The Theory of Production The production process One variable input Two or more variable inputs Optimal combination of inputs Economic region of production

Production Functions Transformation of inputs to output Basis for all cost analysis Short-run (at least one fixed input) vs. long-run (all variable inputs) Q = Q(X1, X2, …, Xn)

Production with One Variable Input Refer to production Table 8.1 on p. 267 Total, average, and marginal product Law of diminishing returns Stages of production

Production with Two Variable Inputs Production isoquants Economic region of production MRTSYX = MPX/MPY = -dY/dX

Production elasticities For any input: EX = %Q/%X MPL > APL, EL > 1

Three Stages of Production Stage 1: AP rising Stage 2: AP falling, but MP positive Stage 3: MP negative and TP falling

Optimal Input Use in Perfect Competition Hire workers, if: addition to revenue > addition to cost TR/ L > TC/ L MRPL > MRCL At optimum, MRPL = wage

Returns to Scale Q = Q(K, L) zQ = Q(hK, hL) CRS when z = h IRS when z > h DRS when z < h

Economic Region of Production Slope of isoquant = MPL/MPK Define region by ridge lines MPL = 0; slope of isoquant = 0 MPK = 0; slope of isoquant = 

Optimal Combination of Inputs Objective: minimize cost for a given Q Isocost: combination of inputs for a given cost Equimarginal principle

Cobb-Douglas Production Function Q = AKaLb a + b = 1, then CRS a + b > 1, then IRS a + b < 1, then DRS MPs depend on both inputs Exponents represent output elasticities Estimated by using log transformation log Q = logA + a logK + b logL

Interpreting Cobb-Douglas Using Q = 100L0.5 K0.5 What is degree of homogeneity? What about returns to scale? What are labor and capital elasticities? What happens to Q, if L increases by 4% and K increases by 2%?

Long-run production with Cobb-Douglas Impact on Q (= h) of proportionate increase in inputs (z) hQ = A(zK)a (zL)b = zazb(AKa Lb) = za+b(AKa Lb) since Q = (AKa Lb), then h = za+b when a+b = 1, h = z => double inputs, double output

Appendix 8A: Lagrangians Maximize Q s.t. cost constraint r is the cost of capital; w is the wage rate

Appendix 8B: Linear programming Manufacturers have alternative production processes, some involving mostly labor, others using machinery more intensively. The objective is to maximize output from these production processes, given constraints on input availability, such as plant capacity or labor constraints. We will discuss linear programming techniques more extensively in Chapter 11.