Economics - General Health Insurance Optimal Ins (Theory)

Self-protection and insurance

Typically, economists when economists look at the health insurance market, they focus on the insurance side of it. By this I mean to define insurance as the purchase of a product which will reimburse the buyer in the case of an adverse event. However, one must also look at the concept of protection. Protection is defined as expending a costly effort to reduce the probability of an adverse event. This costly effort, however, will not effect the amount of the loss, only the probability that it occurs.

A seminal paper by Ehrlich and Becker (JPE 1972) finds the optimal levels of self-protection and how optimal self-protection change when insurance markets are introduced. Let us assume that the probability of a loss is p(e) where e is the effort expended and p'<0. An expected utility maximizer optimizes the following function:

  • maxe [1-p(e)]*U(I -e) + p(e)*U(I – L – e)

The first order condition is:

  • -p’*[U(I -e)-U(I – L – e)]=(1-p)*U'(I -e) + p*U'(I – L – e)

Ehrlich and Becker note that “[t]he term on the left is the marginal gain from the reduction in p; that on the right, the decline in utility due to the decline in both incomes, is the marginal cost.”

When we introduce an insurance market, the expected utility maximizer faces a new objective function.

  • maxe,s ,s [1-p(e)]*U(I-e-s*π(e)) + p(e)*U(I – L – e + s)

Here s is the insurance benefit and π(e) is price of the insurance; s*π(e) is the insurance premium. Let U(0)=U(I-L-e+s) and U(1)=U(I-e-s*π(e)). The first order conditions now become:

  • -(1-p)U'(1) + p*U'(0)=0
  • -p’*[U(1)-U(0)] – (1-p)*U'(1)*[1+s*π’] – p*U'(0)=1

How does self protection change when insurance markets are introduced? According to Ehrlich and Becker “On the one hand, self protection is discouraged because its marginal gain is reduced by the reduction of the difference between the incomes and thus the utilities in different states, on the other hand, it is encouraged if the price of market insurance is negatively related to the amount spent on protection through the effect of these expenditures on the probabilities.”

If insurance companies are actually able to measure self-protection and can price insurance accordingly, then individuals will have some incentive to increase prevention in order to lower their premiums. If insurance is priced in an actuarially fair manner (i.e., π=p(e)/[1-p(e)]) we can show that premiums will drop when self-protection increases:

  • ∂π/∂e=p’/(1-p)2<0

However, if insurance companies are not able to observe self-protection efforts, than it is likely that moral hazard will occur–self protection will decrease. In the words of the authors, “Self-protection would then usually be discouraged by market insurance–moral hazard would exist–because the main effect of introducing market insurance would be to narrow the differences between incomes in different states.”