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We study preferences over lotteries in which both the prize and the payment date are uncertain. In particular, atime lotteryis one in which the prize is fixed but the date is random. With Expected Discounted Utility, individuals must be riskseekingover time lotteries (RSTL). In an incentivized experiment, however, we find that almost all subjects violate this property. Our main contributions are theoretical. We first show that within a very broad class of models, which includes many forms of nonexpected utility and time discounting, it is impossible to accommodate even a single violation of RSTL without also violating a property we termed Stochastic Impatience, a risky counterpart of standard Impatience. We then present two positive results. If one wishes to maintain Stochastic Impatience, violations of RSTL can be accommodated by keeping Independence within periods while relaxing it across periods. If, instead, one is willing to forego Stochastic Impatience, violations of RSTL can be accommodated with a simple generalization of Expected Discounted Utility, obtained by imposing only the behavioral postulates of Discounted Utility and Expected Utility.

Modeling intergenerational altruism is crucial to evaluate the long-term consequences of current decisions, and requires a set of principles guiding such altruism. We axiomatically develop a theory of pure, direct altruism: Altruism is pure if it concerns the total utility (rather than the mere consumption utility) of future generations, and direct if it directly incorporates the utility of all future generations. Our axioms deliver a new class of altruistic, forward-looking preferences, whose weight put on the consumption of a future generation generally depends on the consumption of other generations. The only preferences lacking this dependence correspond to the quasi-hyperbolic discounting model, which our theory characterizes. Our approach provides a framework to analyze welfare in the presence of altruistic preferences and addresses technical challenges stemming from the interdependent nature of such preferences.

This paper extends the long-term factorization of the stochastic discount factor introduced and studied by Alvarez and Jermann (2005) in discrete-time ergodic environments and by Hansen and Scheinkman (2009) and Hansen (2012) in Markovian environments to general semimartingale environments. The transitory component discounts at the stochastic rate of return on the long bond and is factorized into discounting at the long-term yield and a positive semimartingale that extends the principal eigenfunction of Hansen and Scheinkman (2009) to the semimartingale setting. The permanent component is a martingale that accomplishes a change of probabilities to the long forward measure, the limit of T-forward measures. The change of probabilities from the data-generating to the long forward measure absorbs the long-term risk-return trade-off and interprets the latter as the long-term risk-neutral measure.

Levy (2013) presented examples of discounted stochastic games that do not have stationary equilibria. The second named author has pointed out that one of these examples is incorrect. In addition to describing the details of this error, this note presents a new example by the first named author that succeeds in demonstrating that discounted stochastic games with absolutely continuous transitions can fail to have stationary equilibria.

Following Kreps (1979), we consider a decision maker who is uncertain about her future taste for immediate consumption. This uncertainty leaves the decision maker with a preference for flexibility: When choosing among menus containing alternatives for future choice, she weakly prefers menus with additional alternatives. Existing representations accommodating this choice pattern cannot address dynamic decision situations like a consumption savings problem. We provide representations of choice over continuation problems that are recursive and take the form of Bellman equations. Two specific models are axiomatized. They feature stationary and Markovian beliefs over future tastes, respectively. The parameters of the representations, which are relative intensities of tastes, beliefs over those tastes and the discount factor, are uniquely identified from behavior. We characterize a natural notion of 'greater preference for flexibility' in terms of a stochastic order on beliefs and give an example of a Lucas tree economy, where a representative agent with greater preference for flexibility corresponds to larger price volatility in the sense of second order stochastic dominance.

This paper takes the position that an individual’s concern for relative status is expressed in her preferences. We embed this hypothesis in an otherwise conventional model of economic growth, and examine its consequences. The model generates two kinds of equilibrium. In the first, all individuals always follow a deterministic consumption and investment strategy. Such a strategy must be linear in output, is independent of payo s and technology, and only depends on the discount factor. This sort of equilibrium exists when the individual cares only about status and the individual production function is convex in investment. In contrast, the other kind of equilibrium involves convergence to a steady state in which there is persistent and endogenously generated gambling. This kind of equilibrium exists in more plausible circumstances. The individual has a utility function that may depend on consumption per se as well as implied status and individual production functions are concave. We characterize the unique steady state in such a situation, and prove that any dynamic equilibrium path must converge to it. Risk-taking arises not from the presumption of a utility function with strictly convex segments (as in Friedman-Savage), but naturally from a view of utility as depending on relative status. Our steady state is broadly consistent with the stylized facts that individuals both insure downside risk and to gamble over upside risk, and generates similar patterns of risk-taking and avoidance across environments with quite di erent overall wealth levels. Finally, in contrast to Friedman (1953), endogenous risk-taking here is generally Pareto-ine cient.

Experimental studies have found that a decision maker prefers spreading good and bad outcomes evenly over time. We propose, in an axiomatic framework, a new model of discount factors that captures this preference for spread. The model provides a refinement of the discounted utility model while maintaining dynamic consistency. The derived discount factors incorporate gain/loss asymmetry recursively: the difference between average future utility and current utility defines a gain or a loss, and gains are discounted more than losses. This notion of utility smoothing can induce a preference for spread: if bad outcomes are concentrated on future periods, moving one of the bad outcomes to today would be beneficial because such an operation eliminates a large loss and replaces it with a small gain.

In standard auctions resale creates a role for a speculator—a bidder who is commonly known to have no use value for the good on sale. We study this issue in environments with symmetric independent private-value bidders. For second-price and English auctions the efficient value-bidding equilibrium coexists with a continuum of inefficient equilibria in which the speculator wins the auction and makes positive profits. First-price and Dutch auctions have an essentially unique equilibrium, and whether or not the speculator wins the auction and distorts the final allocation depends on the number of bidders, the value distribution, and the discount factor. Speculators do not make profits in first-price or Dutch auctions.

In this paper, I analyze a decentralized search and matching economy with transferable utility composed of heterogeneous agents. I explore whether Becker's assortative matching result generalizes to an economy where agents engage in costly search. In an economy with explicit additive search costs, complementarities in joint production (supermodularity of the joint production function) lead to assortative matching. This is in contrast to previous literature, which had shown that in a search economy with discounting, assortative matching may fail even when the joint production function is supermodular.

We assume that an impatient decision maker (DM) runs variable-size experiments at an increasing, strictly convex cost before choosing an irreversible action. We introduce and solve a tractable continuous time version of this problem --- a control of variance of a diffusion with uncertain mean. Assuming two states and two actions, we prove: (a) the optimal experimentation level rises in the Bellman value; and deduce testable implications, like (b) experimentation costs drift up; and (c) a more impatient decision maker may experiment more, given lump-sum final payoffs. We show that (a) and (b) are robust to finitely many states and actions, and we also extend an R&D interpretation of the model, where experimentation is monotonic not only in the value, but also in beliefs. Our intuition for our key monotonicity finding (a) is very economic. There are two decisions at each instant: stop or experiment, and then at what level n. The second choice equates the marginal costs and benefits of information: c'(n) = MB(n). In our diffusion setting, the marginal benefit of experimentation is constant, and so the total benefit TB is linear in the level: TB = n*MB = nc'(n). So the DM acts like a neoclassic competitive firm, producing information at an increasing marginal cost and selling it to himself at the fixed price c'(n). Since postponing the final decision entails a discounting cost, optimal stopping demands that the DM equate his producer surplus from experimentation nc'(n)-c(n) to the delay cost rV (given the interest rate r). Intuitively, the DM closes down his information firm (i.e. acts) when he cannot generate profits (producer surplus) to justify his capital rental (his deferred action). Since this surplus rises in quantity with convex costs, greater experimentation is needed to generate the higher surplus for a higher value V.

We show that, in repeated common interest games without discounting, strong ‘perturbation implies efficiency’ results require that the perturbations must include strategies that are ‘draconian’ in the sense that they are prepared to punish to the maximum extent possible. Moreover, there is a draconian strategy whose presence in the perturbations guarantees that any equilibrium is efficient. We also argue that the results of Anderlini and Sabourian (1995) using perturbation strategies that are cooperative (and hence nondraconian) are not due to computability per se but to the further restrictions they impose on allowable beliefs.

Blume and Easley (1992) show that if agents' have the same savings rule, those who maximize the expected logarithm of next period's outcomes will eventually hold all wealth (i.e. are "most prosperous"). However, if no agent adopts this rule then the most prosperous are not necessarily those who make the most accurate predictions. Thus, agents who make inaccurate predictions need not be driven out of the market. In this paper, it is shown that, among agents who have the same intertemporal discount factor (and who choose savings endogenously), the most prosperous are those who make accurate predictions. Hence, convergence to rational expectations obtains because agents who make inaccurate predictions ar1e driven out of the market.

Gul (Econometrica, 1989) introduces a non-cooperative bargaining procedure and claims that the payoffs of the resulting efficient stationary subgame perfect equilibria are close to the Shapley value of the underlying transferable utility game (when the discount factor is close to 1). We exhibit here an example showing that efficiency, even for strictly super-additive games, does not imply that all meetings end in agreement. Thus efficiency does not suffice to get Gul's result.

This paper studies the class of denumerable-armed (i.e. finite- or countably infinitearmed) bandit problems with independent arms and geometric discounting over an infinite horizon, in which each arm generates rewards according to one of a finite number of distributions, or "types." The number of types in the support of an arm, as also the types themselves, are allowed to vary across the arms. We derive certain continuity and curvature properties of the dynamic allocation (or Gittins) index of Gittins and Jones (1974), and provide necessary and sufficient conditions under which the Gittins-Jones result identifying all optimal strategies for finite-armed bandits may be extended to infinite-armed bandits. We then establish our central result: at each point in time, the arm selected by an optimal strategy will, with strictly positive probability, remain an optimal selection forever. More specifically, for every such arm, there exists (at least) one type of that arm such that, when conditioned on that type being the arm's "true" type, the arm will survive forever and continuously with nonzero probability. When the reward distributions of an arm satisfy the monotone likelihood ratio property (MLRP), the survival prospects of an arm improve when conditioned on types generating higher expected rewards; however, we show how this need not be the case in the absence of MLRP. Implications of these results are derived for the theories of job search and matching, as well as other applications of the bandit paradigm.

Two-person repeated games with no discounting are considered where there is uncertainty about the type of the players. If there is a possibility that a player is an automaton committed to a particular pure or mixed stage -game action, then this provides a lower bound on the Nash equilibrium payoffs to a normal type of this player. The lower bound is the best available and is robust to the existence of other types. The results are extended to the case of two-sided uncertainty. This work extends Schmidt (1993) who analysed the restricted class of conflicting interest games.

The authors study repeated games in which players observe a public outcome that imperfectly signals the actions played. They provide conditions guaranteeing that any feasible, individually rational payoff vector of the stage game can arise as a perfect equilibrium of the repeated game with sufficiently little discounting. The central condition requires that there exist action profiles with the property that, for any two players, no two deviations--one by either player--give rise to the same probability distribution over public outcomes. The results apply to principal-agent, partnership, oligopoly, and mechanism-design models, and to one-shot games with transferable utilities. Copyright 1994 by The Econometric Society.

The authors demonstrate existence of a perfect foresight equilibrium under borrowing constraints in a one-sector model with infinitely-lived heterogeneous agents. The class of admissible preferences includes, but is not limited to, recursive preferences. Existence is proven using a t$2Ctonnement argument under appropriate conditions on preferences and technology. A new measure of discounting, the norm of marginal impatience, is used to determine which technologies are admissible. Depending on the norm of marginal impatience, the admissible technology may either allow for permanent growth or have a maximum sustainable stock. Copyright 1991 by The Econometric Society.

A dynamic programming model of job exit behavior and retirement is constructed and estimated using the method of simulated moments. The model and estimation method allow for both unobserved individual effects and unobserved job-specific "match" effects. The model is estimated using two different assumptions about individual discount factors. First, a static model, with the discount factor equal to zero, is estimated. Then a dynamic model, with the discount factor equal to .95 is estimated. In both models, it is found that bad health, age, and lack of education increase the probability of retirement. The dynamic model performs better than the static model and has different implications for retirement behavior. The job-specific effects are an important source of unobserved heterogeneity. Copyright 1991 by The Econometric Society.

The problem of controlling a stochastic process, with unknown parameters over an infinite horizon, with discounting is considered. Agents express beliefs about unknown parameters in terms of distributions. Under general conditions, the sequence of beliefs converges to a limit distribution. The limit distribution may or may not be concentrated at the true parameter value. In some cases, complete learning is optimal; in others, the optimal strategy does not imply complete learning. The paper concludes with examination of some special cases and a discussion of a procedure for generating examples in which incomplete learning is optimal. Copyright 1988 by The Econometric Society.

In a repeated principal-agent game (supergame) in which each player's criterion is his long-run average expected utility, efficient behavior can be sustained by a Nash equilibrium if it is Pareto-superior to a one-period Nash equilibrium. Furthermore, if the players discount future expected utilities, then for every positive epsilon, and every pair of discount factors sufficiently close to unity (given epsilon), there exists a supergame equilibrium that is within epsilon (in normalized discounted expected utility) of the target efficient behavior. These supergame equilibria are explicitly constructed with simple "review strategies."

This paper treats dynamic optimization problems from the point of view of programming in infinite dimensional spaces. Sufficient conditions (of a dominant diagonal nature) for smooth dependence of optimal paths on initial conditions and other parameters are given. It is also shown that the smooth dependence and the so called "turnpike property" are very closely related. This relationship is used to prove a turnpike theorem for a model that may be interpreted as a multisectoral model of optimal growth with positive discounting and to show that the turnpike property is kept under "small" perturbations.

This paper presents a systematic framework for studying infinitely-repeated games with discounting, focusing on pure strategy (subgame) perfect equilibria. It introduces a number of concepts whi ch organize the theory in a natural way. These include the idea of an optimal penal code, and the related notions of simple penal codes an d simple strategy profiles. Copyright 1988 by The Econometric Society.

We provide game theoretic foundations for the classic kinked demand curve equilibrium and Edgeworth cycle. We analyze a model in which firms take turns choosing prices; the model is intended to capture the idea of reactions based on short-run commitment. In a Markov perfect equilibrium (MPE), a firm's move in any period depends only on the other firm's current price. There are multiple MPE's, consisting of both kinked demand curve equilibria and Edgeworth cycles. In any MPE, profit is bounded away from the Bertrand equilibrium level. We show that a kinked demand curve at the monopoly price is the unique symmetric "renegotiation proof" equilibrium when there is little discounting. We then endogenize the timing by allowing firms to move at any time subject to short-run commitments. We find that firms end up alternating, thus vindicating the ad hoc timing assumption of our simpler model. We also discuss how the model can be enriched to provide explanations for excess capacity and market sharing. KEiiwoRDs: Tacit collusion, Markov perfect equilibrium, kinked demand curve, Edgeworth cycle, excess capacity, market sharing, endogenous timing.

We consider optimal capital accumulation in a nonlinear activity analysis model in which production and primary resource supplies are affected by a stationary stochastic process of exogenous shocks; the optimality criterion is the sum of discounted expected future social utilities. Under various "neoclassical" conditions on technology and preferences, (i) there exists an optimal policy of investment and consumption expressible as a continuous time-invariant function of the capital stocks and the history of stochastic shocks, and (ii) there is a stationary stochastic process of capital stocks that is consistent with the optimal policy.

For several years the Federal Reserve Board and the Federal Reserve banks have employed a method of presenting figures derived from Federal Reserve bank condition statements and from Treasury circulation statements organized in such a manner as to define the channels connecting banking and monetary conditions of the country with the Federal Reserve banks. These figures are presented in the form of a balanced statement, which shows the various currency and credit elements that correspond to increases or decreases in the supply of and in the demand for reserves of member banks. The importance of factors of supply and use of member bank reserves lies in the fact that the ability of member banks to make loans or investments and their attitude in the matter are influenced by the availability to them of reserves and by the method through which these reserves are obtained. There is an important difference in cost, in liability, and in attitude of the banks between reserves obtained at the banks' initiative through discounting paper and reserves obtained either through open-market operations by the Reserve baniks or through the inflow of gold from abroad or of currency from circulation.