In the model considered so far the fixed cost of offshoring consists of labor input in the South (for setting up a plant, establishing a network of suppliers, training workers, etc.) only. This ignores that a large part of the planning process for setting up a foreign subsidiary is carried out in the headquarter. Furthermore, the success or failure of offshoring in practice is often determined by issues that relate to the cooperation between parent and subsidiary (e.g., bridging cultural gaps, ensuring stability of production chains, quality controls, etc.) and, therefore, require inputs both in the headquarter and in the subsidiary. So, as a robustness check, the present section assumes that the fixed cost of offshoring consists of labor inputs both in the North and in the South. The analysis confirms the pessimistic assessment of the employment and welfare effects of offshoring based on the one-factor model without headquarter services.
Suppose, in addition to the ¯LN (unskilled) workers, there are ¯HN (> 0) skilled workers in the North, each supplying one unit of skilled labor (while unskilled labor is the only factor of production in the South). Setting up a subsidiary in the South requires the input of fN (≥0) units of skilled labor in the North andfM (≥0) units of labor in the South (fN >0 iffM = 0). We assume that
n−nS < L¯S
fM < H¯N
fN . (8.25)
That is, there are enough labor in the South and enough skilled labor in the North to pay the fixed cost of offshoring all varieties. The production function for varieties of differentiated goods in the North is x(i, j) = Ah(i, j)ζl(i, j)1−ζ (0< ζ < 1, A = {[ζ/(1−ζ)]1−ζ+ [(1− ζ)/ζ]ζ}), where h(i, j) and l(i, j) are the inputs of skilled and unskilled labor, respectively.
The wage rate for skilled labor is denotedvN. The definition of an equilibrium is the same as in Section 8.3 except thatvN is included in the list of variables and the market clearing condition for skilled labor in the North is included in the set of conditions.20
The price of each variety produced in the North (vN)ζ(wN)1−ζ/α is the usual markup on unit cost. The price setting equation becomes
wN
(cf. (8.4)). The condition that it does not pay to move further varieties abroad is (ε−1) with at most one strict inequality (cf. (8.5)). The Nash product for the wage bargain in industryiis
is the demand for unskilled labor per Northern firm (for the sake of convenience, here and in what follows, we do not specify domain and range of functions). The constant wage elasticity of labor demand is ε−ζ(ε−1). The union and the employers in each industry take the economy-wide wage rate for skilled labor vN as given. Maximization of the Nash product subject to the labor demand equation yields
wN
if there is unemployment (cf. (8.6)). Market clearing for skilled labor implies H¯N = (n−nS−nM)B(aN)ε
20The assumption that manufacturing is the only use of unskilled labor (as headquarter services in the North are provided by a different factor, whose price is determined via perfect competition) allows us to sidestep the question of how union wage setting in manufacturing interacts with the wage for, and the employment probability of, unskilled labor in different tasks.
The labor market clearing condition for the South (8.7), the price setting equation (8.26), the arbitrage condition (8.27), the wage setting rule for unskilled labor in the North (8.28), and the labor market clearing condition for skilled labor in the North (8.29) jointly determine the equilibrium values of wN/P,vN/wN,nM,wN/(wSaS), andxN.
with at most one strict inequality. From (8.30), the South has a cost advantage (i.e.,wSaS <
(vN)ζ(wN)1−ζ) exactly if wN
wSaS > ωˆN
[nS+αε−1(n−nS)]ε−11
= ˆνN. (8.33)
The following inequality ensures that this condition is satisfied in equilibrium:
B For ζ= 0, (8.34) boils down to the corresponding condition (8.9) in Section 8.3. An equilib-rium exists if, given fM and fN, there exists (nM,(wN/(wSaS)) with 0≤nM < n−nS and wN/(wSaS)>νˆN that satisfies (8.31) and (8.32).
nS
n−
wN
nM
f
νN
g f M
nS
n− f
g fM
wN
nM
νN S S
a w
νN S S
a w
νN
Figure 8.5: Equilibrium with headquarter services.
Proposition 8.8: Suppose (8.25) and (8.34) hold and L¯N is sufficiently large. Then a symmetric equilibrium with a cost advantage for the South and unemployment exists.
Proof: See Appendix 8.10.2. ||
The determination of the equilibrium values of wN/(wSaS) and nM via (8.31) and (8.32) is illustrated in Figure 8.5. Equation (8.31) has a unique solutionwN/(wSaS) = ˆf(nM, fM, fN).
f(nˆ M, fM, fN) starts at ˆf(0, fM, fN) > νˆN and goes to ˆνN with infinite slope as nM → n−nS. A solution to equation (8.32) holding with equality may or may not exist. If so, solutions come in pairs, and the smaller solution ˆg(nM, fM, fN) satisfies ˆg(nM, fM, fN)>νˆN for all nM. In the main text, we assume that ˆg(nM, fM, fN) exists for all nM and that ˆf and ˆg intersect at some positive nM, with ∂f /∂nˆ M < ∂ˆg/∂nM at the point of intersection.
For ¯LN sufficiently large, the intersection represents a symmetric equilibrium with a cost advantage for the South, unemployment, and offshoring.
Having established existence of equilibrium, we address the comparative statics effects of changes in the labor requirement for offshoring on the amount of offshoring, employment, and the wages of skilled and unskilled workers in the North. Broadly speaking, the results reinforce the pessimistic conclusions drawn from the one-factor model: a decrease in the cost of offshoring raises Southern workers’ utility, but this gain comes at the expense of unskilled Northern workers’ expected utility and possibly skilled workers’ expected utility as well.
While general analytical results are hard to come by, the model is tractable enough so that we can substantiate these claims analytically for the case of low offshoring cost. Since changes infM are easier to deal with than changes infN, we focus on the former. Throughout, it is understood that variables refer to equilibrium values.
Proposition 8.9: Suppose nM > 0 and ∂f /∂nˆ M < ∂ˆg/∂nM. Then dnM/dfM < 0 and d(wN/(wSaS))/dfM >0.
Proof: See Appendix 8.10.2. ||
An increase in fM shifts ˆf to the left and ˆg to the right. For ˆg downward-sloping at the equilibrium point, the comparative statics effects are obvious from the right panel of Figure 8.5. For the opposite case, the proof requires some tedious algebra.
Numerical analysis shows that generally dLN/dfM > 0. That is, employment falls when the labor requirement for offshoring in the South fM falls. This is easy to see for a small offshoring cost:
Proposition 8.10: dLN/dfM >0 for fM andfN small enough.
Proof: See Appendix 8.10.2. ||
Starting from a low level of offshoring, the welfare effects of a decrease in the labor requirement for offshoring in the South are similar as in the one-factor model. The fact that the relative wagewN/(wSaS) and employment fall implies that Southern workers’ utility wS/P goes up, but Northern unskilled workers’ expected utility goes down, and the same holds true for the skilled workers:
Proposition 8.11: ForfM andfN small enough, a decrease infM raises Southern workers’
utility but reduces both unskilled Northern workers’ expected utility and skilled workers’ utility.
Proof: See Appendix 8.10.2. ||
Evidently, the commitment problem analyzed in Section 8.5 remains present in the two-factor model: each industry union would commit to agree to a wage rate ˆωN+dwN/P (dwN/P <0) marginally below the RTM wage ˆωN in (8.28) if it could, since this would completely eliminate the incentive to offshore at the cost of a negligible loss in the indirect utility of an employed worker.