Concavifiability and constructions of concave utility functions

Concavifiable convex preference orderings are characterized and minimally concave utilities are constructed, using three different approaches. One involves the intersection of arbitrary lines with the three indifference surfaces, another involves conditions on the normals of two indifference surfaces and is related to the super-gradient map of a possible concave utility. In the third approach it is assumed that the ordering is induced by a twice-differentiable utility and Perror's integral of a certain expression formed from the derivatives is used. A possible economic interpretation of minimally concave utilities is suggested, and it is shown that one cannot select concave utilities so that they depend continuously on the ordering.     

Evaluating the Adequacy of Parametric Functional Forms in Estimating Monotone and Concave Production Functions

We consider situations where the a priori guidance provided by theoretical considerations indicates only that the function linking the endogenous and exogenous variables is monotone and concave (or convex). We present methods to evaluate the adequacy of a parametric functional form to represent the relationship given the minimal maintained assumption of monotonicity and concavity (or convexity). We evaluate the adequacy of an assumed parametric form by comparing the deviations of the fitted parametric form from the observed data with the corresponding deviations estimated under DEA. We illustrate the application of our proposed methods using data collected from school districts in Texas. Specifically, we examine whether the Cobb–Douglas and translog specifications commonly employed in studies of education production are appropriate characterizations. Our tests reject the hypotheses that either the Cobb–Douglas or the translog specification is an adequate approximation to the general monotone and concave production function for the Texas school districts.     

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.     

On games without side payments arising from a general class of markets

The class of games without side payments obtainable from finite trader markets having possibly infinite dimensional commodity spaces, individual compact, convex consumption and production sets, and concave upper-semicontinuous utility functions is considered. It is shown that these market games are precisely the totally balanced games. In fact, each totally balanced game is shown to have both a finite commodity representation and an infinite commodity ‘simple’ representation.     

Nonparametric Bayes inference for concave distribution functions

Bayesian inference for concave distribution functions is investigated. This is made by transforming a mixture of Dirichlet processes on the space of distribution functions to the space of concave distribution functions. We give a method for sampling from the posterior distribution using a Pólya urn scheme in combination with a Markov chain Monte Carlo algorithm. The methods are extended to estimation of concave distribution functions for incompletely observed data.     

Proportional distribution schemes

An amount of income can be obtained jointly by m agents, the ith agent's share of income being θ_i. The income and the utilities of each agent are functions of the state of nature. Each agent has a probability measure over the states of nature. An efficient proportional distribution is one which is (1) Pareto optimal and for which (2) the expected proportion of income agent i recieves divided by θ_i is independent of i. It is shown that if the attitudes are strictly concave then there exists exactly one proportional distribution scheme. Furthermore, in special cases, each agent expects to recieve an income that exceeds his share.     

Consumer Risk‐reduction Behavior and New Product Purchases

Consumers purchase lower quantities of new products compared with those they have purchased in the past. We explain this observation as a result of risk‐averting behavior by utility‐maximizing consumers. If a new product involves a higher degree of risk that quality expectations will not be met compared with an incumbent product, we show that utility will be more concave for the new product. We test this prediction using a multiple‐discrete/continuous extreme value (MDCEV) model of demand. We show that utility is indeed more concave for new products relative to previously purchased products. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

Production planning and inventories optimization: A backward approach in the convex storage cost case

We study the deterministic optimization problem of a profit-maximizing firm which plans its sales/production schedule. The firm controls both its production and sales rates and knows the revenue associated to a given level of sales, as well as its production and storage costs. The revenue and the production cost are assumed to be respectively concave and convex. In Chazal et al. [Chazal, M., Jouini, E., Tahraoui, R., 2003. Production planning and inventories optimization with a general storage cost function. Nonlinear Analysis 54, 1365–1395], we provide an existence result and derive some necessary conditions of optimality. Here, we further assume that the storage cost is convex. This allows us to relate the optimal planning problem to the study of a backward integro-differential equation, from which we obtain an explicit construction of the optimal plan.     

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 .     

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.     

Hölder continuity of the policy function approximation in the value function approximation

We show that given a value function approximation V of a strongly concave stochastic dynamic programming problem (SDDP), the associated policy function approximation is Hölder continuous in V.     

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.     

Periodic learning about a hidden state variable

In active learning models the value function is necessarily convex in the priors. Hence, in combination with a concave objective, the decision problem need not become concave so that nonregularity problems are inherent. This paper considers an objective that unambiguously implies a quasi-convex decision problem and highlights the effect of the inherent nonregularities on active learning. A trigger policy for learning is shown to be optimal: the minimum amount of learning is optimal until uncertainty surpasses a critical value. At this trigger point the maximum amount of learning is chosen, uncertainty falls temporarily, and the cycle then repeats itself.     

A simple proof of the nonconcavifiability of functions with linear not-all-parallel contour sets

Consider a real-valued function that, on a convex subset of a real vector space, is continuous on line segments and has convex contour sets. Inspired by a compelling intuitive argument due to Aumann (1975), we provide a simple proof that no strictly increasing transformation of such a function can be concave unless all contour sets are parallel, i.e., unless for every pair of contour sets, either their affine hulls are disjoint or one of their affine hulls contains the other.     

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.     

Afriat’s Theorem and Samuelson’s ‘Eternal Darkness’

Suppose that we have access to a finite set of expenditure data drawn from an individual consumer, i.e., how much of each good has been purchased and at what prices. Afriat (1967) was the first to establish necessary and sufficient conditions on such a data set for rationalizability by utility maximization. In this note, we provide a new and simple proof of Afriat’s Theorem, the explicit steps of which help to more deeply understand the driving force behind one of the more curious features of the result itself, namely that a concave rationalization is without loss of generality in a classical finite data setting. Our proof stresses the importance of the non-uniqueness of a utility representation along with the finiteness of the data set in ensuring the existence of a concave utility function that rationalizes the data.     

Antitrust Compliance: Managerial Incentives and Collusive Behavior

This article analyzes a manager's incentives to establish and sustain an illegal collusive agreement if her firm is subject to profit shocks, if her utility function is concave in profits (e.g., because of risk aversion), and if she incurs opportunity costs (e.g., by violating a social norm). The model supports the empirical observation that if collusion is to be established and sustained in a state with low profits, then this state must be quite persistent. It also indicates that compliance with antitrust laws can be ensured best by combining a zero tolerance policy with a strategy of forgiveness. Copyright © 2017 John Wiley & Sons, Ltd.     

Dense families of low-complexity attainable sets of markets

The complexity of the attainable set of utility outcomes of a market (with finitely many traders) is defined as the least number of commodities involved in any market giving the same set. This notion is investigated both for the case of quasiconcave and concave utility functions. It is shown that, in either case, there is a dense collection of attainable sets, each having complexity at most n(n?1)/2.     

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.