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Fixed vs. Variable Proportions

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With a fixed-proportions production process, vertical integration cannot increase profits, whereas with variable proportions profits can rise. This difference between fixed proportions and variable proportions can be illustrated using an example with the following properties:
  1. Demand: The final goods market has a linear inverse demand curve of

    Equation 1
    p = 10 - Q

  2. Input costs: The marginal cost of the two inputs, E and L, are m = w = 1.

  3. Production processes: We consider a fixed-proportions production function and a variable-proportions production function, both of which have two properties: (1) constant returns to scale, and (2) 1 unit of E and 1 unit of L produces 1 unit of Q.
We start by considering the outcome if all markets are competitive. Competitive markets are socially desirable because welfare, defined as consumer surplus plus profits, is maximized. We then compare monopoly of E with fixed proportions and variable proportions to competition and to each other. The fixed- and variable-proportions production functions are chosen so that the integrated monopoly outputs are the same for both. The table (click here to open table) shows the properties of the three cases. [Vertical Integration Under Fixed and Variable Proportions shows how these values are derived formally.]

The first column of the table shows what happens if all the markets are competitive. The factor markets are competitive, so the price of E, e, equals its social cost, c. Because 1 unit of E and 1 unit of L produce 1 unit of Q, the marginal cost and hence the price of Q is 2 (= mE + wL = 1 x 1 + 1 x 1). At that price, 8 units of Q are produced using 8 units each of E and L. Under competition, there is no deadweight loss nor any losses from inefficient production, and consumer surplus and welfare are maximized (at 32). The rest of this example concerns a monopoly in E.

If the production process uses fixed proportions, so that the input proportions used are insensitive to change in the relative factor prices, m/w, the profit of the upstream monopoly is the same whether or not it vertically integrates. As shown in the second and third columns of the table, final good price and quantity, consumer surplus, and deadweight loss are the same in either case. The consumer surplus is the triangle under the demand curve and above the price: (10 - p)Q/2. The deadweight loss is also a triangle: (p - 2)(8 - Q)/2, where one side is the difference between the price and the social marginal cost, 2, and another side is the difference between the quantity and the quantity that would have been produced if price equaled the social marginal cost, 8. The price, p = 6, is the one that maximizes monopoly profits. The only difference in the two columns is that when the firm is vertically integrated, the monopoly charges itself e = m = 1, whereas in the absence of vertical integration, the monopoly charges the final goods industry e = 5 (see Figure 12.1).

Figure 12.1 Fixed-Proportions Production Function

(a) Profits of an integrated firm(b) Profits of a monopoly supplying a competitive industry

Whether or not the firm integrates, production is efficient, because the cost-minimizing choices of E and L are used (E* = L* = Q* = 4). A deadweight loss results from setting the monopoly price, p* = 6, above the true social marginal cost (competitive price), 2. Total welfare falls from 32 under competition to 24 under monopoly (the difference equals the deadweight loss of 8). Consumer surplus falls from 32 in the competitive case to 8, so consumers are much worse off due to monopoly pricing.

With a variable-proportions production function, most of these variables differ in the integrated and nonintegrated market organizations. In the table, a Cobb-Douglas production function is used:

Equation 2
Q = L½ E½

Here, L and E are (imperfect) substitutes. Corresponding isoquants look like the one in Figure 12.4. This production process was chosen so that if the firm is integrated, price, quantity, profits, consumer surplus, and so forth are the same as with a fixed-proportions integrated monopoly.

If the monopoly vertically integrates, it behaves as if it charges itself e = m = 1, and produces efficiently (E* = L* = Q* = 4), so that costs are minimized. The results are different, however, if the monopoly does not vertically integrate. Without integration, if the monopoly increases its price, m, the downstream firms substitute away from E and towards L, as shown in Figure 12.4 in your textbook. As a result, the marginal cost does not rise as much as e does. The final goods producers face a marginal cost of

Equation 3
MCQ = 2(we)½ = 2e½

because w = 1. Thus, if e doubles, the marginal cost increases by only 41 percent.

Because of the substitution by downstream firms, E and L are no longer used in equal proportions. Rather, at the profit-maximizing choice of e, 7.93, nearly 8 times as much L as E is used. The marginal cost, and hence price, of the downstream competitive firms is 5.63, so consumers are better off than in the integrated case: their consumer surplus is 19 percent higher.

There is now a second social loss, beyond that from monopoly pricing, due to inefficient production. The least expensive way to produce the quantity sold, Q = 4.37, is to use 4.37 units of E and L at a social cost of 8.74. Instead, E = 1.55 and L = 12.30, with a social cost of 13.85. Thus, the social loss from inefficient production is 5.11 = 13.85 - 8.74. The sum of consumer surplus, profits to the owner of E, deadweight loss, and loss from inefficient production is the same in all five cases. The reason is that they collectively equal the area above the social marginal cost (= 2) and below the demand curve. That is, they equal the consumer surplus in the competitive case. This social loss represents 16 percent of the consumer surplus in the competitive case, and 54 percent of the consumer surplus in this case. Because the integrated firm can avoid this social loss and charges a higher price, its profits are larger: 16 versus 10.75. Thus, the upstream monopoly has a substantial incentive to vertically integrate when the final goods market uses a variable-proportions production function. It integrates if this increase in profits exceeds the cost of integrating.

Consumer surplus is higher (9.55 versus 8) without integration than with it, because the nonintegrated price, 5.63, is less than the integrated price, 6, although it is still above the competitive level, 2. Society's welfare is lower without integration than with it (20.3 = 9.55 + 10.75 versus 24 = 8 + 16). This drop in welfare of 3.7 reflects the loss from inefficient production of 5.11 that outweighs the reduction in deadweight loss due to monopoly pricing of 1.41 (= 8 - 6.59).

Thus, welfare is highest if all markets are competitive. In this example, assuming no cost of integrating, welfare is the same with fixed proportions whether or not an energy monopoly vertically integrates. With variable proportions, if integration does not take place (perhaps because of a legal prohibition), social welfare is lower because of the additional loss from inefficient production in addition to the loss from monopoly pricing.

The results of this example do not always hold, however. It is possible for both consumer surplus and welfare to rise when a monopoly vertically integrates to extend its monopoly power. For example, it may happen that price could fall because of the gains in efficiency (Mallela and Nahata 1980).


Abiru, Masahiro. 1988. "Vertical Integration, Variable Proportions and Successive Oligopolies." Journal of Industrial Economics 36:315-25.

Mallela, Parthasaradhi, and Babu Nahata. 1980. "Theory of Vertical Control with Variable Proportions." Journal of Political Economy 88:1009-25.

Vernon, John M., and Daniel A. Graham. 1971. "Profitability of Monopolization by Vertical Integration." Journal of Political Economy 79:924-5.

Warren-Boulton, Frederick R. 1974. "Vertical Control with Variable Proportions." Journal of Political Economy 82:783-802.

© 2000 Dennis W. Carlton and Jeffrey M. Perloff. Reprinted by permission.

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