prelims_Micro Prelim ANSWERS August 2006

prelims_Micro Prelim ANSWERS August 2006 - Microeconomics...

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Microeconomics Prelim August 31, 2006 Answer Keys 1. 1(a). Case 1: σ > 1. In this case the utility function is strictly quasiconcave, and, because all prices are equal and the utility function is symmetric in x 1 , …, x L , the consumer demands the same amount of every commodity or variety, i.e., her Walrasian demand for variety j is Lp w , j = 1,…, L . Hence she reaches the utility level p w L p w L Lp w L Lp w L Lp w L i 1 1 1 1 1 1 1 1 1 1 - σ - - σ σ - σ σ - σ σ σ - σ - σ σ = σ - σ = = = = , an expression that is increasing in L when σ > 1. Hence, in this case more variety is good for the consumer. Case 2: σ (0,1). As in Case 1, the utility function is strictly quasiconcave and the consumer reaches the utility level p w L 1 1 - σ , an expression that is now decreasing in L . Hence, in this case more variety hurts the consumer. Case 3: σ < 0. Now the utility function is strictly quasiconvex, and all the solutions to the utility maximization problem are of the corner variety, with the consumer buying only one good. Thus, she reaches the utility level p w p w = - σ σ σ - σ 1 1 , independent of L . Hence, in this case more variety neither benefits nor hurts the consumer. 1(b). The solution to the utility maximization problem for (1) is the same as the one for the usual CES function 1 1 1 - σ σ = σ - σ L i i x , (2) because (1) is an increasing transformation of (2). Hence, ( σ - - = σ - = j L i i L j p p w w p p x 1 1 1 1 ) , ,..., ( ~ , with elasticity
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( ( ( 29 σ - - = σ - - σ - - = σ - σ - - σ - - = σ - σ - + σ - - = j L i i j j L i i j j L i i j j j j p p w p p p p p p w x p p x 1 1 1 1 1 1 1 1 1 2 1 1 ) ( ) 1 ( ) 1 ( ~ ~ = ( ( 29 ( ( 29 ( 29 σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ - - - = - - - - = - - - - = - - - - = - - - = - - - - = - 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 ) 1 ( ) 1 ( L i i j j L i i j j L i i j L i i j j L i i p p p p p p p p p p p p . If p 1 = p 2 = … = p L = p , then this expression becomes ( σ σ σ σ σ σ σ σ σ σ - - - = - - - = - - - - - + - - - - - 1 1 1 1 1 1 1 ) 1 ( ) 1 ( ) 1 ( L L p Lp p , which tends to - σ as L .
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