2. Theoretical Model
2.2 Outsourcing to Another Country
2.1 Producing in Home Country
First, let’s presuppose that outsourcing does not exist. The final-good producer has the intention to maximize Ri−wLi, and it has to hire the local labor at the local wage rate w. The optimal hired amount of the labor and the sales revenue for the final-good producer are as following: Now we have the optimal amount of labor and the sales revenue. The net profit for the producer can be represented as follow,
πi= Ri−wLi= (1 − Φ) Ri , Φ = α (1 − η) (2.8) Equation (2.8) implies that the higher capital share is, the higher profit the producer will earn3. Since labor cost is the only cost that the final-good producer has to care, we can see a clear and negative relationship between wage rate w and net profit πi from equation (2.7) and (2.8). In other words, an increase in local wage rate will result in a decrease in the net profit for the producer. Thus, if the local wage rate in country B is relatively lower, the final-good producer might not keep hiring labor in local market but outsource the procedure to country B.
2.2 Outsourcing to Another Country
If outsourcing happens, both parties,the final-good producer and the foreign supplier, would have to negotiate how to cooperate during production. Figure 2.1 summarizes the
3This can be easily examined by taking one simple derivatives as following, ∂ ln πi/∂η = α (1 − Φ)−1ln (K/L) > 0 iff K > L. This derivative tells us that a higher capital share will accom-pany with higher level of profit given capital intensity is larger than one.
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2. Theoretical Model 8
Fig. 2.1: Decision tree for final-good producer
sequence of both parties’ decisions and activities. Due to the contracts incompleteness, both parties will start to bargain the output after production. Moreover, the output manufactured through the outsourcing in country B is valueless to other producers in country A in any sector or industry. This setting reflects a very common phenomenon in OEM and ODM industries. Because the foreign suppliers have to adjust their design to meet the needs of every order, their products are nearly impossible to resell to another buyer. This setting is also one of the features derived from incomplete contracts. Since the output is useless outside this relationship, the contract incompleteness gives rise to a standard holdup problem.
After production, both parties will negotiate how to divide the output xi. Because production process is divided among two parties, final-good producers do not necessarily own the property right of the intermediate input. It might be more reasonable to assume that the outside option for both parties is zero. To generalize our analysis, we assume that once the negotiation fails a third party, court for instance will arbitrate and there will be a probability δ ∈ (0, 1) which the final-good producer will obtain xi through arbitration. We also assume both parties are risk-neutral. If the final-good producer fails
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to negotiate and wants to take over all output, it will on average receive a fraction δ of the amount of xi produced. Since the final-good producer will lose a fraction (1 − δ) of original producing amount, bargaining with the foreign supplier would be an adoptable policy if δ is significant.
By substituting δxi into equation (2.5), we can translate the output of δxi into δα times of sales revenue. Following the most common setting in the literature, we assume the ex-post bargaining as a Generalized Nash Bargaining, and that gives the final-good producer a fraction β ∈ (0, 1) of the ex-post gains from trade. For the final-good producer, it will on average receive δαRbi if it fails to bargain, so revenue for bargaining will be only (1 − δα)Rbi, where Rbi is the sales revenue under outsourcing relationship. The total revenue for final-good producer will be
δαRbi+ β (1 − δα)Rbi (2.9)
and that will also be the profit because the final-good producer is not obligated to pay for labor input. On the other side, the foreign supplier set its labor input to maximize (1 − β) (1 − δα)Rbi−w1Lbi, where w1is the wage rate in foreign country and w1 is strictly smaller than w by assumption. By solving the first order condition, we obtain the optimal amount of labor and sales revenue under outsourcing:
Lbi =
Checking equation (2.6), (2.10) and comparing these two first order conditions, we can discover that (1 − β) (1 − δα) is less than one, and we can find out an underinvestment in labor input. Such underinvestment stems from the fact that, with incomplete contracts, the foreign supplier receives only (1 − β) (1 − δα) of the marginal return to its ex ante investment. From equation (2.7) and (2.11), the sales revenue would also be lower for the same reason. Similar to the previous section, we obtain the net profit for the foreign
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The net profit of the foreign supplier increases with higher capital share4. Higher capital share induces the outsourcing firm to hire less labor under profit-maximizing principle, so the labor cost for the foreign supplier will be much relieved; therefore, the foreign supplier can make greater profit under a higher capital share industry. Moreover, with incomplete contracts, the foreign supplier can capture the contribution of capital of the final-good producer, so the net profit of the foreign supplier increases with higher capital share.
For the final-good producer, outsourcing would be adopted if its net profit is greater under outsourcing. Combine equation (2.8) and (2.9), outsourcing will take place if the following inequality holds
δαRbi+ β (1 − δα)Rbi> (1 − Φ) Ri,
and this inequality can be simplified to
D ≥ w1
Notice that D is a positive constant. The disadvantage of outsourcing is the reduction of efficiency due to contracts incompleteness. Therefore, outsourcing takes place only when the labor cost is significantly low in country B. If outsourcing does not happen, the final-good producer has to pay for labor input itself. Thus, it would bear a higher manufacturing cost if domestic wage rate is significantly high. Although incomplete con-tracts will reduce the labor input of the foreign supplier and the revenue of the final-good producer, the final-good producer may still adopt outsourcing policy if wage difference is significantly high.
4It could be rigorously showed by footnote 3.
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In order to bring R&D into our discussion, we consider a labor biased production function and rewrite it as
xi= Kiη(θLi)1−η,
where θ is the efficiency index of the final-good producer. Suppose that the final-good producer hires its local labor, and we could rewrite its optimal amount of labor and sales revenue as: On the other hand, there exists a lot of foreign suppliers with different θ1m(θ1m is the efficiency index of the foreign supplier m). To simplify our discussion, we assume that the information of efficiency index is prevalent, and there is no cost to search a suitable foreign supplier. If outsourcing takes place, the final-good producer will screen out its partner with the greatest θ1m, and the foreign supplier hires its local labor to maximize its profit. To simplify notation, we drop out subscript m. We have the optimal amount of labor and sales revenue from first order condition:
Lbi =
In order to capture the efficiency gap between two different countries, we assume that the labor hired by the final-good producer are more skilled than that hired by the foreign supplier (θ > θ1). Again, the final-good producer has to make a decision whether to adopt outsourcing policy. Outsourcing takes place if and only if
D ≥ w1
w θ θ1
. (2.18)