The term hs,tα_ H denotes the effective hours that a labour firm produces from hours hs,t supplied by the worker from household s. xs,t is what the labour packer pays for such unit of labour. The first term in equation 52 therefore measures earnings of the labour firm from selling hours worked. But for these hours it has to pay to the household hourly wage, which is in this case newly renegotiated, ws,t*. Because we assume that labour firms pay some labour taxes (social security contributions), the cost for the labour firm is increased by taxes paid, at the rate τtwf. In the next period, if the firm and the worker do not separate, which occurs with probability (1 - δx,s), two cases can arise. In the first case, which occurs with the probability (1 - ξw,s), wages are renegotiated and the value of the worker for the labour firm is again the value of a worker with a renegotiated wage, just in the next period, A t+1F(ws,t+1*). With probability ξw,s wages are not renegotiated and the firm is in next period stuck with the worker value at the current wage, A t+1F(ws,t*).

The value at t+1 of the worker with renegotiated wage from time t is

A t+1F(ws,t*) = uprime(cs,t+1)(hs,t+1α_ Hxs,t+1-hs,t+1ws,t*((1+ π)Pt)/(Pt+1)(1+τt+1wf)); +β(1-δx,s)[(1-ξw,s) A t+2F(ws,t+2*)+ξw,s A t+2F(ws,t*)]

(53)

The wage from the previous period has been indexed by the ratio of trend inflation bar π and the price level growth, Pt/Pt+1 = (1 + πt+1).

If we substitute equation 53 into equation 52, and do this for every future period, we arrive at the following expression:

A tF(ws,t*) = sum from k = 0 to ∞ of [β(1-δx,s)ξw,s]kuprime(cs,t+k)hs,t+kα_ Hxs,t+k; -ws,t* sum from k = 0 to ∞ of [β(1-δx,s)ξw,s]kuprime(cs,t+k)((1+ π)kPt)/(Pt+k)hs,t+k(1+τt+kwf); + 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*)

(54)

One can define auxiliary variables and write the infinite sums in recursive form. For the first line in equation 54, define

Ss,tx ≡ sum from k = 0 to ∞ of [β(1-δx,s)ξw,s]kuprime(cs,t+k)hs,t+kα_ Hxs,t+k = uprime(cs,t)hs,tα_ Hxs,t+β(1-δx,s)ξw,sSs,t+1x

(55)

For the second line, define

begin align* Ss,twf ≡ 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+τtwf); = uprime(cs,t)hs,t(1+τtwf)+β(1-δx,s)((1+ π))/((1+πt+1))ξw,sSs,t+1wfend align*

Using these definitions, we can simplify equation 54 to

(56)