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Substituting in for the conditional value functions, we get: <br />ut (x, Bt -1) - ut (x , Bt -1) + �Et [V -t+1 (©(x, Bt -1))J - QEt IVt+1 (8 (x', Bt -r))] (6) <br />In pt (x Bt -r) <br />pt (x 1Bt-1) <br />Now assume x and x' are renewal actions in the sense that 0 (x, Bt -1) = O (x', 0t_1) . <br />Note that we do not require x = x', although this will often be the case. For example, if <br />two households in non -rent controlled housing are living in the same neighborhood j and <br />have the same level of neighborhood tenure, then x = S and x' = j, i.e. one household <br />choosing to stay in the current home and the other moving to another house in the same <br />neighborhood, constitute renewal actions. The key implication is that the future continuation <br />values difference out, leaving: <br />x <br />Bt -1 <br />( ) <br />ut (x Bt -1) - Tit, Bt -1) = In pt + It (Bt -1) - It (01t-1) (7) <br />If Bt -1 0 0't-1, we also need to remove the difference of log sums, which implicitly involves <br />future continuation values as well. <br />To do so, suppose the households move to some neighborhood j* E J, with j* =h x and <br />j* j x'. This always constitutes a renewal action, so we get equation (7) again with x and <br />X' replaced with j*: <br />'at (j*, Bt -1) - ut (j* 0t-1) = In �A(j l Bt 1) +It (Bt -1) - It (Bt -1) (8) <br />t-1 <br />Differencing equations (7) and (8) yields: <br />In A (xlBt-1) _ In pt (j*'Bt-t) _ �u x B _t) - ut �x >Bt -1)] (p) <br />A (x'�Bt-t) A (j*lot-1) ( t <br />25 <br />