4. Utility - Solutions

# 4. Utility - Solutions - Chapter 4 NAME Utility...

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Chapter 4 NAME Utility Introduction. In the previous chapter, you learned about preferences and indiFerence curves. Here we study another way of describing prefer- ences, the utility function . A utility function that represents a person’s preferences is a function that assigns a utility number to each commodity bundle. The numbers are assigned in such a way that commodity bundle ( x, y ) gets a higher utility number than bundle ( x 0 ,y 0 ) if and only if the consumer prefers ( x, y )to( x 0 0 ). If a consumer has the utility function U ( x 1 ,x 2 ), then she will be indiFerent between two bundles if they are assigned the same utility. If you know a consumer’s utility function, then you can ±nd the indiFerence curve passing through any commodity bundle. Recall from the previous chapter that when good 1 is graphed on the horizontal axis and good 2 on the vertical axis, the slope of the indiFerence curve passing through a point ( x 1 2 )isknownasthe marginal rate of substitution .An important and convenient fact is that the slope of an indiFerence curve is minus the ratio of the marginal utility of good 1 to the marginal utility of good 2. ²or those of you who know even a tiny bit of calculus, calculating marginal utilities is easy. To ±nd the marginal utility of either good, you just take the derivative of utility with respect to the amount of that good, treating the amount of the other good as a constant. (If you don’t know any calculus at all, you can calculate an approximation to marginal utility by the method described in your textbook. Also, at the beginning of this section of the workbook, we list the marginal utility functions for commonly encountered utility functions. Even if you can’t compute these yourself, you can refer to this list when later problems require you to use marginal utilities.) Example: Arthur’s utility function is U ( x 1 2 )= x 1 x 2 . Let us ±nd the indiFerence curve for Arthur that passes through the point (3 , 4). ²irst, calculate U (3 , 4) = 3 × 4 = 12. The indiFerence curve through this point consists of all ( x 1 2 ) such that x 1 x 2 = 12. This last equation is equivalent to x 2 =1 2 /x 1 . Therefore to draw Arthur’s indiFerence curve through (3 , 4), just draw the curve with equation x 2 =12 /x 1 .A t the point ( x 1 2 ), the marginal utility of good 1 is x 2 and the marginal utility of good 2 is x 1 . Therefore Arthur’s marginal rate of substitution at the point (3 , 4) is x 2 /x 1 = 4 / 3. Example: Arthur’s uncle, Basil, has the utility function U ( x 1 2 3 x 1 x 2 10. Notice that U ( x 1 2 )=3 U ( x 1 2 ) 10, where U ( x 1 2 )is Arthur’s utility function. Since U is a positive multiple of U minus a con- stant, it must be that any change in consumption that increases U will also increase U (and vice versa). Therefore we say that Basil’s utility function is a monotonic increasing transformation of Arthur’s utility function. Let

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34 UTILITY (Ch. 4) us fnd Basil’s indiFerence curve through the point (3 , 4). ±irst we fnd that U (3 , 4) = 3 × 3 × 4 10 = 26 . The indiFerence curve passing through
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4. Utility - Solutions - Chapter 4 NAME Utility...

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