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Nj In E exp (v (x, j)) I , <br />i xE1S}U9 1 <br />where Nj is the number of people living in neighborhood j. Note that the state variable now <br />does not include rent control status since we are consider the pre -law steady state. Suppose <br />that the law raises rents in zipcode j by San Francisco by a proportional amount equal to <br />d In Rj. Using standard calculations, we find that the local welfare impact of a change in <br />rents is given by: <br />Nj p(xlj)E21�xR�)dInR,, (16) <br />x k <br />where p (xl j) are the pre -law conditional choice probabilities To compute this quantity we <br />thus need to calculate av (x, j) /d 1nRk for all j, x, and k E 9 and we need to determine the <br />zipcode level rent response to the measured reduced form supply reduction. <br />Steady-state in the model is characterized by the equation: <br />vj : Nj (1—p(slj)-p(jh))=ENj,p(jh')- (17) <br />j'0j <br />This simply says that, in steady state, the number of renters flowing out of neighborhood j <br />must be equal to the number of renters flowing into neighborhood j. We now assume that the <br />supply decrease is the same proportionally in each zipcode. Since small multifamily housing <br />constituted 44% of 1994 non rent -controlled housing stock, our reduced form results indicate <br />that rental supply in San Francisco decreased by 6 percent. Letting dIn Nj/d(D denote the <br />supply response, where (1) is simply a convenient notation indicating the impact of the law, <br />we have <br />d In Nj— d In NSF <br />A A) = —.06 for all ,j in SF <br />We determine how much rents have to change by in the new long -run steady state given this <br />41 <br />