are represented by the flow of newly formed job-matches and, therefore, by the
matching-function m(U, V ; λ). Firms create and offer new productive jobs and they
have to fill these vacancies by searching for suitable workers. At the aggregate level,
the filling of vacancies depends on the number of unemployed, the number of offered
vacancies, the search intensities of firms and unemployed and the rate of technical
progress; all determinants are expressed in the matching-function. Taking outflow
and inflow together, the dynamics of employment result as the difference between
outflows and inflows and can be expressed as
• ʌ
(8)
2? = m(U, V ; λ) - vE.
Each vacancy induces search costs of cv with cv := cvQext. Since the newest jobs
contain the latest technology, it is costly for the firm to find unemployed workers
being able to handle most recent technologies. Therefore, search costs grow with
the rate of technical progress.
Taking these aspects into consideration, the representative firm faces the following
intertemporal optimization problem with the current flow of profits as output minus
factor payments minus search expenditures. Denoting r as the discount factor the
firms maximization can be written as
max
ɪ,v
I {F (K,λE)
Jo
— rK — wE — crI — cvV}e rtdt
s.t. E = m(U, V ; λ) — E
Tf = I
K (0),E (0),V (0),U (0) given.
In order to solve the optimization problem, a present-value Hamiltonian function
H(K, E, V, I, μ1μ2) with costate variables μi [2 = 1, 2] is set up. Denoting Fj as the
16