Ch14[1] - Ch14. Consumers Surplus We have discussed how to...

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1 Ch14. Consumer’s Surplus We have discussed how to derive a consumer’s demand function from her underlying preferences or UF. We are now concerned with the reverse problem: how to estimate unobservable preferences or UF from observed demand functions. In fact, we have done this by recovering a C-D UF from average expenditure shares, and the present chapter will introduce more approaches to this problem. 14.1 Demand for a Discrete Good Go back to the example in Ch 6, where UF is quasilinear: u(X) = v (x 1 ) + x 2 , good 1 is a discrete good priced at p, good 2 is a $ amount spent on a composite good, and the BL is x 2 = m - px 1 . Looking at figure 6.12, one sees that for r 1 > r 2 > r 3 > …, the consumer is indifferent between 0 or 1 unit of good 1 at p = r 1 , 1 or 2 units of good 1 at p = r 2 , and so on. That is, = = = ......... ) 3 , 3 ( ) 2 , 2 ( ) 2 , 2 ( ) , 1 ( ) , 1 ( ) , 0 ( 3 3 2 2 1 r m u r m u r m u r m u r m u m u () + = + + = + + = + .......... ) 3 ( ) 3 ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) ( 1 ) ( ) 1 ( ) 0 ( 3 3 2 2 1 r m v r m v r m v r m v r m v m v We have: = = = .......... ) 2 ( ) 3 ( ) 1 ( ) 2 ( ) 0 ( ) 1 ( 3 2 1 v v r v v r v v r ……. . (Ex1) Clearly, the reservation price measures the MU (i.e., the increment in the utility needed to induce one more unit of good 1 for consumption). We want to show the relation between prices and demand that if n units of good 1 are demanded, then 1 + n n r p r . Proof : when she chooses to consume, say, 6 units of good 1 at p , then she prefers ( 6, m-6p ) to ( x 1 , m-px 1 ) for any other x 1
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2 ) 7 ( ) 7 ( ) 6 ( ) 6 ( ) 5 ( ) 5 ( ) 6 ( ) 6 ( p m v p m v p m v p m v + + + + That is, () ) 6 ( 7 ) 5 ( ) 6 ( v v p p v v or 7 6 r p p r This shows that the list of reservation prices contains all info necessary to characterize demand behavior, which is depicted in figure 14.1. 14.2 Constructing Utility from Demand What was just done is constructing the demand “curve” from the reservation prices or UF. We now want to build the UF from the demand “curve” by taking quasilinear utility for example. Setting utility from zero unit of good 1 equal to zero gives v(0) = 0 . Adding up both sides of the first n expressions in (Ex1) yields = = n i i r n v 1 ) ( , that is, v(n) is just the sum of the first n reservation prices. E.g., n = 3, v(3) = r 1 *1 + r 2 *1 + r 3 *1 = shaded area in figure 14.1 A. That is, the utility from consuming n units of good 1 is just the area of the first n bars that make up the demand function. This area is called the gross benefit or the gross consumers’ surplus . If she consumes
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Ch14[1] - Ch14. Consumers Surplus We have discussed how to...

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