Comparative statics of changes in risk on monotonically and partially responsive kinked payoffs

Abstract

This paper derives two sets of necessary and sufficient conditions on the comparative statics of changes in risk under kinked payoff functions that are monotonically responsive and partially responsive to the realization of risk, respectively. The former case includes the newsboy problem with backorder; the latter case includes the newsboy problem without backorder and under some restrictions, also includes the optimal deductible insurance problem. Some relatively non-restrictive conditions derived from the necessary and sufficient conditions reveal that the three problems can be quite different, even though they are often viewed in the literature as being congruous. These conditions lead to simple predictions of the direction of change in any risk-averse agent’s optimal choice upon a change in risk, without assuming specific functional forms for the utility function.

Introduction

Kinked payoff functions arise naturally in operational research problems. A well-known problem involving kinked payoffs is the newsboy (or newsvendor) problem first studied by Horowitz’s, 1970, Baron’s, 1973. This problem has been widely extended and remains a prominent research topic (e.g., Lau’s, 1980, Sankarasubramaian and Kumaraswamy’s, 1983, Li et al.’s, 1990, Gerchak and Mossman’s, 1992, Khouja’s, 1999, Petruzzi and Dada’s, 1999, Eeckhoudt and Treich’s, 2003, Alfares and Elmorra’s, 2005, Benzion et al.’s, 2008, Abdel-Malek and Nathapol Areeratchakul’s, 2007, Li and Liu’s, 2008, Abdel-Malek et al.’s, 2008). To derive close-form solutions, most studies assume risk-neutral newsboys, even though risk aversion is often believed to be the norm for small businesses and for businesses subject to bankruptcy costs. For a risk-averse newsboy, one needs to assume a specific utility function or that the newsboy is a mean–variance optimizer (e.g., Keren and Pliskin, 2006).

Kinked payoffs also occur in many economics problems, such as optimal deductible insurance purchase (e.g., Arrow’s, 1963, Schlesinger’s, 1981, Schlesinger’s, 2000, Braun and Muermann’s, 2004, Cheung’s, 2007), the new Soviet incentive problem (Kanbur, 1982), and the problem of labor supply under switching tax regimes (Kanbur, 1983). Kanbur’s, 1982, Kanbur’s, 1983 argues that the comparative statics of a change in risk given a risk-averse decision maker under a kinked payoff function differs significantly from that under a smooth payoff function. The latter has been analyzed extensively (e.g., Kraus’s, 1979, Katz’s, 1981, Meyer and Ormiston’s, 1983, Meyer and Ormiston’s, 1985, Landsberger and Meilijson’s, 1990, Hadar and Seo’s, 1990, Ormiston and Schlee’s, 1993, Dionne et al.’s, 1993). In fact, Gollier (1995) has come up with the necessary and sufficient conditions for unambiguous comparative statics of changes in risk under a smooth payoff function. However, Kanbur’s, 1982 concern seems to suggest that Gollier’s, 1995 results may not be applied directly to the case of a kinked payoff function.

Without assuming a special functional form for a newsboy’s utility, Eeckhoudt et al. (1995) derive sufficient conditions for generating unambiguous comparative statics of a change in risk under the classic newsboy problem for any risk-averse newsboy. Eeckhoudt et al. (1991) derive sufficient conditions under the optimal deductible insurance purchase problem. These conditions turn out to be quite different. Dionne and Mounsif (1996) impose the condition of mean-preserving transformation and simplify Eeckhoudt et al.’s (1995) model by eliminating salvage value and backorder. Ibarra-Salazar (2005), however, suggests that the assumption of mean-preserving transformation is unnecessary. Unfortunately, the sufficient conditions derived by these papers turn out to be rather restrictive and hence difficult to apply.

This paper derives the necessary and sufficient conditions for unambiguous comparative statics of a change in risk on a risk-averse decision maker under kinked payoff functions that are monotonically responsive and partially responsive, respectively. These results are applied to the newsboy problem with backorder, the newsboy problem without backorder, and the optimal deductible insurance problem. The first purpose is to derive relatively less restrictive and more readily applicable sufficient conditions than those derived in the literature. The second purpose is to show that the conditions for unambiguous comparative statics can be quite different under the three problems that are often treated in the literature as if they are directly compatible (Dionne and Mounsif’s, 1996, Ibarra-Salazar’s, 2005). The newsboy problem with backorder differs from the newsboy problem without backorder and the optimal deductible insurance problem in that the latter problems have payoff functions that are only partially responsive to the realization of the risk concerned. In addition, the deductible insurance problem differs from the two newsboy problems in that the payoff function is decreasing in the realization of the risk concerned and that it is affected by the distribution of the risk through the premium charge. These give rise to different sets of comparative statics conditions for the three problems.