Wage bargaining. The search-and-matching setup developed above is compatible with
a number of different assumptions concerning the wage-setting process. In the largest part
of this paper, we follow Felbermayr, Prat, and Schmerer (2008). We assume that wages are
bargained before production takes place and that every worker is treated as the marginal
worker. This approach is fairly standard now in the literature (see Cosar et al. (2009);
Helpman and Itskhoki (2008)); it’s axiomatic foundation is laid out in Stole and Zwiebel
(1996). In a later section of this paper, we will argue that this formulation implies
too much wage flexibility so that foreign unemployment reacts too little to domestic
institutional changes compared to the empirical evidence. Hence, we also experiment
with the opposite extreme case of a perfectly rigid real wage as proposed by Shimer
(2004).
The total surplus from a successful match is split between the employee and the
intermediate input producer. The worker’s surplus is equal to the difference between the
value of being employed at firm φ, i.e., Ei [и] = (wi [и] + (1 — η)Ei [и] + ηUi) /(1 + r)
and the value of being unemployed Ui = (biΦi + θim[θi]Ei + (1 — θimi[θi])Ui) /(1 + r),
where θimi[θi] is an unemployed worker’s probability to find a new job and EJi is the value
of employment at the average firm. The flow value of unemployment is given by biΦi
with bi ∈ [0, 1] and is proportional to the marginal value product of labor at the average
domestic firm deflated by the price index:10
Φi ≡ Иiipii [Иii] /Pi. (10)
The variable Φi will turn out to be a sufficient statistic for determining the role of changing
productivity distributions on labor market outcomes. In the sequel (with some abuse of
wording) we refer to Φi as a measure of aggregate productivity.
Reformulating the expression for Ei[φ], the advantage of holding a job at firm φ over
searching one can be expressed as:
Ei [и] — Ui = (wi [и] — rUi) / (r + η). (11)
The firms’s surplus is equal to the marginal increase in the firm’s value ∂Ji [и] /∂Lij [и],
which results from the assumption that every worker is treated as the marginal worker.
The outcome of the bargaining process over the division of the surplus follows the “surplus-
splitting” rule:
(1 — βi)(Ei И — Ui) = βi -J⅛L, (12)
dLij [и]
where the parameter βi measures the bargaining power of the workers and belongs to
(0, 1). From (6) and (12) it is already apparent that the value of employment Ei cannot
vary across firms so that heterogeneous firms will pay identical wages.
10The productivity of the average domestic firm is defined as φii and further explained in subsection
2.3. As in Melitz (2003), the upper-tier CES aggregate implies pii[φ]φ = pii[φ']φ' for all values of φ and
φ'. Hence, specifically for φii.