Bilateral Duopoly Efficiency

Most of the products sold on the final market are the result of several production stages. For instance, the food products are firstly packed and branded, then distributed to the stores and retailers and finally sold to the final consumers.

The purpose of this essay is to compare the performance of two different market structures in an industry characterized by two production stages: an upstream and a downstream one.

For the sake of clarity, we will assume that the production of one unit of output at the downstream stage requires the use of one unit of output from the upstream stage. As a consequence, the downstream output Y will equal the upstream one.

The two market structures taken under examination are a bilateral duopoly and a vertically integrated monopoly.

A bilateral duopoly is a market structure where there are two firms at each stage of production, whereas in a vertically integrated monopoly there is just one firm carrying out both processes.

Throughout the essay it will be assumed that consumer demand in the final market is a linear function of the final output Y and is described by the linear equation:

p=a-bY (a), where a is the consumer reservation price.

The performance, or efficiency, of the two models will be measured by the total 'welfare' they entail. From society's point of view, the total gain is the sum of the consumer surplus and producer surplus, which is known as total surplus.

Consumer surplus is defined as "the difference between what a consumer is willing to pay and what she has to pay" (Katz and Rosen, 1998, p. 110) and is calculated simply as the area under the demand curve and above the market price.

FIGURE 1

The shaded area in FIGURE 1 indicates the consumer surplus CS and is given by the formula: CS=(a-p)Y/2 (b).

Instead, the producer surplus PS is simply the producer's profit. For the purpose of this essay it will be also assumed that firms act in order to maximize their profits.

Bilateral Duopoly

In a bilateral duopoly market, there are two firms, U1 and U2, at the upstream stage and two firms, D1 and D2, at the downstream stage, as FIGURE 2 below shows.

FIGURE 2

The analysis of this market holds on the following assumptions. First of all, the two firms at each stage must be identical, i.e. incur in the same costs of production, and must produce homogeneous good, i.e. goods that are perfect substitutes.

This implies that, in equilibrium, the output of the two firms at each stage is equivalent:

YD1=YD2 (c)

YU1=YU2 (d)

Since we are assuming that the upstream output equals the downstream one, the output of all four firms is the same:

YD1=YD2=YU1=YU2

Furthermore, we will assume that at both stages the duopolists compete on quantities and decide the quantity to produce simultaneously, i.e. not knowing the other firm's production. In other words, we base the analysis on the Cournot model.

Suppose further that all four firms produce at constant marginal cost. The letter c will denote the upstream firms' marginal costs. The downstream firms' marginal costs instead will be equal to (d + t), where d indicates the cost per unit of production incurred by the firms, while t indicates the price per unit paid to the upstream firms to buy their output.

The bilateral duopoly can be analyzed as a two-stage game. In Stage 1, the upstream firms choose the price per unit t, known as transfer price, at which to charge the downstream firms. In Stage 2, the downstream firms decide how much output to sell in the final product market.

This game is solved through backward induction, i.e. considering Stage 2 first, because in reality the upstream firms will decide the transfer price t to charge only when they know how much is going to be sold in the final market. In other words, the upstream firms will look ahead when faced with the decision of setting t.

Stage 2

Let us consider the representative downstream firm D1. This firm's profits, πD1, will be given by the difference between its total revenue and total costs:

πD1=pYD1-(d+t)YD1

Substituting (a) in the equation:

πD1=(a-bY)YD1-(d+t)YD1, where Y is the total final output, i.e. the sum of the two downstream firm's outputs YD1 and YD2 and hence:

πD1=(a-bYD1-bYD2-d-t)YD1, which is the expression of the representative downstream firm's profit as a function of the parameters and of the rival firm's output.

D1, as a profit maximising agent will choose its output YD1 in order to maximise its profits, i.e. the level of output that set to zero the profits first derivative:

∂πD1∕∂YD1=a-2bYD1-bYD2-d-t=0

=> YD1=(a-d-t-bYD2)∕2b

Recalling (c):

YD1=YD2=(a-d-t)∕3b

=> Y=YD1+YD2=2(a-d-t)/3b

On the one side, Y represents the output level that the downstream firms will supply to the final market overall. On the other side, it represents the amount of input that the downstream firms will want to buy from the upstream ones. In other words, Y represents the demand equation faced by the upstream firms U1 and U2.

The equation above can be re-written as:

t=(2a-2d-3bY)/2 (e)

Stage 1

Let us now turn to the upstream firms. The upstream representative firm's profits, πU1, will be given by the difference of its total revenue and total costs:

πU1=tYU1-cYU1=(t-c)YU1

Substituting equation (e):

πU1=[(2a-2d-3bY)/2-c]YU1, where Y is the overall input demanded by the downstream firms,