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 fi rms (e.g. inputs demands based on pro fi 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 fi 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 , x 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 , h 2 ) u 0 = 0 which yields h i ( p 1 , p 2 , u 0 ) . (2) Input demands of a pro fi t-maximizing price taking fi rm arise from the problem Max x 1 , x 2 pf ( x 1 , x 2 ) w 1 x 1 w 2 x 2 (3) which yields x i ( w 1 , w 2 , p ) . Input demands of a cost-minimizing fi rm derive from the problem Min x 1 , x 2 w 1 x 1 + w 2 x 2 such that f ( x 1 , x 2 ) y 0 = 0 which yields x i ( w 1 , w 2 , y 0 ) (4) Problems (2) and (4) are clearly the same and thus the associated demands func- tions h i ( p 1 , p 2 , u 0 ) and x i ( w 1 , w 2 , y 0 ) have much in common. The only di ff erence is that utility cannot be measured in the same sense as output. Problems (1) and (3) are di ff erent. The analogue of (1) in producer theory would be a fi rm that maximized sales subject to a constraint on the total costs of its inputs. The problem faced by this kind of fi rm would be
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E frqrplfv 601 2 Max x 1 , x 2 pf ( x 1 , x 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. Max x 1 , x 2 1 λ u ( x 1 , x 2 ) p 1 x 1 p 2 x 2 (5) which yields x i ( p 1 , p 2 , λ ) . Note that the fi 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 , p 2 , u ) is concave in prices ( p 1 , p 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 , p 1 2 ) and ( p 2 1 , p 2 2 ) and the convex combination of them ( p t 1 , p t 2 ) = t ( p 1 1 , p 1 2 ) + (1 t )( p 2 1 , p 2 2 ) . Then e ( p 1 , p 2 , u ) is concave in prices ( p 1 , p 2 ) if e ( p t 1 , p t 2 , u ) te ( p 1 1 , p 1 2 , u ) + (1 t ) e ( p 2 1 , p 2 2 , u ) .
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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
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