Useful Notes on Equilibrium of Multi-plant Monopolist

Here we will discuss about a monopolist, who produces a homogenous product in two or more different plants with different cost conditions.

It is referred to as a case of Multiplan monopoly. To maximise profits, the monopolist has to make two important decisions. Firstly, how much output to produce altogether and at what price to sell it so as to maximise profit? Secondly, how to allocate the optimal (profit maximising) output between the different plants?

Thus, the Multiplan monopolist not only faces the problem of determining the profit maximising price and output levels. He has also to decide a profit maximising way for distributing this output among the various plants, which in turn depends upon the cost conditions prevailing in each plant.

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If the marginal cost of production in any plant is lower than that in the others, then it costs relatively less to produce an additional unit of output in this plant in comparison to the other plants. The concerned monopolist can reduce his total costs and hence raise his profit level by diverting production from other higher cost plants to that specific plant.

Ultimately, the cost minimising or profit maximising allocation of total output among different plants of a Multiplan monopolist can be achieved, when the marginal cost of production is equated across all the plants. Now, no more further redistribution of output from one plant to the other can reduce the overall costs of production.

As far as the determination of the profit maximising output level is concerned, the Multiplan monopolist (just like an ordinary monopolist or any profit maximising firm) applies the marginalist rule of the equality of marginal revenue (MR) and (total) marginal cost (MC) in the market. This is described below in case of a monopolist operating with two plants. It can easily be generalised to any number of plants.

Assume that the monopolist operates two plants ‘A’ and ‘B’ each with a different cost structure as shown in Fig. 14.9 (a) and Fig. 14.9 (b) respectively. AC_A and AC_B represent the average cost curves of plant ‘A’ and plant ‘B’ respectively.

Their corresponding marginal cost curves MC_A and MC_B intersect the average cost curves AC_A and AC_B from below at their respective minimum points. The total marginal cost (MC) curve of the Multiplan monopolist can simply be derived from the horizontal summation of the marginal cost curves (MC_A and MC_B) of the individual plants (Fig. 14.9 (c)). The monopolist is supposed to know the cost structures of the different plants besides the market demand or average revenue curve AR (and the corresponding marginal revenue curve MR).

Given the MR and MC curves, the profit maximising total output will be OQ corresponding to the equilibrium point ‘E’ in Fig. 14.9 (c), where the MR curve intersects the (aggregate) MC curve from below. The firm will sell this output at equilibrium price OP.

The Multiplan monopolist attains maximum profits at equilibrium point ‘E’. The monopolist is now confronted with the problem of allocating the profit maximising output OQ between the two plants in an optimal manner. He will allocate this output in a way such that marginal cost of each plant is equal to the Multiplan marginal cost at the optimal output level, i.e. MC_A = MC_B = MC.

If MC_A > MC_B, the monopolist can reduce his total costs and increase the profits by transferring the output from plant ‘A’ to plant ‘B’ similarly if MC_A, < MC_B, the output would be transferred from plant ‘B’ to plant ‘A’. This process of transfer would continue until MC_A = MC_B.

Graphically, this can be shown by drawing a horizontal line from equilibrium point ‘E’ in Fig. 14.9 (c), parallel to the X-axis, until it intersects the MC_A and MC_B curves in Fig. 14.9 (a) and Fig. 14.9 (b) at points E_A and E₀ respectively. At these points, the equilibrium condition MC_A = MC₀ = MR = MC is satisfied. This is the condition required for efficient allocation of the profit maximising output of the Multiplan monopolist among the two industrial plants ‘A’ and ‘B’.

The output levels corresponding to the equilibrium points ‘E_A‘ and ‘E_B‘ are obtained by dropping perpendiculars to X-axis of Fig. 14.9 (a) and Fig, 14.9 (b) respectively. Out of the total output OQ to be produced by the monopolist under consideration, OQ_A will be produced in plant ‘A’ and OQ_B will be produced in plant ‘B’.

Evidently, OQ_A + OQ_B = OQ, since the aggregate marginal cost curve MC in Fig 14.9 (c) was itself obtained from the horizontal summation of the individual marginal cost curves MC_A and MC_B, as depicted in Fig. 14.9 (a) and Fig. 14.9 (b) respectively.

The profits from the two plants ‘A’ and ‘B’ are shown by the shaded areas DPBC and HPFG respectively. The total profit earned by the monopolist at the equilibrium point ‘E’ is shown by the shaded area JIE in Fig. 14.9 (c), which is equal to the sum of the profits earned in plant ‘A’ and plant ‘B’, i.e., area (DPBC+ HPFG).

From the foregoing discussion, it can be inferred that the Multiplan monopolist maximises his profits and attains equilibrium by utilising each plant up to the level at which the marginal costs are equal to each other and to the common marginal revenue and total marginal cost in the market.