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.postprin

The Incentive To Trade Under Ambiguity Aversion

This paper examines the behavior of an exporting firm that sells in both the home country and a foreign country. The firm makes its optimal production and export decisions when facing ambiguous exchange rate risk. Ambiguity is modeled by a second-order probability distribution that captures the firm's uncertainty about which of the subjective beliefs govern the exchange rate 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 exchange rate risk. Within this framework, we derive necessary and sufficient conditions under which the ambiguity-averse firm optimally sells more in the home country and exports less to the foreign country in response either to the introduction of ambiguity or to greater ambiguity aversion when ambiguity prevails. We further show that ambiguity and ambiguity aversion have adverse effect on the firm's incentive to export to the foreign country.postprin

The banking firm under ambiguity aversion

We examine risk taking when the bank's preferences exhibit smooth ambiguity aversion. Ambiguity is modeled by a second-order probability distribution that captures the bank's uncertainty about which of the subjective beliefs govern the financial asset return 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 return risk. Within this framework, the banking firm finds it less attractive to take risk in the presence than in the absence of ambiguity. This result extends to the case of greater ambiguity aversion. Given that the competitive bank's smooth ambiguity preferences exhibit non-increasing absolute ambiguity aversion, imposing a more stringent capital requirement to the bank reduces the optimal amount of loans, if the bank's coefficient of relative risk aversion does not exceed unity. Ambiguity and ambiguity aversion as such have adverse effect on the bank's risk taking

Ambiguity and the Incentive to Export

This paper examines the optimal production and export decisions of an international firm facing exchange rate 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 exchange rate 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 exchange rate risk. Within this framework, we show that ambiguity has no impact on the firm's propensity to export to a foreign country. Ambiguity and ambiguity aversion, however, are shown to have adverse effect on the firm's incentive to export to the foreign country

Ambiguity and the Value of Hedging

This paper examines the optimal production and hedging decisions of the competitive firm under price uncertainty when the firm's preferences exhibit smooth ambiguity aversion and an unbiased forward hedging opportunity is available. 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, the separation and full-hedging theorems remain intact. Banning the firm from trading its output forward at the unbiased forward price has adverse effect on the firm's production decision. The firm finds the unbiased forward hedging opportunity more valuable in the presence than in the absence of ambiguity. Furthermore, the value of hedging increases when the firm's beliefs are more ambiguous, or when the firm becomes more ambiguity averse.postprin

Contracting in the Presence of Uncertainty

This thesis concerns the enforcement of contracts in the presence of uncertainty. Sometimes uncertainty is exogenously given and the agents cannot influence its existence. Frequently, however, there is strategic uncertainty created by the behavior of other agents. Both kinds of uncertainty have important impacts on contracting. Chapter 1 shows that insurers use uncertainty about auditing strategies to fight insurance fraud. For this purpose, we study a costly state verification model with ambiguity. The insurers abstain from commitment to an auditing strategy, even if commitment is possible without incurring any costs. This contrasts with conventional wisdom, which claims that it is optimal to commit, as the credible announcement of thoroughly auditing claim reports might act as a powerful deterrent to insurance fraud. Yet, empirically it is very unusual for insurers to try to overcome the credibility issue. We prove that it can be optimal for the insurers to maintain the ambiguity and forgo commitment. Thus, strategic ambiguity, i.e., the strategic choice to withhold information about auditing costs and strategies, is an equilibrium outcome. The second chapter considers legal uncertainty in competition law. I show that legal uncertainty can be welfare-enhancing, if the uncertainty is not too large. As an example, consider Article 101 (TFEU) prohibiting vertical restraints with a block exemption excluding companies with market shares below 30%. There are guidelines available how the relevant market shares are to be calculated. Nevertheless, it is extremely difficult to predict correctly the market share that the competition authorities will determine. In addition, there is uncertainty about the size of the fine that firms have to pay in case of a conviction. This exemplifies legal uncertainty as scrutinized in the chapter. The third chapter analyzes a principal-agent model, in which the performance measure of the principal is non-verifiable and unobservable by the agent. Instead, the principal has the possibility to communicate with the agent. The communication occurs at the very end of the interaction and there is no repeated interaction. Nevertheless, it is crucial for the agent’s motivation that the principal gives feedback and justifies her evaluation, in particular, in case of bad outcomes. In addition, it is optimal to pool evaluations and to compress wages at the top. These results fit well with empirical observations, like the leniency bias and the centrality bias. Hence, this pattern of evaluations can be understood as a feature of the optimal contract instead of biased behavior. Corresponding to the distinction between first-order and second-order risk aversion, the fourth chapter defines first-order and second-order ambiguity aversion. With second-order ambiguity aversion, for every ambiguity-averse agent there is an ambiguity-neutral agent so that the set of all improvement directions at an unambiguous endowment is the same for both agents. With first-order ambiguity aversion, in contrast, the set of improvement directions is a strict subset of the improvement directions of an ambiguity-neutral agent. Chapter 4 provides three equivalent definitions for the distinction. For this purpose, I introduce a general ambiguity premium and a notion of reference beliefs of an ambiguity-averse agent. This distinction has direct implications for settings in finance, insurance, and contracting. In particular, I consider the validity of an adapted version of Holmström’s informativeness principle under ambiguity aversion

The continuous-time pre-commitment KMM problem in incomplete markets

This paper studies the continuous-time pre-commitment KMM problem proposed by Klibanoff, Marinacci and Mukerji (2005) in incomplete financial markets, which concerns with the portfolio selection under smooth ambiguity. The decision maker (DM) is uncertain about the dominated priors of the financial market, which are characterized by a second-order distribution (SOD). The KMM model separates risk attitudes and ambiguity attitudes apart and the aim of the DM is to maximize the two-fold utility of terminal wealth, which does not belong to the classical subjective utility maximization problem. By constructing the efficient frontier, the original KMM problem is first simplified as an one-fold expected utility problem on the second-order space. In order to solve the equivalent simplified problem, this paper imposes an assumption and introduces a new distorted Legendre transformation to establish the bipolar relation and the distorted duality theorem. Then, under a further assumption that the asymptotic elasticity of the ambiguous attitude is less than 1, the uniqueness and existence of the solution to the KMM problem are shown and we obtain the semi-explicit forms of the optimal terminal wealth and the optimal strategy. Explicit forms of optimal strategies are presented for CRRA, CARA and HARA utilities in the case of Gaussian SOD in a Black-Scholes financial market, which show that DM with higher ambiguity aversion tends to be more concerned about extreme market conditions with larger bias. In the end of this work, numerical comparisons with the DMs ignoring ambiguity are revealed to illustrate the effects of ambiguity on the optimal strategies and value functions.Comment: 53 pages, 7 figure

The von Neumann/Morgenstern approach to ambiguity

Dumav M, Stinchcombe MB. The von Neumann/Morgenstern approach to ambiguity. Center for Mathematical Economics Working Papers. Vol 480. Bielefeld: Center for Mathematical Economics; 2013.A choice problem is risky (respectively ambiguous) if the decision maker is choosing between probability distributions (respectively sets of probability distributions) over utility relevant consequences. We provide an axiomatic foundation for and a representation of continuous linear preferences over sets of probabilities on consequences. The representation theory delivers: first and second order dominance for ambiguous problems; a utility interval based dominance relation that distinguishes between sources of uncertainty; a complete theory of updating convex sets of priors; a Bayesian theory of the value of ambiguous information structures; complete separations of attitudes toward risk and ambiguity; and new classes of preferences that allow decreasing relative ambiguity aversion and thereby rationalize recent challenges to many of the extant multiple prior models of ambiguity aversion. We also characterize a property of sets of priors, descriptive completeness, that resolves several open problems and allows multiple prior models to model as large a class of problems as the continuous linear preferences presented here

Elicitation of ambiguous beliefs with mixing bets

I consider the elicitation of ambiguous beliefs about an event and show how to identify the interval of relevant probabilities (representing ambiguity perception) for several classes of ambiguity averse preferences. The agent reveals her preference for mixing binarized bets on the uncertain event and its complement under varying betting odds. Under ambiguity aversion, mixing is informative about the interval of beliefs. In particular, the mechanism allows to distinguish ambiguous beliefs from point beliefs, and identifies the belief interval for maxmin preferences. For ambiguity averse smooth second order and variational preferences, the mechanism reveals inner bounds for the belief interval, which are sharp under additional assumptions. In an experimental study, participants perceive almost as much ambiguity for natural events (generated by the stock exchange and by a prisoners dilemma game) as for the Ellsberg Urn, indicating that ambiguity may play a role in real-world decision making