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Equilibrium in Competitive Insurance Markets:

A Model An individual has income W if he has no accident. He has income of W − d in case he is unlucky. He can insure himself against the accident by paying a premium α1 , and in case of an accident he will be compensated 2 αˆ by the insurer. Given insurance, his income is ( ) 1 2 W −α ,W − d +α in the case of (no accident, accident), where 2 2 1 α =αˆ −α . The vector ( ) 1 2 α = α ,α completely describes the insurance contract. 4 Individual i’s expected utility may be described as ( ) ( ) ( ) 1 2 1− pi U W −α + piU W − d +α . We will assume that all individuals are identical in all respects except for their probability of having an accident, which may be either pL or pH > pL . The proportion of high-risk individuals in the population is λ ∈(0,1) . Insurers are assumed to be risk-neutral. Thus a contract α when sold to individual i has expected value to the firm of ( ) α1 − i α1 +α 2 p . Insurers are assumed to have sufficient financial resources so that they are willing and able to sell contracts they expect to be profitable. Further, the market is competitive in that there is free entry. These assumptions, as in Rothschild and Stiglitz, ensure that any contract which is demanded and which is expected to be profitable will be supplied.