Ecnomics D10-1: Lecture 11 - Northwestern University

Ecnomics D10-1: Lecture 11 Profit maximization and the profit function Profit maximization by the price-taking competitive firm The firm is assumed to choose feasible input/output vectors to maximize the excess of revenues over expenditures under the assumption that it takes market prices as given. There are 3 equivalent approaches to the profitmaximization problem (and associated comparative statics) The algebraic approach using netput notation The dual approach using the properties of the profit function. The Neoclassical (calculus) approach using FONCs and the Implicit Function Theorem. The algebraic approach to the profit maximization problem

The problem of the firm is to maxyY p.y Define the profit function (p) as the value function Let y(p) = argmaxyY p.y denote the solution set CONVEXITY Let y0y(p0), y1y(p1), and yty(pt), with pt = tp0 + (1-t)p1 (pt) = ptyt = tp0yt + (1-t)p1yt tp0y0 + (1-t)p1y1 = t(p0) + (1-t)(p1) LAW OF OUTPUT SUPPLY/INPUT DEMAND py = (p1-p0)(y1-y0) = (p1y1-p1y0) + (p0y0-p0y1) 0 Implies all own price effects are nonnegative: i.e., yi/pi 0 Results using the profit function

The Derivative Property and Convexity: D=y(p) and D2 is positive semi-definite Proof: Let y0 = y(p0) for some p0>>0. Define the function g(p) = (p) - p.y0. Clearly, g(p) 0 and g(p0) = 0. Therefore, g is minimized at p = p0. If is differentiable, the associated FONC imply that Dg(p0) = D(p0) - y(p0) = 0. Similarly, if is twice differentiable the SONCs imply that D2g(p0) = D2(p0) is a positive semi-definite matrix. LAW OF OUTPUT SUPPLY/INPUT DEMAND Combining the above results, D2(pp) = Dy(p) is a positive semidefinite matrix. This implies that (yj/pj)0: i.e., the physical quantities of ouputs (inputs) increase (decrease) in own prices. The Neoclassical approach to profit maximization: the single output case Problem: maxz pf(z)-w.z Solution: z(p,w) = argmaxz pf(z)-w.z

Assume f is twice continuously differentiable. FONCs: pDf(z(p,w))-w 0; z(p,w) 0; (pDf(z(p,w))-w).z = 0 For z(p,w)>>0, SONCs require pD2f(z(p,w)) negative semi-definite COMPARATIVE STATICS: Assuming z(p,w)>>0,differentiate the FONCs to obtain pD2fDwz = I or Dwz = (1/p)[D2f]-1 when the Hessian matrix of f is nonsingular. In that case, Dwz is negative semi-definite. (Also, Df + pD2fDpz = 0 or Dpz = -(1/p)[D2f]-1Df so that q/p = DfDpz = -(1/p)Df[D2f]-1Df 0) The Neoclassical approach: single output, two input example Max pf(z1,z2) - w1z1 - w2z2 Let (z1(p,w1,w2),z2(p,w1,w2)) = argmax pf(z1,z2)-w1z1 - w2z2 FONCs for interior solution:

pf1(z1(p,w1,w2),z2(p,w1,w2)) - w1 = 0 pf2(z1(p,w1,w2),z2(p,w1,w2)) - w2 = 0 Solving via Cramer' s Rule yields 1 p f12 0 pf22 z1 pf22 2 w1 pf11 pf12 p (f11 f22 f12 f21) p f21 pf22 pf11 pf21 Differentiating with respect to, e.g. z2

w1, yields w1 pf11 p f21 pf11(z1/w1) + pf12(z2/w1) = 1 pf21(z1/w1) + pf22(z2/w1) = 0 1 0 pf21 2 pf12 p (f11 f22 f12 f21) pf22