Essays in Optimal Resource Allocation under Uncertainty with Capacity Constraints Citation Hoffman, Elizabeth (1979) Essays in Optimal Resource Allocation under Uncertainty with Capacity Constraints. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/qw1g-4b75. https://resolver.caltech.edu/CaltechTHESIS:06262014-103044842 Abstract This thesis brings together four papers on optimal resource allocation under uncertainty with capacity constraints. The first is an extension of the Arrow-Debreu contingent claim model to a good subject to supply uncertainty for which delivery capacity has to be chosen before the uncertainty is resolved. The second compares an ex-ante contingent claims market to a dynamic market in which capacity is chosen ex-ante and output and consumption decisions are made ex-post. The third extends the analysis to a storable good subject to random supply. Finally, the fourth examines optimal allocation of water under an appropriative rights system. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Social Science) ; optimal resource allocation ; capacity constraints Degree Grantor: California Institute of Technology Division: Humanities and Social Sciences Major Option: Social Science Thesis Availability: Public (worldwide access) Research Advisor(s): Quirk, James P. Thesis Committee: Unknown, Unknown Defense Date: 23 September 1978 Funders: Funding Agency Grant Number Department of Energy (DOE) EY-76-G-03-1305 Record Number: CaltechTHESIS:06262014-103044842 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06262014-103044842 DOI: 10.7907/qw1g-4b75 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 8525 Collection: CaltechTHESIS Deposited By: Tony Diaz Deposited On: 30 Jun 2014 16:27 Last Modified: 26 Nov 2024 22:32 Thesis Files Preview PDF - Final Version See Usage Policy. 18MB Repository Staff Only: item control page

ESSAYS IN OPTIMAL RESOURCE ALLOCATION UNDER UNCERTAINTY WITH CAPACITY CONSTRAINTS Thesis by Elizabeth Hoffman In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1979 (submitted September 23, 1978) ii ACKNOWLEDGEMENTS The author wishes to thank James P. Quit·k for his assistance and Debra Polsky for doing the typing. Research was funded by U.S. Department of Energy grant No. EY-76-G-03-1305. iii ABSTRACT This thesis brings together four papers on optimal resource allocation under uncertainty with capacity constraints. The first is an extension of the Arrow-Debreu contingent claim model to a good subject to supply uncertainty for which delivery capacity has to be chosen befo!e the uncertainty is resolved. The second compares an ex-ante contingent claims market to a dynamic market in which capacity is chosen ex-ante and output and consumption decisions are made ex-post. The third extends the analysis to a storable good subject to random supply. Finally, the fourth examines optimal allocation of water under an appropriative rights system. iv CONTENTS ACKNOWLEDGEMENTS • ii ABSTRACT . • • . . iii I. INTRODUCTION AND REVIEW OF THE LITERATURE Footnotes . • • . II. III. Sl~Y AND DISCUSSION OF RESULTS A FURTHER NOTE ON THE ROLE OF SECURITIES IN THE OPTIMAL ALLOCATION OF RISK-BEARING: OPTIMALITY UNDER CAPACITY CONSTRAINTS 5 7 11 ................... 11 Review of the Arrow-Debreu Contingent Claim Market •. 12 A Contingent Claim Model With a Capacity Constraint . 14 A Competitive Contingent Claims Market in Which Firms Hold Title to Resources 29 A Stockholder Model With a Capacity Constraint 31 Footnotes •• 40 Introduction IV. 1 .. A DYNAMIC PROGRAMMING MODEL OF THE CONDITIONS FOR OPTIMAL CAPACITY CHOICE UNDER SUPPLY UNCERTAINTY 43 Introduction 43 Review of Optimality Conditions Under Uncertainty With Capacity Constraints 46 A Dynamic Programming Model of Ex-Ante Capacity Choice and Ex-Post Spot Markets 49 1. Consumers Own Rights 49 2. Firms Own Rights 56 Conclusion 66 Footnotes 68 v v. OPTIMAL CAPACITY CHOICE AND INVENTORY POLICY FOR A STORABLE GOOD SUBJECT TO RANDOM SUPPLY Introduction • • • 69 A Two Period Dynamic Progranuning Model With Capacity Constraints and a Storage Facility 72 A Competitive Market . ..... .. 85 1. Consumers Own Rights 85 2. Producers Own Rights 97 ..• Footnotes . . . . .... Conclusion VI. 69 .... • 106 • 107 OPTIMAL RESOURCE ALLOCATION UNDER APPROPRIATIVE WATER RIGHTS Introduction . • 108 Review of Appropriate Optimality Conditions • 109 Allocation Under Appropriative Water Rights • . • 114 Conclusion • • 123 Footnotes • 124 1 CHAPTER I INTRODUCTION AND REVIEW OF THE LITERATURE The four papers brought together in this thesis address a theoretical issue which draws together at least three different economic literatures: optimal resource allocation under uncertainty, peak load pricing and the effect on resource allocation of the legal doctrine of appropriative water rights. The issue is to identify conditions which allow competitive markets to produce ex-ante optimal outcomes when the good being allocated is subject to random supply and delivery or storage capacity has to be built before the random variable is observed. An ex-ante optimal allocation is defined as an allocation which maximizes consumer i's expected utility ex-ante, subject to the constraint that all other consumers' expected utilities are held constant ex-ante. Only ex-ante optimality is considered because the introduction of capacity constraints implies that ex-ante and ex-post optimality ca1mot necessarily be equated. The particular good analyzed specifically is Colorado River water. The literature on optimal resource allocation under uncertainty can be dated from the original work by Arrow Debreu 2 on ex-ante contingent claim markets. 1 and They concluded that if all decisions were made ex-ante, contingent on the observation of the random variable, a competitive market in contingent claims would achieve the same allocation that would be achieved in a competitive 2 market under certainty. Recognizing, however, that such a system of markets does not exist and probably would not come into being if only because of its informational inefficiencies, Arrow suggested that a market in firm securities should achieve that same result. Since the seminal work by Arrow and Debreu this literature has developed along several different lines. These include extensions and qualifications of Arrow's and Debreu's conclusions (Dreze, Radner, 5 3 Starr, 6 Nagatani ), rigorous analysis of the conditions which allow securities markets to achieve optimality (Diamond, 7 l.eland, 8 Ekern . 9 10 11 and W1lson, Radner, Forsythe, ), and informational equilibrium models (Radner, 12 Rubinstein 13 ). The papers presented below extend this literature in two directions. First, ex-ante optimality conditions under capacity constraints are outlined and sufficient conditions for achieving those optimality conditions under a competitive ex-ante contingent claims market and a competitive securities market are explored. Second, two of the papers consider sufficient conditions for ex-ante optimality under a competitive market structure which has not been explored to any great extent in the literature. In this alternative to an ex-ante contingent claims market, capacity decisions are made before the random variable is observed, but marginal production and consumption decisions are made in competitive markets after. The current American literature on peak load pricing under uncertainty (Joskow, 14 Johnson, 15 Crew and Kleindorfer 16 ) owes its 4 3 . 17 18 start to the work of French economists (Bo1teux, Dreze ) who were trying to devise pricing schemes which would both satisfy marginal optimality conditions and cover operating costs for the French national electricity network. The general approach has been to assume a random demand and then use consumer surplus analysis to devise optimal peak and off peak prices and optimal size generation facilities given a probability distribution over demands. While the papers presented below are not specifically concerned with optimal pricing schemes and the pe a k lo a d pricing literature does not deal with competitive markets, the models used and the results obtained are similar. Basically, most of the optimality conditions presented in these papers should not be seen as new. Rather, they are rearrangements or reinterpreta- tions of the optimal pricing conditions from the peak load pricing literature. Where these papers depart from tradition is in asking what conditions would allow ex-ante optimality to be achieved under competition. A recent paper by McKay on consumer surplus analysis. 19 has moved away from the emphasis His welfare model is a weighted sum of individual utility functions and his analysis is similar to that used in the papers collected here, although the questions he addresses are different. . k20 on t h e appropr1at1ve . . Recent papers b y Burness an d Qu1r water rights system form a natural introduction to the third and fourth papers presented below. Their papers consider two issues: 4 the effect on resource allocation of the legal doctrine of appropriative water rights with a prohibition against sale of those rights and optimal dam and reservoir size and release policy under that institution. The papers presented below extend their analysis in two directions. First, conditions for optimal storage size, storage, and releases are analyzed from a social welfare point of view, independent of legal impediments. Second, the effect on resource allocation of simply lifting the ban on the sale of appropriative water rights is explored. Turning now to the four papers presented below, the first one is simply an extension of Arrow's and Debreu's contingent claim and securities model to a good subject to random supply for which delivery capacity has to be chosen before the random variable is observed. The second paper compares an ex-ante contingent claims market to a dynamic market in which capacity is chosen before the random variable is observed and consumption and marginal production decisions are made after. The third paper extends the analysis in the second paper to a storable good subject to random supply. Finally, the fourth paper applies the analysis in the third paper to water allocated under an appropriative water rights system. 5 FOOTNOTES 1. K. J. Arrow, "The Role of Securities in the Optimal Allocation of Risk-Bearing," Review of Economic Studies 31 (April 1964). 2. G. Debreu, The Theory of Value: An Axiomatic Analysis of Economic Equilibrium, Cowles Foundation Monograph No. 17, Yale University Press. 3. J. H. Dreze (ed.), Allocation Under Uncertainty, Equilibrium and Optimality (New York: McMillan, 1974). 4. R. M. Starr, "Optimal Production and Allocation Under Uncertainty," Quarterly Journal of Economics 87 (1973). 5. R. Radner, "Existence of Equilibrium of Plans, Prices, and Price Expectations in a Sequence of Markets," Econometrica 40 (2) (March 1972). 6. K. Nagatoni, "On a Theorem of Arrow," Review of Economic Studies 41 (1974). 7. P. A. Diamond, "The Role of a Stock Market in a General Equilibrium Model with Technological Uncertainty," American Economic Review 57 (4) (September 1967). 8. H. E. Leland, "Production Theory and the Stock Market," The Bell Journa l of Economics and Management Science 5 (1) (Spring 1974). 9. S. Ekern and R. Wilson, "On the Theory of the Firm in an Economy with Incomplete Markets," The Bell Journal of Economics and Management Science 5 (1) (Spring 1974). 10. R. Radner, "A Note on the Unanimity of Stockholder's Preferences Among Alternative Production Plans: a Reformulation of the Ekern- Wilson Model," The Bell Journal of Economics and Management Science 5 (1) (Spring 1974). 11. R. E. Forsythe, "Unanimity and the Theory of the Firm Unde r Multiplicative Uncertainty," California Inst i tute of Technology, Social Science Working Paper No. 90, February 1976. 12. R. Radner, "Competitive Equilibrium Under Uncertainty," Econometrica 36 (1) (January 1968). 6 13. M. Rubinstein, "Securities Market Effi c iency in an Arrow-Debreu Economy," The American Economic Review 65 (5) (December 1975). 14. P. Joskow, "Contributions to the Theory of Regulatory Economics," The Bell Journal of Economics and Management Science 7 (1) (Spring 1976) 15. G. Brown and H. B. Johnson, "Public Utility Pricing and Output Under Risk," The American Economic Review 59 (1) (Spring 1974). 16. M. A. Crew and P. R. Kleindorfer, "Peak Load Pricing With a Diverse Technology," The Bell Journal of Economics and Management Science 7 (1) (Spring 1976). 17. P. Boiteux, "Peak Load Pricing," The Journal of Business (April 1960). 18. J. H. Dreze, "Some Postwar Contributions of French Economists to Theory and Public Policy," The American Economic Review 54 (3) (June 1964, Supplement). 19. D. McKay, "Two Essays on the Economics of Electricity Supply and Pricing. Part I: Has the Averch-Johnson Effect Been Empirically Verified? Part II: Electricity Pricing," Ph.D. Thesis, California Institute of Technology, 1978. 20. H. S. Burness and J. P. Quirk, "Appropriative Water Rights and the Efficient Allocation of Resources," California Institute of Technology, Social Science Working Paper No. 157, Revised March 1978. Forthcoming in The American Economic Review (December 1978). "Appropriative Water Rights and the Theory of the Dam." To appear in festshrift for E. T. Weiler, published by Purdue University Press, 1979. 7 CHAPTER II SUMMARY AND DISCUSSION OF RESULTS The four papers presented below show several results or conclusions which can be seen as either new or new extensions of previous results. The first three papers make clear the limitations of an ex-ante contingent claims market as a model for describing market decisions under uncertainty. The first paper shows that if all choices for all states of the world have to be made before any observation of the random variable, as would happen with complete contingent claims markets, then the introduction of fixed capacity constraints does not alter the Arrow-Debreu result that competitive contingent claims markets are ex-ante Pareto optimal. Arrow's contention that a market in securities achieves the same results as can be achieved with an ex-ante contingent claims market does not necessarily extend to the case where there are ex-ante fixed capacity constraints, however. Further, as the second and third papers show, when capacity or inventory choices have to be made ex-ante, but marginal production and consumption decisions can be made ex-post, then such noncontingent claims competitive markets will not necessarily allow ex-ante optimal capacity and inventory choices. There are at least two general reasons why such noncontingent claims competitive markets might not be ex-ante Pareto optimal when capacity has to be chosen ex-ante. In the first place, a problem 8 may occur because gains from trade exist among consumers when capacity and inventory choices are being made. example. Consider the following simple Suppose there are two individuals, Mr. A and Mr. B, who jointly own a river and wish to dam it, store water, and deliver it to their homes. Suppose in addition that Mr. A thinks there will be a great deal of water in both periods and Mr. B thinks there will only be a moderate amount of water in each period. Consequently, Mr. A thinks a large storage facility unnecessary, but wants to build a large diversion system. Mr. B, in contrast, wants a larger storage facility and a smaller diversion system. capacity is more expensive in each case. Assume larger Clearly, in the absence of transactions costs, Mr. A would be willing to bribe Mr. B to build a larger diversion system and Mr. B would be willing to bribe Mr. A to build a larger storage facility and store more water. Now consider two market means of choosing capacity size and inventory policy. In the first case, firms producing capacity sell shares to consumers who own delivery rights. Consumers then both rent capacity shares and delivery rights to firms and use capacity shares and delivery rights to store water for future use. In this case the market in shares becomes the means by which consumers such as Mr. A bribe those such As Mr. B and visa versa. Alternatively, suppose firms own delivery rights and produce capacity and store water on the basis of their own subjective probability distributions, discount rates and attitudes towards risk. 9 In this case, no mechanism exists to exhaust the gains from trade among consumers. Finding prices which allow an optimal allocation then resembles the problem of designing an optimal tax. However, if all consumers and firms owing capacity have the same probability distribution and discount rate and firms owning capacity are risk neutral (i.e., expected profit maximizing), then these gains from trade do not exist to begin with. The second reason why noncontingent claims competitive markets might not be ex-ante Pareto optimal is related to the fact that more than one kind of uncertainty is introduced when some decisions are allowed after the random variable is observed. The first kind of uncertainty is uncertainty about capacity utilization because of supply uncertainty. The problem generated by that uncertainty can be overcome by giving consumers title to all risky assets, as discussed above. The second kind of uncertainty occurs because demand is uncertain when marginal consumption choices can be made ex-post. Since differences in available supply of one good imply differences in relative prices and incomes in different states of the world, final demand cannot be known with certainty. Returning to the simple example discussed above, suppose Mr. A also thinks it highly probable that if there is a lot of water that prices and his income will be such that he will choose to purchase a great deal of that water if sufficient capacity is built. In this case, if Mr. A could guarantee his intuition, he would be 10 willing to pay Mr. B even more to build a larger delivery capacity than if prices and his income were the same in all states of the world. Otherwise, the gains from trade created by this kind of uncertainty remain unexhausted. Only if such income effects do not exist to begin with can noncontingent claims competitive markets be ex-ante Pareto optimal when some choices are made ex-ante and others are made ex-post. In an analogy with the constant marginal utility of income requirement when consumer surplus analysis is used, these papers adopt the sufficient condition that expected marginal utilities of income be discounted constant functions of consumer subjective probability distributions. This condition eliminates the income effects of demand uncertainty, while still retaining all the effects on conswner choice of the supply uncertainty. After outlining the conditions discussed above in the first three papers, the fourth paper shows that if all the sufficient conditions for an ex-ante optimal competitive allocation, when firms own capacity and inventories hold, then the appropriative rights system can be made efficient. The means for achieving efficiency discussed in that paper is to open a market in state dependent percentage shares of appropriative rights. Such a system would allow firms both to pool risks and thereby eliminate the monopoly element of the appropriative rights system and also to only purchase as many rights as are needed in any particular state of the world. 11 CHAPTER III A FURTHER NOTE ON THE ROLE OF SECURITIES IN THE OPTIMAL ALLOCATION OF RISK-BEARING: OPTIMALITY UNDER CAPACITY CONSTRAINTS INTRODUCTION This paper develops ex-ante optimality conditions under uncertainty for a situation in which delivery capacity of one good has to be chosen before the random supply to be delivered is observed. Examples of goods which satisfy this condition are water delivered by aqueduct from a river of random flow and variable supply goods transported by rail or truck. In each case, the delivery capacity cannot be adjusted optimally in every state of the world, although other inputs can be adjusted optimally for a given capacity. The model developed in this paper is an extension of the l Arrow -Debreu 2 ex-ante contingent claim model to an economy with a capacity constraint on one good. Assuming differentiable utility functions, ex-ante optimality is defined as an ex-ante allocation of contingent claims which maximizes consumer i's expected utility subject to the constraint that the expected utilities of all other consumers are held constant. The addition of the capacity constraint implies that even a market in ex-ante contingent claims no longer eliminates consumer uncertainty from the allocation decision ex-ante. Rather, when the capacity constraint is binding, the ex-ante optimal allocation equates expected marginal rates of substitution and 12 expected price ratios. Thus, in this situation~ ex-ante and ex-post optimality will not necessarily be equivalent even though for a given capacity larger then available delivery supply they are equivalent. After developing ex-ante optimality conditions, this paper shows that an ex-ante optimal allocation can be achieved by an ex-ante contingent claims market even if firms hold title to delivery and capacity rights and consumers have differing probability judgments. If a stockmarket in shares of firms owning capacity is substituted for the contingent claims market, however, an optimal allocation of risk bearing cannot necessarily be achieved under the usual assumptions of stockholder unanimity models. Rather, additional restrictive assumptions are close to being necessary in order to achieve unanimity: Consumers have equal subjective probability distributions and constant marginal utilities of income. REVIEW OF THE ARROW-DEBREU CONTINGENT CLAIM MARKET The original paper by Arrow describes how an ex-post Pareto optimal allocation of resources can be achieved under uncertainty with a complete set of ex-ante contingent claim markets. While the specific optimality conditions he describes are not spelled out in the paper, he says that making each good in each state of the world a separate good makes the uncertainty case exactly analogous to the certainty case. • , xilC'xiZl, . • . , xiSC) be the utility of individual i if he is assigned claims of amount x. 1SC for 13 conunodity c if state s occurs (c = 1, . . . , C; s = 1, . . . , S). This is exactly analogous to the utility function in the case of certainty except that the number of variables has increased from C to CS. We may therefore achieve any optimal allocation of risk-bearing by a competitive system. Let xi* (i=l, • • . , I; s=l, . • • , S; c=l, . • . , C) be any sc optimal allocation; then there exist a set of money incomes y. for individual i and prices 1 psc for a unit claim on connnodity c if state s occurs, such that if each individual i chose values of the variable x. 1SC (s = 1, . • • , S; c = 1, . . • C) subject to the restraint s E c Ep x. s=l c=l sc 1sc yi taking prices as given, the chosen values of the x isc 's will be the given optimal allocation x:'< 1SC (i = 1, . . . , I; s = 1, . . . , S; c = 1, . . . , C) . The argument is a trivial reformation of the usual one 3 in welfare economics. The following problem is equivalent to the one Arrow is stating: maximize: s subject to: c E E p x. s=l c=l sc 1sc y .. 1 (a) 14 Ex-ante, the first order condition is: av.1 ax.1SC Vi,c,d,r,s. av.1 For a given state of the world, this ex-ante condition reduces to: avi ax.1SC Vi,c,d,s. av.1 ax.1S d This is the same optimality condition that would prevail if the choice could be made after the random variable were observed (i.e. ex-post, under certainty). This implies that a complete set of ex-ante contingent claim markets will lead to the same ex-post optimality conditions under uncertainty as under certainty. Let us call this effect removing uncertainty from the allocation decision. A CONTINGENT CLAIM MODEL WITH A CAPACITY CONSTRAINT Turning now to the situation such that capacity has to be chosen before the random variable is observed, let xis denote the amount of good x consumed by individual i in state of the world s = l , . . . ,s. x s is the amount of x which is delivered to consumers n in state s. L X. < X i=l 1S - • s Let z s denote the supply of x available 15 for delivery in state s. z s is a discrete random variable which occurs with subjective probability a. 1S . We identify states of the world s with levels of supply of x, where x There are n individuals in society, the < z s s i~ with strictly quasi-concave, differentiable utility function U . (x. 1 where y. 1S 1S ,y. 1S ,L. ) , 4 1S is the amount of the composite goody consumed by i in state of the world s and L. 18 in s. for all s. is the amount of labor offered by i The delivery capacity for x of the world. s is x and x For a given capacity, x production function x(L xs ) and y s < x in every state s - is delivered according to is produced according to production s n function y(L ys ) , where n L: y. i=l < y , L lS - S XS + L < L: L. • ys - i=l 1S in period 0 according to production function x(LA). X x is produced Labor is supplied in period 0 according to i's utility function U . (L. ) and 1 10 Now we wish to define a Pareto optimal allocation of x and y . s s For this model we adopt the following definition of Pareto optimality. An allocation is said to be ex-ante Pareto optimal if, given consumer subjective probability distributions over states of the world, it is not possible to increase consumer i's expected utility ex-ante without reducing at least one other consumer's expected utility ex-ante. All trading and production are assumed to take place before the random variable is observed. optimality will be considered later. Ex-post Now, given this definition and assuming x and a composite good enter all appropriate utility 16 functions, an ex-ante Pareto optimal state can be characterized as the maximum of a weighted sum of individual expected utility functions. Therefore, we can identify any arbitrary ex-ante Pareto optimum by solving the following problem: n S l: f3.[U.(L. ) + L: a. u . (x. ,y. ,L. )] i=l ~ ~ ~ 0 s=l ~s ~ 1s ~s ~s Max W Subject to the following constraints: Multipliers n l: x. < x(L ) i=l ~s xs Vs ) < z vs x(L xs s n l: y. < y s i=l ~s y(Lys) Ys A xs vs X(L A) X x(L vs X A ) < X Vs n LA < L L. x - i=l ~a n L + L xs < l: L. ys - i=l ~s L. ~s < H - 6 Vs Vi, s A. 3s 17 where: W the social welfare function, and si the weight given to i's utility function, H maximum time an individual can spend working per time period. Forming a Lagrange expression and differentiating, the first order conditions are: aui si - - + Yz aLiO 0 au. S.a. ~ ~s ~ a;z:-- A.ls 0 Vi, s - A.3s = 0 Vi, s 0 Vi, s ~s au. S.a. ~ ~s ~ ay is au. S.a. ~ ~s ~ ~+ A.6s ~s 0 vs Vs 0 0 Vs 18 0 If we consider only effective constraints that are in fact binding, there are three cases: 1. If z 2. If z 3. If z s < x, then "ss s s x, then "ss = 0 and A.2s -1 0. > A. 2s = 0. x, then A. 5 s -1 0 and >.. 2 s 0. These observations allow us to reduce the first order conditions to the following equations: Case 1 {zs < x; A.Ss = 0, A. 2 s -:f 0): au. a. ~s ~ ~ 1S V h, i, s, r au.1 ~ 1r - (3 . a. 1 lS au./ aL. 1 ~s A. ax/ClL + 2s xs au . /aL. 1 1r -6 ~. a 1r . Cly/ClL yr Vi, r 19 Case 2 (z = s x·' ASs =A 2s = 0): V h, i, r au. 1. a -ir ayir au. ;aL. l.S 1. ax/aL xs au./aL. V i,r l.r 1. ay/aL yr au. 1. "' s>x l.S oXis au. 1. a -ir ay. J.r L: a. E a. Vh, i, r aui s>x l.S axis au 1 a -ir ay ir L: s>x a. l.S au. /aL. 1. 1. 0 + ax/aL,_ aui/aLis ax/aL X XS au . /aL a. 1. J.r ay/3L 1r yr where: aw n a( L: L. ) i=l l.S is the marginal social value of labor in s; Vi,r 20 aw n a( t: L. ) i=l 10 is the marginal social value of labor in period 0; >.. 2 s aw is the marginal social value Clz s of z in s.7 Notice that in Cases 1 and 2 the ex-ante optimality conditions are the same as Arrow's. In Case 3, however, these first order conditions imply that at an ex-ante Pareto optimum, for z s > x, only the expected marginal rates of substitution will be the same across all individuals and equal to the expected marginal social values of x s and y • s This deviation from the Arrow-Debreu result that a contingent claim market transforms an uncertainty problem to a certainty one occurs because the capacity constraint makes it impossible for producers of x to optimize productive inputs in every state of the world. These results can be summarized as follows: Proposition 1: If the supply of a good is a random variable and delivery capacity has to be chosen before the random variable is observed, then for those states of the world such that available supply exceeds delivery capacity, an ex-ante optimal allocation of contingent claims equates the expected marginal rates of substitution between the random good and all other goods across all individuals to the expected marginal social values of x and y. Uncertainty has not been removed from the allocation decision, ex-ante. 21 Proposition 2: If the supply of a good is a random variable and delivery capacity has to be chosen before the random variable is observed then for those states of the world s u c h tha t capacity exceeds or is equal to available supply, an ex-ante optimal allocation of contingent claims removes uncertainty from the allocation d e cision. The question now is whether a competitive equilibrium will satisfy these optimality conditions and if so, whether any special conditions will have to be imposed. First, complete contingent claim markets have to exist for labor and delivery rights concurrently with a contingent claim market for delivery of x. Otherwise, a contingent claim market for delivery of x will not correctly evaluate the expected relative values of x and y. Consider now a competitive economy with n consumers and m + k + g firms. The first m firms produce x in period 0, the next k firms produce x and the last g firms produce y. Consumers hold title to delivery rights and capacity units once capacity has been built. Consumer i' s problem is: s Max EU. 1. subject to: U.(L. 1. 1. 0 )+ L:a. U.(x. s=l l.S l l.S ,y . l.S ,L.) l. S 22 Multiplier: A.. 1 s m+k m+k+g 1 1] J L: [p " +w L. +q + L: 8 J1rJ + L: 8 1Tys] _ 1 XS 1 S 1S S 1S . m+l XS XS j=m+k+l ys sJ= x. r. where: e~j X eij Xfl percentage of firm xj' s profits going to i; = percentage of firm xj's profits going to i in s; eij ys percentage of firm yj's profits going to i in s., wo competitive wage in period 0; w s = competitive contingent claim wage in state s·, ni profits of firm "j in period 0; X 1Tj xs profits of firm xj in state s; 1Tj ys profits of firm yj in state s; p,. competitive price of a unit of capacity purchased X X from the firms producing capacity in period 0. PXS competitive contingent claim price of X in state s·, pys competitive contingent claim price of yin state s; Pxs competitive contingent claim price of a. unit of capacity in state s; x.1 number of capacity rights owned by consumer i·, 8 23 competitive contingent claim price of a state s delivery right; 9 initial allocation of state s delivery rights to i; n z: eij i=l xs 1·, 1; and n z: e=:J 1. i=l X Mru<ing the assumption of a decomposible profit function, firm ij's problem is: Max Multipliers nl X s.t. Firm xj 's problem is: s l: [p s=l s.t. . . ,.J xJ - w J) XS S S XS ) XS XS < x(Lj ) xs s - x(Lj ) < ~ S j - p" x - q r ] S S S 24 where: ~j x s rj s amount of capacity rented by firm x. in state s, and J number of delivery rights rented by firm x. in state s. J Firm y. 's problem is' J s L: [p yj s=l ys s w Lj ] s ys s.t. The equilibrium conditions are: n "' X, ~ t... i=l 1 m rx.i j=l n L X, i=l 1S m+k . E XJ j=m+l s n m+k+g L: yj j=m+k+l s L: y, i=l 1S n E L. i=l 10 n L: L i=l is n I r. i=l 1S m . E L{ j=l X m+k m+k+g . L: L j + L: LJ xs j=m+k+l ys j=m+l m+k . I rJ j=m+l s Multipliers 25 Forming Lagrange expressions L. (i = 1, . . . , n), Ll (j = 1, 1 X . . , m), Lj (j = m + 1, . . • , m + k) , and U (j = m+ k + 1, . . . , m + k + g) and y X differentiating, the first order conditions are: au _ _i_+ AiWO aLiO au. 1 a. l.f? ~ 1S Vi 0 - Aipxs = 0 Vi, s au. 1 1S ~1S a. 0 Alpys Vi, s au . a. 1S 1 ar::+ A.w 1 s 1S Vi, s 0 s A. [p A1 X PXO -w L: p A ] s=1 XS - >.iX 0 Vi 0 j=l, . . • ,m ox j=l, . • . ,m + Aj - - = 0 0 X aLi X pxs -w s - Aj1s + ~(Aj aLJ ls xs -pxs + Aj 2s vs, j = m + 1, . . . , m + k 0 0 Aj 2s Aj ) 3s 0 vs, j = m + 1, . . • , m + k vs , j = m + 1, . . . , m + k 26 -qs + >)Js pys - -w s A.j 4s 0 Vs, j = m + 1, . . . , m + k 0 Vs, j =m+k+l, . . . , m+k+g + A.j4s ~= 0 s, j =m+k+l, . . . , m+k+g aLj ys Considering only effective constraints which arebinding, these first order conditions reduce to: au. 1 oLiO au. 1 wo au. 1 a. 1S aL. 1S au. 1 a. 1r ~ 1r w s w r a. 1S ~ 1S a. 1S a. 1r w s (a) Vi, r, s (b) Vi, r, s (c) Vi,s (d) au. 1 --ax.1S PXS au. 1 Pyr ay:1r -aisaui/aLis w s Vi, s A.. 1 27 Case 1 (z < x): s w s Pxs 3x/aLj pyr w + qs xs Vj, a, r (e) vj, a' r (f) Vj, a, r (g) r 3yj()La yr Case 2 (z s x): w s ax/ aLj PXS XS --= - - - w pyr r Cly/3L Case 3 (z s a yr > x): wo w ax; 3L~X ax/ 3L j --~- + --~s__ XS w r ay/3Lys a Now compare (c) with the production conditions (e), and (g) and substitute (d) for w . s The three cases reduce to: (f), 28 Case 1 (z < x): s -ais au.~ 1 aL.1S au. ~ a. 'ax/ aLj axis XS au. =-a. au. /aL. 1 1r 1 1r a.1r ay ir 'dy/aLa yr ~s Case 2 (z + A.q ~ s au. ;aL. 1 1S ~ 1S 'ax/ aLj 1 - - - -xs au:-= au. 1aL. 1 1 1r ay,1r s (h) Vi, a, j,r (i) x): s au. Case 3 (z Vi,a,j,r 'ay/aLa yr > x): au. E a. 1 au./aL. E a - 1 18 au./aL. 1 10 " b 'ax/ aL j 'ax/ aL" xs X au./aL. 1 1r -a.1r 'ay/aLa yr " is s>x 1S ax.1S au. S >X 1 a.1r ay,1r Vi,a,b,j,r (j) Notice that equations (h)-(j) describe an ex-ante Pareto optimum with 1 welfare weights ai = ~ . This result can be summarized as follows: i Proposition 3: If the supply of a good is a random variable and delivery capacity has to be chosen and produced before the random variable is observed, then a competitive contingent claims equilibrium is ex-ante Pareto optimal if consumers hold title to all resources. 29 A COMPETITIVE CONTINGENT CLAIMS MARKET IN WHICH FIRMS HOLD TITLE TO RESOURCES Suppose now we consider a somewhat more likely market situation. In this case consumers participate in a contingent claims market for labor and output, but firms delivering x build and hold title to their own capacity and hold title to delivery . h ts. 10 r1g Firms alone participate in a contingent claim market in delivery rights. The maximization problems now become: s Consumer: MaxU . (L. ) + l:aiU.(x. ,y. ,L.) 1 10 s=l s 1 1s 1s 1s s s.t. s + l: [ p x. + p y . ] < w L . O + l: [ w L . 0 1 s=l xs 1s ys 1s s=l s 1s k+g L: eij n l j=k+l ys ys Vi j Firm x.: J Max -wOLx"' + s.t. Firm y.: J Max ~ [p s . l: [p XJ - wsLJX. s s=l xs s ~ < x(Lj ) S - XS s=l yj - w Lj ] ys s s ys s.t. yj < y(Lj ) s ys . . q (rJ - rJ)] s s s 11 30 The first order conditions for this problem reduce to: Consumers: same as (a)-(d) Firms: Case 1 (z Case 2 (z Case 3 (z s s s < x) = same as (f). x): same as (g). > x): l: A s>x a, j, r w (k) r ay / 'dL a yr The fact that Cases 1 and 2 are identical to the equations describing the allocation when consumers own delivery rights indicates that the conclusions of Proposition 3 generalize to the case such that firms own the state-dependent delivery rights. This result can be generalized as follows. Proposition 4: If the supply of a good is a random variable and delivery capacity is not a binding constraint, then the distribution of delivery rights does not affect the allocation of final output if the market is organized as an ex-ante contingent claims market. Notice that equation (k) describes an optimal allocation a 1 so. . s t atement, T o prove th ~s . sub st~tute . equat~on ( a ) f or--Pxs Pyr 31 wo (b) for w w , (c) for r au. 1 I a. 1S ~ s>x 1S au. 1 a. 1r ay. 1r w s and sum over s > x: r I s>x a. au . /3L. 1S 1 1S + au/aLiO ax/aLj ax/aLi XS X au./aL. 1r 1 a. 1r ay/aL a yr Vi,j,a,r (1) This result can be summarized as follows: Proposition 5: If the supply of a good is a random variable and delivery capacity has to be chosen before the random variable is observed, then the distribution of capacity rights does not affect the final allocation if the market is organized as an ex-ante contingent claims market. A STOCKHOLDER MODEL WITH A CAPACITY CONSTRAINT Since ex-ante contingent claims markets are unwieldy and don't allow consumers to adjust their portfolios of consumption claims after the random variable is observed, much of the literature on optimal risk bearing has focused on outlining the conditions under which a contingent claims allocation can be achieved by allocating risk through a securities market. In such a market consumers need only purchase securities ex-ante and they can alter their portfolios of securities at any time. In this literature, finding conditions which allow an optimal allocation of risk bearing is equivalent to finding conditions which allow consumers to readjust their portfolios 32 of stocks to hedge against a firm's decision they disagree with. Under such conditi.ons holders of a firm's stock could always agree on what decision the fi.rm should make. It turns out, however, that when the decision i .s what size capacity a firm should build, unanimity canno t in general be a c hieved. Consider the following example. Adapting Forsythe, Ekern and Wilson's model, 12 a s interpreted by 13 let individual i have an undiscounted, additive utility function over certain current consumption and uncertain future consumption, su c h that EU . 1 U.(x. ,y. ) 1 10 10 + s 2: a. U.(x. ,y. ), 14 s=l 1s 1 1s 1s which he maximizes subject to budget constraints for current and future consumption. The current budget constraint is k k+g k k+g 2: VJ gJ x1.0pxO + Y1.0pyO + .l: VJx 8 J1·x + 2: VJS~ < wOL1.0 + 2: VJS~ + . ' J= 1 j=k+l Y 1Y j=l X 1X j=k+l y 1Y Multiplier: >.i 0 whe re: value of firm (~,yj) number of (xj,yj) shares outstanding; 33 v . ,v . X] YJ j j -j -j price per share of (xj,yj) stock; s.1X 's.1y s 1X . 's.1y stocks owned by i; j _ 's initi a l endowment of (xj ,yj) shares. The future budget constraints are: k x1.SPXS + Y1· spys < wSL1.S + . . L TIJ sJ j=l xs ix + k+g . . L TIJ sJ j=k+l ys iy vs. Multiplier: Forming a Lagrange expression and differentiating, first order conditions are au. i 1 axiO - A.opxo au. 1 ayiO i A.opyo 0 0 au. 1 a. ~- A.ispxs 1S 1S 0 au. 1 A.i a. lS ayis - spys 0 0 34 0 These reduce to: s au.1 ) 'IT]. ax. xs a. 2:: 1S s=l s a 2:: s=l (m) 1S ( au.1_ ) nJ. __ is ayis ys ( (n) pys Now, let us consider the effect on person i's utility of a change in the amount of labor firm xj devotes to capacity: au. 1 aui axiO ad. axiO aL~ s au' ax. s=l is axis aLi --= ----+ X 2:: a 1 -- 1S - (o) X From the budget constraints: l I xiO = pxO [ wO"iO + k k+g ~ VJ-SJ + j~l x ix 2:: .. ] VJS~ j=k+l y l .y (p) k x. 1S . . 2:: n J S J j=l xs ix + k+g . 2:: nJ sJ . j=k+l ys iy (q) 35 Now making the usual assumptions that the state distribution of prices of outputs and inputs and the other firms' profits are indep endent of Li: X au. 1 --= aLi avm (s~ au. 1 X axiO 1X aLi X - s~1X ) Pxo X k+g + aVU ~ m=k+l aLl L: X s + L: a. s=l 1S s~ au.1 [anjXS 1X] ax.1S "Lj pXS o " (r) X From (m): au . 1 ~Po. s ~~7TJ L: a. s=l 1S au.1 p xs xs axiO A.i Forsythe interprets X.1S ' s as a contingent claim price for consumption of A.i 0 independent of Li. This implies: X avm _J_ = 0 Vm; aLi X avm X aLi = for m # j ; and 0 X s L: A.i arrj xs -s - - s=l A.i aLi 0 X for m j (s) 36 Substituting (s) in (r) gives: au. 1 au. -j 1 s.1X aLi X s ax. anj XS 1S L: Ct. - - - 1S PXS s=l aLi (t) X At this point an assumption is usually made which is that the change being undertaken does not alter the feasible set of profits or the probability distribution of profits over states of the world. This is a sufficient condition for stockholder unanimity. Following Forsythe's spanning formulation: m L: a nj 'k k , k=l J s Let where, the ajk' s are a set of weights such that the change in capacity keeps the profits within the same linear subspace. au. Thus, l --= s au /ax. 1 j L: a.k L: Ct.1S --=---1=-S 7T ks 1X k=l J s= 1 N s~ N -j s. L: a . 1 x k=l J 1<. Since au./ax. 1 (n) 10 aui/axiO vk, by (m) • Pxo > 0 by the assumption of quasi-concavity, - has the same sign for every i for a change in capacity such (u) 37 m that La .k Vk > 0 (<0). k=l J Thus, stockholders would agree about the affect on their utilities of a change in capacity. Such an assumption is not valid in this case, however, because the state distribution of profits depends upon the choice of capacity. Higher or lower capacity means higher or lower profits in all states of the world affected by the capacity change because the states of the world covered by the three cases change. In other words, the feasible set of profits is altered over certain states of the world if capacity is changed. Thus, while there may still exist special conditions which would allow stockholder unanimity, it cannot be proved as a general theorem by employing the usual assumptions of these models. Proposition 7: This result can be summarized as follows: When firms hold title to capacity which must be built before the supply random variable is observed, then a stock market in shares of firms owning capacity does not necessarily allow stockholders to come to agreement on the choice of capacity, given the usual assumptions of stockholder unanimi.ty models. Suppose, however, that we in~ose the following more strict conditions on equation (t): 1. all consumers have the same subjective probability distribution; and 38 2. each consumer has a constant marginal utility of income and an expected marginal utility of income which is a constant function of his subjective prob a bilities (i.e. Ai C. and Ai 0 s ~ C.a. ~ ~s vi, s). 15 Under these conditions an optimal allocation of risk bearing can be achieved through a securities market. Ai = c.a s a 1 s Vi, s, To prove this statement let: (v) where true or agreed upon probability of being s in s; Ci i's constant marginal utility of income. This implies: c.a s p xs (w) Vi, s ~ by the first order conditions. Substituting (w) in (t): au.1 ad.X -j s.1Xc.1 s E a s s=l d1Tj XS (x) ad.X -j Since Six > 0 and Ci > 0 by quasi-concavity (i.e., aui oN > 0), the sign of (x) is independent of i and all consumers will agree on the effect of the change. This result can be summarized as follows: 39 Proposition 8: If supply of a good is a random variable and delivery capacity is owned by firms and has to be chosen before the random variable is observed, then a market in shares of firms owning capacity allows consumers to come to agreement on firms' choices of capacity when the following sufficient conditions hold in addition to the usual assumptions of stockholder unanimity models: 1. consumers have the same probability distributions over states of the world; and 2. consumers' expected marginal utilities of income are constant functions of those subjective probabilities. 40 FOOTNOTES 1. K. J. Arrow, "The Role of Securities in the Optimal Allocation of Risk-Bearing," Review of Economic Studies 31 (April 1964). 2. Gerard Debreu, The Theory of Value: An Axiomatic Analysis of Economic Equilibrium, Cowles Foundation Monograph No. 17, Yale University Press. 3. K. J. Arrow, "The Role of Securities in the Optimal Allocation of Risk-Bearing," Review of Economic Studies 31 (April 1964). 4. Without loss of generality we assume that i's utility function is constant over states of the world. Further, utility functions are au 1 aui assumed to be neoclassical. That is, lim ---- = lim ---- = + oo; ax ay xis+ 0 is y is-+ 0 is au. aui 1 O· lim - -- > 0 and lim lim ~ < 0 so that positive ' y-+ 00 ayis x-+oo Lis-+ 0 is amounts of x and y are consumed in all states, positive amounts of labor are supplied in all states and there is no bliss point. 5. Without loss of generality we assume that no consumption takes place in period 0. 6. We assume 7. To prove this statement use the envelope theorem. - oo so that the constraint L. lim Lis-+H never binding. l.S Let W = W(x) < H is - (a) Subject to gj(x,L) > 0 dW dL N l: i=l aw dxi Clx. dL m L W+ (b) l. .. l: AJ gJ j=l (c) 41 m a i aL - - = aw + L: >) ~= 0 ax. axi a xi j=l ~ (d) Now consider all effective constraints: k a j dx. ac) L: ~ -~ = _::_g_ i=l axi dL aL (e) Substituting (e) and (d) in (b): dW dL k -~ E A.J aL n -= E i=l j=l j=l th agi = :\j since L only enters the j - constraint aL = 0, and i.,; j, agj lf2_ ax = 1 when a:x .,; 0. The same proof applies for dw/dz. 8. A capacity right is a right to use one unit of capacity. Consumers own these rights and rent them to firms who use them to deliver x. 9. A delivery right in state s is a right to use one unit of z in state s. Consumers own such rights and rent them to firms who use them to deliver units of x in s. 10. The results of this model follow whether delivery firms produce their own capacity or a separate set of firms produce capacity and sell it to dalivery firms. 11. Note that firms do not bear any risk in this model even though they own capacity. The reason there is no risk is that all trading takes place before there is any production. Therefore, firms know with certainty their total revenues and expenditures regardless of what state of the world occurs. 12. S. Ekern and R. Willson, "On the Theory of the Firm in an Economy with Incomplete Markets," Bell Journal of Economics and Management Sciences 5 (1) (1974): 171-180. 13. R. E. Forsythe, "Unanimity and the Theory of the Firm Under Multiplicative Uncertainty," California Institute of Technology, Social Science Working Paper No. 90, February 1976. 42 14. This model assumes that some capacity exists and the firm might wish to change it. Without loss of generality we assume that labor does not enter an individual's utility function. 15. See footnote 7 for a proof of the statement that Ai is the s expected marginal ut i lity of income in state s • 43 CHAPTER IV A DYNAMIC PROGRAMMING MODEL OF THE CONDITIONS FOR OPTIMAL CAPACITY CHOICE UNDER SUPPLY UNCERTAINTY INTRODUCTION The previous paper developed ex-ante optimality conditions for the allocation of a good subject to supply uncertainty and capacity constraints. Then it showed that, although an ex-ante contingent claims market would achieve an optimal allocation regardless of who owned the capacity and de l ivery rights, a market simply in securities would not necessarily achieve an optimal allocation unless more strict assumptions were made than is customary in stockholder unanimity models. The particular additional sufficient conditions examined in that paper were: 1. consumers have identical subjective probability distributions over states of the world; and 2. consumers have expected marginal utilities of income which are constant functions of those subjective probabilities. While contingent claim markets are interesting examples of solutions to problems caused by uncertainty, most economists would agree that it is unlikely they will ever develop as a means of allocating goods for final consumption. The information needed for consumer decision-making is too great. Further, the need to 44 make all decisions for future time periods at the birth of an economy does not allow consumers to readjust their portfolios in light of new information. This problem is similar to the one noted by Starr: 1 namely that unless consumers have sufficiently simtlar probability distributions ex-ante contingent claim markets will not allow an efficient allocation between present and future consumption. This paper considers an alternative competitive market structure under both distributions of capacity and delivery rights. In this market capacity is chosen before the random variable is observed, but purchases of goods and marginal production decisions are made via spot markets after the random variable is observed. The object of this exercise is to see if a normal market might allow an ex-ante optimal choice of capacity under uncertainty if less stringent conditions were imposed than were imposed on the securities market in the previous paper . • The technique used to model this market is expected utility maximization subject to random income constraints since consumers must plan purchases subject to receiving different incomes l.n different states of the world. This is mathematically equivalent to a dynamic programming formulation. It is used in this paper in preference to a standard dynamic programming formulation because it allows a more direct comparison between the results in this paper and those in the previous paper. First, we consider conditions which apply when the capacity constraint is not binding. At a pure trade competitive spot market 45 equilibrium, a necessary and sufficient condition for an ex-ante optimum when consumers have different incomes in different states of the world is for the ratios of individual expected marginal utilities of income across states of the world to be constant across consumers. A. A.1r That is ~ is independent of i, where A. 1S is the multiplier associated with i's income in state s. When production is introduced,a sufficient condition for ex-ante optimality of a competitive spot market equilibrium is for each consumer's expected marginal utility of income in each state of the world to be a constant function of his subjective probabilities (i.e. A. 1S = C.a.. 1 1S Vi, s, where a. _1 s is i's subjective _._ probability). This strong condition, which we imposed on the securities market, appears simply because consumers are planning purchases subject to different incomes in different states of the world. When the capacity constraint is binding, the conditions for ex-ante optimality of the dynamic competitive equilibrium depend upon the structure of rights. If consumers hold title to capacity rights a sufficient condition for ex-ante optimality of a dynamic competitive equilibrium is the same as the sufficient condition when the capacity is not binding: for example, >..is = c1 a is Vi, s · If firms hold title to delivery capacity, the following sufficient conditions are close to being necessary: owing capacity are risk neutral; (1) firms (2) all consumers and firms have 46 the same probability distribution; and (3) Ais = c ais Vi,s. 1 A simple example is used to show that even with similar probability distributions ex-ante optimality can fail. In general, therefore, the results presented in this paper indicate that if firms own capacity, the conditions for ex-ante optimality are more strict in a competitive spot market than in a securities market. REVIEW OF OPTIMALITY CONDITIONS UNDER UNCERTAINTY WITH CAPACITY CONSTRAINTS As de scribed in the previous paper, the ex-ante optimal allocation is identified by solving the following problem: s n Max W + L: i3 . [U. (L. ) . 1 1 1 10 1= L: a. U.(x. 8= 1 1S 1 1S ,y. 18 ,L . )]. 18 Subject to the following constraints: Multipliers n L x. < x(L ) i=l 18 X8 V8 x(L vs xs ) < z s n vs L: y. < y i=l 18 - S y x(L xs s = y(Lys ) "' X x(L,...) ) < x vs X vs 47 Multipliers n L ,_ < X - L l: 1. i=l 1 0 n < l: L. ys - i=l 1s + L xs vs where: W the social welfare function, si the weight given to i's utility function, U. i's neo-classical utility function, x variable supply good produced in state s, 1 s z random supply of x available in s, s ys composite good produced in s, X delivery capacity for x. The optimality conditions reduce to three cases: 1. If z 2. If z 3. If z s < s s > x, then Ass 0 and A2s ::f 0 "' then Ass x, A = 0. 2s x, then ASs ::f 0 and A2s o. These observations allow us to reduce the first order conditions to the following equations: 48 Case 1 (z < x·, A A s 0, A2s 1 0): Ss au. 1 a. 1S ax . 1S au. 1 a. 1r ay. 1r a~s auh a -hr ayhr au. 1 a.1S "oX. - S.a. ax/aL 1 1S auh ahs au. /aL . 1 1S a au.1 -ir ay. 1r Case 2 (z Vh,i,r au./aL. ax.1S ax/aL 1 ayir Vi, r yr Vh, i, r au. ~= + A2s 0): au. 1 a.1S "'oX. 1S au. 1 a -ir ayir 1 XS au/aLir - S.a. 1 1r ()y/aL x·• A5s s 1S 1 1S xs au./aL. 1 ay/aL Vi,r 1r yr au. 1 I a. s >x 1S ~ au. 1 a -ir ay. 1r Vh, i, r 49 au. 1aL. au. 1aL . 0 1 l.S + 1 1 axlaLA l.S axiaL XS X S >X au.1 IClL.1r air Clyi()L au. l.S ax. S >X l.S au. ay.l.r L: a. l_ L: a. A A l_ Vi,r yr where: aw n Cl( L: L. ) i=l lS is the marginal social value of labor in s; aw is the marginal social value of labor in period 0; Azs : : is the marginal social value of z in s. 2 s A DYNAMIC PROGRAMMING MODEL OF EX-ANTE CAPACITY CHOICE AND EX-POST SPOT MARKETS 1. Consumers Own Rights Consider now a competitive economy with n consumers and m + k + g firms. The first m firms produce x in period 0, the next k firms produce x and the last g firms produce y. Consumers hold title to de livery rights and capacity units once capacity has been built. Consumption of i ins is limited by i's resources available in s because marginal transactions are made in spot markets. i's problem is: Consumer 50 Multipliers s Max EU , 1 Subject to: p U . (L. 1 px~ Oxi + 10 ) + L: a . U.(x. s=l 1S 1 1S ,y, ,L, ) 1S 18 S S _ m 0 0 L: q r. < L. q r. + w L O + L: e~Jnl 0 i s=l s 18 - s=l s 1s j=l x x m+k m+k+g "'-' e 1 J '""J s + e 1 J .,.J + p "' x~ . Vs x . + p y. < w L . + q r. + '-'" xs 1s ys 1s- s 1s s 1s j=m+l xs x j=m+k+l ys ys xs 1 11 11 where: e X~j percentage of firm x:1 's profits going to i; eij percentage of firm~ 's profits going to i in s; eij ys percentage of firm~ 's profits going to i in s; w 0 competitive wage in period 0; w s competitive wage in state s; ni profits of firm xj in period 0; XS X nj xs profits of firm xj in state s; ~ys profits of firm yj in state s; competitive price of a unit of capacity in period 0; competitive price of x in state s; competitive price of y in state 8; competitive price of a unit of capacity in state s; 51 xi units of capacity owned by cons umer i 0 qs competitive price of a state s delivery right in period 0; qs competitive price of a state s delivery right; r. state s delivery rights owned by person i; r. initial allocation of state s delivery rights 1S lS to i. Making the assumption of a decomposible profit function firm x:i 's problem is: Multipliers Firm xj's problem is: xj < x(Lj ) xs s - vs >) x(Lj ) < xj XS s vs Aj 2s x(Lj ) < rj XS s vs >.j 3s s. t. ls 52 where: firm x1 's subjective probability of being in s; ~X firm ~'s differentiable utility function over profits; x-is amount of c a pacity rented by j from consumers in state s; number of state s capacity rights rented by j consumers. Firm yj's problem is: s.t. yj < y (L ) s ys The equilibrium conditions are: n E x i=l 1 11 I: r. 1S i=l 11 LX i=l is 11 I: y. i=l 1S m ... j I: X j=l n I: r. 1S i=l m+k I: xj s j=m+l m+k+g . I: YJ j=m+k+l s m+k I: rj s j=m+l vs from 3 53 n L: eij i=l XS 1 n L: e1J i=l ys 1 n L: e~J i=l X l. Forming Lagrange equations L. (i = 1, . . . , n); Li(j = 1 . .• , m); 1 X ' Lj(j =rn+l, . . • , m+k); and Lj(j =rn+k+l, . . . , m+k+· g), differentiat- x y ing and considering only effective constraints which are binding, these equations imply the following marginal conditions. au.1 a is ax.1S a au 1 -ir Clyir qs .\i pxs -s - >. 1 Pyr r .\i 0 Oqs Vi (a) Vi, s, r (b) Vi, s (c) vi, s (d) .\i s 54 Case 1 (z s < x): v j, a, r (e) Vj,a,r (f) . a a.YiaL yr Case 2 (z s x): w· s pxs ax/ aL j XS w r Pyr aylaL Case 3 (z s a yr > x) : wo w --~~~ + --~s~--- ax I aLi ax I dLjXS X w yj,a,r (g) r 'ay I()L a yr Consider first equation (b) without any production equations. Notice that a necessary and sufficient condition for (b) to be ex-ante .A i optimal is for~ to be constant across individuals, where .Ai is i's }.1 s r expected marginal utility of income in states (see footnote 1). This is equivalent to saying that marginal rates of substitution for 55 income in different states of the world are equal across consumers. This result can be summarized as follows: Proposition 1: If consumers earn different incomes in different states of the world, then a necessary and sufficient conditton for a pure trade competitive spot market equilibrium to be ex-ante Pareto optimal is for the ratios of consumer expected marginal utilities of income across states of the world to be constant across consumers. Now compare (b) with the production conditions (e), (f), and (g). Assume now that A;= Ciais Vi,s. This is the same sufficient condition used in the stock market model ip the previous paper and it is equivalent to saying that each consumer's expected marginal utility of income is a constant function of his subjective probability distribution. Substituting (c) for q 8 and (d) for ws, the three cases reduce to: Case 1 (z s < x): au.1 a..1S "'oX. 1S a. Case 2 (z s au.1 -ir ay.1r -a.. 1S au.]_ /aLj.s + . ax/aLJXS -a..1r au./aLi 1 r C.q 1 0 s Vi, a, j , r (h) Vi, a, j , r (i) a ay /aL yr x): "aU. au.1 1aL.1S axis ax I aL~s au.1 au.1 taL.1r ay ir ay/aL a 1 - - = ----;-:--- yr 56 Case 3 (z L: s>x where: > x): s au. 1 a. L: 1S ~ 1S au. 1 a. 1r ayir C. " s >x a. au./aL. 1 1S aui/aLiO ax/aLj XS b dx/aL ,., X 1S Vi,a,b,j,r -a. au.1 taLi r 1r ay /aLayr i's constant multiple of his subjective probabilities. 1 Notice that equations (h)-(j) describe an ex-ante Pareto optimum with welfare weights 8 =_!_ and q i c.1 0 s = t.. s. 2 This result can be sunnnarized as follows: Proposition 2: If the supply of a good is a random variable and delivery capacity has to be chosen before the random variable is observed, then a competitive spot market equilibrium is ex-ante Pareto optimal if the following conditions hold: (1) consumers hold title to all resources; and (2) consumers' expected marginal utilities of income are constant functions of their subjective probabilities. 2. Firms 0\.m Rights Suppose now we consider a somewhat more likely market situation. In this case consumers sell lahor and purchase output in an ex-post spot market but firms delivering water build and hold (j) 57 title to their own capacity ex-ante and hold title to d e livery . h ts. 4 r1g Firms alone participate in a spot market in delivery rights. The maximization problems now become: Multipliers s Consumer: Max U . (L. ) + 10 1 L: a U. (x. , y . , L. ) s=l 1 S 1 1S 1S 1S k + + wS L 1S . + r: e J n 1 0 . l XS XS < M. 1 J= s.t. X X (Lj XS (Lj ) XS ) < xj x(Lj ) < rjs xs xJ < X(L~) X s Firm yj: Max L: aj <Pj ( p yj - w L j ) s=l ys y ys s s ys s.t. y (Lj ) . ys k+g >:: i. e J1T j=k+l ys ys 58 Where: MiO i's income earned in period 0 for purchase of consumption goods. The equilibrium conditions are: n L: 1.0 i=l 1. n k L: L. L: LJ . i=l l.S j=1 XS + k+g . LJ j=k+1 ys L: n L X. i=1 l.S k+g n L: L: y. i=1 l.S yj j=k+1 s k . rJ . 1 s L: J= n I: i=1 n I: a1 j = 1 xs 1 i=1 Forming Lagrange expressions Li (i- 1 • • • • • n) •· LjX (J' = 1, . • , k) ; Lj (j = k + 1, • . . • k +g). differentiating, and considering only y effective constraints which are binding, these equations imply the following marginal conditions. 59 au. a ~ is ax. i ASPXS au. A_i rPyr ~s a. ~r ~ \fi,r (k) va,j,r (l) va,j,r (m) ay. ~r Case l (z s < x): w s + qs ax/ aLj PXS XS w r --= pyr a y/aL a yr Case 2 (z s x): w s ax/ aL j Pxs -- = pyr XS w r ay;aLa yr Case 3 (z s > x): l: aj ,.,. 'j "' xs'~'x pxs s>x----Pyr l: s>x w Va,j,r (n) r Notice that equations (k)-(m) are identical to equations (b), (e) and (f). This indicates that the conclusions of Proposition l generalize to the case such that firms own the state-dependent delivery rights. result can be generalized as follows. This 60 Proposition 3: If the supply of a good is a random variable and delivery capacity is not a binding constraint, then the distribution of delivery rights does not affect the allocation of final output when output is sold in an ex-post spot market. It is clear from equation (n), however, that the distribution of capacity rights does affect the allocation of output for states of the world such that capacity is a binding constraint. In particular, when firms assume the risk of the returns from investment in capacity, then the subjective probabilities and relative risk aversion of firms enter the allocation decision. Notice that if the following three conditions hold, equations (k) and (n) describe an ex-ante Pareto optimum: 1. firms owning capacity are risk neutral; 2. all individuals and firms owning capacity have the same subjective probability distribution; and 3. A. lS Cia. lS Vi, s. To prove this statement rewrite (n) as: cp'j X 2: a ( s>x s ws . ) Clx/3L~s a w r r + 0 ~w ax/3L~ yj , a, r (O) 61 where: a true or agreed upon probability of being in s; and s firm xj's constant marginal utility of profits. This reduces to: Ci. l: l: s >x Ci.sp xs Ci.rpyr s >x s (dx~:Lj ) w + 0 ax/aLl XS X V j, a, r. Ci. w r r (p) ay/()L a yr Now, let ,i s A C. Ci. l s Vi, s and consider all equations '.;rhich have a au. . particular a r 'a y~ and a particular A~pys in the de nominators and 1r au. and some a 1 ---- and some s axis . 1 Aspxs in each numerator. Canceling the C 's in each equation and summing over s > x gives the following result: i l: s >x a spxs Vh,i,r. Ci.rpyr aisaui/aLis Substitution of (q) in (p), Ai au./aL. 1 10 Ai 0 s for w implies: 0 - l: s>x for ws, and _ - l: Ci. s>:X s ( au./aL. ) 1 1S (q) au./aL. 0 - 1 1 ai 1 aLl axJ aL j XS X a au./aL. r 1 1s ay /'aLa yr V h,i,j,a,r. (r) 62 which is the desired result. Proposition 4: This can be summarized as follows: If firms delivering a good subject to random supply are risk neutral, all consumers and firms owning capacity have the same subjective probability distribution over states of the world, and consumer expected marginal utilities o f income are constant functions of their subjective probabilities, then a competitive spot market equilibrium is ex-ante Pareto optimal when firms own capacity and capacity is a binding constraint. The obvious next question to ask is what happens if firms owning capacity are not risk neutral or agents have different probability distributions, or consumers have expected marginal utilities of income which are not constant functions of their subjective probabilities. The following example shows that it is easy to construct such situations which are not ex-ante Pareto optimal. Consider an economy with two consumers, one x firm and one y firm, and two states of the world. Le t E¢ X E¢ y 63 Thus, all agents but one have the same subjective probability distribution and one firm is risk neutral. The optimality conditions will be: Case 1 (z s au 1 5-. axll au 2 .25 - a x21 au 2 . 75 - a y22 au 1 5-· ayl2 Case 2 (z s au 5 - 1. axll s - S2 (.25au 2 /aL 21 ) -S1 ( .5 au/aLll ) + A21 ax/aLxl -6 1 (. 5aU/ aL 12 ) ay ;aL 7 Y- + A21 ax/aLxl -S 2 (.75au 2 /aL 22 ) _2_y__ aLY 2 (s) x): au 2 .25 -dx21 au 2 .75 - a y22 au 1 5-. ayl2 Case 3 (z < x): .5au /aL 1 11 ax/aLxl .5au /aL 12 1 ay/aLY 2 .25au /aL 2 21 ax/aLxl .75au /aL 2 22 ay/aLY 2 > x) --Actual items summed depend upon the size of X: (t) .75<3U /aL 2 22 'dy/<3Ly2 Under competition, the marginal conditions will be: au 2 .25 -'\,2 ox21 /\.1 pxl --.,...au-== = >.. 2 2 .75 -ay_2_2 2 Y p--;- Case 1 (z s < x): pxl wl + ql 'dx/<3Lxl --= py2 Case 2 (z s (w) w2 <3y/3Ly 2 (x) x): wl Pxl = dX/dLxl py2 w2 <3y/<3Ly2 (y) 65 Case 3 (z s > x): (z) Clearly, in order for (v) and (w) to satisfy (s) it is necessary to have: This will be true iff: However, even assuming this special form of the expected marginal utilities of income, (v), (w), and (z) would only satisfy (u) in general if A~ = Ai · 25 -l/ 2 and Al = A2 1T xl 2 2 -1/2 .251Tx • 2 Even if firm x were risk neutral (u) would still not in general be satisfied without sufficiently similar probabilities. Thus, while necessary conditions may be difficult to find, the sufficient conditions outlined in Proposition 4 are close to being necessary when firms hold capacity rights. This seemingly bizarre result occurs because consumers with differing probability distributions view the probability of having 66 either unused or insufficient capacity differently . present a problem when co nsumers the~~elves This does not assume the risk of investing in capacity becaus e the market in capacity rights allows consumers with d i ffering probabilities to exhaust the gains from trade created by the diff e ring probabilities. Neither does it cause difficulties in a full ex-ante contingent claims market because firms a s s ume no r is k in such a market. Hm.;~ever, if firms do assume risks in provid i ng capacity these gains from trade among consumers are not exhausted. In a sense, a free rider problem develops if f irms assume the risks because consumers who place a low probabi 1 ity on the event that there will be insufficient cap acity wi s h to pay less for x relative to y on the a ver age when there is n o shortage. If firms are risk neutral and consumers have identical probability distributions such gains from trade d o not exist to begin with. CONCLUSION This paper has considered the effe c t on resource allocation under uncertainty of moving from a complete contingent claim market to a market in which capacity is built ex-ante but marginal production and consumption decisions are made ex-post. The result that th e ex-ante efficiency of the competitive spot equilibrium under uncertainty depends upon the distribution of ex-ante capacity rights provides insight into the real world welfare effects of vario us production rigidities which firms typically impose upon themselves 67 either because of technological rigidities or long lead times for construction, or to reduce output or input price or quantity demanded uncertainty. Future research may extend the analysis in this paper to a more general examination of the welfare effects of supply rigidities. 68 FOOTNOTES 1. R. M. Starr, "Optimal Production and Allocation Under Uncertainty," Journal of Economics 87 (1973). ~arterly 2. For a proof of this statement, see footnote 7 of the previous paper. 3. While the assumption that a firm possesses a utility function is controversial, the use of a firm utility function makes explicit the possibility that firm managers may have attitudes towards risks which affect the firm's decisions. 4. The results follow whether firms delivering x produce capacity or a separate set of firms produce capacity and rent it to x firms. 69 CHAPTER V OPTIMAL CAPACITY CHOICE AND INVENTORY POLICY FOR A STORABLE GOOD SUBJECT TO RANDOM SUPPLY INTRODUCTION The previous two papers have considered ex-ante optimality conditions under uncertainty when capacity alone has to be chosen before the random variable is observed. The results were that ex-ante contingent claims markets led to ex-ante optimal allocations but stock markets in securities alone required strict conditions to achieve optimality. Sufficient conditions were for consumers to have identical subjective probability distributions and constant marginal utilities of income such that expected marginal utilities of income were constant functions of those subjective probabilities. If spot markets in labor and final output were allowed after the random variable was observed, optimality conditions depended on the structure of rights. If consumers owned capacity rights optimality could be achieved if expected marginal utilities of income were constant functions of consumer subjective probabilities. If firms owned capacity sufficient conditions included firm risk neutrality, equal subjective probability distributions of all agents and the above constant function expected marginal utilities of income. 70 A natural extension of the models developed in the previous papers is to a storable good subject to random supply. In this case, the optimal size of the storage facility and an optimal inventory policy have to be determined in addition to the optimal delivery capacity. Water is perhaps the best example of such a good since the storage facilities in the form of dams and reservoirs represent large, long-lived capital investments. The welfare model developed in this paper is an extension of the model developed in the previous paper. In this case the delivery and storage capacities are built before any consumption takes place and consumption plans are made contingent on the observation of the random variable subject to supply constraints. Then, for every observation of the random variable in period 1 there is an ex-ante optimal storage and an optimal portfolio of consumption plans contingent on the observation of the random variable in period 2. The optimality conditions involve eight cases which describe the range of combinations of binding constraints and are too complicated to enumerate here. However, in general they involve the same basic rules developed in the previous papers. Whenever the delivery or capacity constraints are binding, expected marginal rates of substitution equal expected marginal resource values. In addition, whenever there is stored water which does not go to waste in the next period, the discounted expected marginal rate of substitution equals the discounted expected marginal resource value. 71 The competitive model used in this paper is an extension of the dynamic programming model developed in the previous paper. Delivery and storage capacity are built before any consumption takes place and then capacity rights are either sold to consumers or held by firms. Marginal production, consumption and storage decisions for period 1 are made after the random variable is observed. Then, given an amount of storage from period 1, marginal production and consumption decisions for period 2 are made after the second observation of the random variable. The results on the effects of the distribution of delivery capacity rights generalize to the case of a storable good subject to capacity constraints. In this case, if consumers own delivery capacity, storage capacity, and storage rights then a competitive spot market equilibrium will be ex-ante Pareto optimal if consumers have expected marginal uti.lities of income which are discounted constant functions of their conditional subjective probabilities. If firms own any of these risky assets, however, additional sufficient conditions include identical subjective distributions and firm risk neutrality. Further, nonidentical discount factors among consumers and firms cause difficulties when firms assume risks. 72 A TWO PERIOD DYNAMIC PROGRAMMING MODEL WITH CAPACITY CONSTRAINTS AND A STORAGE FACILITY Let X~s be i's consumption of x in period l, states 2 and x. be i's consumption of x in period 2, stater, givens 1rs 2 was observed in period 1. and L are labor devoted to the xrs production of x in periods 1 and 2, states s and r, given s. x(L 1 2 ) and x(L ) are delivered x in periods 1 and 2, states s xs xrs and r, given s. n 1 E xi i=l < x(L s - 1 xs n Vs and ) E x: x(L < i=l 1rs - 2 ) xrs Vr, s 1 2 and z are the random supplies of x available in periods 1 and s r 1 2 2, states s and r. z and z are independently and identically z s r distributed according to i's discrete subjective probabilities a.. 1S and a.. • 1r z 1 s can either be consumed as x(L 1 xs 1 ) or stored as X in s the storage facility F, subject to storage losses of 8e: (0, 1) per unit of storage. capacity Delivery of x(L 1 xs x, such that x (L 1xs ) -< x ) and x(L Vs and 2 xrs x (L ) are through delivery 2 xrs ) < - x vr, s. F is produced according to production function F(LF) in period 0, x is produced where LF is labor used in the production of F. according to production function x(L A) in period 0, where L~ is X labor used in the production of x. X LF +LA < X - n E L. , where LiO i=l 10 is the labor supplied by i in period 0. Now, let y~s and y~rs be i's consumption of a composite good y in periods 1 and 2, states s and r, givens. 1 2 ys and y rs are the amounts of y produced in periods 1 and 2, states s and r, 73 n 1 L: y. given s. i=l 1S and L 2 yrs 1 < ys Vs and n 2 2 L: Y • < Yrs . 1rs Yr, s. 1= 1 are labor devoted to the production of yin period 1 and 2, states s and r, givens. y(L 2 ys ) and y(L 2 yrs ) are the production functions for y in periods 1 and 2, states s and r, L~1S and L 2i rs are labor supplied by given s. and 2 and states s and r, given s. and L n 2 2 + L2 < L: L. yrs xrs i=l 1rs Ll XS in periods 1 i n 1 + I.lys S: i=l L: L. 1S vs Vi, s. Now assume that no consumption takes place in period 0. Consumer i supplies labor to the production of F and to differentiable utility function Ui(Li ). 0 x according In periods 1 and 2, states s and r, given s, consumer i has strictly quasi-concave aui Lim 2 x? +0 a xi 1rs rs aui Lim xl +0 ax.1 1S is and Lim LiO+ H aui ar:-iO aui Lim 1 Ll:n a1 is aui Lim - 1 y~+O ayis 1S aui Lim 2 L? + H aL. 1rs 1rs aui Lim 2 2 +0 ayirs Yirs - oo +co Vi, r, s , where H is the maximum time available for work in all periods. i . au L 1m - - > 0 xl + oo ax~ is Vi, s; Lim x2 + 00 irs aui > 0 a 2 xirs Vi, r, s; Lim yl -TOO is Vi, r, s Vi, s; 1s 74 Lim 2 -r co yirs and > 0 Lim L2 -r 0 ClL irs Vi, r, s; < 2 0 Vi; Lim L~ -rO 1S 0 vi, s; Vi, r, s. irs 1 2 1 2 Now, since xis' xirs' yis' and yirs are always consumed in positive quantities by all consumers, we can express an ex-ante Pareto 1 2 1 2 optimal allocation of xs, xrs' ys' and yrs by maximizing a weighted sum of individual utility functions. w n i L f3i {U (LiO) • 1-=l + Let S i 1 1 1 L ai [U (x. ,y. ,L.) s=l S 1S 1S 1S + R i 2 2 2 o.1 r=l La. U (xi ,yi ,Li )}, 1r rs rs rs where: ai welfare weight assigned to i, and o. i's discount rate. }_ The welfare problem is to maximize W subject to the following constraints: Multipliers F x(L ~ ) X 75 Multipliers n x~ l: < x(1 l i=l l.S n l: . ].= 1 vs ) XS 2 2 x. < x(1 ) 1.rs xrs Vr, s vs x (1 x(1 2 ) < x1s ( 1 - e ) + z r2 vs, r ) < x vs x vs, r xrs 1 xs x(L 2 xrs ) < vs n 1 < y(L L: y. i=l l.S n 2 L: y. i=l 1 ys < y(L 1.rs - vs ) 2 yrs ) vr, s n LF + 1 ~ < X - + 11 l: 1 . i=l l. 0 < n L: Ll . is Ys - ].= 1 vs 76 Multipliers n L2 + L2 < xrs yrs - Vr, s 1: L: i=l 1rs The first order conditions are: aui s 1. - - + y3 Vi 0 aLiO aui S .a. - - + A.l 1 1S 6s aL: 1S S.<\a· 1 1sa.1r aui aL: 1rs vi, s 0 + A.26rs aui S1.a.1S - - - A.lls 1 ax. 1S Vi, r, s 0 0 Vi, s A.2 = 0 lrs Vi, r, s aui S 1.a.1S - 1 - - A.l Ss ayis 0 Vi, s aui S .6 .a. a. 1 1 1s 1r a 2 Yirs A.2 Srs S .6 .a. a. 1 1 1s 1r ~(A. 1 aLl XS ls aui 2 ax irs _ A. 1 _ A. 1 ) 2s 3s A.l 6s Vi, r, s 0 0 Vs A.2 6rs 77 0 -Al + 2s ox(L2 ) xrs - y 1 Vr, s 2 A (1-8)- A s 4 xl(l- e) + z2 2rs s r E = + 0 vs 0 0 -y 2 E + 0 S3X(LlXS )= X 0 Al ~ 5s 8L 1 ys ay A2 5rs 8L 2 yrs Al 6s 0 A2 = 0 6rs vs Vr, s Now if we consider only effective constraints which are binding, the se first order conditions reduce to nine cases: 78 1 => A 2s aui -e.a. 1 1S Clx/aL Cly/ClLl Y9 6 rq ] 2 Cly/aL yr9 o1.a.1sa.1r aui 2 ax irs [either a. - - - o r o a a 1p a 1 i ip 1 9 Yip 2 2rs =A 2 3rs = 0): 2 yr9 Vi, 9, r A2 and aui =A xs 1 xs [either ----=-6~9-:-- or 3s 1 Cly/aL A1 1 o1.a.19a.1r [either a. - -- or 19 Cly.1 19 Clx/aL =A au 1 2 ayi 9p ] (1) 79 -s.1 oia. a.1.r ~s ax/aL 1 xrs aui/aL1 ip or -S . o. a. a . 1 ay/aL 1 1 1p 1q ay/aL2 yp yqp [either -S .ai 1. p ax/ aL 2 xrs :\~p Vi, p, q 2 :\ 6 [either --~- or 9P ] 1 2 ay/aL ay/aL yp yqp 2 1 Case 2 ( z 1 -X1 =X and x (1- 8) + z > s r s s x or 2 1 and x (1- e) + z > s r 2rs A x => :\12s :\1 3s = :\2 z 1 s xl < x, x1 < F, s 0): (1) and o.1 [ 2 a. a. aui ) = X 1.S 1.r ax.2 xrs 1rs aui (either a.1p ---1- or oia.1.p a.1.q ay. s, nx (L l.p -s.o. 1. 1 [2 ) =x s, r3x ( L xrs A a. a. 1s 1r aui /aL~ [either - S.a. or - S.oiai a. 1 1 1P ay/aL P 1.q yp {P ay/aL 2 yqp s (2) 80 ;\2 2 s, ox(L ) = xrs x 6rs 2 ax/3L xrs ---=~- ax/aLA x Al Vi, q, p ;\2 69 P 6 (3) or ] P 2 1 ay I aL 'dy/aL yp yqp [either Case 3 (z Y3 + ---=--- 1 2 1 1 - x > x and X ( 1 - e) + z = x => s s s r A 0): (2) and 1 au) =x i .s "oX.1 a L: S3X (L l A XS 18 = aui [either a. - - or 1p a 1 yip aui /aL ~ -f3.1 L: 1 S3X(L A XS ais ) =X aui/aL ---=1-=..S _ 1 ax/aL xs f3 . 1 10 ax/ aLA X aui /aL: 1 [either -f3.a. 1 1p p or -f3 o.a. a. 1 ay/aL 1 yp Al s 3 x(L 1 ) =x XS ax/aL J.l [either 6E ay/aL~P ay/aL 2 yqp 1 XS + ax/aLA X Vi, p, q ;\2 or 1p 19 y3 6s L: 1 6g,E ay/aL~qp ] (4) 81 1 2 A =A =0): 2s 2sr [ aui [ aui a - - + oi 2 a a S JX ( L1 ) =X~ is "oX,1 s,r3X ( L )=X~ is ir ax.2 xs 1s xrs 1rs 1 aui au [either a. - -- or 2 ] 1p a 1 ayiqp yip 2 1 aui/aLiO aui/aL au /aL ~ -----=i~s [ i irs -Si "~/"L - ei 1'-' ai 1 - S.o. a. a. 2 oX 0 ~ (L ) s ax/ aL 1 1 (L2 ) ~ 1S 1r a I aL x S3X xs =x xs s,r3x xrs =x x xrs ----------------~----~-~-==---------~~--~-------~~= 2 aui/aL 1 aui/aL [either -Sia. ----~iLp S o iqp] 1P ay/aL 1 or - i iaipaiq ay/aL 2 yp yqp A y3 ax/ aL~ X [ + [ + 2 1 1 ) = s , r 3X (L ) =X ax/aL XS xrs XS A1 A2 6E 6gE ] [either or 2 1 ay/aL ay/aL yqp yp S3X(L Case 5 (z x A 6rs 2 ax/ aL xrs Vi, p, q. 1 1 ~ 1 1 2 1 2 1 -X <x, X <F, and X (1-8)+z <x => A =>.. =>.. =0): s s s s r 3s 4s 3 rs aui ais --1-ax. 18 A1 6s A oi (1 - 8) i 2 r3x(L xrs au 2 a is (1 ir 2- L1 )=X (1-8)+z s r aui (either a. ---- or o.a. ai 1p a 1 1 1p q yip aui a2 Yiqp axirs ] (5) 82 aui/aL: 8.6.(1-8) -S.ai 1 s 1 1 nx(L 2 xrs E 1 a. a. 2 18 )=x (1- 8)+z u s HS ax/aL r 2 xrs aui/aL: 1qp] or -~'iuia. a. 1 2 1P 1q ay/aL ay/aL yp yqp D ~ Al A2 6rs _ _6_s-=-- _ (l _ 8 ) E 1 2 2 2 ax/aL r3x(L )=~(l-8)+z ax/aL xs xrs s r xrs Vi, p, q. 1 (6) 2 => A =A = 0): 3s 3rs E 1 S3X = F s [ a. 1S ·~i - 0.1 (1- 8) E 1 2 2 nx(L ) =X (1- 8) + z axis xrs s r 1 aui a. or o.a. a. 2 au [either 1p a 1 1 1p 1q a yip Yiqp a. 1r ·~ ] ax. 1rs J t aui/aL~1p either -S.a. ----=_._or -S.o.a. a. 1 1p ay/aL 1 1 1 1p 1q yp 2 aui/aL iqp ] ay/aL 2 yqp = 83 A16s A26rs ] E _ (1 _ e ) E 1 1 2 2 s 3X = F Cl x/ Cl L r 3x(L )= Clx/ClL [ s xs xrs xrs x1s (1 -8 ) + z r2 vi, p, q (7) [either Case 7 (z 1 -X1 = x X1 = F X1( 1 - 8 ) + z 2 < x => A1 = A2 = A1 = 0) : 3s 3rs 2s s s ' s ' s r A A (1) and <5 • (1 1 8) a. E 18 s 3 Xls = F aui [eith er a . 1p 1 ayip E 2 air n x (L ) = xrs x1 (1- 8)+z 2 s r or <S. a. a i 1 1p aui 2 axi rs aui -2---] q a yiqp = [either - 13 .a. 1 <1 -e) 1P E E 1 2 1 2 (s3 X =F)(r3X(L )=X (1-8)+z) s xrs s r A1 A2 6 qp ] 6 [either P or 2 1 ay/aL ay/aL yp yqp Clx/aL 2 xr Vl., p, q (8) 84 1 1 ~ 1 Case 8 (z s - Xs > x, Xs = F , 2 x1s ( 1- e ) + z r2 < x => A12s = :>- 3rs = 0): A (4) and (8). Case 9 (z 1 2 x, x1s = F and x1s (1- 8) + z r2 > x => A3s =A = 0): 2rs 1 1 - x < s s (3) and 1 S3X s =F aui a·1 [either a. - -- or 1p a 1 yip L: 1 S3X = F s 1 1S P = o1. a.1pa.1q aui/aL: aui/aL: P or -S . o .a. a. 1 1 1p 1q ay/aL 1 ay/aL 2 1 [either -S.a . 1 1 s ax. yp 19.£] yqp vi, p, q (9) 85 A COMPETITIVE MARKET 1. Consumers Own Rights Now let us consider a competitive market in which consumers hold title to delivery capacity, storage capacity, and storage rights. F1. rms xAa , a = 1 , • • • , A produce delivery capacity and sell capacity rights to conswners. Firms Fb, b = 1, ••• , B produce storage capacity and sell rights to consumers. Consumers own delivery rights which they can use either to rent to firms or to store water in the storage facility. Each consumer can store as many units of water as the number of storage units he or she owns. All consumers' assets are liquidated during period 2. Conswner i's problem is: i Max U (L1.0) + S R i 1 1 1 + O· ~~ a1.r Ui( xirs'yirs' 2 2 L2 )] a. [U (x . ,yi ,L. ) 1·rs s=l 1s 1s s 1s 1 r=l l: subject to: Multipliers OA 0 0 PxAx1. + pFF1. + ! ~~ q 0 ri0 < woL1·o + ~~ qs0 ris + ~ 6 ia a B ib b A n A + l: SF rrF. s=l s 8 s=l a=l x x b=l w 1L1• S 1S + cl xi + A A XS ~ q 1 r ? + c1qs ( r 1ss ~ - X~1s ) + r=l rs 1r ,o. 1 A 86 Multipliers 1 Fo + PFs i : 8 ij 7rlj + . 1 xs xs £. m+k L: . i. 1. vs 8 J,r J. +l ys ys J=m J= 2 1 r: ) + q rs (X i s (1 - 8) + 1rs m+k + + .. 2. 81Jn J • j=m+l yr yrs L: X:1s-< Flis vr, s vs Where: selling price for a unit of delivery capacity in period 0, and period 2, stater, given s in period 1; units of delivery capacity owned by i; price of a unit of storage in period 0 and period 1, state s; 0 1 1 1S F. ,F. units of storage capacity owned by i in period 0 and period 1, state s; selling price for a state r delivery right in period 0, period 1, given s in period 1 and in period 2, given s in period 1; 87 state s delivery right purchased by i in period 0, and state r delivery right purchased in period 1, given s in period 1; r. l.S i's initial endowment of states delivery rights; 8~a X percentage Of firm Xa IS a 1T A firm 8~b percentage of firm Fb's profits going to i; X b profits; b' 1TF = firm F 1 2 wO,ws,wrs profitS going tO i; s profits; wage rate in period 0, period 1, state s and period 2, state r, given s; 2 1 Pxs'Pxrs price of X in period 1, state s and period 2 state r, given s; 2 1 pys'Pyrs price of y in period 1, state s and period 2, state r, given s; 88 1 = rental rate for a unit of capacity in period 1, xs state s; 1 c~ c 1 qs rental rate for a state s delivery right in period 1; amount stored by i in period 1 and state s; • I percentage of x 3 s profits going to i in state s; 1T 1"J 1T 2"J xs' xrs profits of firm xj in period 1, state s and period 2, state r, given s; percentage of yj's profits going to i in states; 1" 2" 1TJ1TJ ys' yrs profits o f firm yj in period 1, state s and period 2, state r, given s. ~ a' Firm x s problem is: Multipliers Max s.t. xCL~> X where: ~a x L~ X ~a output of firm x ; labor hired by firm xa. 89 Multipliers Firm F b' s problem is: s.t. where: Fb output of firm Fb Lb labor hired by firm Fb F Firm xj's problem is: Max s . L: aJ s=l XS . 1 X XS S [ cj>J (p 1" 1 1" x J - wi. J - c S XS 1 1" 1 1" r J - cA x J) + qs S XS S subject to: Multipliers x 1"J < x(L 1"J) X S XS 2" 2" J < x(L J ) rs xrs vs V r, s >.. lj ls 90 Multipliers x(L lj) < r 1"J XS s vs 2. 2 x(L j ) < r J xrs rs V s, r Alj x(L lj) < X xs s vs A2j 2 x(L j ) < X xrs rs v r, s where: aj XS ~j X xj's subjective probability of being ins; firm xj's differentiable utility function over profits; X 1" 2. J X J s ' rs output of xj in period 1, state s and period 2, state r, given s; L lj L 2j xs' xrs labor hired by xj in period 1, state s and period 2, state r, given s; 1" 2. s ' rs r J r J state s delivery rights rented by xj in period 1 and state r rights purcha s e d in period 2, given s in period 1; capacity rights rented by xj in period 1, state s and purchased in period 2, state r, given s. A. lj 2s 91 Firm yj's problem is: Max s j . 1 1" 1" a. [ <j>J (p y J - w L J) + s=l ys y ys s s ys L subject to: Multipliers y 1"J < y(L 1"J) s ys vs 2j 2. yrs ~ y(L yrs J ) vs, r where: a. j ys firm yj's subjective probability of being in s; <j>j y firm yj's differentiable utility function over profits; lj 2j ys ,yrs output of firm yj in period 1, state s and period 2, state r, given s; labor hired by yj in period 1, state s and period 2, state r, given s. A.lj ys 92 The equilibrium conditions are: n L X1 i=l is n m 1" 2 L X J' L X . 1 s i=l irs J= n m+k m 2 y 2j , l: y i=l irs j=m+l s 1 i=l is l: A X a=l m l: + XS (10) l: vr, s (11) n B l: Lb (12) L: L.o i=l l. b=l F Llj + j=l 2. y J j=m+l rs vr, s m+k :E y l: L~ m 2. l: X J . rs J= 1 m m+k 1 L2j L2j + :E l: l: Li , . xrs . +l yrs i=l s J= 1 J=m m+k n Llj .J=m+1 ys l: V r, s n l: A ~a ~ x. a=l m ~lj n l: X j=l n l: l: s i=l F~ = i=l l. i=l is (13) xi, Fb m ~ 2j l: X j=l rs n ~ l: X. i=l l. V r, s (15) vs (16) vs (17) b=l n n :E F B l: L: i=l J.rs (14) l: X l. i=l n l: = :E F. i=l l. 93 n ia 1 va (18) 1 Vb (19) 1 vs, j 1, ••• , m (20) e ij L: i=1 ys 1 vs, j m + 1, ••. , m + k (21) n - n i=1 1.r i=1 L: e ~ i =1 X n ib L: e F i=1 n e ij L: i=1 xs n L: r. m 1. L: r J = s j=1 m 2j L: r r s j=1 n 0 :tr L: r. n L: i= 1 1 1rs L: r. i=1 1 1 (r. -xi ) l.SS n 1 i=1 1s L: ( x. vr, s (22) Vs (23) Vs, r (24) S 1 (1 - e) + r i ] rs a Forming Lagrange functions L . (i = 1, ••• , n), LA(a= 1, ••• , A), l. X L ~ (b = 1, . . . , B) , L ~ (j = 1, . . . , m) , and L; (j = m + 1, . . . , m + k) , the first order conditions are: 94 1 1 0 - AispFs + yis ClUi - - + ~.oiw o 0 ClLiO Vi, s (25) Vi (26) Vi, s (27) Vi, r, s (28) vi, s (29) Vi, r, s (30) vi, s (31) Vi, r, s (32) Vi (33) Vi (34) 1 a. lS ~u + ,1 w 1 ClL 1 is 1\. lS s 0 aui 2 2 a - - + ),, w 1rs rs i is ir aL2 irs 0 6 a a ClUi - - - >. 1 is ax~ 1 ispxs 0 lS 0 o.a. a. 1 1s 1r 0 6 .a. a. 1 1s 1r 0 0 0 ->..p ~ l X + 95 0 0 -A 1 1 1 2 2 (q - c ) + A q is ss qs iss ss 1 1 - A·s c qs + (1- 8) 1 + A~ -w 0 X 0 2 2 A1·rsqrs - Yis = 0 2~ r 3q > 0 rs ax = 0 a L~ Vi, s (35) Vi, s i r (36) Vi, s (37) vi, (38) s Va (39) va (40) Vb (41) Vb (42) X b ap w +A - - = 0 - 0 F a Lb F • • I aJ q,J p XS X 1 0 XS 2. AJ 1rs • • I - a J q,J w 1 XS X S • • + 0 0 • I 2 - o.J JxsaJxr q,Jx wrs + ax aL 2 j vs, j =1, • • . ,m (43) vr, s, j 1, . . . , m (44) v s, j 1, . . . , m (45) 1, . . . , m (46) (A2j - A2j - A2j ) · 1rs 2rs 3rs 0 xrs Vr, s, j 96 . .' 1 + Alj c qs 2s -a] <fJJ XS X vs; j = 1, ••• , m 0 Vr , s ; j 0 vs; j 0 . . .' 2 2. + AJ xrs 3rs -o aJ aJ <fJJ pA 0 j xs xr x 0 • • • I o .aJ aJ p <jl.l J ys yr y 2 yr s 2• -A J yrs 0 a =0 -aJ· <fJJ·' w1 + A.1·J ~ ys y s . . ys aLlj ys .' 2 -o.aJ aJ <fJJ w J ys yr y rs ay aL 2 j yrs 0 (47) =1 , . . . , m = 1, (48) ... , m (49) vr , s ; j = 1 , . . . , m (SO) Vs, j =m+l, .•• , m+k (51) Vr, s, j = m + 1, •.• , m + k (52) vs,j=m+l, . . . ,m+k (53) Vr , s , j = m + 1 , . . • , m + k (54) Now if we consider only cases such that prices are positive 1 and we let A. l.S = C.a. 1. l.S 2 Vi, sand A. = o.C.a. a. Vi, r, s, these 1.rs 1. 1. 1.s 1.r first order conditions reduce to nine cases, which are identical to equations (1)-(9), given the equilibrium conditions (10)-(24). This result can be summarized as follows. Proposition 1: If consumers hold title to delivery, capacity, storage capacity, and storage rights and each consumer's discounted expected marginal utility of income in each state of the world is a 97 constant function of his conditional subjective probabilities then a competitive spot market equilibrium is ex-ante Pareto optimal when storage capacity and delivery capacity constraints are binding. 2. Producers Own Rights If producers own capacity, storage, and storage facility rights instead of consumers the competitive market can be described as follows: Consumer i: s i Max U (LiO) + R ~ ~ ~ u.ctir 1 r=l i 1 1 1 2 L2 E cti [U (x. ,y. ,L. ) + s=l s 1s 1s 1s Ui( 2 xirs'yirs' irs )) • Multipliers s. t. 1 1 1 1 + w 1L 1 Pxsxis + Pysy~s ~ MiO s is + .L m i" 1" m+k j · 1" E 8 J1T J + E 8 -J .rr J xs xs ys ys j=1 j=m+1 98 · Firm xJ: · · Max - w (Lx~ +LJF) - 0 s q o(r o·J- r]) · + sL: a.J· [cpJ· (p 1 x1·J L: s=l S S S s=l XS X XS S Multipliers s.t. 1'J 1' Vs Alj ls 2. 2'J < x(L J ) rs xrs vr, s A2j lrs x(Llj) < XAj xs - vs Alj 2s x(L 2 j ) < XAj xrs - Vr, s A2j 2rs x(L 1 j) < r 1'J - xlj s xs - ss vs Alj 3s x(L 2 j ) < x 1 j ( 1 - e) + r 2 j rs xrs - s vr, s A2j 3rs xlj < Fj s vs Alj 4s X X s < x(L J) xs < x<L1) X j yl Fj < F(Lj) F yj 2 ~j X 99 where: rjs j's initial endowment of states delivery rights; state r delivery rights purchased by j in period 0, period 1, given s in period 1, and period 2, given s in period 1; xj delivery capacity owned by j; Fj storage capacity owned by j; L1 labor hired by j to build delivery capacity; L~ labor hired by j to build storage capacity; X xs1 j =amount of x stored by j, givens in period 1. Max S . . 1 1· 11· s=l ys y ys s s ys 1: a.J [<j>J(p y J -WL J) + Multipliers s.t. 1. 1. s ys 2j 2. y J < y(L J) Yrs :: y(L yrs J ) A2j 5rs 100 The equilibrium conditions are: n E LiO i=l n E L~ i=l l.S n E L i=l n L X 2 irs 1 i=l is ~ L 1j + m+k J= j=m+l ys . 1 xs ~ L2j + i= 1 xrs m 1. L J L: vs m+k 2. L: L J vr, s j=m+l yrs 1. vs E X J s j=l L X 2 i=l irs m 2. E X J rs i=l vr, s n 1 E y. i=l l.S m+k 1. E y J j=m+l s vs m 2 E y. i=l l.rs m+k j=m+l m m n 1. E r J . rs J= 1 L: 2. J j=l rs L: r 2. y J rs yr, s m L: r o·J Forming Lagrange equations Li (i = 1, . • . , n), L; vr, s j=l r L~(j = 1, . • • , m); and (j = m + 1, • . . , m + K) and differentiating, the first order conditions are: 101 aui 0 - - + \wo 0 aui 1..1 w1 is s aLiO a is - - + aL: 1S 0 aui + 1..2irsw2rs 2 1 1S 1r aL 1rs o.a. a. aui ,_1 1 1S - 1 - - ispxs ax. 1S 0 a. aui 2 1 1s 1r ax. 1rs ,_2 2 irspxrs o .a. a. aui ;>...1 1 l.S - 1 - - ispys ay. l.S aui 1..2 2 irspyrs 1 1s 1r a 2 Yirs 0 0 a. o .a. a. 0 ax -wo + yj1 - - = 0 ad 0 Vi (52) Vi, s (53) vi, r, s (54) vi, s (55) Vi, r, s (56) Vi, s (57) Vi, r, s (58) Vj = 1, . • . , m (59) Vj=1, . . • ,m (60) vs; j = 1, . . . , m (61) X -wo + yj2 aF n) 0 F • • I 1 - ,_1j xs X PXS ls a J <PJ 0 102 0 - a j cp j lw1 + ~ xs X s aL1j xs p . 1j- >..1j- >..1j) 1s 2s 0 3s vr,s;j=1, • • . ,m (62) vs;j=1, ••• ,m (63) - 6 a j a j w2 + ax ( >. 2j ->..2j ->..2j ) =0 1rs 2rs 3rs j xs xr rs aL2j xrs -q -a -a p j j I 1 L a xp cpx qsp 0 + s j xs j xs 0 vr' s; j = 1, ... ' m (64) vs;j=1, • • . ,m (65) vrls; j=1, ••• ,m (66) vs; j=1, ••• ,m (67) Vr,s;j=1, ••• ,m (68) p=1 [ cp [ ,., j I 1 x q rs j 1 1 ~x q ss • •I 2 - 6 aJ ,~, J q ] j xr~x rs . •I 0 2 1. 6 aJ ,~,J q ] + ' J j xs~x ss A3s - 0 0 1· L >.. S3X(L 1 j)=xj xs J + L Zs r,S3X(L2j )=xj xrs 2· >.. J Zrs 0 (69) j=l, ••• ,m 1. 3s L ->.. J + 1. 0 2. (70) r,s 3x(L2j ) =X J(l-e) +r J xrs s rs vs;j=1, ••• ,m 0 j = 1, . . . , m (71) 103 • • I 1 aJ <j>J p ys y ys 0 • • j 2 oj aJys aJ <j> p yr y yrs I 0 a -aJ· <j>J·' w1 + Alj ~ ys y s 5s <lLlj ys + A2j 0 ay 5rs <lL2 yrs 0 vs; j =m+l, ••• , m+k (72) vs;j=m+l, ••. ,m+k (73) vs;j=m+l, •.• ,m+k (74) vr, s; j =m+l, .•• , m+k (75) If we compare these equations with equations (25)-(51) which we know describe an ex-ante Pareto optimum, it is clear that the same difficulties arise in this problem as arise in the problem in the previous paper. In particular, equations (69)-(71), which describe a firm's capacity choice and inventory policy, ensure that when equations (61) and (62) are evaluated firms owning capacity and inventory will evaluate expected prices and returns for states of the world such that capacity and storage constraints are binding on the basis of their own discount rates, utility functions, and subjective probability distributions. An optimal allocation uses consumer discounted expected utilities to evaluate the returns from capacity choices and inventory policies. On the other hand, if firms are risk neutral, consumers and firms owning capacity have the same discount rates and subjective p-obability distributions, A~s = Ciais Vi, s and A~rs = oiCiaisair 104 Vi, r, s (i.e. expected marginal utilities of income are discounted constant functions of consumer conditional subjective probabilities), firms and consumers will agree about the expected utility of different To prove this statement let A.= C , 1. i capacity and inventory policies. A~l.S = oc .a a Vi, s, A2 irs 1. s r C. a 1. s aj a. a XS l.S s • I Vi, r, s, <j>J = c., j = 1, ..• , m; 0. = 0 Vi, j = 1, ••• ' m. X Vi,s;j=l, • • • ' m; and 0. 1. J J The first order conditions (52)-(68) which have consumer and x firm probability distributions, now can be rewritten as follows. aui - - + CiwO = 0 aLiO a aui s aL <Sa a s r a 1 w -+ C.a 1 1. s s 0 Vi (76) Vi, s (77) Vi, r, s (78) Vi, s (79) Vi, r, s (80) Vi, s (81) is aui aL: 1.rs aui 1 s ax. 2 + oCia s a r wrs - - - C .a p 1 1. s xs 0 0 l.S <Sa a s r a aui 2 ax. 1.rs 2 oC.a a p 1. s r xrs aui 1 C.a p s - 1 - - 1. s ys ay. l.S 0 0 105 2 - oC i a s a r p yrs oa a s r -w o 0 Vi, r, s + yJ· -ax- = 0 ad 1 (82) j=1, ••• ,m (83) j=1, • • . ,m (84) X -w aF 0 + yj2 - - = 0 aLj F 1 aspxs A1j 1s 0 Vs; j 2 oa a p s r xrs 0 = 1, . . . , m Vr, s; j =1, ••• , m 0 -oa a w 2 + ~ (A 2 j - A2 j - A2 j ) aL2j 1rs 2rs 3rs s r rs xrs 0 -qs + p L: a q 1 0 p=1 p sp 1 2 -a [~ - oa q ] s s r rs 0 1 2 -a [q - oa q ] + A1j s ss s ss 3s 2 A2j -oa a q s r rs + 3rs 0 0 0 (85) (86) Vs;j=1, ••• ,m (87) vr,s;j=1, ••• ,m (88) vs (89) vr 1 s (90) Vs; j=1, ••• ,m (91) Vr, s; j =1, . • • , m (92) 106 Equations (76)-(92) together with equations (69)-(75), reduce to conditions which are equivalent to the ex-ante Pareto optimal conditions. Proposition 2: This result can be summarized as follows. If delivery and storage capacity have to be chosen before the random variable is observed, the planned life of those facilities exceeds one year, and firms hold title to delivery, storage capacity, and inventory, then the following conditions are sufficient to guarantee an ex-ante optimal choice of capacity. 1. Consumers expected marginal utilities of income are discounted constant functions of their conditional subjective probability distributions. 2. Consumers and firms owning capacity have identical probability distributions and discount rates. 3. Firms owning capacity are risk neutral. CONCLUSION This paper has extended the analysis of the welfare effects of supply rigidities to the case of a storable good subject to random supply. The important result of this analysis is that the ownership of rights affects ex-ante resource allocation not only when firms own capacity, but when they have to hold inventory as well. This suggests that any risk-taking by firms makes it difficult to achieve an ex-ante Pareto optimal allocation of resources. 107 FOOTNOTES 1. Note that consumers can only rent out delivery capacity in period 1. This restriction is imposed to keep this model consistent with the welfare model since the welfare model does not allow any adjustments in capacity after the observation of the random variable in period 1. A more general model would allow delivery capacity to be changed in period 1 for use in period 2. The results presented in this paper generalize to that model. 108 CHAPTER VI OPTIMAL RESOURCE ALLOCATION UNDER APPROPRIATIVE WATER RIGHTS INTRODUCTION The previous three papers have developed conditions which allow competi.tive markets to make ex-ante Pareto optimal choices of capacity and inventory policies for a storable good subject to random supply. The result was that,under the normal condition that firms own storage and delivery capacity and make inventory decisions,sufficient conditions for optimality were quite strict. In addition to the general sufficiency condition whenever there are ex-post spot markets (namely that consumers have constant marginal utilities of income such that expected marginal utilities of income are those constants, discounted, times the subjective probabilities), sufficient conditions included identical discount rates and subjective probability distributions among consumers and firms owning capacity and risk neutrality for firms owning capacity. This paper analyzes the allocation of one such good, water. The focus is on the allocation of Colorado River Water. The object is to determine whether it is possible under the current structure of property rights in water to reach an optimal allocation. 109 In brief, the current system of appropriative water rights works as follows. Each firm has a legal right to a certain flow of water per year, subject to the constraint that senior claimants have absolute priority over the available flow. The water right is obtained by diverting water and putting it to "beneficial consumptive use." Seniority (priority) is based on the chronological order of appropriation: "first in time means first in right." Under current Bureau of Reclamation policy, water rights can not be sold by firms owing money to the Bureau of Reclamation for capital projects, such as diversion works. This creates inefficiencies both because prices do not reflect the scarcity value of water and because risks are shared unequally among appropriators. 1 The question is whether the introduction of a market in state dependent percentage shares of appropriative water rights can correct these inefficiencies if the general conditions for optimality under firm ownership of rights are already satisfied. REVIEW OF APPROPRIATE OPTIMALITY CONDITIONS Recall from the previous paper that an ex-ante optimal allocation of water can be described by solving the following welfare maximization problem. MaxW 110 s.t. F(L ) F F 1 n L x. i=1 < x(L l.S 1 XS ) 2 2 < x(L ) x. xrs i=1 I.rs n L x(L x(L x(L x(L 1 xs ) < z 1 s vs, r vs A xs vr, vs s 2 ) xrs 1 vs ) < X 2 ) <:X xrs vr, s Vs n 1 L y. i=1 l.S < y (L1ys ) vs Vr, s n L L. i=1 1. 0 111 n + Ll 1: Ll ys - i=l is < vs n 2 2 < 1: L yrs - i=l irs +L vr, s where: i's social welfare weight; i's neo-classical utility function; labor supplied by i in period 0; i's discount rate; i's subjective probability of being in states s and r; i's consumption of water in period 1 state s, and period 2, state r, given s; i's consumption of a composite good in period 1 state s, and period 2, state r, given s; labor supplied by i period 1 in state s, and period 2, state r, ·given s; F size of water storage capacity production function for storage capacity; 112 labor devoted to building storage capacity; size of water delivery capacity; production function for delivery capacity; labor devoted to building delivery capacity; x(L 1 2 ),x(L ) xs xrs production function for delivery of water in period 1, state s, and period 2, state r, given s; labor devoted to delivery of water in period 1, state s, and period 2, state r, given s; random supply of water available in period 1, state s, and period 2, state r; x1s amount of water stored after period 1, given state s in period 1. production function for the composite good in period 1, state s, and period 2, state r, given s; labor devoted to the production of the composite good in period 1, state s, and period 2, state r, given s; and 8 rate of evaporation loss. 2 113 Since we are assuming for the purposes of this paper that all the general conditions for ex-ante optimality are satisfied by this market, we need only focus on the first order conditions which describe the allocation of water rights when capacity constraints are not binding. The appropriate condition is case 5 from the previous paper. This is a situation such that all water flowing down the river in the two time periods is used, but neither the delivery capacity nor the storage capacity is fully utilized. Assuming consumers and firms delivering water have identical discount rates and subjective probability distributions, the ex-ante optimality condition is as follows: aui -1- - a s -B.a 1 s (1- e) Cl xis Clx/aL 2 r 3x(L El as 2 )=X s (1-8) + z r xrs 1 XS aui/aL: [either -B.a 1 1 p ----~~P~ or -S.oa a ay/aL 1 yp 1 P q Cly/ClL 2 ypq Vi, p,q (1) 114 ALLOCATION UNDER APPROPRIATIVE WATER RIGHTS The current structure of appropriative water rights is clearly inefficient. First, the restriction on the sale of rights does not allow a means of equilibrating the marginal scarcity value of water across appropriators. Further the distinction between senior and junior appropriators creates monopoly rights in segments of the probability distribution. If sale of rights is prohibited, this situation results in an unequal sharing of risk among otherwise identical firms. Even if rights could be sold, some sort of monopoly power would remain as long as rights were defined over segments of the probability distribution. This problem would persist because prices for water rights would reflect differing probabilities of receiving water. Therefore, a particular priority water right would be a unique commodity owned by one firm. If institutions for pooling risk could be developed at the same time as a market in water rights, however, it might be possible for prices of marginal water rights to reflect marginal scarcity values even though lump sum monopoly profits would be earned during the first round of trading. institutional arrangement. Consider the following Suppose firms delivering water could purchase percentage shares of other firms' appropriative rights and the shares could vary according to what state of the world occurred. Now, if we number states of the world in order of 115 increasing river flows and segment the probability distribution into m + 1 blocks, each of the first m corresponding to a firm's original appropriative right and the m +1st being the unappropriated portion of the distribution, the decision problem under appropriative rights of the jth firm delivering water can be stated similarly to that under state-dependent rights. Assuming risk neutrality and a known discount rate and probability distribution, firm xj's problem is: m+l ,.t.. o=l -<- o s.e_ ,.t.. a { p 1 x lj -wlLlj S XS S S XS s=s.t-l m+l l: ri a r [p2xrs x2jrs - w2rs Lxrs 2j q=l r=r 2j lj 2 [Sk - Sk ]Rkqk ]} k=l rs rs rs q/m+l l: (2) q-1 Multipliers s.t. 1. 1. x J < x(L J) s xs 2. X J < rs - 2. x(L J ) xrs vs V r, s vs X (L 2j xrs ) < Xj Vr, s >.lj ls 116 Multipliers .e. 1' x(Llj ) < E f3 J ~ xsl k=l kss xlj s vs q 2j x(L 2 j ) < x 1 j (1 -e) + E f3 ~ s xr s k=l krqs q vr, s xlj < Fj s vs Aj X x<Ll) X Fj = F(L~) where: [s.e__ ,s.e_) = interval in the cumulative probability 1 distribution encompassing firm i.'s original appropriative rights; state r dependent percentage of firm k's original appropriative right owned by j as an initial endowment, in period o, in period 1, given s in period 1, and in period 2, given s in period 1·, Rk acre-feet of water in firm k's original appropriative right; 117 0 1 2 qkr'qkrs'qkrs = state r dependent price per acre-foot for water from k's original appropriative right in period 0, period 1, state s and period 2, given s in period 1. Equation (2) essentially says that if state of the world s occurs it implies that priority rights 1 to l- 1 are satisfied and 1. l can be at least partially satisfied. In order to deliver x J s 1 s water in states and store x j, firm j has to amass enough shares in rights 1 -l: 1 1· l 1· i.e. shares such that x j + X J < E S J Rk• sl sl - k=l ksls Turning now to consumers and firms producing the composite good, the remainder of the competitive market can be described as follows: Consmner: i Max U (LiO) + s i 1 1 1 E as[U (xi ,yi ,L. ) s=l R s i 2 s 1s 2 + 2 or=l La U (x. ,yi ,Li )] r 1rs rs rs Multipliers s.t. m L e ij lJ. 1r j=l XS XS + m+k ij 1. E 8 1r J + MiO j=m+l ys ys Vs 118 Multipliers m+k I: . i" . e J1T2J +l yr yrs vr, s J=nl Firm ; : vs s.t. y 2]" 2. < y(L J ) rs yrs vr, s Where: Income earned by i in period 0; i's percentage of ~'s profits in state s; 1" 2. 1T J 1T J xs' xrs xj's profits in period 1, states, and period 2, state r, given s; eij ys i's percentage of ; ' s profits ins; 119 1T lj 2j 1T ys' yrs yj's profits in period 1, states, and period 2, state r, given s; aj ys yj's subjective probability of being ins; and ~jy yj's differentiable utility function over profits. The equilibrium conditions are: n m 2. m 1. 2 . 1 = L: X J = L: X J' L: X rs i=l irs i=l is j=l s j=l vr, s n . 1 2 y J' L: y l=l irs j=m+l s vr, s n L X n L: m+k 1 Y. L: i=l 1S m+k . y 2J j=m+l rs L: n L: LiO i=l m 1. L: L J j=l xs lj m+k + L: ~t.. Lli ' L j=m+l ys i=l s ~t.. L2j j=l xrs + L: vr, s 1 vs m m m . kr J= 1 . kr J= 1 . krs J= 1 L: gj i=l irs vs n i=l ys n L: L2 1 i=l XS L: eij L j=m+l yrs n L: eij 2j m+k L: SOj r: slj m r: s2j . krs J= 1 1 V k, r, s 120 Forming Lagrange functions Li (i = 1, .•• , n); Lj (j = 1, . . . , m); and Lj (j = m + 1, ••• , m + k) and differentiating, y X the first order conditions for this problem are: 0 aui 1 1 a s --1-- Ais p xs = 0 axis aui 2 axi rs oa s a r A2 p2 =0 irs xrs a 0 s oa a s r 0 Vi (3) vi, s (4) Vi, r, s (5) vi, s (6) Vi, r, s (7) vi, s (8) Vi, r, s (9) i a ~ + Al w 1 = 0 s aL 1 is oa a s r -w 0 is s aui + A2irsw2rs 2 aL irs + yj1 a:X aL~ = X 0 0 j = 1, . . . , m (10) 121 0 0 j = 1; . . . , m (11) vs; j =1, ••• ,m (12) Vr, s; j =1, ••• , m (13) Vs;j=1, ••• ,m - cxw s s -ocx ex w2 + s r rs ax (A 2 j -A 2 j -A 2 j ) aL2j 1rs 2rs 3rs xrs 0 (14) vr,s;j=1, ••• ,m (15) 0 Vk, r (16) V k, r Is (17) Vk,s;j=1, ••• ,m (18) Vk,r,s;j=1, ••• ,m (19) 1j 1. . A2 8 s3x(L J)=xJ xs E 2j j 2. . A2rs - Y1 s,nx(L J )=XJ xrs + E 0 j = 1, ••• 'm 1. 4s A J (20) 0 Vs; j =1, ••• ,m (21) 122 j=l, ••• ,m 0 Consider first equations (18) and (19). 1 2 a s [qk ss + oa s qk ss ] A.2j 3rs 2 oa a q s r krs (22) Rearranging terms: vk,j=l, ••• ,m (23) Vk, j =1, ••• , m (24) (23) and (24) imply that given a state of the world s, or r, given s, all water rights which can be satisfied will be priced at the same marginal cost. Thus, despite possible early monopoly profits, a competitive shares market will develop. 1 Now, if we assume that A. = C.ai 1S 1 s 2 Vi, s and A. = oCia. a. 1rs 1s 1r Vi, r, s and that capacity constraints are not binding, equations (4)-(9), (12)-(15), (21) and (23)-(24) reduce to: a aui aui L: a - o(l-8) q a 2. 1" 1 2 s r r3x(L J )=X J (1-8)+ I: s 2j ax. ax 1S s xrqs Skr qS Rk irqs k=l [either a [either a p p aui aui ] - -- or oa a 1 p v a 2 ay. yipv 1p a /aL 2.t Y ypv v i ' j '.e.' p 'v ( 25 ) 123 Since (25) is the same as (1), given 2 ~ a j = 1, the resulting . krs J= 1 allocation is ex-ante Pareto optimal. CONCLUSION This paper has analyzed the effect of opening a market in percentage shares of appropriative water rights, assuming that the general conditions for ex-ante Pareto optimality are already satisfied. Depending on one's point of view the results could be seen as either encouraging or discouraging about the possibilities for improving the efficiency of the market for Colorado River water. On the one hand, it is reasonable to expect that a percentage share market in water rights might develop if the prohibition against sale were lifted. However, a state dependent share market is perhaps more unlikely because of the information costs involved. On the other hand, the general optimality conditions are quite restrictive and competition among water companies is very limited because of the locationally fixed nature of distribution systems. Consumers can only change water companies by moving and prices are regulated by city governments or public utilities commissions. Despite these caveats, however, the results presented in this paper do suggest that the appropriative rights system can at least be made a better allocative mechanism with little tampering with existing legal structures. 124 FOOTNOTES 1. H. S. Burness and J. P. Quirk, "Appropriative Water Rights and the Efficient Allocation of Resources," Social Science Working Paper No. 157, California Institute of Technology, Revised March 1978. Forthcoming in The American Economic Review. "Appropriative Water Rights and the Theory of the Dam," December 1977. To appear in Festshrift for E. T. Weiler, published by Purdue University Press, 1979. 2. We recognize that this model does not exactly describe the allocation of water since it is both a consumption good and an intermediate good. However, even if another final good which used water were included, the marginal conditions describing the allocation of water for final consumption would not change. To prove this statement, consider the following simple welfare model with one consumer: Max U(x ,y,z,L) c Multipliers s.t. x(L ) "1 y = y(L ) y >..2 z = z(x ) z "3 X X X c L X +x +L < X < L z - y - "4 "s The first order conditions are: _1Q_ - f. Cly 2 0 (1) 0 (2) 12S au A.3 az-~+ A. 31 0 (3) s =0 (4) ox >..1 a"L - A.s 0 (S) 0 (6) X _h_ a1 - A.s A.2 y -A. A. 1 + A. 4 = 0 3 ~ax z A.4 (7) 0 (8) These reduce to: au axc au ay _h_ 31y ax aL X au Clz azaxz au ay (9) It is clear from (9) that the addition of good z has not affected the marginal rate of substitution or the price ratio between x and y.