prelims_Micro Prelim ANSWERS Sept 2004

# prelims_Micro Prelim ANSWERS Sept 2004 - Answer Key...

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Answer Key, Question 2 2(a). There is a production externality, because the physical cost of firm i depends on the aggregate output q = = N j j q 1 through the term ( ) α + i i q q . The parameter α reflects the externality. If α > 0, then the externality is negative , because an increase in aggregate output increases the cost of firm i : perhaps all these firms are using an exhaustible natural resource, or there is some form of congestion among them. If, on the contrary, α < 0, then the externality is positive (because, say, knowledge spillovers). Note that firm i contributes to these effects not only on the other firms but also on itself, and that the externality across firms depends only on aggregate output q , and not in the manner in which q is allocated among firms. The value α = 0 corresponds to the traditional case of absence of production externalities. When α = 0, the parameter β depicts the scale economies within the firm, with β > 1 corresponding to decreasing returns to scale (the cost function is strictly convex), and β < 1 corresponding to increasing returns to scale (cost function strictly concave). When 0 α this interpretation is not exact, because q i appears in the term ( ) α + i i q q , and needs to be made conditional to keeping aggregate output constant. 2(b). C ( q ) is the value of the problem ( ) β α = + + i N i N q q q q q N ) ... ( min 1 1 ) ,..., ( 1 subject to q q N i i = 1 , (2.2) i.e., () = β α N i i q q q q N 1 ) ,..., ( 1 min subject to q q N i i = 1 . Its solutions are those of the problem ( ) = β N i i q q q N 1 ) ,..., ( 1 min subject to q q N i i = 1 . (2.3) Case 1 : β < 1. For fixed q , β < 1 entails economies of scale within each plant. Intuitively, the merged firm will then concentrate its production in one plant. Formally, the objective function in Problem (2.3) is strictly quasiconcave, with corner solutions, see Figure 2.1, i. e., any vector of the form q i = q , q j = 0 for j i solves Problem (2.3), and hence, Problem 2.2. The value function of Problem (2.2), or, in other words, the physical cost function of the merged firm is then
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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 ANSWERS Sept 2004 - Answer Key...

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