potential to distinguish wage stickiness of new hires and existing workers, as in Bodart et al. (2006) and De Walque et al. (2009). In addition, it distinguishes between labour market segments for Ricardian and non-Ricardian households.
The number of workers in segment s (s = i for Ricardian and s = j for non-Ricardian households) that are employed after the matching process has been completed, ndes,t, is:
ndes,t = (1-δx,s)ndes,t-1+Ms,t,
(47)
where Ms,t is the number of new matches formed in a period, and δx,s is the fraction of existing employment relationships that have (exogenously) separated. The number of matches Ms,t is:
Ms,t = φs,M(uns,t)μ_ s(vacs,t)1-μ_ s = ps,tWuns,t = ps,tFvacs,t,
(48)
where φs,M is matching efficiency, uns,t is the number of searching workers in each segment, vacs,t is the number of vacancies, ps,tW is the matching probability for workers of each type, ps,tF is the matching probability for firms, and μs is the elasticity of the matching function with respect to unemployment.
The probability for a searching worker to find a job is
ps,tW = (Mt)/(uns,t) = φs,M((vacs,t)/(uns,t))1-μ_ s
and the probability of a firm finding a worker is
(49)
ps,tF = (Ms,t)/(vacs,t) = φs,M((vacs,t)/(uns,t))-μ_ s
(50)
The number of unemployed workers who search for work at the beginning of period t is equal to those who were unemployed at the end of period t-1 after the t-1 matching has been completed plus the newly separated workers:
uns,t = unes,t-1+δx,sndes,t-1,
(51)
Total unemployment in a bloc is a weighted unemployment across labour market segments, where ω is the share of non-Ricardian households: unet = ω unej,t + (1 - ω)unei,t.
Value functions for a labour firm Let AF(ws,t*) denote the value of a job for a firm employing a worker from household type s ∈ [i, j], where ws,j* is the renegotiated wage, and i stands for a Ricardian and j for a non-Ricardian household. Following Bodart et al. (2006), it will be convenient to use this value in marginal utility terms, so we define A F(ws,t*) ≡ u'(cs,t)AF(ws,t*), where u'(cs,t) is the marginal utility of consumption of household s. The value of a job with a renegotiated wage for a labour firm can then be written as
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*)]
(52)