Basic form of optimization of design Combines: Production function - Technical efficiency Input cost function, c(x) Economic efficiency

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1 Marginal Analysis Outline 1. Definition 2. Assumptions 3. Optimality criteria Analysis Interpretation Application 4. Expansion path 5. Cost function 6. Economies of scale Massachusetts Institute of Technology Marginal Analysis Slide 1 of 16 Marginal Analysis Basic form of optimization of design Combines: Production function - Technical efficiency Input cost function, c(x) Economic efficiency Massachusetts Institute of Technology Marginal Analysis Slide 2 of 16 Page 1

2 Assumptions of Marginal Analysis Feasible region is convex (over relevant portion) No constraints on resources Models are analytic (needed only for derivation) Massachusetts Institute of Technology Marginal Analysis Slide 3 of 16 Optimality Conditions for Design, by Marginal Analysis The Problem: Min C(Y ) = c(x) s.t. g(x) = Y cost function production function vector of resources output The Lagrangean: L = c(x) - λ [g(x) - Y ] Massachusetts Institute of Technology Marginal Analysis Slide 4 of 16 Page 2

3 Optimality Conditions for Design, by Marginal Analysis (cont d) Key Result: c(x) / X i = λ g(x) / X i marginal cost marginal product Optimality Conditions: MP i / MC i = 1 / λ = MP j / MC j or MP i / MP j = MC i / MC j A balanced design Each X i contributes same bang for buck Massachusetts Institute of Technology Marginal Analysis Slide 5 of 16 Graphical Interpretation of Optimality Conditions (A) Input Cost Function X 2 B/P 2 P 2 P 1 B = Budget c(x) = Σp i X i B Linear case: In general, non-linear (as in curved line) B/P 1 X 1 Massachusetts Institute of Technology Marginal Analysis Slide 6 of 16 Page 3

4 Graphical Interpretation of Optimality Conditions (cont d) (B) Conditions X 2 Slope = MRS ij = - MP 1 / MP 2 = - MC 1 / MC 2 Isoquant $ X 1 Massachusetts Institute of Technology Marginal Analysis Slide 7 of 16 Application of Optimality Conditions Problem: Y = a 0 X 1 a1 X 2 a2 c(x) = Σp i X i Note: Linearity of Input Cost Function - typically assumed by economists - in general, not valid z prices rise with demand z wholesale, volume discounts Solution: [a 1 / X 1 *] Y / p 1 = [a 2 / X 2 *] Y / p 2 ( * denotes an optimum value) Massachusetts Institute of Technology Marginal Analysis Slide 8 of 16 Page 4

5 Expansion Path Locus of all optimal designs X* Not a property of technical system alone Depends on local prices Optimal designs do not, in general, maintain constant ratios between optimal X i * e.g.: crew of 20,000 ton ship crew of 200,000 ton ship Massachusetts Institute of Technology Marginal Analysis Slide 9 of 16 Calculation of Expansion Path Assume: Y = 2X X c(x) = X 1 + X (increasing RTS) Optimality Conditions: (0.48 / X 1 ) Y / 1 = (0.72 / X 2 ) Y / (1.5X ) = MP i / MC i => X 1 * = (X 2 *) 1.5 Graphically: Y X 2 Expansion Path X 1 Massachusetts Institute of Technology Marginal Analysis Slide 10 of 16 Page 5

6 Cost Functions C(Y) = C(X*) = f (Y) Not same as input cost function optimal cost of Y vs. cost of any X Massachusetts Institute of Technology Marginal Analysis Slide 11 of 16 Cost Functions (cont d) Graphically: C(Y) Feasible Economist s view Y Feasible Engineer s view Cost-Effectiveness Y C(Y) Great practical use: How much Y for budget? Y for B? Cost effectiveness, B / Y Massachusetts Institute of Technology Marginal Analysis Slide 12 of 16 Page 6

7 Calculation of Cost Function Cobb-Douglas production function Y = a 0 πx i ai Linear input cost function Result c(x) = Σp i X i C(Y) = A(πp i ai/ r )Y 1/r where r = Σa i Easy to estimate statistically => Solution for a i => Estimate of production function Y = a 0 πx ai i Massachusetts Institute of Technology Marginal Analysis Slide 13 of 16 Calculation of Cost Function (cont d) Assume Again: Y = 2X X c(x) = X 1 + X Expansion Path: X 1 * = (X 2 *) 1.5 Thus: Y = 2(X 2 *) 1.44 c(x*) = 2(X 2 *) 1.5 => X 2 * = (Y/2) 0.7 c(y) = c(x*) = ( )Y 1.05 Massachusetts Institute of Technology Marginal Analysis Slide 14 of 16 Page 7

8 Economies of Scale A possible characteristic of cost function Concept similar to returns to scale, except ratio of X i not constant refers to costs (economies) Economies of scale exist if costs increase slower than product Total cost = C(Y) = Y < 1.0 Massachusetts Institute of Technology Marginal Analysis Slide 15 of 16 Economies of Scale (cont d) Note: If Cobb-Douglas, linear input costs Increasing RTS => Economies of scale r = Σa i > 1.0 => C(Y) = fcn Y 1/r Not necessarily true in general See other example!! Massachusetts Institute of Technology Marginal Analysis Slide 16 of 16 Page 8