begin align* A tF(ws,t*) = (Ss,tx-Ss,twfws,t*)+ sum from k = 0 to ∞ of β(1-δx,s)(1-ξw,s)[β(1-δx,s)ξw,s]k A t+k+1F(ws,t+k+1*); = (Ss,tx-Ss,twfws,t*)-β(1-δx,s)ξw,s(Ss,t+1x-Ss,t+1wfws,t+1*)+β(1-δx,s) A t+1F(ws,t+1*)end align*

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After rearranging, we obtain the second line.

We can then similarly define the value of a worker with an average wage for a labour firm:

A tF(ws,t) = uprime(cs,t)(hs,tα_ Hxs,t-hs,tws,t(1+τtwf)); +β(1-δx,s)[(1-ξw,s) A t+1F(ws,t+1*)+ξw,s A t+1F(ws,t)]

Following the same steps as above, we obtain, after some algebra

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A tF(ws,t) = (Ss,tx-Ss,twfws,t)-β(1-δx,s)ξw,s(Ss,t+1x-Ss,t+1wfws,t+1)+β(1-δx,s) A t+1F(ws,t+1)

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Free entry condition A firm posting a vacancy for household type s must pay a per-period constant cost ψs for having a vacancy open. If κw,s denotes the probability that a firm cannot renegotiate the wage for a newly hired worker from household type s, then the value of employing a new worker is, in monetary terms (recall, A F(ws,t*) ≡ u'(cs,t)AF(ws,t*), and the same for the value at average wage), equal to the weighted average of the value of a worker at a newly-renegotiated job and the value of a worker hired at average wage. The free-entry condition is:

ψs = ptFβ(uprime(cs,t+1))/(uprime(cs,t))[(1-κw,s) A tF(ws,t+1*)+κw,s A tF(ws,t+1)].

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Value functions for a worker We have two types of value functions, one for a newly-renegotiated wage and one for the average wage, for each type of household. The value of a job, net of the value of unemployment, for a worker with newly-renegotiated wage is

A tH(ws,t*) = uprime(cs,t)(hs,tws,t*(1-τtwh)-bs,t)-χ(hs,t1+φ)/(1+φ); +β(1-δx,s)[(1-ξw,s) A t+1H(ws,t+1*)+ξw,s A t+1H(ws,t*)]; -β ps,tW[(1-κw,s) A t+1H(ws,t+1*)+κw,s A t+1H(ws,t+1)]

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We can define additional auxiliary variables to sum the utility obtained from wages, unemployment benefits, and the disutility of labour terms,