601-finalexamquestions

601-finalexamquestions - E 601 1 Final Examination...

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E frqrplfv 601 1 Final Examination questions 1. Textbooks introduce various kinds of systems of demand equations for con- sumers (e.g. Walrasian and Hicksian demands) and f rms (e.g. inputs demands based on pro f t maximization or on cost minimization). How are the various consumer demand systems related to the various producer demand systems? If a particular consumer (producer) demand system has no counterpart in producer (consumer) the- ory can you design a “new” demand system that would f ll this gap? Explain. ANSWER Assume 2 goods or 2 inputs. Marshallian demands derive from the problem Max x 1 ,x 2 u ( x 1 2 ) such that w p 1 x 1 p 2 x 2 =0 which yields x i ( p 1 ,p 2 ,w ) . (1) Hicksian demands derive from the problem Min h 1 ,h 2 p 1 h 1 + p 2 h 2 such that u ( h 1 2 ) u 0 which yields h i ( p 1 2 ,u 0 ) . (2) Input demands of a pro f t-maximizing price taking f rm arise from the problem x 1 2 pf ( x 1 2 ) w 1 x 1 w 2 x 2 (3) which yields x i ( w 1 2 ) . Input demands of a cost-minimizing f rm derive from the problem x 1 2 w 1 x 1 + w 2 x 2 such that f ( x 1 2 ) y 0 which yields x i ( w 1 2 ,y 0 ) (4) Problems (2) and (4) are clearly the same and thus the associated demands func- tions h i ( p 1 2 0 ) and x i ( w 1 2 0 ) have much in common. The only di f erence is that utility cannot be measured in the same sense as output. Problems (1) and (3) are di f erent. The analogue of (1) in producer theory would be a f rm that maximized sales subject to a constraint on the total costs of its inputs. The problem faced by this kind of f rm would be
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E frqrplfv 601 2 Max x 1 ,x 2 pf ( x 1 2 ) such that C 0 w 1 x 1 w 2 x 2 =0 which would yield x i ( w 1 ,w 2 ,C 0 ,p ) , which are much like the Marshallian demands of consumer theory. One could produce the analogue of (3) in consumer theory by using the concept of the marginal utility of money - the extra utility yielded by one more dollar of expenditure. Denote the marginal utility of money by λ . The inverse of λ is the “price” of utility - the number of dollars it takes to buy one unit of utility. x 1 2 1 λ u ( x 1 2 ) p 1 x 1 p 2 x 2 (5) which yields x i ( p 1 2 , λ ) . Note that the f rst-order conditions in (5) are the same as those in (3). 2. Answer both parts of this question. (a) Prove that the expenditure function, e ( p 1 2 ,u ) is concave in prices ( p 1 2 ) . (b) Consider a three-good demand system where x 1 = a 1 + a 2 p 1 /p 3 + a 3 p 2 /p 3 + a 4 p 1 p 2 /p 2 3 x 2 = b 1 + b 2 p 1 /p 3 + b 3 p 2 /p 3 + b 4 p 1 p 2 /p 2 3 , and where the demand function for the third good follows from the person’s budget constraint. If this system is derived from a well-behaved utility-maximizing problem what are the restrictions on ( a j ,b j ) 4 j =1 ? ANSWER Consider two sets of prices ( p 1 1 1 2 ) and ( p 2 1 2 2 ) and the convex combination of them ( p t 1 t 2 )= t ( p 1 1 1 2 )+(1 t )( p 2 1 2 2 ) . Then e ( p 1 2 ) is concave in prices ( p 1 2 ) if e ( p t 1 t 2 ) te ( p 1 1 1 2 t ) e ( p 2 1 2 2 ) .
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E frqrplfv 601 3 Let ( h 1 ,h 2 ) be the expenditure-minimizing bundle for prices ( p t 1 ,p t 2 ) and utility level u. Then e ( p t 1 t 2 ,u )= p t 1 h 1 + p t 2 h 2 =( tp 1 1 +(1 t ) p 2 1 ) h 1
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601-finalexamquestions - E 601 1 Final Examination...

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