Moral Hazard and Moral Imperative

Abstract

In a recent article in this Journal, Wu and Colwell 1988) seek to describe how the optimal level of safety production chosen by an individual is determined. Their objective is to . . . handle questions of moral hazard, and moral imperative in a simpler and more heuristic manner than the previous literature (p. 101). The authors attempt to determine how the marginal benefit of safety production is impacted by various parameters for the cases of full and partial insurance coverage. The conditions under which the presence of insurance will lead to more safety production (moral imperative), and when it will lead to less safety production (moral hazard), are examined. While the questions explored by Wu and Colwell are of interest, many of their conclusions are incorrect, as several of their proofs contain a fundamental error. In particular, the discussion of partial insurance (section Ill of their paper) is fatally flawed, as the proofs of Propositions 1 and 2 are both invalid. Thus, the comparative statics results which follow (section IV of the paper) do not necessarily hold for the case of partial insurance coverage. Furthermore, a portion of their comparative statics results is also invalid for the case of full insurance coverage. The problem in their analysis occurs when equalities are incorrectly treated as identities. The mistake first appears in the proof of Proposition 1, which is contained in Appendix A (p. 116). The incorrect demonstration begins with equation A.1, which is reproduced below using the authors' notation. p'(U[.sub.1] - U[.sub.o]) = [(1 - p)U[.sub.1]' + pUo'](1 - [alpha])L[pi]' The function U[.sub.o] represents utility in the event that a fixed loss (L) occurs, and thus can be expressed as U[.sub.o] = Uo[W - [alpha][pi](s)L - (1 - [alpha])L - C(s)], where W is initial wealth, a is the insurance coverage ratio, C(s) is the cost of safety production, 7r(s) is the price of insurance, and s is the amount of safety production. If a loss does not occur, utility is U[.sub.1] = U[.sub.1](W - [alpha][pi](s)L - C(s)). The probability of a loss (p) depends on the level of safety production, and is written as p = p(s). Wu and Colwell state that the above . . . equality implies that the derivatives of both sides of (A. 1) with respect to a are (p. I 1 6). Taking the derivative Wu and Colwell obtain equation A.2 (p. 116), which is repeated below. [Mathematical Expression Omitted] This equation is not correct. Both sides of equation A.1 depend on s, [alpha], L, and W, and for a particular vector (s, [alpha], L, W) the equality may hold. However, and this is the important point, equation A. I is not an identity, and it does not hold for all [alpha]. Therefore, it is wrong to claim that the derivative of the left-hand side with respect to a will be equal to the derivative of the right-hand side. …