On the nature of certainty equivalent functionals

We explore connections between the certainty equivalent return (CER) functional and the underlying utility function. Curvature properties of the functional depend upon how utility function attributes relate to hyperbolic absolute risk aversion (HARA) type utility functions. If the CER functional is concave, i.e., if risk tolerance is concave in wealth, then preferences are standard. The CER functional is linear in lotteries if utility is HARA and lottery payoffs are on a line in state space. Implications for the optimality of portfolio diversification are given. When utility is concave and non-increasing relative risk averse, then the CER functional is superadditive in lotteries. Depending upon the nature of association among lottery payoffs, CERs for constant absolute risk averse utility functions may be subadditive or superadditive in lotteries. Our approach lends itself to straightforward experiments to elicit higher order attributes on risk preferences.     

On the nature of certainty equivalent functionals

We explore connections between the certainty equivalent return (CER) functional and the underlying utility function. Curvature properties of the functional depend upon how utility function attributes relate to hyperbolic absolute risk aversion (HARA) type utility functions. If the CER functional is concave, i.e., if risk tolerance is concave in wealth, then preferences are standard. The CER functional is linear in lotteries if utility is HARA and lottery payoffs are on a line in state space. Implications for the optimality of portfolio diversification are given. When utility is concave and non-increasing relative risk averse, then the CER functional is superadditive in lotteries. Depending upon the nature of association among lottery payoffs, CERs for constant absolute risk averse utility functions may be subadditive or superadditive in lotteries. Our approach lends itself to straightforward experiments to elicit higher order attributes on risk preferences.     

A polyhedral approximation approach to concave numerical dynamic programming

This paper introduces a numerical method for solving concave continuous state dynamic programming problems which is based on a pair of polyhedral approximations of concave functions. The method is globally convergent and produces computable upper and lower bounds on the value function which can in theory be made arbitrarily tight. This is true regardless of the pattern of binding constraints, the smoothness of model primitives, and the dimensionality and rectangularity of the state space. We illustrate the method's performance using an optimal firm management problem subject to credit constraints and partial investment irreversibilities.     

Dynamic theory of preferences: Habit formation and taste for variety

We analyze individual preferences over infinite horizon consumption choices. Our axioms provide the foundation for a recursive representation of the utility function that contains as particular cases the classical Koopmans representation (Koopmans (1960)) as well as the habit formation specification.We examine some of the consequences of our axiomatization by considering a standard consumer choice problem, and show that typically in the space of concave utility functions satisfying our axioms the consumer displays a taste for variety. The latter means that such a consumer selects optimally time variant consumption programs for any given time invariant sequence of commodities’ relative prices and for all possible sequences of market discount factors. In contrast, if a concave utility function satisfies Koopmans’ axioms the consumer does not display a taste for variety.     

Family size and social utility: income distribution dominance criteria

"This paper generalizes previous results on income distribution dominance in the case where the population of income recipients is broken down into groups with distinct utility functions. The example taken here is that of income redistribution across families of different sizes. The paper first investigates the simplest assumptions that can be made about family utility functions. A simple dominance criterion is then derived under the only assumptions that family functions are increasing and concave with income and the marginal utility of income increases with family size."     

Optimal insurance under moral hazard in loss reduction

This study investigates the optimal insurance when moral hazard exists in loss reduction. We identify that the optimal insurance is full insurance up to a limit and partial insurance above that limit. In case of partial insurance, the indemnity schedule for prudent individual is convex, linear, or concave in loss, depending on the shapes of the utility and loss distribution. The optimal insurance may include a deductible for large losses only when the indemnity schedule is convex. It may also include a fixed reimbursement when the schedule is convex or concave. When the loss distribution belongs to the one dimensional exponential family with canonical form, the indemnity schedule is concave under IARA and CARA, whereas it can be concave or convex under DARA.     

On the Consistency of Maximum Likelihood Estimation of Monotone and Concave Production Frontiers

Banker and Maindiratta (1992) provides a method for the estimation of a stochastic production frontier from the class of all monotone and concave functions. A key aspect of their procedure is that the arguments in the log-likelihood function are the fitted frontier outputs themselves rather than the parameters of some assumed parametric functional form. Estimation from the desired class of functions is ensured by constraining the fitted points to lie on some monotone and concave surface via a set of inequality restrictions. In this paper, we establish that this procedure yields consistent estimates of the fitted outputs and the composed error density function parameters.     

Approximation of convex preferences

It is shown geometrically that a monotone concave preference order can be approximated by orders representable by a concave utility function. This is applied to proving that preferences with ‘desirable’ properties (such as inducing smooth excess demand functions, analyticity, strict convexity) are dense.     

Subdifferentiability and lipschitz continuity in experimental design problems

The calculus of concave functions is a widely accepted tool for optimum experimental design problems. However, as a function of the support points and the weights the design problem fails to be concave. In this paper we make use of generalized gradients in the sense of Rockafellar (1980) and Clarke (1983). A chain rule is presented for the subdifferential of the composition of an information function with the moment matrix mapping. Lipschitz continuity of the global design function is proved and conditions for strict differentiability are given.     

On the separability of the quasi concave closure of an additively separable function

This paper gives a characterization of the additively separable functions whose quasi concave closure also is additively separable.     

Semi-nonparametric estimation of independently and identically repeated first-price auctions via an integrated simulated moments method

In this paper we propose to estimate the value distribution of independently and identically repeated first-price auctions directly via a semi-nonparametric integrated simulated moments sieve approach. Given a candidate value distribution function in a sieve space, we simulate bids according to the equilibrium bid function involved. We take the difference of the empirical characteristic functions of the actual and simulated bids as the moment function. The objective function is then the integral of the squared moment function over an interval. Minimizing this integral to the distribution functions in the sieve space involved and letting the sieve order increase to infinity with the sample size then yields a uniformly consistent semi-nonparametric estimator of the actual value distribution. Also, we propose an integrated moment test for the validity of the first-price auction model, and an data-driven method for the choice of the sieve order. Finally, we conduct a few numerical experiments to check the performance of our approach.     

Necessity of hyperbolic absolute risk aversion for the concavity of consumption functions

Carroll and Kimball (1996) have shown that, in the class of utility functions that are strictly increasing, strictly concave, and have nonnegative third derivatives, hyperbolic absolute risk aversion (HARA) is sufficient for the concavity of consumption functions in general consumption-saving problems. This paper shows that HARA is necessary, implying the concavity of consumption is not a robust prediction outside the HARA class.     

Isotonic estimators of monotone densities and distribution functions: basic facts

We present short proofs of some basic results from isotonic regression theory. A straightforward argument is given to show that the left continuous version of the concave majorant of the empirical distribution function maximizes the likelihood function f↦f (X,)… f (X _n ) within the class of non-increasing densities. Similarly, it is shown that the nonparametric maximum likelihood estimator (NPMLE) of the distribution function of interval censored data has an interpretation in terms of the left derivative of a convex minor ant. Finally, a short proof is given to show that the number of vertices of the concave major ant of the uniform empirical distribution function is asymptotically normal with asymptotic mean and variance both equal to log n .     

Risk neutrality regions

An Expected Utility maximizer can be risk neutral over a set of nondegenerate multivariate distributions even though her NM (von Neumann Morgenstern) index is not linear. We provide necessary and sufficient conditions for an individual with a concave NM utility to exhibit risk neutral behavior and characterize the regions of the choice space over which risk neutrality is exhibited. The least concave decomposition of the NM index introduced by Debreu (1976) plays an important role in our analysis as do the notions of minimum concavity points and minimum concavity directions. For the special case where one choice variable is certain, the analysis of risk neutrality requires modification of the Debreu decomposition. The existence of risk neutrality regions is shown to have important implications for the classic consumption–savings and representative agent equilibrium asset pricing models.     

Perron's stability theorem for non-linear mappings

Perron's theorem on positive matrices including its stability statement is extended to non-linear mappings which need neither additive nor homogeneous nor primitive. This generalizes known results and yields in particular a concave version of Perron's theorem. The theorem may be applied to balanced growth in non-linear systems and also to obtain a dynamic non-substitution theorem for general cost functions.     

Some unresolved issues in the location theory of the firm

This paper presents a generalized location theory of the firm in linear space. Location outcomes are examined by utilizing a general as well as a particular transport rate structure for output and inputs and using a general concave production as well as a homogeneous production function, of the firm. The effect of a change in demand on the location is also investigated.     

Multidimensional inequality and inframodular order

Motivated by the pertinence of Pigou–Dalton (PD) transfers for inequality measurement when only one attribute is involved, we show that inframodular functions are consistent with multidimensional PD transfers and that weakly inframodular functions fit more accurately with the traditional notion of PD transfers. We emphasize, for inequality rankings of allocations of multiple attributes in a population, the similarities of the inframodular order, defined using inframodular functions, with the concave order in the unidimensional framework.     

Differentiated oligopolistic markets with concave cost functions via Ky Fan inequalities

A Nash–Cournot model for oligopolistic markets with concave cost functions and a differentiated commodity is analyzed. Equilibrium states are characterized through Ky Fan inequalities. Relying on the minimization of a suitable merit function, a general algorithmic scheme for solving them is provided. Two concrete algorithms are therefore designed that converge under suitable convexity and monotonicity assumptions. The results of some numerical tests on randomly generated markets are also reported.     

Nonparametric estimation of concave production technologies by entropic methods

An econometric methodology is developed for nonparametric estimation of concave production technologies. The methodology, based on the principle of maximum likelihood, uses entropic distance and convex programming techniques to estimate production functions. Empirical applications are presented to demonstrate the feasibility of the methodology in small and large datasets. Copyright © 2007 John Wiley & Sons, Ltd.     

Lipschitz continuous policy functions for strongly concave optimization problems

We prove that the policy function, obtained by optimizing a discounted infinite sum of stationary return functions, is Lipschitz continuous when the instantaneous function is strongly concave. Moreover, by using the notion of α-concavity, we provide an estimate of the Lipschitz constant which turns out to be a decreasing function of the discount factor.