prelims_Micro Prelim June 2008

prelims_Micro Prelim June 2008 - University of California,...

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Unformatted text preview: University of California, Davis Date: June 23, 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 consumer with wealth w buys goods 1 and 2 in the market, at prices p 1 and p 2 , facing the budget constraint p 1 x 1 + p 2 x 2 < w . (1) The twist is that her consumption of good 2 creates a positive externality from which she, and everybody else in society, benefits. Conversely, she benefits from the positive externality generated by every other consumer. Formally, her utility function, assumed to be differentiable, concave and with strictly positive gradient in the interior of its domain, is written u ( x 1 , x 2 , E ), with the three arguments x 1 (the amount of good 1 that she consumes, x 2 (the amount of good 2 that she consumes) and E , the total amount of the externality generated, to be also called the level of the public good . The dependence of E on both her consumption x 2 of good 2 and the amount E- i of positive externality aggregately generated by everybody else is defined by E < E- i + g x 2 , (2) where g > 0 is a parameter. Our consumer treats E- i (as well as p 1 , p 2 and w ) parametrically. The maximization of her utility function subject to the constraints (1) and (2) yields her market demand functions 1 2 ( , , , ), 1, 2 j i x p p w E j- = for goods 1 and 2, as well as her desired amount of the public good 1 2 ( , , , ) j i E p p w E- . 1(a). Write the maximization problem the solution of which is 1 1 2 2 1 2 1 2 ( ( , , , ), ( , , , ), ( , , , )) i i i x p p w E x p p w E E p p w E--- , and call it Problem PG . Write its Kuhn-Tucker conditions. Assuming that the solution is interior and unique, and that constraints (1) and (2) are satisfied with equality, write a system of equations, not involving the multipliers, the solution of which defines 1 1 2 2 1 2 1 2 ( ( , , , ), ( , , , ), ( , , , )) i i i x p p w E x p p w E E p p w E--- . (Just write the system of equations: there is no need to work towards its solution.) 1(b). For the rest of this part we consider the utility function 2 2 2 1 2 1 2 2 1 : : ( , , ) [ , ] [ , ] 2 E x x u u x x E x a a x E B E E + = +- , where 2 ( , ) E a a and 22 2 2 E E EE b b B b b is a symmetric, positive definite matrix. We are interested in the comparative statics expressed by the partial derivatives 2 i x E- and i E E- . Again, assume that the solutions are interior and the constraints satisfied with equality. University of California, Davis Date: June 23, 2008 Department of Economics Time: 5 hours Microeconomics Reading Time: 20 minutes Page 2 of 6 1(b) (i). Specialize to this utility function the system of equations obtained in 1(a)....
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prelims_Micro Prelim June 2008 - University of California,...

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