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Ecnomics D10-1: Lecture 11

Ecnomics D10-1: Lecture 11. Profit maximization and the profit function. Profit maximization by the price-taking competitive firm.

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Ecnomics D10-1: Lecture 11

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  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) approachusing FONCs and the Implicit Function Theorem.

  3. The algebraic approach to the profit maximization problem • The problem of the firm is to maxyYp.y • Define the profit function π(p) as the value function • Let y(p) = argmaxyYp.y denote the solution set • CONVEXITY Let y0y(p0), y1y(p1), and yty(pt), withpt= tp0 + (1-t)p1 π(pt) = ptyt = tp0yt + (1-t)p1yt ≤ tp0y0 + (1-t)p1y1 = tπ(p0) + (1-t)π(p1) • LAW OF OUTPUT SUPPLY/INPUT DEMAND py = (p1-p0)(y1-y0) = (p1y1-p1y0) + (p0y0-p0y1) ≥ 0 • Implies all own price effects are nonnegative: i.e., yi/pi≥ 0

  4. Results using the profit function • The Derivative Property and Convexity: Dπ=y(p) and D2π is positive semi-definite Proof: Let y0 = y(p0) for some p0>>0. Define the function g(p) = π(p) - p.y0. Clearly, g(p) ≥ 0 and g(p0) = 0. Therefore, g is minimized at p = p0. If π is differentiable, the associated FONC imply that Dg(p0) = Dπ(p0) - y(p0) = 0. Similarly, if π is twice differentiable the SONCs imply that D2g(p0) = D2π(p0) is a positive semi-definite matrix. • LAW OF OUTPUT SUPPLY/INPUT DEMAND • Combining the above results, D2π(p) = Dy(p) is a positive semi-definite matrix. This implies that (yj/pj)≥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: 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)

  6. 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 • Differentiating with respect to, e.g. w1, yields pf11(∂z1/∂w1) + pf12(∂z2/∂w1) = 1 pf21(∂z1/∂w1) + pf22(∂z2/∂w1) = 0

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