Egwald Economics: Microeconomics

Oligopoly / Government Firm | Differentiated Oligopoly | Monopolistic Competition

Imperfect Competition Models

Differentiated Oligopoly

by

Elmer G. Wiens

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Each seller in an imperfectly competitive market faces a negatively sloped demand curve for his product, permitting him some control of the price of his product. In an oligopoly, a few firms produce the same product, while in monopolistic competition, many firms produce differentiated but similar products. In a differentiated oligopoly, a few firms produce products different enough for each firm to have its own downward sloping demand curve.

Barriers to entry, like economies obtained from producing at a large scale, prevent entry of new firms into an oligopoly.

On the other hand, each of the many sellers in a perfectly competitive market must sell its product at the market determined price. A firm's demand curve is a straight line at the prevailing price. There are no barriers to entry.

The differentiated oligopoly model on this page is in some ways similar to the model of monopolistic competition of Henderson and Quandt (235-239). It uses a functional form for the demand functions of the firms that can be found in R.G.D. Allen (114).

While the firms of my model produce differentiated products, at least in the minds of purchasers (like personal computers or television sets), the products are perfect substitutes in the sense that if a consumer buys an item from one firm, she will not also buy an item from another firm (usually one buys just one PC).

Let Q_i be the sum of the quantities of products sold by all firms excluding firm i. Then the demand schedule (relation between price, p, and quantity sold, q) of a typical firm i is:

p_i = p_i(q_i) = A_i / (q_i + Q_i + B_i) - C_i

The total cost schedule of a typical firm is:

cost_i = cost_i(q_i) = a_i q_i² + b_i q_i + c_i

So, the firms are interdependent through their demand schedules, but not through their cost schedules. These schedules can differ between firms. It may be the case that established firms have captured a large market share, advertising to retain brand loyalty, with consequent greater fixed cost. Also, the established firms may have access to advanced technology (through R&D), permitting them to increase production (at a given level of production), at lower marginal cost than their smaller competitors.

I tried to capture the essence of these features in the model of five firms. All the coefficients A_i, B_i, C_i, a_i, b_i, and c_i are positive. I have arbitrarily set the coefficients A_i to vary from 1400 for the largest firm, firm₀, to 1000 for the smallest, firm₄, b_i to vary from 1.6 to 2.2, and c_i to vary from 30 to 14. You can shift these distributions up or down by filling in the form at the bottom of the web page.

Each firm attempts to exploit its downward sloping demand schedule, maximizing its profit by producing that amount of output where marginal revenue, mr, equals marginal cost, mc.

To see this, notice that a firm's profit equals its revenues less its costs:

profit_i(q_i) = rev_i(q_i) - cost_i(q_i),

where revenues are:

rev_i(q_i) = q_i * p_i(q_i).

A firm will maximize its profit when its output q_i is set so that the derivative of its profit equation with respect to q_i is zero. The derivative of revenue with respect to q_i is marginal revenue, mr_i; the derivative of cost is marginal cost, mc_i. Profit maximization requires:

mr_i(q_i) - mc_i(q_i) = 0,

with the marginal cost curve intersecting the marginal revenue curve from below. Thus at the point of intersection, the slope of the marginal revenue function is less than the slope of the marginal cost function, or:

mr_i'(q_i) < mc_i'(q_i).

Firms adjust their output, interactively, until each firm is in equilibrium, when mr_i = mc_i. That is, each firm attempts to set the level of output where mr = mc, given its demand schedule, i.e., given the output levels of its competitors. In equilibrium, in the absence of collusion, there is no incentive for firms to change their levels of output.

Differentiated Oligopoly
firm0 equilibrium demand p0 = (1400 / (q0 + 56.03 + 10) - 5 cost cost0 = 0.15 q02 + 1.6 q0 + 30 p0 = 11.31   q0 = 19.8 rev0 = 223.95   cost0 = 120.49 profit0 = 103.46
firm1 equilibrium demand p1 = (1300 / (q1 + 58.23 + 10) - 5 cost cost1 = 0.15 q12 + 1.75 q1 + 26 p1 = 10.15   q1 = 17.61 rev1 = 178.62   cost1 = 103.31 profit1 = 75.31
firm2 equilibrium demand p2 = (1200 / (q2 + 60.54 + 10) - 5 cost cost2 = 0.15 q22 + 1.9 q2 + 22 p2 = 8.98   q2 = 15.29 rev2 = 137.33   cost2 = 86.13 profit2 = 51.2
firm3 equilibrium demand p3 = (1100 / (q3 + 62.98 + 10) - 5 cost cost3 = 0.15 q32 + 2.05 q3 + 18 p3 = 7.82   q3 = 12.85 rev3 = 100.45   cost3 = 69.13 profit3 = 31.32
firm4 equilibrium demand p4 = (1000 / (q4 + 65.55 + 10) - 5 cost cost4 = 0.15 q42 + 2.2 q4 + 14 p4 = 6.65   q4 = 10.28 rev4 = 68.39   cost4 = 52.49 profit4 = 15.9

Solving the Model

Works Cited and Consulted

• Allen, R.G.D. Mathematical Analysis for Economists. London: MacMillan, 1938.



• Archibald, G.C. The Theory of the Firm. Middlesex: Penguin, 1971



• Bain, Joe S. Price Theory. New York: Wiley, 1952



• Henderson, James M. and Richard E. Quandt. Microeconomic Theory: A Mathematical Approach. New York: McGraw-Hill, 1971.