Notes-05_and_06_Choice_and_Demand-1

Notes-05_and_06_Choice_and_Demand-1 - Econ 3130, Spring...

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Econ 3130, Spring 2012, R. Masson Chapter 5 and 6: p. 1 Chapters 5 and 6: Choice and Demand I. Finally , Putting the two sides together A. In these two chapters we more systematically look at the budget line and either indifference curves or utility functions to characterize choices 1. First we look at types of individual maximizing equilibria using indifference curves, Chapter 5. 2. Second we aggregate across individuals and look at Demand and market equilibrium comparative statics, Chapter 6. 3. Third we return to Chapter 5 (Appendix) and look at individual choice behavior using calculus to solve the utility maximization problem 4. Fourth we return to chapter 6 using utility function analysis II. Budget Lines and Indifference Curve Analysis A. Two space, indifference curves are enough for most undergraduate purposes 1. We first look at differentiable (smooth) “well behaved” indifference curves and parametric income and prices 2. We then look at issues like corner solutions, kinked indifference curves, and wiggly, non-convex, indifference curves III. The Most Conventional, Most Common, Models A. Recall the Budget line: 1. Graphically, with x 2 on the vertical axis, M/p 2 is the vertical intercept of the line and -(p 1 /p 2 ) is the slope of the line
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Econ 3130, Spring 2012, R. Masson Chapter 5 and 6: p. 2 2. Mathematically, 3. Just as I did for the marginal rate of substitution of x 2 for x 1 , most of us commonly look at the absolute value of the slope, or | -p 1 /p 2 | = p 1 /p 2 4. This is sometimes called the Marginal Rate of Transformation of x 2 for x 1 B. In this common model, indifference curves are smoothly downward sloping and convex and one “prefers” being on the highest (furthest up to the right) indifference curve 1. The result is that the “utility maximizing” solution has an indifference curve tangent to the budget line C. The consumer is assumed to want to reach the highest possible indifference curve consistent with the budget 1. If utility is U(x 1 ,x 2 ) and we totally differentiate a. dU = U 1 dx 1 + U 2 dx 2 (1) The change in utility is given by the rate of change from expanding x 1 times the expansion of x 1 plus the rate of change from expanding x 2 times the change in x 2 (2) Indifference means that if I change x i I must change x j to keep utility from changing, or dU=0 (3) If dU=0 we have , and this is the slope of an indifference curve b. For those of you with little math, a simplification follows (1) Suppose initial utility is U 0 =100 (2) Suppose (locally) U goes up by 2 units if x 1 goes up by one unit and U goes up by 1 unit if x 2 goes up by one unit.
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Econ 3130, Spring 2012, R. Masson Chapter 5 and 6: p. 3 (3) Expand x 1 by one unit and U goes up to 102. So, to get U back down to U 0 =100 we can contract x 2 by two units 2. In equilibrium, the slope of the indifference curve equals the slope of the budget line or or minus the price ratio equals Varian’s definition of MRS a. Or or the price ratio is equal to my definition of MRS
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Notes-05_and_06_Choice_and_Demand-1 - Econ 3130, Spring...

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