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Marginal Analysis 07



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Marginal Analysis Purposes 1 To present a basic application of constrained optimization 2 Apply to Production Function to get criteria and patterns of optimal system design Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Marginal Analysis Slide 1 of 25 Marginal Analysis Outline 1 Definition and Assumptions 2 Optimality criteria Analysis Interpretation Application 3 Key concepts Expansion path Cost function Economies of scale 4 Summary Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Marginal Analysis Page 1 Slide 2 of 25 Marginal Analysis Concept Basic form of optimization of design Combines Production function Technical efficiency Input cost function c X Economic efficiency Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Marginal Analysis Slide 3 of 25 Marginal Analysis Assumptions Feasible region is convex over relevant portion This is key Why To guarantee no other optimum missed No constraints on resources To define a general solution Models are analytic continuously differentiable Finds optimum by looking at margins derivatives Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Marginal Analysis Page 2 Slide 4 of 25 Optimality Conditions for Design by Marginal Analysis The Problem Min C Y c X cost of inputs for any level of output Y production function s t g X Y Fixed level of output vector of resources The Lagrangean L c X g X Y Engineering Systems Analysis for Design Massachusetts Institute of Technology Richard de Neufville Marginal Analysis Slide 5 of 25 Optimality Conditions for Design Results Key Result c X Xi g X Xi marginal cost marginal product Optimality Conditions MPi MCi MPj MCj 1 MCj MPj MCi MPi Shadow Price on Product A balanced design Each Xi contributes same bang for buck Engineering Systems Analysis for Design Massachusetts Institute of Technology



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