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Q search or enter website name ? ! problem 2: (auto insurance 20 points recall that in an expected utility model with a good g and bad b state of nature, preferences over state-contingent consumption bundles (c,c) are represented by the utility function where p is the probability of the bad state occurring. naturally, this means the good state occurs at probability 1-p assume that the price of e is si in both states. suppose that bob has preferences given by the expected utility function above and that tr(c)- . suppose that regardless of the state of the world, bob has an income of sw. in the bad state, he incurs a cost of 86 due to medical bills. a competitive insurance company approaches bob and offers him the following type of contract: bob pays 0 < m < 1 cents per dollar of insurance for a total premium of smb. if the bad state happens, the nsurne firm pays bob sb. bob must select b (1) (5 pts) set up the household problem where bob buys b amount of insur- ance, (hint: there will be two budget constraints for each state of the world) (2) (5 pts) set up the firms problem where it is maximz expected profit. what is the actuarially-fair price m*? (hint: the firm is competitive and will have zero profit). (3) (5 pts) find bobs choice of insurance b when m is actuarially-fair (4) (5 pts) what is eg and when m is actuarially fair and bob chooses b? (just plug in bobs choice.) does bob desire full insurance?Recall that in an expected utility model with a good g and bad b state of nature, preferences over state-contingent consumption bundles (c_g,c_b) are represented by the utility function

Eleft [ u(c) right ]=

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where pi is the probability of the bad state occurring. Naturally, this means the good state occurs at probability 1-pi Assume that the price of c is $1 in both states. Suppose that Bob has preferences given by the expected utility function above and that u(c)=frac{c^{1-sigma }}{1-sigma } . Suppose that regardless of the state of the world, Bob has an income of $W. In the bad state, he incurs a cost of $$delta due to medical bills.

A competitive insurance company approaches Bob and offers him the following type of contract: Bob pays 0 < m < 1 cents per dollar of insurance for a total premium of $mb. If the bad state happens, the insurance firm pays Bob $b. Bob must select b.

(1) Set up the household problem where Bob buys b amount of insurance. (Hint: there will be two budget constraints for each state of the world)

(2) Set up the firm’s problem where it is maximizing expected profit. What is the actuarially-fair price m*? (Hint: The firm is competitive and will have zero profit).

(3) Find Bob’s choice of insurance b* when m is actuarially-fair.

(4) What is c^*_g~ and ~c^*_b when m is actuarially fair and Bob chooses b*? (Just plug in Bob’s choice.) Does Bob desire full insurance?

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