Weitzman, Martin L.

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•\;';IS THE PRICE SYSTEM OR RATIONING MORE EFFECTIVEIN GETTING A COMMODITY TO THOSE WHO NEED IT MOST?Martin L. Weitzmanmassachusettsinstitute of.technology50 memorial driveCambridge, mass. 02139IS THE PRICE SYSTEM OR RATIONING MORE EFFECTIVEIN GETTING A COMMODITY TO THOSE WHO NEED IT MOST?Martin L. WeitzmanNumber 140 October, 1974The views expressed herein are the author's sole responsibility and do notreflect those of the Department of Economics or the Massachusetts Instituteof Technology.Digitized by the Internet Archivein 2011 with funding fromBoston Library Consortium Member Librarieshttp://www.archive.org/details/ispricesystemorrOOweit1. IntroductionThe question of whether it would be better in various circumstancesto use quotas or market clearing prices to allocate resources is a debateof long standing. From time to time it has flared up as a policy issueof genuine importance.While each specific debate in this series is about a particularissue, and therefore has its own special features, it does seem to me thatthe same general themes reappear again and again. The purpose of thepresent paper is to single out and clarify one of the common strands.A favorite argument for relying on the market to allocate a partic-ular good or service concerns what might be called its built in selectivity.The price system, it has been said, is really quite a sophisticated mech-anism for matching up a scarce commodity with those who need it most.And this is done automatically, simply by giving consumers a chance toexpress their preferences in the market place. By contrast, rationingis seen as a crude allocation device which cannot effectively take accountof individual differences. Any rationing scheme typically ends up over-delivering goods to some people who don't really want them so much, atthe same time that it will be withholding from others with a genuine needfor more.The rejoinder is that using rationing, not the price mechanism, is infact the better way of ensuring that true needs are met. If a marketclearing price is used, this guarantees only that it will get driven upuntil those with more money end up with more of the deficit commodity.How can it honestly be said of such a system that it selects out andfulfills real needs when awards are being made as much on the basis of-2-income as anything else? One fair way to make sure that everyone has anequal chance to satisfy his wants would be to give more or less the sameshare to each consumer independent of his budget size.Both of these arguments are right, or at least each contains a strongelement of truth. With the aid of a very simple model, I hope to indicatehow the two effects just described interact in determining which allocationsystem is actually more effective for meeting real needs.2. A Way of Formulating the ProblemIn what follows, the abstraction is going to be on an heroic scale.Because the primary goal of this paper is to capture sharply the interplayof issues discussed in the introduction, a high premium will be placed onthe use of analytically convenient functional forms which do not at thesame time grossly violate reality. I am well aware that an especiallysimple case is being treated and that is precisely my purpose. Many essen-tial features of the model would remain under more general representationsof the economic environment. But since the basic message would then tend toget diluted, such an approach is not taken.Suppose the population under consideration consists of n consumers.Any person in this population is endowed with a particular set of tastesand a certain level of income. Let e be a variable which quantifies inten-sity of preferences toward the deficit commodity, e is intended to be somemeasure of how much a consumer "wants", "needs", or "enjoys" the deficitcommodity. It is operationally measured by how much the consumer purchasesrelative to what other consumers would be buying if_ they belonged to hissame income category. Levels of income will be indirectly quantified byX, a variable meant to represent the marginal utility of income. As an-3-aggregate approximation, it is being assumed that the marginal value ofan extra dollar is identical for all consumers having the same income.Each consumer endowed with traits (e,X) is postulated to demandx(p;e,X) = A - BAp + e (1)units of the deficit commodity when it is offered for sale at price p. Theabove form might be defended as a first order approximation to a demandcurve if incomes and tastes did not vary significantly. At any rate, choosinga demand function of this type will considerably simplify the analysis.Different tastes are parameterized in expression (1) by various valuesof e. A higher value of e denotes a more intense desire for x and is repre-sented by a rightward shift in the underlying linear demand curve. Note thatthe demand schedule is written as a function of Xp, the real price of thecommodity. It seems to me an allowable partial equilibrium assumption in thepresent context is that people with the same preferences but different incomesshould have the same demand for x when it is expressed as a function of aprice normalized to measure the opportunity loss of income foregone. In ef-fect I am assuming that the consumer (e,X) picks x(p;e,X) to maximizeU(x;£) - Xpxwhere U(*) is a quadratic utility function of the form2U(x;e) = C +(A +B£)X- If ' (2)Let f(e) denote the number of consumers of preference type e and letg(X) be the number with marginal utility of income equal to X. Naturally-4-Z f(e) = Z g(X) = n£ XAs a norm of sorts, and to make the distinction between their roles es-pecially clear, it will be assumed that tastes and income are independentlydistributed. In other words, the number of consumers possessing the traitcombination (e,X) ish(e,X) - f < e > «W . (3)nPurely for notational convenience and without loss of generality, eand X are normalized in (1) so that their average values are respectivelyzero and one:E[e] = Z £ f^ = (4)neE[X] = Z X g (X) = 1 . (5)XnThe variance of e2V[e] = E[(e - E[e]) 2 ] = Z £fJz)(6)ecan be interpreted from equation (1) as the mean square deviation in thedemand for x when income is held constant . This is by contrast with themean square deviation in the overall demand for x (without controlling forincome). The latter variance will be denotedIt should not be difficult for the reader to carry through the analysiswhen e and X have a non-zero correlation. Formula (18) or (19) will justend up with a covariance term tacked on.-5-o2(p) = E[(x(p;e,A) - E[x(p;e,A)])2] , (7)where the symbol E[*] as usual indicates the average per capita value of thevariable in question. From (1), (3), (4), (5),E[x(p; £ ,A)l E I E *<P? e >*> MM) = A _ Bp . (8)e AUsing (8), (7) becomeso2= E E(BP (A - 1) + £ )2h( £? A)E ABy (3), (4), (5), this reduces toa2(p) = B2p2V[A] + V[e],(9)where2V[A] e E[(A - E[A]) 2 ] = Z (X " ^ S^ } . (10)A3. Alternative Allocation MechanismsSuppose there is a fixed supply X of the deficit commodity which is2available to be allocated among the n consumers. The average x availableper capita is-_ Xx = —n2This is also an abstraction since the supply may be variable and mightitself depend on the allocation system, e.g., through the price. Tobin [ ],stresses that the elasticity of supply can be an important determinant ofwhether to use a price or quota system.-6-A distribution plan or allocation mechanism which gives x( £ »^) toeach person of type (e,A) is feasible ifEtxl = x , (IDwhereE[x] = z z X(e^) MM>e AnNote that without loss of generality we are excluding trivial inefficiencyfrom our definition of feasibility by only considering feasible plans ofthe equality rather than inequality form in (11).An allocation of x to a person of type (e,A) yields him benefits inmoney terms 3(x;e,A) equal to the shaded Jp dq area of figure 1:,, , N (A+e) x x,(x;e,A) = 2il (12).- BApFigure 1The effectiveness of a plan {x(e 5 ^)} in meeting true needs for the deficitcommodity is defined as the weighted (by the marginal utility of income)average benefit which it yields,-7-E[B(x(e.X);e,X).X] = E z P(x(e,*)?c,*> Xh( £ ,X) . (n)e XnIt seems to me that expression (13) provides a not unreasonable partialequilibrium measure of the ceteris partibus social welfare obtainable fromany given distribution of the scarce commodity.The most effective possible distribution plan (x*(e»X)} would maximize(13) subject to (11). Such an ideal plan is not in general attainable becausethe government typically lacks the information, authority, or inclination toidentify people of type (e,X) for all e and X. Even when approximate groupingdistinctions can be made, the same problem of not being able to accuratelydemarcate types will crop up again within each group.Our point of departure is a collection of consumers with similar enoughobservable characteristics to make further subdivision of types too expensiveor downright infeasible. In this context only allocation mechanisms canbe employed which do not depend on the ability of anyone to screen out3types. Furthermore, we limit ourselves to simple distribution rules.One simple rule to follow in the absence of identifiable type-features4is to give every consumer the same ration x. The effectiveness of rationingin meeting true needs for the deficit commodity is the weighted averagebenefit which the allocation x( e >^) = x yields3More complicated rules than we will consider are certainly possible, forexample an entire schedule of prices as a function of the quantity purchased,or even just a two-tiered version. Naturally an optimal schedule can't helpbut be better than a single price or quantity, which are just special casesof it. But a schedule is hard to institute and lacks the important qualityof simplicity, which the two special cases provide.4This is what might be called a pure rationing system. We are abstractingaway from prices doing any of the allocating by in effect assuming that theprice charged is so low it deters no one from purchasing his allotted quota.-8-E[X.6(x;e,A)] = I E ^^^ * MM?e AThe price or market system gives a consumer with traits (e,A) thatallocationx(e,A) = x(p;e,A)which he demands by equation (1) . In the above expression p is the competi-tive price of x, satisfying the market clearing conditionE[x(p;6,A)] = x .Substituting from (8) , the above expression becomesA - Bp = x . (14)The effectiveness of the price system in meeting real needs for the deficitcommodity is the weighted average benefit it deliversE[X.6(x(p; £ ,X); E ,A)] = Z E X B<*(P?e,A); E ,A) MM) .e AThe above expression points up how incomplete, limited, and partial isour treatment of welfare. In the price allocation system, and this shouldnever be lost from view, revenues of pX are collected by someone or other.A full treatment would take account of where those funds are going and whois benefitting from them. But since anything is possible here, we merelyevade the more general issue after warning that what happens to collected-9-revenues might just be the most relevant consideration of all. Instead,we continue to concentrate on the narrower technical question of how wellthe deficit commodity in and of itself alone is being allocated, abstractingaway from all else that might be happening.4. Rationing vs. the Price SystemThe comparative effectiveness of the price system over rationing inmeeting true needs for the deficit commodity is defined as6 e E[Xe(x(p;e,X);e,X)] - E[X&(x;e,X)] . (15)The loss function implicit in the definition of 6 is the difference in theweighted average benefit obtained under the two simple allocation mechanisms.Setting p = p in (1) and substituting into (12),3(x(p; e ,X); e ,X) = (A ± O (A - BAp + e) _ (A -»* ± ^ . (16)Similarly, plugging x from (14) into (12) yields3(x; £,X)= (A+0(A-Bp) -&^&L. (17)After employing (3), (4), (5), (6), (10), (16), (17), and cancellingout terms, expression (15) reduces to6 = fl We] " B2P2V[A]) . (18)As an example, it certainly matters whether the funds are taxed by thegovernment and go to help relieve flood victims or they end up in the bankaccount of an oil billionaire. Diamond and Mirrlees [1971] develop a gen-eral equilibrium taxation framework which sheds light on the nature of thegeneral problem.This criterion is analogous to the one employed in Weitzman [1974].-10-An equivalent but somewhat more useful (because it is more operational)expression for 6 is obtained by substituting from (9) into (18) to obtain6 = ^ (2V[e] - a2) . (19)2 2Here a = a (p) is the mean square deviation in demand for x at its marketclearing price p.Expression (18) or (19) is the fundamental result of this paper. Thelast section is devoted to examining it in some detail.5. Analyzing the Coefficient of Comparative Effectivenessvr e 1Starting with an examination of (18), from (2) . is interpretable asBthe variance of the marginal utility of an extra unit of the deficit commodity.Thus, it is a measure of the heterogeneity of tastes. The larger is V[e], themore widely dispersed are "true needs" for the deficit commodity.»2Analogously, p V[A] is the variance of the marginal utility of incomewhere the basis of account is the money value of one unit of the deficit2-2commodity. Thus, V[e] and B p V[X] are both expressed in terms of the samenumeraire, and are therefore comparable. The smaller is V[A], the moreuniform is the distribution of income.1 2-2The multiplicative factor — merely translates (V[e] - B p V[X]) intoZJounits of marginal utility. Expression (18) is essentially the differenceof two terms — a "taste distribution effect", and an "income distributioneffect". Other things being equal, the price system has greater compar-ative effectiveness in sorting out the deficit commodity and getting it .to those who need it most when wants are more widely dispersed or whenthe society is relatively equalitarian in its income distribution. Con--11-versely, rationing is more effective as tastes for the deficit commodityare more uniform or as there is greater income inequality.Expression (19) is really the same as (18), except that in (19) thevariables can be operationally interpreted, whereas (18) is defined in termsof variances of marginal utilities. From (19), 6 is essentially a differenceof two observable mean square deviations - twice the variance of demand amonga sub-population controlled for income (see (6)) minus the uncontrolled vari-ance of demand (7).Thus, if the mean square deviation of demand by the entire consumerpopulation is not much larger than it is by fixed income subgroup, the pricemechanism has greater effectiveness in screening the deficit commodity andfunneling more of it to those who need it relatively more. Conversely, thegreater the dependence of demand on income (as measured by (19)), the largerthe comparative effectiveness of a quota system because it essentially pre-vents those with larger incomes from monopolizing consumption of the com-2modity in question. Naturally if both variances a and V[e] are small, itdoesn't make much difference which system is used.6. Concluding RemarksIt might be appropriate to end this paper by commenting on its relationto some more standard approaches. Economists sometimes maintain or implythat the market system is a superior mechanism for distributing resources.After all, the argument goes, consider any other allocation. There is always-some corresponding way of combining the price system with a specific lump sumNote that a ceteris paribus increase in the market clearing price makesrationing relatively more effective because the income distribution effecttakes on added importance.-12-transfer arrangement which will make everyone better off (or at least no^8worse).This is true enough in principle, but not typically very useful forpolicy prescriptions because the necessary compensation is practically neverpaid. When dispensation jLs made, the point deserves emphasis. For example,by the standard argument of consumer sovereignty, it certainly makeseveryone better off if ration tickets can be sold (and bought) than ifresale is not allowed. The "white market" is superior to strict rationingbecause in redistributing property rights, the losers are being adequatelycompensated or "bought off". However, this does not necessarily meanthat pure direct wants for the deficit commodity are better served inthe sense that those who "need" it most actually end up receiving more.Besides, why stop here? Surely the status quo income distribution isnon-optimal. An even better system from an overall social welfare positionmight be to give all the rationing tickets to the infirm (who maybe don'teven consume the commodity in question) or to Bangladesh, or so forth.There is no end to what could be done on the income distribution side.This is one reason why I have preferred to leave in abeyance incomeconsiderations and concentrate in this paper on inquiring about the puredistributive effectiveness of an allocation system in getting the deficitcommodity to those who need it most. The other and more substantive reasonis that I am not sure but that for many situations such a formulation mightnot be the most appropriate way of posing the question in the first place.There is a class of commodities whose just distribution is sometimesviewed as a desirable end in itself, independent of how society may beGiven the usual general equilibrium assumptions. See, for example,Arrow and Hahn [1971].-13-allocating its other resources. Such "natural right goods" as basic foodand shelter, security, legal aid, military service, medical assistance,education, justice, or even many others are frequently deemed to be suffi-ciently vital in some sense to give them a special status. The principleof limited dimensional equity in the distribution of a commodity is anopen violation of consumer sovereignty. Yet society doesn't seem to mind,at least sometimes. In cases when this is so, our model may have somerelevant things to say about whether prices or quotas are better allocationinstruments.-14-ReferencesArrow, K.J., and F. Hahn, General Competitive Analysis , San Francisco, 1971.Diamond, P. A., and Mirrlees, J. A., "Optimal Taxation and Public Production",American Economic Review,March 1971 (pp. 8-27) and June 1971 (pp. 261-278).Samuelson, P. A., Foundations of Economic Analysis , Cambridge, 1947.Tobin, J., "On Limiting the Domain of Inequality", Journal of Law andEconomics .Weitzman, M. , "Prices vs. Quantities", Review of Economic Studies , forth-coming.Date Duep^%W&&Qhov a Oft "m*a&A3«a8NOV 2 o '#0JMB.ftL3g/fc ' ;MPft-U^zyMAR 2 6 ?5S!ApR 8 78JUL .: "|Vfcf 7^Lib-26-67WGl i i tff>rMIT LIBRARIES3 TDflO D03 TbD EM'iMIT LIBRARIES3 TDflD DD3 TET 251 X rl.TMIT LIBRARIES3 TOAD DD3 TE'l EDEMIT LIBRARIES3 IDfiD DD3 TEfi fifilMIT LIBRARIES3 TOfiD DD3 TET EML|MIT LIBRARIES3 TDflD DD3 TET 41flIDflD DD3 TET M3MMTJ LIBRARIES3 TDfiO DD3 ThO 371TbO 3T7

Is the price system or rationing more effective in getting a commodity to those who need it most?

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