Unemployment in an Interdependent World

Labor market equilibrium. We can use the shadow value of labor as given in equation
(8) and the expression for the advantage of holding a job over searching as given in
equation (11) in the bargaining solution (12) to obtain:

_{r 1} _ ^{d d}Ri[M]    ^d ∂wi[φ]

wi[^M] = ^{β i} ∂L_ij M   ^{β i} ∂L_ij M Li^{j [м] + (}1 ^{β i})^rUi'                (1³)

Using qij [m] = MLij [m] in equation (4) and differentiating with respect to labor input
Lij [M] (assuming that Iij [M] > 0), leads to

^σ - 1 r 1 —¹ r pʌ- ^τij Y^j

σ ^qi^{j [m] σ m} (P^j ) ^σ     mm1-ν J ^,

which allows to solve the wage differential equation (13):¹¹

^wi[M] = ^βi( ^σ Λ ^dR^{i m +} (1 ^{- β}i)^rUi-                   (15)

σ - β_i ∂Lij[M]

Using equation (3) inequation (14) and noting that ^dRiij ^] = ( ^σσ1 ) φτ _ij¹p_ij = ( ^σσ1 ) MPii,
where the last equality follows from equalization of marginal costs between markets, leads
to the job creation curve

Wi = σ - 1 φ -  Ci  r + η

Pi    σ - βi i   mi[θi] 1 - δ'

The job creation curve slopes downward in θ since a higher degree of labor market tightness
makes it more costly to fill vacancies so that a smaller share of the surplus Φ can accrue
to the worker. Hence, the real wage falls in θ. Importantly, the wage rate depends only
on aggregate variables such as P_i , Φ_i or θ and does, therefore, not vary across firms. The
intuition is that firms with high productivity are larger and move their marginal revenue
functions further down by exactly the amount that equalizes the value of a filled vacancy.

Combining equations (3), (9), and (15) shows that the wage rate is given by the sum
of the value of non-employment (rUi) and the rent that the worker can extract from the

Using the expression for U_i, we can write rU_i = b_iΦ_i + θ_im[θ_i] (jEi — Ui). Using equation
(11) and noting that wi [M] - rUi is equal for all firms (see equation (17)), one can derive
the following wage curve:

¹¹The solution can be checked by reinserting
= _β f σ λ ∕∂Ri[^]∖ /-1∖ 1 '

^βi V - βi     ∂Li₁     σ    Li₃)

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