When the asset market is incomplete, the role of prices extends beyond conveying the aggregate scarcity of commodities. In conjunction with the asset structure, they determine the attainable reallocations of revenue. This affects nontrivially the existence, optimality, and determinacy of competitive equilibrium allocations, as well as the revelation of information by prices. Further, it accounts for diverse phenomena, among them the preservation of memory in macroeconomics aggregates. A simple exchange economy extends over two periods. Uncertainty, indexed by finitely many states of nature, s = 1,..., S, is resolved in the second period. Commodities, 1 = 1, .. ., L, are traded in spot markets in the second period after the uncertainty has been resolved and assets have paid off. A commodity bundle is x = ( . . ., x(s), . . . ) = (...,x1(s) ...). Commodity prices are p = (-,P(s) ....) (. (---, Pi(s), -.). Assets, a= 19 . . ., A, are traded in the first period and pay off in the second. A portfolio is y = (..., Ya. ). Assets are real: the payoff of an asset is a commodity bundle ra = (..., ra(s), .. .). At commodity prices p, the payoff of an asset, more precisely the payoff of an asset in terms of revenue, is

I. Introduction, 465.—II. The model and results, 466.—III. Conclusion, 469.

In a general equilibrium model with uncertain second period demand, incomplete markets, and costly labor mobility, the authors analyze the feasibility and optimality of alternative employment contracts. For the case where layoffs are prohibited, they demonstrate that both the fixed wage--constant employment contract, as well as the flexible wage--variable employment contract are equilibria in firm behavior, while the latter is preferable from society's point of view. In the case with layoffs, they show that the competitive mechanism leads to a less than optimal number of layoffs, and demonstrate that unemployment insurance with less than complete experience rating lowers the cost of layoffs to the firm and encourages labor mobility. In the context of the model, a properly designed unemployment insurance program yields a fully efficient allocation.

Philip Dybvig, Heraklis Polemarchakis; Recovering Cardinal Utility, The Review of Economic Studies, Volume 48, Issue 1, 1 January 1981, Pages 159–166, https://d

In this paper we address the question of whether a price-setting monopolist can improve his welfare by imposing quantity constraints on buyers. We show first that if all buyers are identical, quantity rations are not useful for the monopolist. We then show by means of an example that if buyers are diverse, quantity rations may be desirable. It is shown that if there are only two commodities, the only constraint that may be useful to the monopolist is a zero constraint on one of the two commodities. An example shows that this property does not hold for more than two commodities.