prelims_Micro Prelim August 2008

# prelims_Micro Prelim August 2008 - University of California...

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University of California, Davis Date: August 28, 2008 Department of Economics Time: 5 hours Microeconomics Reading Time: 20 minutes Page 1 of 6 PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE Answer FOUR questions Question 1. (a) . Let P the set of relevant price-wealth vectors ( p , w ), a subset of 1 L + ++ , and denote by : j xP →ℜ ± a consumer’s Walrasian demand for good j , j = 1,… L , assumed to be strictly positive and differentiable on P . (a).1 . What do we mean when we say that good j is a necessity for the consumer at ( p , w )? Same for luxury and for borderline necessity-luxury . (a).2 . Show that the concepts of luxury and borderline necessity-luxury can be characterized by a property of the budget share function ( , ) j bpw of the good. Can you do the same with the concept of necessity ? Explain For the rest of this question we consider the indirect utility function 1 () :: ( , ) ( ) ln( / ( )) Fp vP vpw Gp wCp ⎡⎤ = + ⎢⎥ ⎣⎦ , (1) where C ( p ) >> 0, and the functions C , F and G are such that v ( p , w ) has the properties of an indirect utility function on P . (b) . For j = 1, …, L , obtain the Walrasian demand function (,) j x pw ± and the budget share function j corresponding to (1). (c). Consider first the case of (1) with G ( p ) = 0, all p . Show that if good j is a luxury at some (,) 0 pw >> , then it is a luxury at p w , for all w > 0. (d). Consider now the general case of (1) where G ( p ) is not always zero. (d).1 . Suppose that good j is a luxury at some >> . Does it follow that it is a luxury at p w , for all w > 0? Explain. (d).2 . Suppose that all consumers in the economy have identical preferences, of the type represented by (1). Under which conditions on the functions C , F and G can the consumers’ aggregate demand be a function of (only) prices and aggregate wealth? Argue clearly. Discuss the possibility of a positive representative consumer with these preferences.

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University of California, Davis Date: August 28, 2008 Department of Economics Time: 5 hours Microeconomics Reading Time: 20 minutes Page 2 of 6 Question 2. This question considers the profit-maximizing assumption for firms with or without market power. (a). Price-taking firm. Consider a price-taking firm with production set L Y ⊂ℜ and facing a strictly positive price vector p . (a).1. Write the firm’s profit-maximizing problem. (a).2. Normalize all prices with good j as numeraire, and write the profit- maximizing problem under this normalization. Show that the same solution (or set of solutions) obtains no matter which good j is chosen as numeraire. (b). Firm with market power . Now we specialize to two goods ( L = 2), where good 1 is an input for the firm, and good 2 is its output. The production set of the firm is 2 12 1 2 1 {( , ) : 0, / } y yy y y c ∈ℜ ≤− , where c is a positive parameter. The firm is a price setter, and faces the following demand function for its output: 1 212 (, ) 1 p xpp p cp ω = α + −α ± , where the parameters ω and
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## This note was uploaded on 02/09/2010 for the course ECON 200D taught by Professor Pontusrendahl during the Winter '06 term at UC Davis.

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prelims_Micro Prelim August 2008 - University of California...

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