Constraints - CHAPTER 9 Constrained Optimisation Rational...

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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 well-behaved (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 ....
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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.

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Constraints - CHAPTER 9 Constrained Optimisation Rational...

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