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prelims_Micro Prelim August 2006

prelims_Micro Prelim August 2006 - University of California...

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University of California, Davis Date: August 31, 2006 Department of Economics Time: 4 hours Microeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE- ================================================================================== Please answer four four four of the following equally weighted five questions 1. Variations on variable varieties We consider a model of consumer preferences where the different commodities are viewed as different varieties of a good, and the number L of varieties is variable. Variations of this model have been extensively used in industrial organization and international trade. Formally, we postulate that the preferences of our consumer are represented by the following family of utility functions, indexed by the number of varieties present in the market: ) , 1 ( ) 1 , 0 ( ) 0 , ( , , ) ,..., ( 1 1 1 1 -∞ σ γ = - σ σ = σ - σ γ L i i L L x L x x u . (1) In words, ) ,..., ( 1 L L x x u is the utility that our consumer gets when, in a L -variety world, she consumes x j units of variety j , j = 1,…, L . Our consumer has wealth w > 0, takes prices ( p 1 , …, p L ) as given, and is always able to satisfy her Walrasian demand. (a). Let γ = 0. We consider different worlds characterized by different numbers of varieties, but we assume that p 1 = p 2 = … = p L = p , for every L , i. e., every price is always equal to a given positive number p , or, in other words, prices are equal across worlds (or numbers of varieties) and across varieties. Does a higher L benefit the consumer? Argue clearly, separately considering the following three cases. Case 1 : σ > 1 (this is the standard case); Case 2 : σ (0,1); Case 3 : σ < 0. (b). Now γ is unrestricted, but we let σ > 1 (as in Case 1 of part (a)). Compute the own- price elasticity of demand for variety j ( j = 1,…, L ) for the utility function given in (1). (Hint . Recall that the Walrasian demand function for good j for the CES function 1 1 1 - σ σ = σ - σ L i i x ,
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University of California, Davis Date: August 31, 2006 Department of Economics Time: 4 hours Microeconomics Reading Time: 20 minutes Page 2 of 7 σ > 1, is ( σ - - = σ - j L i i p p w 1 1 1 .) What is the limit of this elasticity as L if every price is always equal to p ? Interpret. Compare this limit elasticity with the elasticity of substitution of (1). (c). Again, σ > 1 and γ is unrestricted. We define the love-of-variety function ) ( ) ,..., , ( ) , ( ˆ 1 Lx u x x x u x L L β . Interpret it in words, and argue that, for the utility function (1), ) , ( ˆ x L β is constant with respect to x and, hence, it can be written as a function β ( L ) of only L . Compute its elasticity ) ( ) ( ' ) ( L L L L β β η . What can you say about the sign of η ( L )? Compare this sign with your answer in Case 1 of part (a) above. (d). Once more, σ > 1, and γ is unrestricted. It is assumed in some applications that the love-of-variety elasticity η ( L ) actually coincides with the ratio cost marginal cost marginal - price of a monopolist who supplies variety j to our consumer. Please compute this ratio as a function of the elasticity of the demand function faced by the monopolist, under the assumption that this
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