So by doing this the firms output cannot decrease hence its revenue cannot de

So by doing this the firms output cannot decrease

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So by doing this the firm’s output cannot decrease, hence its revenue cannot de- crease. Since the original ¯ z is profit maximizing by assumption, the revenue must not increase with the same cost. Therefor the new input combination must be another profit maximizing combination. 4. Q4 (a) The CES function exhibits constant returns to scale, i.e. is homogeneous of degree 1 since f ( tz ) = " K X k =1 a k ( tz k ) α # 1 α = " t α K X k =1 a k ( z k ) α # 1 α = t " K X k =1 a k ( z k ) α # 1 α = tf ( z ) (b) Let us first show that a CES function is concave 2 (thus quasi-concave in particular) to justify the FOC analysis we do later. 3 Since it exhibits CRS, it suffices to show that f is quasi concave, and so it is enough to show that a monotonic transformation of f is a concave function. If α > 0, then t 7→ t α is increasing so it is enough to show that [ f ( z )] α = K k =1 a k ( z k ) α is concave. But this is immediate since this is a sum of concave functions. If α < 0, t 7→ - t α is increasing, thus it suffices to show that - [ f ( z )] α = - K k =1 a k ( z k ) α is concave. This follows since for each k , - a k ( z k ) α is concave and so again this is a sum of concave functions. 4 2 When α > 1, this function is convex, which is a consequence of the Minkowski inequality. 3 This property can be shown by a careful examination of the Hessian matrix but I hope that you appreciate the following argument based on general observations on concave functions. 4 The role of assumption of α 1 is now also clear: it guarantees that the quasi-concavity of the production function: if α > 1, the function is quasi-convex and the fator demand will be determined by corner solutions. 6
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(c) Since this is a CRS technology, it suffices to analyze the minimum cost function given factor prices w = ( w 1 , ..., w K ) and the required output level y = 1, since the cost function is homogeneous of degree one in output y . So the cost minimization problem we are interested in is: min z X k w k z k subject to f ( z ) 1 , which is equivalent to if α > 0 , min z X k w k z k subject to K X k =1 a k ( z k ) α 1 , and if α < 0 min z X k w k z k subject to 0 > - K X k =1 a k ( z k ) α ≥ - 1 , It is easy to see (check!) that the constraint is binding at the minimum. Also the objective function is convex, and the constraint is quasi-concave, so the following first order condition is necessary and sufficient condition for cost minimization: λw k = αa k ( z k ) α - 1 , k = 1 , ..., K K X k =1 a k ( z k ) α = 1 , where λ is a positive constant. From the first set of equations, we get w k w l = a k a l z k z l α - 1 , for any k and l . So, ˆ w k ˆ w l 1 α - 1 = z k z l , where ˆ w k = w k /a k for k = 1 , ..., K . Then, from the last equation, 1 = K X k =1 a k ( z k ) α = a k ( z k ) α + X l 6 = k a l z k ˆ w k ˆ w l - 1 α - 1 ! α = ( z k ) α " K X l =1 a l ˆ w k ˆ w l ρ # , where ρ = α 1 - α . So we get the minimizer as z k ( w, 1) = h K l =1 a l ˆ w k ˆ w l ρ i - 1 α , thus 7
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the factor demand for input k given output level y is, for k = 1 , ..., K , z k ( w, y ) = y h K l =1 a l w k /a k w l /a l ρ i 1 α = " w k a k ρ K X l =1 a l w l a l - ρ # - 1 α y, = w k a k - 1 1 - α " K X l =1 a l w l a l - α 1 - α # - 1 α y. (5)
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