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MIT14_30s09_lec14

# MIT14_30s09_lec14 - MIT OpenCourseWare http/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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1 14.30 Introduction to Statistical Methods in Economics Lecture Notes 14 Konrad Menzel April 2, 2009 Conditional Expectations Example 1 Each year, a firm’s R&D department produces X innovations according to some random process, where E [ X ] = 2 and Var( X ) = 2 . Each invention is a commercial success with probability p = 0 . 2 (assume independence). The number of commercial successes in a given year are denoted by S . Since we know that the mean of S B ( x, p ) = xp , conditional on X = x innovations in a given year, xp of them should be successful on average. The conditional expectation of X given Y is the expectation of X taken over the conditional p.d.f.: Definition 1 E [ Y X ] = yf Y | X ( y | X ) if Y is discrete y yf Y | X ( y X ) dy if Y is continuous | −∞ | Note that since f Y | X ( y | X ) carries the random variable X as its argument, the conditional expectation is also a random variable. However, we can also define the conditional expectation of Y given a particular value of X , yf Y | X ( y x ) if Y is discrete y E [ Y X = x ] = yf Y | X ( y | | x ) dy if Y is continuous | −∞ which is just a number for any given value of x as long as the conditional density is defined. Since the calculation goes exactly like before, only that we now integrate over the conditional distribution, won’t do a numerical example (for the problem set, just apply definition). Instead let’s discuss more qualitative examples to illustrate the difference between conditional and unconditional examples: Example 2 (The Market for ”Lemons”) The following is a simplified version of a famous model for the market for used cars by the economist George Akerlof. Suppose that there are three types X of used cars: cars in an excellent state (”melons”), average-quality cars (”average” not in a strict, statistical, sense), and cars in a poor condition (”lemons”). Each type of car is equally frequent, i.e. 1 P (” lemon ”) = P (” average ”) = P (” melon ”) = 3 The seller and a buyer have the following (dollar) valuations Y S and Y B , respectively, for each type of cars: Type Seller Buyer ”Lemon” 5,000 \$ 6,000 \$ ”Average” 6,000 \$ 10,000 \$ ”Melon” 10,000 \$ 11,000 \$ 1
The first thing to notice is that for every type of car, the buyer’s valuation is higher than the seller’s, so for each type of car, trade should take place at a price between the buyer’s and the seller’s valuations. However, for used cars, quality is typically not evident at first sight, so if neither the seller nor the buyer know the type X of a car in question, their expected valuations are, by the law of iterated expectations E [ Y S ] = E [ Y S lemon ”] P (” lemon ”) + E [ Y S average ”] P (” average ”) + E [ Y S melon ”] P (” melon ”) | | | 1 = (5 , 000 + 6 , 000 + 10 , 000) = 7 , 000 3 E [ Y S ] = E [ Y B lemon ”] P (” lemon ”) + E [ Y B average ”] P (” average ”) + E [ Y B melon ”] P (” melon ”) | | | 1 = (6 , 000 + 10 , 000 + 11 , 000) = 9 , 000 3 so trade should

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MIT14_30s09_lec14 - MIT OpenCourseWare http/ocw.mit.edu...

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