prelims_Micro Prelim June 2006

prelims_Micro Prelim June 2006 - University of California,...

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University of California, Davis Date: June 26, 2006 Department of Economics Time: 4 hours Microeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE Question 1. Fancy scotch in Scotland and in Australia Is the ratio of fancy scotch relative to ordinary scotch consumed in Australia higher than in Scotland? This is of course an empirical question. But it has been claimed that this should be so on theoretical ground. To fix the ideas, let good 1 be fancy scotch and good 2 be ordinary scotch, and assume that there is a representative consumer in Scotland, as well as one in Australia, and that the two representative consumers have identical preferences and wealth. (All prices and wealth are expressed in US$). We assume that these preferences are represented by a smooth utility function, and that the consumption of all goods is always positive. Let p 1 (resp. p 2 ) be the price of a bottle of fancy (resp. ordinary) scotch in Scotland, and assume throughout this question that p 1 > p 2 . Shipping one bottle of scotch from Scotland to Australia costs c . Accordingly, the price of fancy (resp. ordinary) scotch in Australia is p 1 + c (resp. p 2 + c ). The claim is that the ratio x 1 / x 2 in the consumption of a representative agent is an increasing function of c , and is verbally defended by noting that, because p 1 > p 2 , an increase in c decreases the price of fancy scotch relative to that of ordinary scotch. We examine the claim in various forms. 1.1 . Walrasian demand, two goods The claim here takes the form 0 ) , , ( ~ ) , , ( ~ 2 1 2 2 1 1 > + + + + dc w c p c p x w c p c p x d , where j x ~ is the representative consumer’s Walrasian demand for good j , j = 1,2. Is the claim correct? A graphical argument suffices. 1.2 . Hicksian demand, two goods The claim now takes the form 0 ) , , ( ) , , ( 2 1 2 2 1 1 > + + + + dc u c p c p h u c p c p h d , where h j is the representative consumer’s Hicksian demand for good j , j = 1,2. Is the claim correct? Argue both graphically and analytically. 1.3 . Hicksian demand, L goods ( L > 3) 1
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Last, the claim takes the form 0 ) , ,..., , , ( ) , ,..., , , ( 3 2 1 2 3 2 1 1 > + + + + dc u p p c p c p h u p p c p c p h d L L , where h j is the representative consumer’s Hicksian demand for good j , j = 1,2. Examine the validity of the claim for the following two cases. Comment, comparing 1.3 with 1.2 above. Case 1. Quasilinear utility function L L j L j j j L x x b x b a x x x u + = ∑∑ = = 1 1 1 1 2 2 1 2 1 ) ,..., , ( ~ , where a > 0, and b > 0 (defined on the domain where all marginal utilities are positive). Case 2 . CES utility function 1 1 1 2 1 ) ,..., , ( ~ ~ σ σ = σ σ = L j j L x x x x u , where σ > 0, σ 1. Hint . The corresponding expenditure function is ( ) σ = σ = 1 1 1 1 ) , ( J j j p u u p e . Question 2. Human capital externalities in production There are J identical firms, each producing the same good (called output) by using M inputs, which are interpreted as labor of various skills due to varying amounts of human capital. For m = 1, …, M , skill type m is defined by a positive real number h m
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prelims_Micro Prelim June 2006 - University of California,...

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