cost_func - 5 Cost Function c(w, y) = min wx x s.t. f (x) =...

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Unformatted text preview: 5 Cost Function c(w, y) = min wx x s.t. f (x) = y. 5.1 Properties c(w0, y) c(w, y). 1. Nondecreasing in w. For w0 w, 2. Homogeneous of degree 1 in w: for t > 0, c(tw, y) = tc(w, y). 3. Concave in w: for w00 = tw +(1-t)w0, t [0, 1], c(w00, y) tc(w, y) + (1 - t)c(w0, y). 4. Continuous in w for w 0 (when defined). 5.2 Conditional Demand Application: c(w, y) = shadow cost. y Sheppard's Lemma: For all i = 1, . . . , n, for wi > 0, x(w) = i c(w, y) . wi Proof: Special case of Envelope Theorem. Envelope Theorem (constrained). For the problem M (a) = max f (z, a), z 5.3 Comparative statics. (revisited?) 1. c(w, y) is nondecreasing in w, x(w) = i c(w, y) 0. wi s.t. h(z, a) = 0. we have L = f (z, a) - h(z, a), dM(a) L = = da a z=z(a) f (x, a) h(x, a) - . a a x=x(a) (a) x=x 2. x(w) are homogeneous of degree 0. i 3. c(w, y) is concave, so xi(w, y) wj ! = 2c(w, y) symm., neg.-def. wiwj ! 5.4 Average and Marginal costs. 5.4.1 Short-Run vs Long-Run. Example: Boston-Chicago flight. Fixed costs (F C): Airport charges, plane maintenance, fuel for the body, ... Per person costs: Check-in, security, food, fuel, luggage, . . . . These sum up to Variable costs (V C). Define Define T C = F C + V C, TC , AC = y T C , MC = y FC , AF C = y VC , AV C = y V C = MC. MV C = y ST C c(w, y, z) = , y y ST C c(w, y, z) SMC = = , y y wxx(w, y, z) SV C SAV C = = , y y wzz SF C = . SAF C = y y SAC = In the Long-Run? Suppose, in S-R some of the factors, z, are fixed. | c(w, y, z) = wxx(w, y, z) + wzz . | {z } {z } ST C | {z SV C } SF C ...
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cost_func - 5 Cost Function c(w, y) = min wx x s.t. f (x) =...

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