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On an intermediate goods market with asymmetric production technologies as well as vertical and horizontal product differentiation, we analyze the influence of simultaneous competition for resources and customers. The intermediaries face either price or quantity competition on the output market and a monopolistic, strategically acting supplier on the input market. We find that there exist quality and productivity differences such that for quantity competition only one intermediary is willing to procure inputs from the input supplier, while for price competition both intermediaries are willing to purchase inputs. Moreover, the well-known welfare advantage of price competition can in general be no longer confirmed in our model with an endogenous input market and asymmetric intermediaries.

We consider a differentiated intermediate goods market in which two intermediaries compete for customers on the sales side as well as for inputs on the procurement side. Hereby, competition is on the sales side, either carried out by strategically choosing production quantities or by setting sales prices. In our context, differentiation on the one hand refers to horizontal product differentiation, i.e., products offered at the market can either be substitutes or complements. On the other hand we allow for quality differences between the intermediaries’ products and therefore products may also be vertically differentiated. In addition, asymmetries can also arise from efficiency differences of the intermediaries’ production technologies. A simple example of such a differentiated intermediate goods market that we have in mind is companies who produce furniture. They need to procure identical resources such as wood in order to produce chairs or tables and, therefore, face competition on the input market. In addition, they sell their products on a competitive output market, on which they compete for customers. Hereby, the companies may be differentiated horizontally and either offer substitute products, i.e., both producing tables, or both producing chairs, or complementary products, with one company offering chairs and the other one tables. We investigate the impact of simultaneous competition for resources and customer demand on the market equilibrium and highlight the influence of the market power of a monopolistic input supplier.

In the literature, two fundamental contributions in oligopoly theory are made by Cournot and Bertrand, in which firms produce a homogenous good and select quantities in the former and prices in the latter model. In the years after, these settings were extended in various directions. Referring to [

While considering vertical and horizontal product differentiation, the focus of our analysis is put on the impact of simultaneous competition for resources and customer demand on the market outcome under Cournot and Bertrand competition with two intermediaries. The input prices are determined first and intermediaries compete by choosing quantities or prices afterwards. For the competition stage, we analyze the effect of the intermediaries’ asymmetries on the market outcome. In this context we find that there exist quality and productivity differences such that for quantity competition only one intermediary is willing to procure inputs from the input supplier, while for price competition both intermediaries are willing to purchase inputs.

The formal analysis in the forthcoming sections is backwards and proceeds as follows: Section 2 presents the basic model. We begin with the assumptions on the customer’s demand, followed by the analysis of two different modes of competition between the intermediaries. As proposed in [

In order to allow for substitutable as well as for complementary products of the intermediaries, we assume that a representative customer has a utility function, which is of the following standard form as for instance in [

Hereby,

As in [

describe these asymmetries we refer to

Within Cournot competition the two intermediaries compete by strategically choosing the quantities they produce. The price that intermediary

where

where

Within Bertrand competition the two intermediaries compete by strategically choosing the prices of their products. The according quantities intermediaries are producing depend on the prices that both charge at the market. Given both intermediaries compete, by using Equation (2) and assuming

Thus, intermediary

By using Equation (7) we compute intermediary

Given the intermediaries best reply functions for Cournot and Bertrand competition we compute the equilibrium quantities and prices for both modes of competition in Appendix 1.1. We make the following restriction on the asymmetries that we consider in our analysis.

Assumption 1. We assume

With this assumption we exclude the range of parameters for which asymmetries drive the inefficient intermediary immediately out of the market in at least one mode of competition. However as long as the intermediaries’ products are complementary, these assumptions are always satisfied and, thus, this effectively restricts our analysis only for substitutable products.

Taking the intermediaries’ equilibrium quantities and prices as given, we next determine the optimal input price the monopolistic input supplier chooses. The total market demand on the input market is

Lemma 1. Suppose the intermediaries compete either in choosing quantities or prices and Assumption 1 holds. If the two intermediaries are sufficiently asymmetric and there exists an inner solution for the monopolistic input price, then it is optimal for the input supplier to choose an input price such that he sells his inputs to just one intermediary and thus excludes the other intermediary from the input market.

As the proof Lemma 1 shows the asymmetries required for the statement in Lemma 1 are actually depending on the mode of competition, denoted by

contrast, if the intermediaries are sufficiently symmetric, which is

For this section we assume

Proposition 1. Suppose Assumption 1 holds. There exist parameters

Proposition 1 makes clear that there exist quality and productivity differences such that in Cournot competition only one intermediary is willing to procure inputs from the input supplier while in Bertrand competition still both intermediaries are willing to purchase inputs. This is illustrated in

Hence, in Proposition 1 we establish that there is no situation in which Bertrand competition leads to an exclusion of one intermediary from the input market while Cournot competition does not. The reason is that the profits of the input supplier, given he sells to both intermediaries, are always higher when intermediaries compete in Bertrand compared to Cournot competition. Therefore, if the asymmetries between the two intermediaries increase, the input price of a profit-maximizing input supplier is raised earlier under Cournot competition than under Bertrand competition. This has the consequence that under Cournot competition the input price is earlier too high for the intermediary with quality and/or productivity disadvantages. Thus, he is no longer

willing to purchase inputs. Thus, the observation of Proposition 1 divides the range of asymmetries into different regions of large, intermediate and small asymmetries between the two intermediaries. The next proposition explicitly compares the input prices for the three scenarios from Proposition 1 and

Proposition 2 (Input Prices). Suppose Assumption 1 holds.

1. For large asymmetries between the intermediaries the input prices are equal under Cournot and Bertrand competition.

2. For intermediate asymmetries between the intermediaries the input prices are always higher under Cournot than under Bertrand competition.

3. Consider small asymmetries between the intermediaries. If input productivities are identical, both intermediaries act as monopolists or both intermediaries produce identical relative qualities, then under Cournot and Bertrand competition the input prices are equal. If the goods are substitutes and the relative quality of the less productive intermediary is higher, then input prices are higher under Bertrand than under Cournot competition. If the goods are complements and the relative quality of the less productive intermediary is higher, then input prices are higher under Cournot than under Bertrand competition.

Proposition 2 indeed confirms the intuition behind Proposition 1. The first statement for large asymmetries is obvious. As for Bertrand and Cournot competition the input supplier sets an input price that excludes one intermediary from the market and sells just to the other intermediary, there is no longer competition on the output market between the intermediaries. Thus, there is no difference in input prices. The second statement for intermediate asymmetries confirms that the input price under Cournot competition is indeed too high for one intermediary to purchase positive quantities on the input market, while the input price is always lower under Bertrand competition. However, for the last constellation of parameters with small asymmetries the comparison of the input prices for Cournot and Bertrand competition depends on several parameters. In the proof of Proposition 2 we establish that for small asymmetries, i.e.,

In order to analyze the intermediaries’ incentives to invest in product quality, the next two Propositions now compare the intermediaries’ equilibrium prices, quantities and profits for Cournot and Bertrand competition.

Proposition 3. Suppose both intermediaries have equal productivities and product qualities and

However, this observation does not carry over to arbitrary degrees of asymmetry as the next proposition establishes.

Proposition 4. Suppose Assumption 1 holds.

1. There exist asymmetries such that there is an intermediary who sets strictly lower prices in Cournot competition than in Bertrand competition.

2. There exist asymmetries such that there is an intermediary who produces a strictly greater quantity in Cournot competition than in Bertrand competition.

3. There exist asymmetries for substitutable products such that the profits for both intermediaries are strictly lower in Cournot competition than in Bertrand competition.

For our model with symmetric intermediaries, Proposition 3 confirms a prominent result from the literature. [

We analyzed the influence of asymmetries between intermediaries on an intermediate goods market with horizontal and vertical product differentiation. We established that introducing a strategically acting input supplier may lead to exclusion of one intermediary from the input market if the asymmetries are sufficiently large (Lemma 1). There exists a range of asymmetries where in Cournot competition the input prices are too high for one intermediary, while in Bertrand competition both intermediaries demand positive quantities on the input market (Propositions 1 and 2). Including a strategically acting input supplier into the model tends to make the competition framework more realistic. It highlights the differences between price and quantity competition depending on the degree of product differentiation. As a managerial implication, firms should also pay attention to the degrees of asymmetries in the market since a strategic input supplier adjusts his price accordingly. From a theoretical point of view and in comparison to [

We are grateful to Claus-Jochen Haake for constructive suggestions and comments. This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Centre “On-The-Fly Computing” (SFB 901). An earlier version can be found as Working Paper No. 2015-04, CIE Center for International Economics Working Papers, Paderborn University. An extended version of this article is going to be part of the dissertation of Jochen Manegold.

Brangewitz, S. and Manegold, J. (2016) Competition of Intermediaries in a Differentiated Duopoly. Theoreti- cal Economics Letters, 6, 1341-1362. http://dx.doi.org/10.4236/tel.2016.66124

We start with Cournot Competition. By using the best reply functions of both intermediaries we compute intermediary

The according equilibrium price

In order to guarantee the quantity to be non-decreasing in ‘‘weighted’’ qualities given by

Assumption 2. We assume

We restrict our analysis to parameter choices

For Bertrand competition the Nash equilibrium price is given by

and the equilibrium quantity is given by

For similar reasons as in Cournot competition we impose an analogue assumption on the parameter choices

Assumption 1.We assume

We restrict our analysis to parameter choices

Taking the intermediaries’ equilibrium quantities and prices as given, we determine the total market demand on the input market given by

and

We now summarize their profits using Lemma 1. The profits for the scenarios identified in Proposition 1 and

and

Proof of Lemma 1. For both modes of competition the proof proceeds in two steps. In a first step we determine the prices the input supplier charges. In doing so we distinguish different cases depending on the number of intermediaries demanding positive quantities. Afterwards the according profits are compared in step two.

We start with Cournot Competition.

Step 1 (Input Prices and Profits)

Case 1:The input price is chosen such that both intermediaries produce strictly positive quantities. Therefore, the following must hold:

Maximizing the profit of the input supplier

Note that for this input price both intermediaries indeed purchase positive quantities. Suppose

and thus,

By using

this implies that

Rearranging yields

showing that the input price

Case 2: The input price is chosen such that intermediary 1 produces a strictly positive quantity while intermediary 2 produces zero. Thus, we require

which implies

Note that by definition this input price is indeed too high for intermediary 2 to purchase positive quantities. The analogous argument holds if intermediary 2 produces a strictly positive quantity while intermediary 1 produces zero. The profit of the input supplier is

Case 3: The input price is too high for at least one intermediary to purchase strictly positive quantities.

Step 2 (Comparing Profits)

An input price as suggested in Case 3 cannot be optimal. In this scenario the input price is too high for the intermediaries to demand goods and the input supplier’s profit is zero. In contrast, Case 1 as well as Case 2 yields positive profits. Therefore, we analyze the input supplier’s profits for Cases 1 and 2. We compare the two monopoly profits for selling to intermediary 1 with the duopoly profits for selling to both intermediaries.

Thus, suppose

Step 2.1: We compare the two monopoly profits for selling to intermediary 1 and obtain by observing

that

Step 2.2: We compare the duopoly profit with the monopoly profit for selling to intermediary 1. If the duopoly profit is greater than or equal to both monopoly profits, then the input supplier decides to sell to both intermediaries. Therefore, we obtain that if

with

then

holds. Thus, whenever

the input supplier prefers to set a price for which both intermediaries procure inputs. Note that the following is always true:

Step 2.1 and Step 2.2 show that for

In summary, if for Cournot competition

with

then it is optimal for a profit-maximizing input supplier to only serve intermediary 1 and with analogous conditions to serve intermediary 2.

We now turn to the analysis of Bertrand Competition.

Step 1 (Input Prices and Profits)

Case 1: The input price is chosen such that both intermediaries produce strictly positive quantities. Therefore, the following must hold:

Maximizing the profit of the input supplier

Note that for this input price both intermediaries indeed purchase positive quantities. Suppose

and thus,

which implies when using

that

Rearranging yields

showing that the input price

Case 2: The input price is chosen such that intermediary 1 produces a strictly positive quantity while intermediary 2 produces zero. Thus, we require

which implies

Note that by definition this input price is indeed too high for intermediary 2 to purchase positive quantities. The analogous argument holds if intermediary 2 produces a strictly positive quantity while intermediary 1 produces zero. The profit of the input supplier is

Case 3: The input price is too high for at least one intermediary to purchase strictly positive quantities.

Step 2 (Comparing Profits)

As in Cournot competition, it suffices to compare the profits of the input supplier for Cases 1 and 2. We compare the two monopoly profits for selling to intermediary 1 with the duopoly profits for selling to both intermediaries. Thus, suppose

Step 2.1: We compare the two monopoly profits for selling to intermediary 1. We obtain by observing

that

Step 2.2: We compare the duopoly profit with the monopoly profit for selling to intermediary 1. If the duopoly profit is greater than or equal to both monopoly profits, then the input supplier decides to sell to both intermediaries. Therefore, we obtain that if

with

then

holds. Thus, whenever

the input supplier prefers to set a price to sell to both intermediaries. Note that the following is always true:

Step 2.1 and Step 2.2 show that for

In summary, if for Bertrand competition

with

then it is optimal for a profit-maximizing input supplier to only serve intermediary 1 and with analogous conditions to serve intermediary 2.

Proof of Proposition 1Proof of Proposition 1. The proof proceeds in two steps. First we compare the profits of the input supplier in the case of intermediaries competing in quantities with the setting in which intermediaries compete in prices.

Step 1 (Duopoly Profit of the Input Supplier)

Suppose the input supplier sells inputs to both intermediaries, which is

We show that

can be rewritten as

Assumption 1 and 2 (from Appendix 1.1), which is

implies for

Thus, we obtain

The analogous argument holds for

and with Assumption 1 (from Appendix 1.1) for

Step 2 (Comparing

Note that

With the observation that the input supplier’s profit when selling to both intermediaries is non-decreasing in

Case 1: Large asymmetries between the intermediaries, i.e.,

In Bertrand as well as in Cournot competition intermediary 2 is excluded from the input market. As the input demand of the intermediary is identical for both modes of competition, the input price is identical also.

Case 2: Intermediate asymmetries between the intermediaries, i.e.,

In Bertrand competition both intermediaries purchase inputs, while in Cournot competition only intermediary 1 procures on the input market. We have

Therefore,

Case 3: Small asymmetries between the intermediaries, i.e.,

Both intermediaries purchase inputs on the input market in Bertrand as well as in Cournot competition. We have

Therefore,

Proof of Proposition 3. For

Thus,

Proof of Proposition 4. For this proof consider small asymmetries between the intermediaries, i.e.,

1. We have

2. We have

3. We have