The literature on optimal resource allocation under uncertainty can be dated from the original work by Arrow 1 and Debreu 2 on ex ante conditional demand markets. The second paper compares an ex-ante market for conditional demands with a dynamic market in which capacity is chosen before the random variable is observed and consumption and marginal production decisions are made afterwards.
CHAPTER II
SUMMARY AND DISCUSSION OF RESULTS
Since changes in the available supply of a good imply changes in prices and relative incomes in different countries of the world, the final demand cannot be known with certainty. B even more to build a greater distribution capacity than if his prices and incomes were the same in all countries of the world.
CHAPTER III
A FURTHER NOTE ON THE ROLE OF SECURITIES IN THE OPTIMAL ALLOCATION OF RISK-BEARING
OPTIMALITY UNDER CAPACITY CONSTRAINTS
INTRODUCTION
Case 2
Proposition 4: If the supply of a good is a random variable and the supply capacity is not a binding constraint, the distribution of supply rights does not affect the allocation of final output if the market is organized as an ex-ante conditional claims market. Proposition 5: If the supply of a good is a random variable and the supply capacity must 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 .
- This model assumes that some capacity exists and the firm might wish to change it. Without loss of generality we assume that
- See footnote 7 for a proof of the statement that Ai is the expected marginal ut i lity of income in state s • s
At this point an assumption is usually made which is that the change undertaken does not change the feasible set of profits or the probability distribution of profits across states of the world. 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 changed across certain states of the world if capacity is changed.
Proposition 8: If supply of a good is a random variable and supply capacity is owned by firms and must be chosen before the random variable is observed, then a market for shares of firms that own capacity allows consumers to agree on firms' choices of capacity when the following sufficient conditions hold beyond the ordinary. assumptions about shareholder unanimity models:. consumers have the same probability distributions over the states of the world; and. consumers' expected marginal income utility are constant functions of these subjective probabilities. Gerard Debreu, The Theory of Value: An Axiomatic Analysis of Economic Equilibrium, Cowles Foundation Monograph No. Without loss of generality, we assume that i's utility function is constant over the states of the world. 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. Therefore, companies know with certainty their total income and expenses, regardless of the state of the world that occurs.
Forsythe, “Unanimity and the Theory of the Firm under Multiplicative Uncertainty,” California Institute of Technology. See footnote 7 for a proof of the statement that Ai is the expected marginal utility of income in the states. The expected marginal utility of income in the states s.
CHAPTER IV
This problem is similar to the one Starr:1 noted, namely that unless consumers have sufficiently uniform probability distributions, ex ante conditional demands will not allow for an efficient allocation between current and future consumption. The purpose of this exercise is to see if a normal market can allow an ex ante optimal choice of capacity under uncertainty, if less. When production is introduced, a sufficient condition for prior optimality of a competitive spot market equilibrium is that each consumer's expected marginal income utility in each state of the world is a constant function of his subjective probabilities (i.e.
This strong condition, which we have imposed on the bond market, comes simply because consumers plan purchases that are 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 on the structure of rights. If consumers hold property rights 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.
A simple example is used to show that even with similar probability distributions ex-ante optimality can fail. In general, the results presented in this paper therefore indicate that if firms have capacity, the conditions for ex-ante optimality are stricter in a competitive spot market than in a securities market.
REVIEW OF OPTIMALITY CONDITIONS UNDER UNCERTAINTY WITH CAPACITY CONSTRAINTS
The first m firms produce x in period 0, the next k firms produce x and the last g firms produce y. Consumption of i i n s is limited by i's resources available in s because marginal transactions are made in spot markets.
2 • Even if firm x were risk neutral (u) it would still not be satisfied overall without sufficiently similar probabilities. This seemingly strange result occurs because consumers with different probability distributions view the probability of having. This does not pose a problem when consumers bear the risk of investing in capacity, because the market for capacity rights allows consumers with different probabilities to exhaust the gains from the trade created by the different probabilities.
It also does not cause problems in a full ex-ante contingent claims market because firms do not assume any risk in such a market. Um.; if firms take risks in providing capacity, these benefits from trade between consumers are not exhausted. In a sense, the free rider problem develops when firms take risks because consumers, who attribute a low probability of insufficient capacity, want to pay less for x relative to y on average when there is no shortage.
If firms are risk neutral and consumers have identical probability distributions, these gains from trade do not exist to begin with. This paper has considered the effect on resource allocation uncertainty of moving from a full contingent damages market to one in which capacity is built ex-ante but marginal production and consumption decisions are made ex-post.
CHAPTER V
OPTIMAL CAPACITY CHOICE AND INVENTORY POLICY FOR A STORABLE GOOD SUBJECT TO RANDOM SUPPLY
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 must be determined in addition to the optimal delivery capacity. The welfare model developed in this paper is an extension of the model developed in the previous paper.
In this case, delivery and storage capacities are built before any consumption occurs, and consumption plans are conditioned by observing the random variable subject to supply constraints. Then, for each observation of the random variable in period 1 there is an optimal ex-ante saving and an optimal portfolio of consumption plans conditional on the observation of the random variable in period 2. The competitive model used in this paper is an extension e dynamic programming model developed in the previous paper.
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 rights on supply capacity can be generalized to the case of a storable good that is subject to capacity constraints.
A TWO PERIOD DYNAMIC PROGRAMMING MODEL WITH CAPACITY CONSTRAINTS AND A STORAGE FACILITY
If we now consider only the effective constraints that are binding, the first-order conditions are reduced to nine cases:.
A COMPETITIVE MARKET 1. Consumers Own Rights
If consumers have the right to supply, capacity, storage capacity and storage rights and the discounted expected marginal utility of income of each consumer in each state of the world is a. If there is a constant function of its conditional subjective probabilities, then a competitive equilibrium in the spot market is ex ante Pareto optimal when storage capacity and supply capacity constraints are binding. If producers own capacity, storage and warehousing facility rights rather than consumers, the competitive market can be described as follows: rj j's initial endowment of states supply s. 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;. If we compare these equations with equations (25)-(51) which we know describe an ex-ante Pareto optimum, it is clear that this problem raises the same problems as the problem in the previous article.
Equations (76)-(92) together with equations reduce to conditions equivalent to the ex-ante Pareto optimal conditions. The important result of this analysis is that the ownership of rights affects ex ante resource allocation not only when firms own capacity, but also when they must hold inventory. This suggests that any risk taking by firms makes it difficult to achieve an ex-ante Pareto optimal allocation of resources.
FOOTNOTES
Note that consumers can only rent out delivery capacity in period
CHAPTER VI
Each company has a statutory right to a certain flow of water per year, subject to the restriction that older beneficiaries have absolute priority over the available flow. Under current Bureau of Reclamation policy, water rights cannot be sold by companies that owe money to the Bureau of Reclamation for capital projects such as diversion works. 1 The question is whether the introduction of a market in percentages of state-dependent appropriation water rights can correct this.
REVIEW OF APPROPRIATE OPTIMALITY CONDITIONS
Since, for the purposes of this paper, we assume that this market satisfies all general conditions for ex-ante optimality, we only need to focus on the first-order conditions that describe the allocation of water rights when capacity constraints are not binding. This is a situation in which all the water flowing down the river is used in both time periods, but neither the delivery capacity nor the storage capacity are fully utilized.
ALLOCATION UNDER APPROPRIATIVE WATER RIGHTS
Equation (2) basically says that if the state of the world occurs, it implies that priority rights 1 to l - 1 are satisfied, and l may be at least partially satisfied.
CONCLUSION
However, even if another final good that uses water were included, the marginal conditions describing the allocation of water to final consumption would not change. 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.
