601-uncertqr-10f

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

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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 f 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 thep r iceo fin su rancea tthepo in twhe re 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 f rm’s expected pro f tonea chdo l la ro f insuranceiszero ,so π p +(1 π )( p 1) = 0 or p =1 π . To f ndtheopt ima lva lueo f x given the parameters of the problem set f ( x, p, π 0 ) x =0 and solve for x at p π .Thu s ( p ) π u ± ( w 0 px 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 ) ,wh e r e 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 f rst-order condition for this problem is f ( ) x =0 which implicitly de f nes the optimal level of x as a function of a . Taking a total di f erential of this equation obtain 2 f ( ) x 2 dx + 2 f ( ) a x da or dx da = 2 f ( x, a )/ a x 2
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601-uncertqr-10f - E 601 1 Uncertainty questions 1....

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