Excerpt

## Contents

1 Introduction

2 The Model

2.1 Assumptions

2.2 Profits

2.3 Effects

3 Laussel (2008)

3.1 Exogenous partial backward integration

3.2 Endogenous backward integration

4 Matsushima and Mizuno (2009)

4.1 Separation under full bargaining power

4.2 Separation under variable bargaining power

4.3 Separation with multiple periods

5 Laussel and Van Long (2011)

5.1 Separation when the downstream firm can commit

5.2 Markov-perfect equilibrium

6 Conclusion

## 1 Introduction

In our industrialized world we are confronted with very complex goods. Most of them became part of our everyday life like cars, planes and so on. These products consist of several components with high technological requirements to guarantee the quality of the final product

For example, the production of a high developed plane, as the Airbus A380, needs many special single components like turbines, wings, etc. These vital intermediate goods could either be purchased by foreign firms or be produced by the downstream assembler itself, respectively by an owned subunit.

A view on the vertical structure among different downstream firms of different prod- ucts shows that the share in purchased goods from foreign upstream firms varies widely.^{1}

Since the components are very specific, the number of upstream firms which produce one vital intermediate input is of course limited. In turn, the specific component could be purchased by less, mostly just one, consumer. A turbine manufactured for the A380, for instance, could not be used by another assembler than Airbus. A downstream firm is faced by deciding whether it should either integrate upstream units and become the owner of them or to sell some upstream firms, respectively let them stay independent, with respect to maximize its own profit.

To provide an answer to the optimal behavior of an assembler, with respect to in- tegrate or to separate upstream units, I will use a model where the downstream firm is a monopolist in the final good market and needs a fixed number of vital intermediate inputs in fixed proportions to produce the final product. This model is relatively new and not being discussed by many authors. It was first examined in 2008 by Laussel. We will see that there exist two effects that influence the decision of the downstream firm. If the independent upstream suppliers have some positive bargaining power, they will sell their produced goods to a price above their costs. That causes the *dou- ble marginalization effect*, because the upstream firms do not account for the effect of its mark-up on the downstream firm. Furthermore an independent supplier, with a positive bargaining power, has a decreasing effect on the prices of the components supplied by the other independent upstream firms, because of the perfect complemen- tary character of the intermediate goods.^{2} A high number of independent suppliers yields to relatively low input prices of the particular upstream firms.

We will see from Laussel (2008) that an exogenous merger is only profitable if at least two thirds of all upstream suppliers are integrated. Furthermore he shows that inte- gration of suppliers by the assembler never occurs. Although the *double marginaliza- tion effect* would be eliminated by integrating a supplier, the decreasing competition among the remaining independent upstream firms reduces the revenue of the down- stream firm. This yields to a marginal increase of the profit of the downstream firm that is strictly smaller than the profit of the upstream unit by staying independent.

Since it is not optimal to integrate upstream units we will take a view on possible separation of integrated upstream suppliers.

Matsushima and Mizuno (2009) provide the optimal degree of separated units by the assembler in a static view depending on the initial number of integrated firms. We will also see that, in a case of variable bargaining power, the incentives of the assembler to separate an upstream unit increase when the independent upstream firms have strong bargaining power or the separated unit has a low bargaining power. In a further step we can regard from Matsushima and Mizuno the profit maximizing behavior of the assembler in a situation of multiple periods. Especially the price expectations of possible buyers of the upstream units play a crucial role.

This issue is picked up by Laussel and Van Long (2011). In a first step they assume that the assembler could make a binding commitment for its sale policy. This leads to a determination of the asset price of the upstream units, but does not satisfy the property of time-consistency, because the assembler is not able to reoptimize in the future. For this Laussel and Van Long develop a Markov-perfect equilibrium based on a rational market expectation function of the asset price, which shows us a very detailed picture of the optimal behavior of the downstream firm. This contains either of an immediately separation of all suppliers or dictates a gradual sale policy for a longer time.

## 2 The Model

To show the optimal behavior of a monopolistic downstream firm with respect to separation (or possibly integration) of its upstream suppliers I will introduce a general model. This model first was examined in the paper *Buying back Subcontractors: The strategic limits of backward integration* (Laussel 2008).

### 2.1 Assumptions

The model considers a vertical structure between a monopolistic downstream assembler *D* and *n* upstream suppliers.^{3} Every upstream firm supplies one vital intermediate input to the downstream firm. Exactly one unit of each component is necessary for *D* to produce one unit of the final product. The assemblers (*Leontief-*) production function *F* is determined by

illustration not visible in this excerpt

where *xi* is the amount of the input from supplier *i* and *q* is the output level. The demand in the final good market is given by a linear function,

illustration not visible in this excerpt

where *p ∈* [0 *,* 1] is the price of the final good.

The set of upstream suppliers is divided into two subsets. *k* (with [illustration not visible in this excerpt]) of the *n * upstream entities are exclusively owned by *D*, the other *n − k* firms are independent.

We denote the independent firms with[illustration not visible in this excerpt]and the integrated ones with [illustration not visible in this excerpt]. The only costs of *D* originate from the prices of the *n − k* purchased goods from the independent suppliers and are given by

illustration not visible in this excerpt

with *c ∈* [0 *,* 1] as the unit costs of the final product. *wi* is the negotiated price between *D* and supplier *i* for the vital input *xi*, where *i ∈ I*.

The downstream firms maximization problem is then given by

illustration not visible in this excerpt

where the first order condition yields to an profit-maximizing price of

illustration not visible in this excerpt

with a resulting quantity of

illustration not visible in this excerpt

From the characteristic of the production function follows that each component *xi* is asked in the same amount equal to the output level of the final product. The resulting profit of the downstream assembler as a function of the costs is

illustration not visible in this excerpt

It is assumed that the upstream suppliers produce their vital intermediate goods with costs of zero. So the profit of each independent supplier *i ∈ I* with dependency on the negotiated price *wi* is

illustration not visible in this excerpt

where

illustration not visible in this excerpt

The *n − k* price negotiations over the price *wi* between *D* with all independent suppliers *i ∈ I* take place simultaneously and do not effect each other. The assembler has a bargaining power of 1 *− α*, with *α ∈* [0 *,* 1]^{4}, which influences the outcome *wi* of the bargaining problem with supplier *i*

illustration not visible in this excerpt

where *di* and *D* are the disagreement payoffs of the parties involved. Since there is no further assembler firm next to *D* the upstream firms have no alternative purchaser of their goods, it follows that *di* = 0. Furthermore the goods supplied by the upstream firms are vital to produce the final product so that *D* = 0. With respect to this, the maximization problem can be rewritten as

illustration not visible in this excerpt

with the first order condition

illustration not visible in this excerpt

This equation shows the negative dependency from *wi* to the sum of the other input prices *W − i*. By considering a symmetric case with same bargaining power and consequently identical input prices *w* = *wi* among independent suppliers *i ∈ I*, it arises an equilibrium input price of

illustration not visible in this excerpt

For a bargaining power *α >* 0, the independent suppliers achieve a strictly positive price for their supplied input goods which is greater than their costs of zero. If the suppliers have no bargaining power, *α* = 0, the price *w* equals the costs of zero. In case of full bargaining power of the suppliers, *α* = 1, it occurs a *Cournot equilibrium solution*.

Furthermore the input price is increasing in the number of integrated upstream suppliers *k* by the assembler firm.^{5}

### 2.2 Profits

In consideration of the equilibrium input price *w*, the profit of each independent upstream supplier is

illustration not visible in this excerpt

with the first derivative with respect to the number of integrated firms *k*

illustration not visible in this excerpt

This implies that by reducing the number of independent suppliers the profits of the firms who remain independent rise.

The profit of the assembler *D* with its *k* integrated firms is

illustration not visible in this excerpt

with an increase in *k*

illustration not visible in this excerpt

and a limit value in fully integrated industry

illustration not visible in this excerpt

From this particular profits the joint profit Ω can be derived.

illustration not visible in this excerpt

This inequation implies that the most preferable situation with respect to maximal welfare, i.e. joint profits, is a fully integrated industry with no independent supplier. Also the profit of the assembler is maximal in this situation as we can see from equation (4). But we will see in the following that this situation (nearly) does not occur. To investigate this, we have to take a look on the effects by changing the number of integrated upstream units.

### 2.3 Effects

One effect that causes welfare losses in a situation with independent upstream suppliers is the *double-marginalization effect*,^{6} which occurs when the suppliers of the vital inputs have some positive bargaining power and negotiate a positive input price *w*. The other effect is caused by the perfect complementary character of inputs and the resulting competition between independent upstream firms.^{7} By a decreasing number of independent suppliers the competition also decreases, which induces higher prices and profits for remaining independent suppliers.

The effects could be shown by comparing the marginal increase in the profit of the merger Π and the rise in the equilibrium profit of an independent supplier followed from a reduction of independent suppliers:

illustration not visible in this excerpt

The right-hand side is negative for *k < n*. That implies that the profit of an in-

dependent supplier is greater than the marginal increase of Π when this supplier is

integrated. For a more intuitive view (5) could be rewritten as

illustration not visible in this excerpt

The first term on the right-hand side shows the direct effect on Π by integrating an independent supplier.^{8} The second term is the indirect effect and reduces the marginal revenue on Π. It is caused by an increase in the equilibrium price *w* as a result of a decreasing number of independent suppliers and shown in Figure 1.

illustration not visible in this excerpt

Figure 1: Marginal increase from integrate an extra supplier

The fact that the marginal revenue on Π by integrating a supplier is smaller than its profit by staying independent, gives an intuitive explanation why vertical integration does not occur, because *D* is not willing to pay a price above its revenue and the independent supplier will not accept a price below his profit.

To examine whether there exists any equilibrium where the downstream buys upstream suppliers, I will pick up next the analysis in Laussel’s paper *” Buying back Subcontractors: The strategic limits of backward integration ”* (2008).

## 3 Laussel (2008)

### 3.1 Exogenous partial backward integration

As already seen, in the industry with one monopolistic downstream assembler and *n * upstream suppliers, the welfare and *D* ’s profit are maximal in a complete integrated case. Does this mean that the higher the number of integrated upstream suppliers the higher *D* ’s profit? Is it optimal to integrate some independent suppliers? To provide an answer to these questions, Laussel examines in his paper a number of suppliers *k ∈* [0 *, n* ] and the downstream assembler, with regarding whether a merger leads to a more profitable situation for them. This could be obtained by following equation:

illustration not visible in this excerpt

The first term on the right-hand side equals the profit in a merged situation whereas the second term is according to the sum of individual profits in a disintegrated situa- tion.^{9}

**[...]**

^{1} see Cusumano and Takeishi (1991) for a comparison between the U.S. and the Japanes car industry

^{2} see Cournot (1838) for a model of perfect complements

^{3} *n* is a fixed number of upstream suppliers

^{4} This implies a common bargaining power *α* for all independent upstream firms. Matsushima and Mizuno (2009) investigate a case with variable bargaining power between upstream firms.

^{5} because[illustration not visible in this excerpt]

^{6} first examined by Stigler (1951)

^{7} overpricing of perfect complements by oligopolists was discussed in Cournot (1838)

^{8} caused by the elimination of the double-marginalization effect

^{9} the first derivative is[illustration not visible in this excerpt]where the first term is positive and the second one negative

- Quote paper
- Alexander Max (Author), 2013, Optimal separation of upstream suppliers of vital intermediate inputs by a monopolistic assembler, Munich, GRIN Verlag, https://www.grin.com/document/214540

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