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which removes the log sums. Intuitively, equation (9) compares the difference in utility <br />between two different actions a household in state Ot-1 could take versus a household in state <br />0/ <br />t_1. This "differences -in -differences" approach removes all long-term utility differences since <br />actions are selected to create renewals. <br />5.2.2 One Period Ahead Renewals <br />Now suppose that x and x' are not renewal actions in period t. Following Scott (2013), we <br />substitute the expected difference in continuation values in equation (6) with its realization <br />and expectational errors: <br />where <br />a (x, Ot-1) - ut W, Bt-.1� - In (Ptt(7 1) - [It (Ot-1) - It (Oi-1)� <br />t-1 <br />l� M+1 (0- (� , 0't-1)) - Vt+1(O (x, Ot-1))) + t (x�, Bt -1) - t (x, Ot-1) <br />�(x, Ot-1) _ �3 (Et [Vt+1 (O (x, Ot-i))] - Vt+1 (O 1)) <br />(x, Ot-) <br />is the expectational error. <br />We now again make use of renewals. Suppose that at time t + 1, both households move <br />to the same neighborhood, that is xt+1 = xt+1 = j* E J. To see the effects of this, first <br />substitute out the realized ex -ante value functions using equations (4) and (5). We have: <br />ut (x,Ot-1) - at W, 0t-1) - In (Ptt(91� O1� [It (Ot-1) - It (0t-1)[ <br />t-1 <br />1) - <br />(vt+1 (j*, (x,, 0,t-1)) - vt+1 W, O (x, Ot-0)) <br />-a In (Pt+1 (I *, 0 (x� 0t-1) ) / + di (X1, 0t-1) - ci (x, 0t-1) . <br />Pt+1 (j*, O (xI0t <br />Since j* is a renewal action, the time t+ 2 expected value functions difference out and this <br />26 <br />