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Unformatted text preview: 3 Profit Function
(p) = max py.
yY 3.1 Properties 1. If p0 pi, p0 po for all i (inputs) and o (outo i puts), then (p0) (p). 2. Homogeneous of degree 1: for t > 0, (tp) = t(p). 3. Convex in p: for p00 = tp + (1 - t)p0, t [0, 1], (p00) t(p) + (1 - t)(p0). 4. Continuous in p for p 0 (when defined). 3.2 Supply and Demand 3.2.1 Example: Hotelling's Lemma: For all i = 1, . . . , n, for pi > 0, yi (p) = The LeChatelier principle: (p, wx, wz ) = max py - wxx - wz z.
y,x,z (p) . pi (1) wx, wz are fixed. Short-run: z is fixed.
Proof: Special case of Envelope Theorem. Suppose z = z(p)L-R level, Envelope Theorem: For the problem M(a) = max f (z, a)
z h(p) = LR(p) - SR(p, z ) = SR(p, z(p)) - SR(p, z ) 0. We have p = arg min h(p),
p we have dM (a) f (x, a) . = da a x=x(a) Substitution matrix: Eq. (1) implies h(p) = 0, (SOC): y(p) = D(p) = Dy(p) = D2(p), (p) convex = Dy(p)symm., pos. semi-def.. 2 LR(p) 2 SR(p, z ) - 0, p2 p2 dy(p) dy(p, z ) - 0. dp dp ...
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- Spring '10