Unemployment in an Interdependent World

that across every market where the firm is active, it will equalize marginal revenues, i.e.,
∂R_ij [и] /∂L_ij [и] = ∂R_i [и] /∂L_i [и] for all j, where Li [и] is firm φ^,s total employment.
This rule determines the distribution of sales across markets given the total output of the
firm (which is, in turn, determined through the choice of ν_i).

Vacancy posting. The optimal value of an intermediate input producer is given by:

^N

-Pi     Iij [^]^fij ^{+ (1 - δ})Jii ^[и] ) ^,

j=1

s.t. (i) R_i [и]   given in equation (4),

(ii) Li M = (1 ^- X^)Li M ⁺ mi[θi]vi ∣φl^,

where r denotes the interest rate, w_i [и] is the wage rate in country i paid by firm φ, J'_i [и]
is the value of an intermediate input producer next period, and L_i is firm φ's employment
in the next period. Constraint (i) is the revenue function and constraint (ii) gives the law
of motion of employment at the firm level. The first order condition for vacancy posting
can be stated as follows:

ciPi            ^dJ'i И                                _tr∖

= ^{(1 - δ}) LT'   ■                               ⁽⁶⁾

mi[θi]             ∂Li [и]

It shows that the firm equalizes marginal recruitment costs (given on the left hand side)
and the shadow value of labor (given on the right hand side). Note that firms with
different φ face identical expected recruitment costs; hence, the shadow value of labor is
the same across firms, too.

From the equalization of marginal revenues across markets, it follows that the shadow
value of labor does not depend on the market where the additional output is actually
sold. Hence, ∂J_i [и] /∂L_i [и] = ∂J_i [и] /∂Lij [и] ■ Differentiating the objective function of
the firm (5) with respect to Lij yields:

^dJi ^[и^] _ ^{1 d d}Ri[φ∖ _{r 1} ^dwi ^[и^{]                            d}Ji ^[и^]

SLj[^i = 1+r (ад ^{- w}i ^{И -} ∂L~W]^{Lij M + (1 - δ})(^{1 - χ}) ∂LζM ) ■   ⁽⁷⁾

Employing the steady-state condition ∂J_i [и] /∂L_ij [^] = ∂J_i' [^] /∂L'_ij [^] we obtain:

^dJi M _   ¹   ∂ ^dRi[^φ] -   [ ] _ ^dwi ^[^^] _L [ A                ₍₈₎

∂Lij M   r + η∖∂Lij [_φ] w^{i M} ∂Lij ИL^ijmJ ■            ⁽⁸⁾

Using (6) and ∂J_i [и] /∂L_i [и] = ∂J_i [и] /∂L_ij [и] , we can solve for ∂R_i[φ∖∕∂L_ij [^] and
obtain an expression that implicitly determines the optimal pricing behavior of the inter-
mediate input producer:

^dRi ^[И^]       _{r 1 l} ^dwi ^[И^]           ciPi Λ ⁺ η

= wi ^[И^]+        Lij ^[И^]+                 ■               ⁽⁹⁾

∂Lij [и]              ∂Lij [и]           mi [θi]   1 - δ

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