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Unformatted text preview: Problem Set 4 SOLUTIONS Due Thursday, Oct. 22nd, by 10pm 1. Consider the production function f ( l,k ) = l + k . (a) For what values of > is the production function quasi-concave? Quasi-concave production functions imply that the isoquants have convex upper contour sets, or that averages are at least as good as extremes. If > 1 then the isoquants bend away from the origin, implying that averages are worst than extremes. If = 1 then the isoquants are linear lines, which means that averages are always as good as extremes. If < 1 then the isoquants bend towards the origin, producing convex upper contour sets, which imply the production function is quasi-concave. So the production function is quasi-concave if < = 1. For the rest of this problem, assume is such that the production function is quasi-concave. (b) Find the conditional input demands for l and k . If = 1 then the firm will use either only labor or only capital, whichever is cheaper (except for the case where they have the same price, then the firm is indifferent between mixing). The answers to the rest of the problem when = 1 are trivial, so for the remainder of the problem we will focus on when < 1. When < 1 we have to use lagrangians. To find the conditional input demands we set up the expenditure minimization problem. min wl + rk s.t. x = l + k L = wl + rk + ( x- l - k ) L l = w- l - 1 = 0 L k = r- k - 1 = 0 L = x- l - k = 0 1 Since we know the production function is quasi-concave, we know the first order condi-...
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- Fall '09