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fin-401-07fa - 1 7th December 2007 Instructions Answer 6 of...

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1 7th December 2007 Instructions : Answer 6 of the following 8 questions. All questions are of equal weight. Indicate clearly on the fi rst page which questions you want marked. 1. A job candidate from a big US university is presenting his work which empha- sizes a “wonderful” new functional form for estimating ordinary demand equations, based on the utility maximizing approach. Two of the observations on one of his households are that the household bought: (a) (1 , 1) at prices (1 , 2) and (b) (3 / 2 , 1 / 2) at prices (2 , 2) . What question would you ask him? ANSWER Cost of these bundles at these prices 1 2 Deductions 1 3 5 / 2 1 " 2 2 4 4 2 " 1 These data are inconsistent with WARP so it is nonsensical to fi t demand equa- tions based on utility-maxmizing approach to the data. The natural question is “Why are you doing this – it makes no sense?” 2. Consider a risk-averse individual with initial wealth w 0 and VNM utility func- tion u ( · ) 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 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. If the loss ( L ) increased would this individual buy more or less insurance at the point where insurance is fair? Justify your answer carefully. ANSWER The individual’s expected utility will be 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
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2 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. The question asks for the sign of dx/dL at the point where insurance is fair. Since we have just proved that, with fair insurance, x = L , dx dL = 1 > 0 . In words, if the person’s loss in the event of an accident increases he or she will buy one more dollar of insurance for each dollar loss increases. 3. Consider the two-state world of Andy and Brian. In the good state each has a wealth of 100 ; in the bad state each has a wealth of 50 , assuming they do not trade with each other. Let the probability of the good state be 2 / 3 . The only way in which they di ff er is Andy is risk averse and Brian is risk neutral. Describe as precisely as you can the Pareto e cient allocations for these two people. Now suppose there are many Andys and an equal number Brians. Describe as precisely as you can the
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