Stochastically independent randomization and uncertainty aversion

Summary.   This paper proposes a preference-based condition for stochastic independence of a randomizing device in a product state space. This condition is applied to investigate some classes of preferences that allow for both independent randomization and uncertainty or ambiguity aversion (a la Ellsberg). For example, when imposed on Choquet Expected Utility (CEU) preferences in a Savage framework displaying uncertainty aversion in the spirit of Schmeidler [27], it results in a collapse to Expected Utility (EU). This shows that CEU preferences that are uncertainty averse in the sense of Schmeidler should not be used in settings where independent randomization is to be allowed. In contrast, Maxmin EU with multiple priors preferences continue to allow for a very wide variety of uncertainty averse preferences when stochastic independence is imposed. Additionally, these points are used to reexamine some recent arguments against preference for randomization with uncertainty averse preferences. In particular, these arguments are shown to rely on preferences that do not treat randomization as a stochastically independent event. Received: February 10, 2000; revised version: March 30, 2000     

Unintended hedging in ambiguity experiments

We describe an ambiguity hedging problem in Ellsberg experiments, where combinations of individually ambiguous bets eliminate aggregate ambiguity, and which may yield incorrect classifications of ambiguity averse subjects. We propose a new classification consistent with this hedging possibility.     

Cobb-Douglas preferences under uncertainty

This paper axiomatizes Cobb-Douglas preferences under uncertainty. First, we extend the original Trockel (Econ Lett 30:7–10, 1989)’s axiomatic foundation to a general state space framework based on the Strong Homotheticity Axiom, obtaining also the incomplete case a la Bewley (Decis Econ Financ 25:79–110, 2002). We show that this key axiom for the Cobb-Douglas expected utility specification is refuted by Ellsberg’s uncertainty aversion behavioral pattern. Our main result provides a set of meaningful axioms characterizing Cobb-Douglas min-expected utility preferences, an important class of uncertainty averse preferences for studying the consequences of ambiguity in finance and other fields. Finally, we present briefly how to obtain more general representations like the variational case.     

Regret theory and the competitive firm

This paper examines the production decision of the competitive firm under uncertainty when the firm is not only risk averse but also regret averse. Regret-averse preferences are characterized by a modified utility function that includes disutility from having chosen ex-post suboptimal alternatives. The extent of regret depends on the difference between the actual profit and the maximum profit attained by making the optimal production decision had the firm observed the true realization of the random output price. If the firm is not too regret averse, we show that the conventional result that the optimal output level under uncertainty is less than that under certainty holds. Using a simple binary model wherein the random output price can take on either a low value or a high value with positive probability, we show the possibility that the firm may optimally produce more, not less, under uncertainty than under certainty, particularly when the firm is sufficiently regret averse and the low output price is very likely to prevail.     

Optimal sequential contract with a risk‐averse supplier

We study optimal contracts in environments where a risk‐averse supplier discovers cost information privately and gradually over time: the supplier is privately informed about its cost uncertainty at the time of contracting and discovers the realization of cost condition privately after contracting and before production. We show that both the buyer and the supplier prefer more cost uncertainty when the supplier is not very risk‐averse but less cost uncertainty when the supplier is sufficiently risk‐averse. However, the buyer always prefers to contract before the cost uncertainty resolves regardless of the supplier's degree of risk aversion. The nature of the optimal contract also depends on the supplier's risk preference. A separating contract is optimal when the supplier is not very risk‐averse; however, a pooling contract, which offers the same contract terms regardless of the cost uncertainty, can be optimal when the supplier becomes sufficiently risk‐averse. Moreover, the optimal production schedule is often characterized by “inflexible rules.”     

International Trade,Hedging, and the Demand for Forward Contracts*

One of the main results of the literature on the effects of uncertainty on trade states that uncertainty should not matter in the presence of well‐developed forward markets. Empirical studies, however, do not support this result. We derive the demand for forward cover in a small open economy with terms‐of‐trade uncertainty. Adopting a standard and more realistic decision structure than the one usually used in this literature, we find that risk‐averse agents will not buy forwards at an unbiased price. Agents treat forward contracts as an asset rather than as an insurance. This is the reason why, when calibrating the model, only 17% of imports are covered by forwards.     

DISPLAYING UNCERTAIN INFORMATION ABOUT PROBABILITY: EXPERIMENTAL EVIDENCE

Different from previous experiments that used three representations of uncertain information for probabilities: best estimate, interval and sets of probabilities, we use visual display to represent different levels of uncertainty through varying amount of probabilistic information provided to subjects. Results confirm that the individuals’ willingness to pay is higher when a larger amount of information is available. Further, individuals are found uncertainty averse for high probability of gain and uncertainty seekers for low probability of gain. Similarities in results across different representations of uncertainty indicate that the representation tested here is a viable method for communicating uncertainty to decision makers.     

UNCERTAINTY AND THE THEORY OF MARK-UP PRICING

This paper examines the role of demand uncertainty in influencing a firm's mark-up pricing decision. With no uncertainty the marginalist approach represents this mark-up as inversely and solely determined by the elasticity of demand. Here it is shown that the introduction of uncertainty does not alter this simple dependency for a risk neutral firm. For a risk averse firm, however, the mark-up is shown to depend on a range of factors, including the level of fixed and variable costs and the level of expected demand. It is argued that such variability of margin is more in keeping with observed behaviour.     

Preservation of “more risk averse” under expectations

The preservation of risk aversion properties of utility functions under expectation operations is of interest in the study of derived utility functions in sequential decision problems and in problems with multiple (contemporaneous) sources of uncertainty. The purpose of this note is to present sufficient conditions for the Pratt relation “more risk averse” to be preserved under expectations.     

Take a risk: Preferring prevention over control of biological invaders

Scientists have argued that invasive species can be managed most cost effectively with greater investments in prevention. Further, under ideas like the precautionary principle it is reasonable to expect that a cautious manager would use more prevention relative to control because it keeps more invaders out. Yet, this is not typically done. In many cases, private and public resources are invested primarily to control existing invaders rather than to prevent new invasions. Managers frequently wait until after invaders have arrived and then scramble to limit the damages. We believe these paradoxical decisions can be understood by recognizing the link between typical human preferences for risk bearing and the technology of risk reduction. We demonstrate quantitatively how managers perceived to be cautious or averse to risk tend to shy away from prevention relative to control. This counterintuitive result arises because control is a safer choice than prevention because its productivity is relatively less risky: it works to remove existing invaders from the system. In contrast, the productivity of prevention is more uncertain because prevention only reduces the chance of invasion, it does not eliminate it, and invasion may not occur even in the absence of prevention. Managers' averse to risk will inherently avoid as much uncertainty as possible, whether the source of uncertainty regards ecological outcomes or economic productivity. Implications for environmental decision making are clear. In invasive species management, if managers act as though they are risk averse, their caution can backfire when it leads to more control rather than prevention. The social consequences of this choice are a greater probability of future invasions and lower social welfare. Our results suggest that social welfare is highest when managers were willing to “take a risk” with prevention.     

A theory of subjective compound lotteries

We develop a Savage-type model of choice under uncertainty in which agents identify uncertain prospects with subjective compound lotteries. Our theory permits issue preference; that is, agents may not be indifferent among gambles that yield the same probability distribution if they depend on different issues. Hence, we establish subjective foundations for the Anscombe-Aumann framework and other models with two different types of probabilities. We define second-order risk as risk that resolves in the first stage of the compound lottery and show that uncertainty aversion implies aversion to second-order risk which implies issue preference and behavior consistent with the Ellsberg paradox.     

Knightian uncertainty and poverty trap in a model of economic growth

This paper explores how Knightian uncertainty affects dynamic properties in an economic growth model. The decision-making theory employed in the analysis is the theory of expected utility under a non-additive probability measure, i.e., the Choquet expected utility model of preference. We apply this decision-making theory to an overlapping generations model where producers face “uncertainty” in their technologies. When the producer is averse to uncertainty, the firm's profit function may not be differentiable. Therefore, the firm's decision to invest and hire labor becomes rigid for a certain measurable range of real interest rates. In dynamic equilibrium, the existence of firm-level rigidity causes discontinuity in the wage function; this makes multiple equilibria the more likely outcomes under the log utility and Cobb–Douglas production functions. In this paper, we show that even if aversion to uncertainty is small, the “poverty trap” can arise for a wide range of parameter values.     

Knightian decision theory and econometric inferences

An uncertainty averse Knightian decision maker has a set of probability distributions over outcomes and chooses something other than the status quo only if the change increases the expected payoff according to all the distributions. It is possible to define a standardized degree of uncertainty aversion. To each such degree, there corresponds a set of prior distributions over the parameters of a Gaussian linear regression model, these priors being centered on a uniform prior. The set of posterior means corresponding to this set of priors has the same properties as a standard confidence region.     

The Impact of Regret on Exports

We examine optimal production and export decisions of a firm facing exchange rate uncertainty, where the firm's management is not only risk averse but also regret averse, i.e., is characterized by a utility function that includes disutility from having chosen ex post suboptimal alternatives. Experimental and empirical results support the view that managers tend to be regret averse. Under regret aversion a negative risk premium need not preclude the firm from exporting which would be the case if the firm were only risk averse. Exporting creates an implicit hedge against the possibility of regret when the realized spot exchange rate turns out to be high. The regret‐averse firm as such has a greater ex ante incentive to export than the purely risk averse firm. Finally, we use a two‐state example to illustrate that the firm optimally exports more (less) to the foreign country than in the case of pure risk aversion if the low (high) spot exchange rate is more likely to prevail. Regret aversion as such plays a crucial role in determining the firm's optimal allocation between domestic sales and foreign exports.     

A SMOOTH AMBIGUITY MODEL OF THE COMPETITIVE FIRM

This paper examines the optimal production decision of the competitive firm under price uncertainty when the firm's preferences exhibit smooth ambiguity aversion. Ambiguity is modeled by a second‐order probability distribution that captures the firm's uncertainty about which of the subjective beliefs govern the price risk. Ambiguity preferences are modeled by the (second‐order) expectation of a concave transformation of the (first‐order) expected utility of profit conditional on each plausible subjective distribution of the price risk. Within this framework, we derive necessary and sufficient conditions under which the ambiguity‐averse firm optimally produces less in response either to the introduction of ambiguity or to greater ambiguity aversion when ambiguity prevails. In the case that the price risk is binary, we show that ambiguity and greater ambiguity aversion always adversely affect the firm's production decision.     

Long-run industry equilibrium with uncertainty

In this paper we consider the effects of uncertainty on industry equilibrium when firms must commit themselves to production before prices are revealed. We show that (a) an increase in demand uncertainty will (i) not affect the equilibrium number and size of firms if they are risk neutral, (ii) reduce the equilibrium number of firms if they are risk averse, but will have an ambiguous effect on their size. (b) In equilibrium, firms operate at capacity if they are risk neutral, but at excess capacity if they are risk averse.     

Event-Separability in the Ellsberg urn

Ellsberg??s three-color urn involves two distinct sources of uncertainty??the color composition of the urn (which is subjective) and the identity of the drawn ball (which is objective)??and bets on it can involve mixed objective/subjective uncertainty. It is known that typical betting preferences on this urn violate both the Sure-Thing Principle and probabilistic sophistication over its mixed uncertainty but are consistent with both of them over its purely subjective uncertainty. In this paper, we show that the standard Ellsberg-type preference reversal is actually implied by the Independence Axiom over its purely objective uncertainty.     

Risk, ambiguity, and state-preference theory

The state-preference framework for modeling choice under uncertainty, in which objects of choice are allocations of wealth or commodities across states of the world, is a natural one for modeling ??smooth?? ambiguity-averse preferences. It does not require reference to objective probabilities, personalistic consequences, or counterfactual acts, and it allows for state dependence of utility and unobservable background risk. The decision maker??s local revealed beliefs are encoded in her risk-neutral probabilities (her relative marginal rates of substitution between states) and her local risk preferences are encoded in the matrix of derivatives of the risk-neutral probabilities. This matrix plays a central but generally unappreciated role in the modeling of risk attitudes in the state-preference framework. It can be computed by inverting a bordered Slutsky matrix and vice versa, it generalizes the Arrow?CPratt measure for approximating local risk premia, and its structure reveals whether the decision maker??s risk preferences are ambiguity averse as well as risk-averse. Two versions of the smooth ambiguity model are analyzed??the source-dependent risk aversion model and the second-order uncertainty (KMM) model??and it is shown that in both cases, the overall premium for local uncertainty can be decomposed as the sum of a risk premium and an ambiguity premium.     

Uncertainty averse preferences

We study uncertainty averse preferences, that is, complete and transitive preferences that are convex and monotone. We establish a representation result, which is at the same time general and rich in structure. Many objective functions commonly used in applications are special cases of this representation.     

Fixed versus variable rate loans under regret aversion

This paper examines the optimal mix of fixed and variable rate loans of a competitive bank facing funding cost uncertainty, where the bank is not only risk averse but also regret averse. Regret aversion is characterized by a utility function that includes disutility from having chosen ex-post suboptimal alternatives. We show that a negative spread between fixed and variable rate loans is a necessary but not sufficient condition for the dominance of variable rate loans over fixed rate loans. If the bank optimally extends both fixed and variable rate loans, the total amount of loans depends neither on the bank's regret aversion nor on the funding cost uncertainty. The bank, however, optimally lends less should it be forced to assume the entire funding cost uncertainty by exclusively extending fixed rate loans. Finally, using a two-state example, we show that the bank optimally extends more (less) fixed rate loans than in the case of pure risk aversion if the high (low) marginal cost of funds is more likely to prevail. Regret aversion as such plays a crucial role in determining the bank's optimal choice between fixed and variable rate loans.