Consider a person with a current wealth W0 = 100 , 000 who faces the prospect of p = . 25 chance of losing 20 , 000 through car theft during the next year. Suppose that his utility function is U ( W )=ln W where W is the persons wealth at the end of the year. This person can buy an antitheft device that costs $1 , 950 and reduces the probability of theft to p =0 . 15 . (a) Will this person buy the antitheft device? Show mathematically why this is the case. Now suppose there is an insurance company that sells full insurance coverage. The premium is determined as the actuarially fair premium plus $200 of admin- istrative charges. The company cannot monitor whether or not the individual installed a device, so the probability of theft used by the insurance company to determine the premium is p = . 25 . (b) What premium does the insurer charge? Does the consumer buy insurance? Does the consumer buy the device? Show how you de- termine this. Assume now that the insurance company can monitor whether or not the in- dividual installed an anti-theft device and charge a premium based on whether consumer uses the device or not. If a consumer doesnt use the device, the pre- mium is the actuarial fair price (for p = . 25 )plus$200tocoveradministrative costs. If a consumer does use the device, the premium will be the actuarially fair premium (for p 0 = . 15 )plus$200ofadministrativecharges and an additional $10 to cover the monitoring costs (the cost of verifying that the consumer is using the device). Assume that the insurer does not cover the cost of buying the device and does not cover theft of the device itself (i.e. the insurer only covers the loss of 20,000). (c) How much does insurance cost with and without the device installed? What will the individual do? Will the individual install an antitheft device? Will he buy insurance? What type of insurance will the person buy: with or without anti-theft device