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Unformatted text preview: 4 Cost minimization SOC: f (x) is quasiconcave, V (y) is convex suffice. Somewhat complicated in general. FOC imply: 4.1 Problem of the firm. f (x ) wi xi = f (x) . wj xj c(w, y) = min wx
x s.t. f (x) = y. Lagrangian: When f (x1, x2) SOC becomes htD2f (x)h 0,
4.1.1 Difficulties h, wh = 0. L(, x) = wx - (f (x) - y). FOC: wi - f (x) = 0, xi f (x) = y. i = 1, . . . , n, differentiability border solutions non-existence multiplicity w = Df (x). Leontief: 4.1.2 Examples f (x1, x2) = min [ax1, bx2] , w1 w2 . c(w1, w2, y) = y + a b
x1,x2 1-a s.t. xax2 = y. 1 Cobb-Douglas: c(w, y) = min w1x1 + w2x2, Linear c(w1, w2, y) = min w1x1 + w2x2,
x1,x2 FOC: w1 a x2 = . w2 1 - a x1 Solution depends on w1 a . w2 b s.t. ax1 + bx2 = y, x1 0, x2 0. a Express x1 = 1-a w2 x2, substitute into f , obtain w1 x2 = !a 1 - a a w1 , a w2 4.2 Weak Axiom of Cost Minimization t, y t, x(wt, y t) for some t = 1, . . . , T . w 1-a w a 2 . x1 = 1-a w1 " 1-a# 1-a a a a 1-a c(w1, w2, y) = + w1 w2 y. a 1-a !1-a Observe WACM: for all t and s, wtxt wtxs. Implications: wt - ws xt - xs 0 or wx 0. RECOVERABILITY: V I, V O, Read the book. ...
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This note was uploaded on 03/18/2012 for the course ECON 201 taught by Professor Çakmak during the Spring '10 term at Middle East Technical University.
- Spring '10