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 f 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 f 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 } ) .S ince 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 f ts; denote the price of $1 worth of insurance coverage by p.
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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 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 ,w 0 ,L ) x =0 and solve for x at p =1 π .Thu s ( 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
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This note was uploaded on 01/26/2012 for the course ECON 401 taught by Professor Burbidge,john during the Fall '08 term at Waterloo.

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

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