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supply response. Taking a derivative of equation (17) with respect to (P gives: <br />dInNj dp(xIj) = dInNy dp(j Y) <br />dI N' ClE p(xlj) —Nj 1:d� d4, N3'p(jlj)+N, <br />ad� <br />E{S,j} xE{S,il YAJ <br />(18) <br />for all j <br />So: <br />Now, in steady state, the conditional probabilities are given by: <br />exp (v (x, j)) <br />P (x j) _ E., exp (v (x', j)) <br />dp(xlj) _ ap(xlj)dlnRk (19) <br />d(D aInRk d(P <br />x (MLL _� av(x"A dInRk <br />k p( j) aInRk X p( j) alnRk d(P <br />To finish the calculation, we therefore need to determine av (x, j) /a In Rk. With these in <br />place, we can plug equation (19) into equation (18) and solve the resulting system of equations <br />for the rent responses dInRk/d(h. <br />We note that in steady-state: <br />v (x, j) = u (x, j) + a In I exp (v V, j (x))) <br />Taking derivatives with respect to log rents, we get: <br />av(x,j)="Yexp (Rj)RjID(x)=k1+/�� p(xlj(x))av(x'j(x)) <br />a In Rk lX,,' a In Rk <br />This is a system of equations which can be numerically solved for the partial derivatives. The <br />system for young renters is similar, but takes into account the possibility of transitioning to <br />a mature renter. <br />42 <br />