Indivisible labor implies chaos

Summary. We study a simple infinite horizon model with indivisible labor. We characterize the optimal plans under the assumptions that and that , where R is the gross interest rate and is the discount factor. We show that under those assumptions, optimal plans are almost always asymptotically nonperiodic unless initial wealth is excessively small or large. Received: May 25, 1999; revised version: June 24, 1999     

Heterogeneous labor skills,the median voter and labor taxes

This paper analyzes the median voter?s most preferred sequences of labor taxes in the standard neoclassical growth model. We consider an infinite horizon economy in which agents are heterogeneous with respect to both initial wealth and labor skills. We start by providing a set of sufficient conditions for the existence of a Condorcet winner. We then characterize the most preferred tax sequence by the median agent. First, we show that marginal labor taxes depend directly on the absolute value of the distance between the median and the mean value of the skills? distribution. Second, we find that in contrast to the intuition stemming from standard representative agent economies, labor taxes are more volatile and counter-cyclical taxation (e.g., increasing taxes in recession) might be optimal depending on the correlation between inequality and TFP. To assess the quantitative relevance of these findings, we calibrate the model economy to six countries and find that counter-cyclical labor taxation is optimal for all but the US.     

Credibility and endogenous societal discounting

We consider a dynamic moral hazard economy inhabited by a planner and a population of privately informed agents. We assume that the planner and the agents share the same discount factor, but that the planner cannot commit. We show that optimal allocations in such settings solve the problems of committed planners who discount the future less heavily than agents. Thus, we provide micro-foundations for dynamic moral hazard models that assume a societal discount factor in excess of the private one. We extend the analysis to allocations that are reconsideration-proof in the sense of Kocherlakota [Kocherlakota, N., 1996. Reconsideration-proofness: A refinement for infinite horizon time inconsistency. Games and Economic Behavior 15, 33–54]. We show that these allocations solve the choice problem of a committed planner with a unit discount factor.     

How long to eat a cake of unknown size? Optimal time horizon under uncertainty

This paper is concerned with the determination of the optimal time horizon for the cake–eating problem under uncertainty. It is shown that if the uncertain exhaustible resource stock is a discrete random variable admitting at most a finite number of values, the optimal planning horizon is infinite (finite) according as the marginal utility of extraction–cum–consumption is infinite (a finite positive value) as the latter approaches zero, thereby extending the scope of the similar result under perfect certainty. Other results show that uncertainty will generally lengthen the planning horizon, implying a more conservative extraction policy under uncertainty, and that the extraction policy aimed at extracting an amount equal to the expected value of the uncertain resource stock takes longer than the expected value of the optimal planning horizon. JEL Classification: D81 and Q31 Combien de temps pour manger un gâteau de taille inconnue? L’horizon temporel optimal en régime d’incertitude. Ce mémoire s’attaque à la détermination de l’horizon temporel optimal dans le cas du problème du gâteau–à–manger en régime d’incertitude. On montre que si le stock incertain de la ressource épuisable est une variable aléatoire discontinue qui ne peut prendre qu’un nombre fini de valeurs, l’horizon temporel est infini (fini) selon que l’utilité marginale de l’extraction–cum–consommation est infinie (prend une value finie positive) quand celle–ci approche zéro, et ce faisant élargit la portée d’un résultat similaire obtenu en régime de certitude parfaite. D’autres résultats montrent que l’incertitude accroît généralement l’horizon temporel, ce qui suggère qu’une politique d’extraction plus conservatrice va prévaloir en régime d’incertitude, et que la politique d’extraction visant à extraire une quantitéégale à la valeur anticipée d’un stock de ressource incertain prend plus de temps que la valeur anticipée de l’horizon temporel optimal.     

Uniqueness of asset prices in an exchange economy with unbounded utility

Summary. This paper studies conditions under which the price of an asset is uniquely determined by its fundamental value – i.e., no bubbles can arise – in Lucas-type asset pricing models with unbounded utility. After discussing Gilles and LeRoy's (1992) example, we construct an example of a two-period, representative agent economy to demonstrate that bubbles can arise in a standard model if utility is unbounded below, in which case the stochastic Euler equation may be violated. In an infinite horizon framework, we show that bubbles cannot arise if the optimal sequence of asset holdings can be lowered uniformly without incurring an infinite utility loss. Using this result, we develop conditions for the nonexistence of bubbles. The conditions depend exclusively on the asymptotic behavior of marginal utility at zero and infinity. They are satisfied by many unbounded utility functions, including the entire CRRA (constant relative risk aversion) class. The Appendix provides a complete market version of our two-period example. Received: January 22, 1996; revised version: February 18, 1997     

Decreasing absolute risk aversion and utility indices derived from cake-eating problems

This paper establishes the following characterization of decreasing absolute risk aversion (DARA) utility indices: J exhibits DARA if and only if it is the indirect function corresponding to an infinite horizon cake-eating problem for some nondecreasing and concave utility function of consumption. The characterization is applied to the analysis of resource extraction under uncertainty and to an inverse optimal problem.     

Decentralized learning from failure

We study decentralized learning in organizations. Decentralization is captured through Crawford and Haller's [Learning how to cooperate: optimal play in repeated coordination games, Econometrica 58 (1990) 571-595] attainability constraints on strategies. We analyze a repeated game with imperfectly observable actions. A fixed subset of action profiles are successes and all others are failures. The location of successes is unknown. The game is played until either there is a success or the time horizon is reached. We partially characterize optimal attainable strategies in the infinite horizon game by showing that after any fixed time, agents will occasionally randomize while at the same time mixing probabilities cannot be uniformly bounded away from zero.     

On optimal steady states of n-sector growth models when utility is discounted

This paper contains results on local and global stability of n-sector growth models when utility is discounted mostly for small rates of discount. It is well known that when future utility is not discounted one can prove precise results about optimal steady states (OSS's) under fairly general assumptions. In particular, existence, uniqueness, and turnpike properties have been established by several authors. The counter examples presented by Kurz, Sutherland, and Weitzman, however, show that when utility is discounted, additional assumptions are required to obtain turnpike results. In general, it would be interesting to know how the submanifolds of stability change as δ changes. One hopes that certain conditions on the utility function would be sufficient to “classify” the submanifolds of stability and instability. Such a question is apparently very difficult to answer, but we think that the results obtained here will help in this task.The proof that the turnpike theorem holds for discount factors near one is divided in two parts. First, we prove that optimal paths “visit” neighborhoods of the modified OSS's. Then, we prove that local stability holds for such neighborhoods.In order to show this fact, we must prove that the local “stable manifold” varies continuously with the discount factor. This roundabout method is necessary since our problem is similar to proving uniform continuity with respect to a parameter of solutions of a differential equation in a noncompact interval of time.Other problems analyzed here include uniqueness and continuity of OSS's. We also discuss the relation between the saddle-point property and the local stability of infinite horizon optimal growth paths.     

Renegotiation-proof contract in repeated agency

Renegotiation-proof contracts are studied in infinitely repeated principal–agent problem. Contracts satisfying a weaker notion of renegotiation-proofness always exist. The renegotiation-proof value function has a simple characterization: It is the principal's optimal value function when an appropriate lower bound is placed on the agent's expected utility. Sufficient conditions are provided for renegotiation-proof value function in finite horizon to converge to renegotiation-proof value function in infinite horizon as time goes to infinity.     

Simple Markovian equilibria in dynamic spatial legislative bargaining

The paper proves, by construction, the existence of Markovian equilibria in a dynamic spatial legislative bargaining model. Players bargain over policies in an infinite horizon. In each period, a sequential protocol of proposal-making and voting, with random proposer recognitions and a simple majority, produces a policy that becomes the next period's status-quo; the status-quo is endogenous. The construction relies on simple strategies determined by strategic bliss points computed by the algorithm we provide. A strategic bliss point, the dynamic utility ideal, is a moderate policy relative to a bliss point, the static utility ideal. Moderation is strategic and germane to the dynamic environment; players moderate in order to constrain the future proposals of opponents. Moderation is a strategic substitute; when a player's opponents do moderate, she does not, and when they do not moderate, she does. We provide conditions under which the simple strategies induced by the strategic bliss points computed by the algorithm deliver a Stationary Markov Perfect equilibrium, and we prove its existence in generic games with impatient players and in symmetric games. Because the algorithm constructs all equilibria in simple strategies, we provide their general characterization, and we show their generic uniqueness.     

The growth of relative wealth and the Kelly criterion

We propose an evolutionary framework for optimal portfolio growth theory in which investors subject to environmental pressures allocate their wealth between two assets. By considering both absolute wealth and relative wealth between investors, we show that different investor behaviors survive in different environments. When investors maximize their relative wealth, the Kelly criterion is optimal only under certain conditions, which are identified. The initial relative wealth plays a critical role in determining the deviation of optimal behavior from the Kelly criterion regardless of whether the investor is myopic across a single time period or maximizing wealth over an infinite horizon. We relate these results to population genetics, and discuss testable consequences of these findings using experimental evolution.     

Entry and Vertical Differentiation

This paper analyzes the entry of new products into vertically differentiated markets where an entrant and an incumbent compete in quantities. The value of the new product is initially uncertain and new information is generated through purchases in the market. We derive the (unique) Markov perfect equilibrium of the infinite horizon game under the strong long run average payoff criterion. The qualitative features of the optimal entry strategy are shown to depend exclusively on the relative ranking of established and new products based on current beliefs. Superior products are launched relatively slowly and at high initial prices whereas substitutes for existing products are launched aggressively at low initial prices. The robustness of these results with respect to different model specifications is discussed. Journal of Economic Literature Classification Numbers: C72, C73, D43, D83.     

Indeterminacy in dynamic models: When Diamond meets Ramsey

We consider an aggregate two-periods overlapping generations model with endogenous labor, consumption in both periods of life, homothetic preferences and productive external effects coming from the average capital and labor. We show that under realistic calibrations of the parameters, in particular a large enough share of first period consumption over the wage income, local indeterminacy of equilibria cannot occur with capital externalities alone. It can nevertheless occur when there are only, even very small, vanishing labor externalities provided that the elasticity of capital-labor substitution and the wage elasticity of the labor supply are large enough. We also show that if labor externalities are slightly stronger, but still small enough to be plausible, and the elasticity of labor supply is larger, local indeterminacy occurs in a Cobb-Douglas economy. Finally, we show that a locally indeterminate steady state is generically characterized by an under-accumulation of capital. It follows therefore that while agents live over a finite number of periods, the conditions for the existence of locally indeterminate equilibria are very similar to those obtained within infinite horizon models and that from this point of view, Diamond meets Ramsey.     

A political economy theory of the soft budget constraint

Why do soft budget constraints exist and persist? In this paper we argue that the prevalence of soft budget constraints can be best explained by the political desirability of softness. We develop an infinite horizon political economy model where neither democratic nor autocratic politicians can commit to policies that are not ex post optimal. We show that because of the dynamic commitment problem inherent in the soft budget constraint, politicians can in essence commit to make transfers to entrepreneurs which otherwise they would not be able to do. This encourages such entrepreneurs to support them politically. Though the soft budget constraint may induce economic inefficiency, it may be politically rational because it influences the probability of political survival. In consequence, even when information is complete, politicians may fund bad projects which they anticipate they will have to bail out in the future. We show that, maybe somewhat surprisingly, dictators who are less likely to lose power, are more likely to use the soft budget constraint as a strategy to gain political support.     

Chores

We analyze situations where the provision of each of c public goods must be voluntarily assumed by exactly one of n private agents in the absence of transfer schemes or binding contracts. We model this problem as a complete information, potentially infinite horizon game where n agents simultaneously wage c wars of attrition. Providing a public good commits an agent not to take on the provision of another public good for a fixed period. We explore the strategic trade-offs that this commitment ability and the multiplicity of tasks provide. Subgame perfect equilibria (SPEs) are characterized completely for games with two agents and two public goods. For games with two identical agents and c > 1 identical public goods, we establish that an equilibrium that yields a surplus-maximizing outcome always exists and we provide sufficient conditions under which it is the unique equilibrium outcome. We show that under mild conditions, the surplus-maximizing SPE is the unique symmetric SPE. Journal of Economic Literature Classification Number: H41, C72, D13.     

Multi-period portfolio optimization under possibility measures

A single-period portfolio selection theory provides optimal tradeoff between the mean and the variance of the portfolio return for a future period. However, in a real investment process, the investment horizon is usually multi-period and the investor needs to rebalance his position from time to time. Hence it is natural to extend the single-period fuzzy portfolio selection to the multi-period case based on the possibility theory. In this paper, we propose the possibilistic expected value and variance for the terminal wealth with fuzzy forms after T periods by using the central value operator. Classes of multi-period possibilistic mean-variance models are formulated originally under the assumption that the proceeds of risky assets are fuzzy variables. Besides, we apply a particle swarm optimization algorithm to solve the proposed multi-period fuzzy portfolio selection models. A numerical example is given to illustrate the performance of the proposed models and algorithm.     

WHEN DOES DETERMINACY IMPLY EXPECTATIONAL STABILITY?

Since the introduction of rational expectations, there have been issues with multiple equilibria and equilibrium selection. We study the connections between determinacy of rational expectations equilibrium and learnability of that equilibrium in a general class of purely forward‐looking models. Our framework is sufficiently flexible to encompass lags in agents' information and either finite horizon or infinite horizon approaches to learning. We are able to isolate conditions under which determinacy does and does not imply learnability and also conditions under which long‐horizon forecasts make a clear difference for learnability. Finally, we apply our result to a relatively general New Keynesian model.     

Greater prudence and greater downside risk aversion

Variation in the degree of downside risk aversion across decision makers has implications for efficient risk sharing. However, except for small differences in risk preferences, there is no index, analogous to the Arrow-Pratt index of risk aversion, that depends only on local properties of the utility function and indicates the degree of aversion to downside risk. A measure that does depend only on local properties of the utility function u, the index of prudence p=−u^?/u^″, is related to downside risk aversion, which is indicated by a positive value for u^?. Although we show that the degree of prudence is not an accurate indicator of the degree of downside risk aversion, we nonetheless demonstrate that a uniform increase in prudence accompanied by a uniform increase (decrease) in risk aversion is sufficient to indicate greater downside risk aversion, provided prudence is greater (less) than three times the degree of risk aversion.     

On locally optimal programs in the Robinson–Solow–Srinivasan model

In the context of a model due to Robinson, Solow and Srinivasan (the RSS model), we report results on the existence and characterization of locally optimal programs, a concept taken from theoretical physics. In particular, we propose a (new) transversality condition under which all locally optimal programs are good. An extended introduction places our theorems in the context of previous work on the existence question, including that on agreeable programs. It appears that there is no completely rational way to attack [the] problem without considering development programmes over an infinite horizon (Gale in Rev Econ Stud 34:1–8, 1967). The analysis of simple models is essential if we are to understand the corresponding situation for more complex models of the economy (Mirrlees and Stern in J Econ Theory 4:268–288, 1972). The technical convenience, for clear and quantitative results, of using an infinite time horizon is rather great (Hammond and Mirrlees in Models of economic growth, Wiley, New York, pp 283–299, 1973).     

Efficient allocations with moral hazard and hidden borrowing and lending: A recursive formulation

We propose a tractable recursive framework to study the optimal allocation of consumption and effort in a dynamic setting with moral hazard where agents have secret access to the credit market or to storage. The recursive structure is based on a generalized first-order approach, whose validity must be verified ex post. Thanks to the recursive formulation of the optimal contract, the verification procedure turns out to be numerically parsimonious as it can be performed using standard dynamic programming techniques with only one endogenous state variable: The agent's level of assets. We study the performance of our ex post verification test in practice by solving numerically three representative infinite horizon examples.