301-note-pp16-35 - CHAPTER 5 CHOICE The optimization...

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CHAPTER 5: CHOICE The optimization principle: A rational consumer chooses a consumption bundle that is preferred the most in the budget constraint. The optimal choice is a consumption bundle, also called a demand bundle , in the budget constraint that is weakly preferred to any other bundle in the budget constraint. In other words, there is no consumption bundle in the budget constraint that is strictly prefered to a demand bundle. In our notation, consumption bundle ( x 1 , x 2 ) is a demand bundle if p 1 x 1 + p 2 x 2 m , and for any consumption bundle ( x 1 , x 2 ) such that p 1 x 1 + p 2 x 2 m , ( x 1 , x 2 ) º ( x 1 , x 2 ) ⇐⇒ u ( x 1 , x 2 ) u ( x 1 , x 2 ) . This means that u ( x 1 , x 2 ) is the maximum of the utility function over the budget constraint. That is, u ( x 1 , x 2 ) = max x 1 ,x 2 u ( x 1 , x 2 ) , subject to p 1 x 1 + p 2 x 2 m. This constrained optimization problem is referred as the consumer problem . Under certain conditions, the budget line is tangent to an indi ff erence curve at a demand bundle: - 6 x 1 x 2 0 Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z s x 2 x 1 An interior solution to the consumer problem. The budget line and such an di ff erence curve have the same slope at an interior demand bundle. A necessary condition is slope of indi ff erence curve = MRS = p 1 p 2 = slope of budget line. That is, the marginal rate of substitution is equal to the economical rate of substitution (internal rate of substitution equals external rate of substitution). 16
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On the graph, it is easy to argue that if these two rates of substitution are di ff erent, then the consumption bundle cannot be a demand bundle. Suppose that MRS is 1 but p 1 p 2 = 2. The consumer is willing to exchange one unit of good 1 with one unit of good 2. Good 1 costs twice as much as good 2. Consider the following procedure which will make the consumer better o ff : the con- sumer buys one unit less of good 1, with the money saved, the consumer will be able to buy two more units of good 2. Note that just one more unit of good 2 will make the consumer indi ff erent for giving up one unit of good 1. The additional one unit of good 2 will make the consumer better o ff . So the consumption bundle where MRS 6 = p 1 p 2 cannot be an optimal choice. A few exceptions to this necessary condition. 1. MRS does not exist, such as perfect complements; 2. Non-interior solution, corner solution, such as perfect substitutes; - 6 x 1 x 2 0 @ @ @ @ @ @ @ @ C C C C C C C C X X X X X X X X s Kinky tastes. - 6 x 1 x 2 0 H H H H H H H H s Corner solution. 3. Non-convex preference, - 6 x 1 x 2 0 @ @ @ @ @ @ @ @ c s s Multiple interior solutions. - 6 x 1 x 2 0 @ @ @ @ @ @ @ @ c Boundary solution. A demand bundle is a solution to the consumer’s utility maximization problem. ( x 1 , x 2 ) arg max ( x 1 ,x 2 ) u ( x 1 , x 2 ) , subject to p 1 x 1 + p 2 x 2 m.
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