Each firm produces the final good Y with labor only, taking wages and prices as given. Production requires two intermediate stages and, depending on the structure of the process (seeFigure 3.2), subsequent assembling. Abstracting from any contracting issues7we stay agnostic with respect to the ownership structure.
3.2.1
Optimal Firm-Level Production Structure
The intermediate productxais manufactured with probability 1−λusingaunits of labor
implying that 1−aλ units of labor are required to produce one unit of xa. Similarly, 1−bλ
units of labor are needed to complete one unit of intermediate input xb. While the labor
requirements of the two intermediate stages are identical across production structures the process differs with respect to the input xb. Within a sequential structure, one unit of
intermediate product xa is processed into the final product YS using b units of labor and
exposing the value added of the firstandsecond stage to a loss that occurs with probability
λ. In contrast, if a firm chooses a parallel production structure intermediates xa and xb
are produced within independent processes with the respective unit labor requirements of
a
1−λ and b
1−λ. Subsequently, they have to be assembled to the final product. Although we
assume that labor input of assembly is negligible compared to intermediate stages (and therefore set labor requirements to zero), the final product will only be accomplished with probability 1−σλ. We impose 0 < σ <1, as assembly involves usually less far-reaching activities than actual production. Nevertheless, the final product will be identical across production structures.
Since we assume perfect competition on output markets, a firm’s profit maximization implies setting prices equal to marginal cost. Furthermore, a firm’s linear cost struc- ture involves that marginal cost equal minimum unit cost to produce the final good Y. Since either production process would deliver an identical final output, a firm’s decision is reduced to choose between a sequential or parallel production process. The implied
minimum unit cost functions are kS = aw 1−λ +bw 1−λ (3.1) kP = aw 1−λ + bw 1−λ 1−σλ (3.2)
where kS are minimum unit cost of sequential and kP minimum unit cost of parallel
production. An optimal choice involves to minimize costs, i.e. to compare minimum unit cost of a parallel or sequential production process: kS ≷kP.
Proposition 3.1 The higher the relative value added of the first intermediate production
step the more inclined is a firm to choose parallel production. A smaller probability of making mistakes during intermediate stages and a higher risk of failures during assembling imply that firms rather choose sequential processes.
The proof simply involves comparing minimum unit costs (3.1) and (3.2). As a result, the firm chooses a parallel production process if and only if kS > kP implying that
a b >
σ(1−λ)
1−σ . (3.3)
Otherwise, it chooses a sequential structure. In the case of ab = σ(1−1−σλ), the firm is in- different between a sequential and a parallel production structure. Since this case does not add insight to our analysis, we abstract from it throughout this study8. Figure 3.3 illustrates the fundamental trade-offs driving a firm’s choices9. A decrease in the value added within the second intermediate stage (i.e. a lower labor requirementb) in graph(I) induces an increase of the relative value added of the first step, ab. Accordingly, the range for which sequential production is optimal, becomes smaller. In (I), this implies a shift of the threshold level of value added towards lower values ofa. Graph (II) illustrates the impact of a lower probability of making mistakes, λ0 < λ, involving a higher threshold value of failure rates σ(1−1−σλ0) > σ(1−1−σλ). As a consequence, a sequential production process is optimal for higher relative value added and the respective threshold is shifted towards 8The most simple remedy to an exclusion would be to assume that whenever a firm is indifferent it
chooses a parallel (or, equivalently, a sequential) production process.
9Note that we do comparative statics with respect toλandb. The choice ofainstead ofbwould not
Figure 3.3: Firm-Level Comparative Statics σ λ σ − − 1 ) 1 ( σ λ σ − − 1 ) 1 ( σ λ σ − − 1 ) ' 1 ( λ a a b a b a ' b a b sequ.‘ sequ. sequ. sequ.‘ para.‘ para. para. para.‘ (I) (II)
Impact of higher relative value added and a lower probability of making mistakes on a firm’s choice.
higher labor requirements a. Or, although the failure rate of assembling decreases pro-
portionally to the intermediates’ probability of making mistakes, its absolute decrease is smaller since λ−λ0 > σλ−σλ0 ⇐⇒ 1 > σ. Consequently, the risk of losing products during assembling outweighs the risk of destruction during the second intermediate step for higher relative value added ab, implying a higher threshold value ofasincebis assumed to be constant in this case.
3.2.2
An Application to the Aviation Industry
We pick up our example of the aviation industry from the introduction to illustrate firms’ endogenous choices on the structure of production processes. In the introduction, we established that Boeing’s production of the 787 (Dreamliner) exhibits a more sequential structure than the manufacturing of Airbus’ A380. This directly relates to our firm- level analysis of optimal production processes from above. It implies that either Boeing’s production structure involves lower relative value added within later intermediate stages or that it is exposed to higher failure rates within intermediates’ production.
Since the 787 and the A380 are rather close substitutes10, the relative value added within the production of comparable parts (e.g. wings, engine) should not differ significantly. 10This is true from our global perspective. We are aware that both aircraft differ with e.g. respect to
the maximum of passenger numbers and, though less, range (787-9: 290, 15700km; A380-800: 853, 15200 km).
However, the economic literature tends to acknowledge that firms are more able to reduce mistakes in production processes if their organizational structures are more integrated11. Moreover, while Airbus adds about 75%12 of the work done in the manufacturing of the A380, Boeing contributes only 30% in the manufacture of the 787. Thus, we can sensibly assume that the probability of making mistakes in the Dreamliner’s intermediate production stages is higher than that within the intermediate steps in the process of making A380s. Or, Boeing’s tendency to outsource large chunks of 787’s production requires to structure production in a rather parallel manner as it faces higher failure rates from external suppliers. In contrast, Airbus opts for a more sequential structure of its production process since its more integrated value chain implies lower probabilities of making mistakes in intermediated stages.
Note that we abstract in this example from different failure rates across countries since assembly as well as intermediate production stages are carried out to a large extend in countries with a similar level of productivity, i.e. similar probabilities of making mistakes. Here, the ownership structure13, and not the location of plants within different countries, implies disparities in the probability of making mistakes.