     Egwald Economics: Microeconomics

Oligopoly / Government Firm | Differentiated Oligopoly | Monopolistic Competition

Imperfect Competition Models

Differentiated Oligopoly

by

Elmer G. Wiens

Egwald's popular web pages are provided without cost to users.
Please show your support by joining Egwald Web Services as a Facebook Fan: Follow Elmer Wiens on Twitter: 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 Qi 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:

pi = pi(qi) = Ai / (qi + Qi + Bi) - Ci

The total cost schedule of a typical firm is:

costi = costi(qi) = ai qi2 + bi qi + ci

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 Ai, Bi, Ci, ai, bi, and ci are positive. I have arbitrarily set the coefficients Ai to vary from 1400 for the largest firm, firm0, to 1000 for the smallest, firm4, bi to vary from 1.6 to 2.2, and ci 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:

profiti(qi) = revi(qi) - costi(qi),

where revenues are:

revi(qi) = qi * pi(qi).

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

mri(qi) - mci(qi) = 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:

mri'(qi) < mci'(qi).

Firms adjust their output, interactively, until each firm is in equilibrium, when mri = mci. 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 Adjust Coefficients Reset value of A0 [1200, 1600] 1400 Reset value of b0 [ 1.2, 2.0] 1.6 Reset value of c0 [26, 38] 30

Solving the Model

Qi = sum of the quantities of products sold by all firms, excluding firm i

Firm i's demand schedule: pi(qi) = Ai / (qi + Qi + Bi) - Ci

Firm i's cost schedule: costi(qi) = ai qi2 + bi qi + ci

Firm i's revenue schedule: revi(qi) = qi * pi(qi)

Firm i's profit schedule: profiti(qi) = revi(qi) - costi(qi)

Firm i's marginal demand schedule: mdi(qi) = d pi(qi) / d qi = -Ai / (qi + Qi + Bi)2

Firm i's marginal revenue schedule: mri(qi) = d revi(qi) / d qi = pi(qi) + qi * mdi(qi)

Firm i's marginal cost schedule: mci(qi) = d costi(qi) / d qi = 2 * ai qi + bi

First Order Conditions

Profit Maximization: Li(qi) = d profiti(qi) / d qi = mri(qi) - mci(qi) = 0

Jacobian

Note: d Qi / d qi = 0;   d Qi / d qj = 1,  i ≠ j

The Jacobian matrix: for all i and j,

 || d Li(qi) / d qj ||

The gradient and Newton methods use the first order equations and the Jacobian matrix
to solve for the equilibrium - profit maximizing - values of qi.

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. Copyright © Elmer G. Wiens:   Egwald Web Services All Rights Reserved.    Inquiries 