading out" the same qualitative features but would be effect occurs. Buyers arriving to market now harder to analyze. are each carrying away larger stocks than In what follows, the consumer's basic de- before (s' instead of s), but these same cision variable is x, the rate of consumption buyers are arriving less frequently because from t = 0 until the inventory stock u runs they are running down existing stock at a out, at which time it is optimal to follow the rate of d', which is lower than d. It turns new steady-state policy. Assume that the out that the two effects exactly cancel each money cost of consuming any good is pay- other, and in aggregate buyers are taking as-you-go. It is proposed that consumption away the same amount per unit time as in is at rate x for 0 ? t o/x. The utility cost of following In the old steady state at t < 0, a con- this policy over the interval [0, T'] is sumer with stock af at time 0 would have planned on next picking up stock s at time (45) A(px()+ Apd( T x) 0r (43) Assume further that inventory storage After the shock, a consumer with stock ar at cost is essentially the storage-cost coeffi- time 0 will next pick up stock s' at time t' cient times the new inventory level times defined by (42). The ratio of the frequency the interval over which it operates. Some of buyer arrivals in the old equilibrium to stories could be told to support this in- the frequency of buyer arrivals in the transi- terpretation, but it is more important to tion interval is thus emphasize that virtually any reasonable for- mulation would yield qualitatively similar t' s' results. The inventory-carrying cost of the (44) -= 1r s proposed policy over the interval [0, T'] is VOL. 81 NO. 3 WEITZMAN:PRICE DISTORTIONAND SHORTAGES 413

then [0, T']. Since the right-hand side of (50) does not contain oa, the slope of the effort func- (46) hs'(T'--) tion, hereafter denoted e', must be constant throughout [0,T']. In other words, e(t) is a For notational convenience, assume that w straight line, as depicted in Figure 1. and v are normalized to unity. From applying (32) to (50), With e(t) representing effort as a func- tion of time, the total net utility of the (51) e'=0 for s'=s. proposed policy over the interval [0,T'] is then In other words, a transition from an old steady state to a new steady state that is the (47) U(x) + T - U(d)-e same as the old steady state is no transition at all. The concave function - [Apx(1) + Apd(T'--)] hs'y (52) U(y)-Apy- d

-hs'(T'-(J X has a unique maximum, where its derivative is zero, at y = d defined by the first-order The consumer chooses x to maximize (47), condition (32). In particular, this means that, yielding the first-order condition for y = ds / s', hs' (48 J a(J lU(d) (48 -U(x) - 2 U(x) + 2 (53) U(d)-Apd- d x x2 d

a(JaJ hs'a U ads ads hs' ads Apdar > si -Ap si + _ x 2 x2 X 2=? d s'J

The expression e'(o-/x) stands for the time Substituting (53) into (50) and rearranging derivative of the effort function evaluated at yields t = a/x. to fol- ds ds hs' For the consumer to be choosing (54) e'-<- - U. s' -A p -- low the prescribed path of Figure 1, condi- tion (48) must hold for the value From concavity of expression (52), the (49) x=d' derivative with respect to y of (52) is greater than zero for y < d. Letting y = ds/s', this where d' is defined by (41). Substituting (49) observation applied to (54) yields and (41) into (48) and rearranging yields (55) e'<0 for s'>s. as \ ds (50) e' ds U(d) Since e' is continuously differentiable in s', (51) and (55) imply that, when s' is suffi- ds d ciently close to s, - , +Apd+hs'. (56) e'>0 for s'

As the parameter a- varies from 0 to s, It has been shown, then, that in undergo- condition (50) defines the slope of the re- ing a change from an old steady state with a quired effort function e'(t) on the interval lower degree of price distortion to a new 414 THE AMERICAN ECONOMIC REVIEW JUNE 1991 steady state of higher price distortion, there A clear theme of the paper is that price will occur a transition phase in which effort distortion and monetary overhang can pre- is greater and consumption is lower than in sent very severe threats to the normal func- either steady state. Conversely, during a tioning of an economy. The essential inade- transition to a state of less price distortion, quacy is in the monetary domain of prices effort will be lower, and consumption will and income, not in the real economy of be higher. production and distribution. The seeming All of this means that increased price paradox of the missing Soviet soap must be distortion has particularly painful immedi- resolved not by scapegoating distributors or ate effects as goods are squeezed out of by making token increases in soap produc- consumption and into inventory stocks. The tion, which do little to alleviate the underly- situation may stabilize later of its own ac- ing problem. Instead, prices on soap must cord to a still unsatisfactory level, but the be increased, or incomes lowered, so that period immediately after an increase in consumers can move toward a better state monetary overhang is likely to be one of where they are not impelled to hoard large especially acute shortage symptoms. inventories. The essential issue is to remove The good news is that if price distortion the incentives that lead to excessive inven- can be lessened then society can go on a tory stocks blocking what should be a direct temporary binge as unwanted inventories flow of goods from production to consump- are worked off into extra consumption. Re- tion. versing the shortage process-by raising prices or lowering money incomes-results in a "consumption dividend" from un- REFERENCES needed stocks coming out of storage into general circulation. Buchanan, James M., Tollison, Robert D. and Tullock, Gordon, eds., Toward a Theory of V. Conclusion the Rent-Seeking Society, College Station, TX: Texas A&M Press, 1980. This paper has presented a formal model Kornai, Janos and Weibull, Jorgen W., "The of consumer behavior under conditions of Normal State of the Market in a Shortage shortage. A critical role in allocating short- Economy: A Queue Model," Scandina- age goods is played by the directly unpro- vian Journal of Economics, March 1978, ductive activities of search and storage. It is 80, 375-98. possible to express directly these two key Osband, Kent, "Economic Crisis in a Re- variables as simple functions of the underly- forming Socialist Economy," unpublished ing degree of price distortion and, there- manuscript, November 1989. fore, to analyze easily the relation between Polterovich, Victor M., "Problema Izmereniya price distortion, search effort, and goods Defitsitnosti Blag" [The Problem of stockpiling. "Hoarding psychology," or what Measuring Welfare Under Shortage], might better be called "defensive hoarding," Ekonomika i Matematicheskie Metodi, can be thoroughly analyzed as an economic May 1983, 19 (5), 878-91. phenomenon by extensions of standard eco- Stahl, Dale 0. and Alexeev, Michael, "The In- nomic theory. In shortage equilibrium, ev- fluence of Black Markets on a Queue- eryone must expend effort to locate and Rationed Centrally ," hoard goods because everyone else is ex- Journal of Economic Theory, April 1985, pending effort to locate and hoard goods. 35, 234-50.