1. Ecnomics D10-1: Lecture 11 Profit maximization and the profit function

2. 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 profit- maximization 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.

3. The algebraic approach to the profit maximization problem The problem of the firm is to max y  Y p. y Define the profit function π(p) as the value function Let y(p) = argmax y  Y p. y denote the solution set CONVEXITY Let y 0  y(p 0 ), y 1  y(p 1 ), and y t  y(p t ), with p t = tp 0 + (1-t)p 1 π(p t ) = p t y t = tp 0 y t + (1-t)p 1 y t ≤ tp 0 y 0 + (1-t)p 1 y 1 = tπ(p 0 ) + (1-t)π(p 1 ) LAW OF OUTPUT SUPPLY/INPUT DEMAND  p  y = (p 1 -p 0 )(y 1 -y 0 ) = (p 1 y 1 -p 1 y 0 ) + (p 0 y 0 -p 0 y 1 ) ≥ 0 – Implies all own price effects are nonnegative: i.e.,  y i /  p i ≥ 0

4. Results using the profit function The Derivative Property and Convexity: Dπ=y(p) and D 2 π is positive semi-definite Proof: Let y 0 = y(p 0 ) for some p 0 >>0. Define the function g(p) = π(p) - p. y 0. Clearly, g(p) ≥ 0 and g(p 0 ) = 0. Therefore, g is minimized at p = p 0. If π is differentiable, the associated FONC imply that Dg(p 0 ) = Dπ(p 0 ) - y(p 0 ) = 0. Similarly, if π is twice differentiable the SONCs imply that D 2 g(p 0 ) = D 2 π(p 0 ) is a positive semi-definite matrix. LAW OF OUTPUT SUPPLY/INPUT DEMAND –Combining the above results, D 2 π(p) = Dy(p) is a positive semi- definite matrix. This implies that (  y j /  p j )≥0: i.e., the physical quantities of ouputs (inputs) increase (decrease) in own prices.

5. The Neoclassical approach to profit maximization: the single output case Problem: max z pf(z)-w. z Solution: z(p,w) = argmax z 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 pD 2 f(z(p,w)) negative semi-definite COMPARATIVE STATICS: Assuming z(p,w)>>0,differentiate the FONCs to obtain pD 2 fD w z = I or D w z = (1/p)[D 2 f] -1 when the Hessian matrix of f is nonsingular. In that case, D w z is negative semi-definite. (Also, Df + pD 2 fD p z = 0 or D p z = -(1/p)[D 2 f] -1 Df so that  q/  p = DfD p z = -(1/p)Df[D 2 f] -1 Df ≥ 0)

6. The Neoclassical approach: single output, two input example Max pf(z 1,z 2 ) - w 1 z 1 - w 2 z 2 Let (z 1 (p,w 1,w 2 ),z 2 (p,w 1,w 2 )) = argmax pf(z 1,z 2 )-w 1 z 1 - w 2 z 2 FONCs for interior solution: pf 1 (z 1 (p,w 1,w 2 ),z 2 (p,w 1,w 2 )) - w 1 = 0 pf 2 (z 1 (p,w 1,w 2 ),z 2 (p,w 1,w 2 )) - w 2 = 0 Differentiating with respect to, e.g. w 1, yields pf 11 (∂z 1 /∂w 1 ) + pf 12 (∂z 2 /∂w 1 ) = 1 pf 21 (∂z 1 /∂w 1 ) + pf 22 (∂z 2 /∂w 1 ) = 0