Suppose an expected utility maximizing consumer with Bernoulli utility on consumption given by 10 ln(c) has wealth $10 and faces a loss of $2. The probability of loss depends on the consumer's effort in avoiding it: P r(e) = 0.2 0.01e. The consumer faces a utility loss from exerting this effort. Suppose this effort cost subtracts from the expected utility and is given by the function c(e) = 0.1e2. (So total expected utility will be P r(e)10 ln(cL) + (1 P r(e))10 ln(cN ) 0.1e2.)

1. What is the optimal level of effort the consumer should exert?

2. Suppose the consumer can purchase insurance at some price q that is actually fair at the optimal level of effort. Verify that the consumer would not actually exert the effort and that the firm would consequently loose money (have negative profits).