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where wj is a time -invariant fixed effect and w t is a per -period neighborhood specific shock. <br />We impose no structure on the distribution of w t beyond requiring that F(wj,t+llcjt, xit) = <br />F(w ,t+l lcjjt). That is, the decision of any individual agent has no impact on the distribution <br />of the neighborhood amenity value next period. <br />Letting R denote the common discount factor, the household's dynamic optimization <br />problem at time t is given by: <br />V (0i,t-1, wt, Eit) = max E Qs—t'U' (`r*, wt, Eit, 9i,t-1) 10i,t-1, wt, Eit� <br />x" <br />s>t <br />We next define the ex -ante value function V (Bit, wt) by integrating over the idiosyncratic <br />errors: <br />Vt (Bt -1) = J ... / V (Bt -1, wt, lElr ..., EJ+1)) dF (El) ... dF (EJ+1) , <br />where J is the number of neighborhoods and EJ+l it the logit error associated with staying <br />in the current home. From this we can define the value function conditional on actions: <br />vt (x, Bt -1) = ut (x, Bt -1) +OEt [Vt+1 (n (x, et -1))] , <br />where ut (x, Bt -1) = u (x, wt, 0, Bt -1), © (x, Bt -1) denotes the state transition function, and <br />Et [•] denotes expectations conditional on time t information. <br />Since the idiosyncratic taste shocks follow a logit specification, we get the standard results <br />(see e.g. Hotz and Miller (1993)) relating conditional value functions to conditional choice <br />probabilities pt (XjBt_1): <br />pt(xIA-1) = exp (vt(x,Bt-1)) (4) <br />E�, exp (vt (xt, Bt -1)) <br />In what follows, we denote the log of the denominator of this expression as: <br />It (Bt -1) = In ��, eXp (vt (x', dt-1)) <br />X, <br />/ <br />23 <br />