begin align* Ss,twh ≡ sum from k = 0 to ∞ of [β(1-δx,s)ξw,s]k uprime(cs,t+k)hs,t((1+ π)k Pt)/(Pt+k)(1-τtwh); = uprime(cs,t)hs,t(1-τtwh)+β(1-δx,s)((1+ π))/((1+πt+1))ξw,sSs,t+1wh,end align*
(62)
Ss,tb = uprime(cs,t+k)bs,t+β(1-δx,s)ξw,s((1+ π))/((1+πt+1))Ss,t+1b,
(63)
Ss,th = χ(hs,t1+φ)/(1+φ)+β(1-δx,s)ξw,sSs,t+1h,
(64)
then the net value of a job for a new wage for the household is:
A tH(ws,t*) = Ss,twws,t*-Ss,tb-β(1-δx,s)ξw,s(St+1wws,t+1*-Ss,t+1b); -Ss,th+β(1-δx,s)ξw,sSs,t+1h; +β[1-δx,s-(1-κw,s)ps,tW] A t+1H(ws,t+1*)-βκw,sps,tW A t+1H(ws,t+1)
(65)
The value of a job for an average wage for the household is:
A tH(ws,t) = Ss,twws,t-Ss,tb-β(1-δx,s)ξw,s(Ss,t+1wws,t+1-Ss,t+1b); -Ss,th+β(1-δx,s)ξw,sSs,t+1h; +β[(1-δx,s)(1-ξw,s)-(1-κw,s)ps,tW] A t+1H(ws,t+1*); +β[(1-δx,s)ξw,s-κw,sps,tW] A t+1H(ws,t+1);
(66)
Wages and hours are determined using efficient Nash bargaining, so that every period, wages and hours worked are determined by maximising the following expression, where ηs is the bargaining power of workers of type s:
max over ws,t*,hs,t of (AtH(ws,t*))η_ s(AtF(ws,t*))1-η_ s.
The Nash sharing rule is:
(67)
ηs(1-τtwh)AtF(ws,t*) = (1-ηs)(1+τtwf)AtH(ws,t*).
(68)
This equation thus determines wt,s*.
The average wage is determined by using the law of motion of labour as follows:
begin align* ndes,ths,tws,t = (1-δx,s)ndes,t-1[ξw,s(1+ π)/(1+πt-1)ws,t-1hs,t+(1-ξw,s)w*s,ths,t]; +Ms,t[κw,sws,ths,t+(1-κw,s)w*s,ths,t],end align*
(69)
where the first row is the average wage of existing workers and the second row is the average wage of new hires, with the number of new matches Ms,t counting the number of new hires.