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601-uncertqr-10f - E 601 1 Uncertainty questions 1 Consider...

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E frqrplfv 601 1 Uncertainty questions 1. Consider a risk-averse individual with initial wealth w 0 and a Bernoulli utility function u ( w ) who must decide whether and for how much to insure his car. Assume the probability that he will not have an accident is π . In the event of an accident, he incurs a loss of $ L < w 0 in damages. Suppose that insurance is available at an actuarially fair price, that is, one that yields insurance companies zero expected pro fi ts; denote the price of $1 worth of insurance coverage by p. Write down the individual’s expected utility if he purchases x dollars of insurance. How much insurance will this individual purchase? Does this this individual’s demand for insurance slope downward with respect to the price of insurance at the point where insurance is priced fairly? Defend your answer carefully. ANSWER The individual’s expected utility is f ( x, p, π , w 0 , L ) π u ( w 0 px ) + (1 π ) u ( w 0 px L + x ) If the insurance is priced “fairly” the fi rm’s expected pro fi t on each dollar of insurance is zero, so π p + (1 π ) ( p 1) = 0 or p = 1 π . To fi nd the optimal value of x given the parameters of the problem set f ( x, p, π , w 0 , L ) x = 0 and solve for x at p = 1 π . Thus ( p ) π u ± ( w 0 px ) + (1 π ) (1 p ) u ± ( w 0 px L + x ) = 0 or u ± ( w 0 px ) = u ± ( w 0 px L + x ) and thus w 0 px = w 0 px L + x or x = L.
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E frqrplfv 601 2 We want to know Sign dx dp ± ± ± ± ± p =1 π ******************************************************************** Aside on the comparative statics method: Suppose an agent is maximizing her objective function f ( x, a ) , where x is the choice variable and a is a parameter. Assuming the maximization problem is well- behaved, and all we are interested in is the sign of dx/da then the method of com- parative statics (the implicit function theorem) tells us that Sign dx da = Sign 2 f ( x, a ) a x . Why? The fi rst-order condition for this problem is f ( x, a ) x = 0 which implicitly de fi nes the optimal level of x as a function of a . Taking a total di ff erential of this equation obtain 2 f ( x, a ) x 2 dx + 2 f ( x, a ) a x da = 0 or dx da = 2 f ( x, a ) / a
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