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fin-401-07wa

# fin-401-07wa - 1 21st April 2007 ANSWERS Instructions...

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1 21st April 2007 ANSWERS Instructions : Answer 7 of the following 9 questions. All questions are of equal weight. Indicate clearly on the fi rst page which questions you want marked. 1. Let X be a choice set and ( B , C ( · )) be a choice structure on X . (a) Using mathematics, de fi ne the weak axiom of revealed preference (WARP). ***** Let B 1 , B 2 X , and { x, y } B 1 B 2 . Then WARP says x C ( B 1 ) and y C ( B 2 ) x C ( B 2 ) . (b) Suppose X = { a, b, c } , B = {{ a, b } , { b, c } , { c, a } , { a, b, c }} , C ( { a, b } ) = { a } , C ( { b, c } ) = { b } , and C ( { c, a } ) = { c } . Assuming the choice rule must pick at least one element from each budget set so that C ( { a, b, c } ) cannot be the empty set, can this choice structure satisfy WARP? Justify your answer. ***** Suppose a were amongst the best elements of { a, b, c } , that is, a C ( { a, b, c } ) . Then, according to WARP, a would have to be amongst the best elements of { c, a } but this is contradicted by our assumption above that C ( { c, a } ) = { c } , so a / C ( { a, b, c } ) . Given the perfect symmetry in this question between a and b and c , it is clear b / C ( { a, b, c } ) and c / C ( { a, b, c } ) . Since C ( { a, b, c } ) cannot be the empty set, there is no way to make this choice structure consistent with WARP. 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. Is insurance a normal good for this individual? Justify your answer carefully.

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2 ***** 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 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. If insurnce is a “normal” good then dx/dw 0 would be positive.Using the compar- ative statics method Sign dx dw 0 = Sign 2 f w 0 x . Since f ( x, p, π , w 0 , L ) x = ( p ) π u ± ( w 0 px ) + (1 π ) (1 p ) u ± ( w 0 px L + x ) 2 f ( x, p, π , w 0 , L ) w 0 x = ( p ) π u ±± ( w 0 px ) (1 π ) (1 p ) u ±± ( w 0 px L + x ) With fair insurance p = 1 π and x = L , so
3 2 f ∂π∂ x = 0 . In words, if the person’s wealth increases he or she will not change the amount of insurance purchased. Insurance is neither normal nor inferior. 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 3 / 4 . The only way in which they di ff er is Andy is risk averse and Brian is risk neutral.

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fin-401-07wa - 1 21st April 2007 ANSWERS Instructions...

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