This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: CHAPTER 9 Constrained Optimisation Rational economic agents are assumed to make choices that maximise their utility or profit. But their choices are usually constrained – for example the consumer’s choice of consumption bundle is constrained by his income. In this chapter we look at methods for solving optimisa tion problems with constraints: in particular the method of Lagrange multipliers . We apply them to consumer choice , cost minimisa tion , and other economic problems. — ./ — 1. Consumer Choice Suppose there are two goods available, and a consumer has preferences represented by the utility function u ( x 1 ,x 2 ) where x 1 and x 2 are the amounts of the goods consumed. The prices of the goods are p 1 and p 2 , and the consumer has a fixed income m . He wants to choose his consumption bundle ( x 1 ,x 2 ) to maximise his utility, subject to the constraint that the total cost of the bundle does not exceed his income. Provided that the utility function is strictly increasing in x 1 and x 2 , we know that he will want to use all his income. The consumer’s optimisation problem is: max x 1 ,x 2 u ( x 1 ,x 2 ) subject to p 1 x 1 + p 2 x 2 = m • The objective function is u ( x 1 ,x 2 ) • The choice variables are x 1 and x 2 • The constraint is p 1 x 1 + p 2 x 2 = m There are three methods for solving this type of problem. 1.1. Method 1: Draw a Diagram and Think About the Economics x 1 x 2 P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Provided the utility function is wellbehaved (increasing in x 1 and x 2 with strictly convex indifference curves), the highest utility is obtained at the point P where the budget constraint is tangent to an indifference curve. The slope of the budget constraint is: p 1 p 2 and the slope of the indifference curve is: MRS = MU 1 MU 2 (See Chapters 2 and 7 for these results.) 153 154 9. CONSTRAINED OPTIMISATION Hence we can find the point P : (1) It is on the budget constraint: p 1 x 1 + p 2 x 2 = m (2) where the slope of the budget constraint equals the slope of the indifference curve: p 1 p 2 = MU 1 MU 2 In general, this gives us two equations that we can solve to find x 1 and x 2 ....
View
Full
Document
This note was uploaded on 04/12/2010 for the course ECON DEAM taught by Professor Vines during the Spring '10 term at Oxford University.
 Spring '10
 Vines
 Economics, Utility

Click to edit the document details