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 (235239). 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_{i}^{2} + 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, firm_{0}, to 1000 for the smallest, firm_{4}, 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 
firm_{0} equilibrium
demand
p_{0} = (1400 / (q_{0} + 56.03 + 10)  5
cost cost_{0} = 0.15 q_{0}^{2} + 1.6 q_{0} + 30
p_{0} = 11.31 q_{0} = 19.8 rev_{0} = 223.95 cost_{0} = 120.49 profit_{0} = 103.46  
firm_{1} equilibrium
demand
p_{1} = (1300 / (q_{1} + 58.23 + 10)  5
cost cost_{1} = 0.15 q_{1}^{2} + 1.75 q_{1} + 26
p_{1} = 10.15 q_{1} = 17.61 rev_{1} = 178.62 cost_{1} = 103.31 profit_{1} = 75.31  
firm_{2} equilibrium
demand
p_{2} = (1200 / (q_{2} + 60.54 + 10)  5
cost cost_{2} = 0.15 q_{2}^{2} + 1.9 q_{2} + 22
p_{2} = 8.98 q_{2} = 15.29 rev_{2} = 137.33 cost_{2} = 86.13 profit_{2} = 51.2  
firm_{3} equilibrium
demand
p_{3} = (1100 / (q_{3} + 62.98 + 10)  5
cost cost_{3} = 0.15 q_{3}^{2} + 2.05 q_{3} + 18
p_{3} = 7.82 q_{3} = 12.85 rev_{3} = 100.45 cost_{3} = 69.13 profit_{3} = 31.32  
firm_{4} equilibrium
demand
p_{4} = (1000 / (q_{4} + 65.55 + 10)  5
cost cost_{4} = 0.15 q_{4}^{2} + 2.2 q_{4} + 14
p_{4} = 6.65 q_{4} = 10.28 rev_{4} = 68.39 cost_{4} = 52.49 profit_{4} = 15.9  
Solving the Model
Q_{i} = sum of the quantities of products sold by all firms, excluding firm i
Firm i's demand schedule: p_{i}(q_{i}) = A_{i} / (q_{i} + Q_{i} + B_{i})  C_{i}
Firm i's cost schedule: cost_{i}(q_{i}) = a_{i} q_{i}^{2} + b_{i} q_{i} + c_{i}
Firm i's revenue schedule: rev_{i}(q_{i}) = q_{i} * p_{i}(q_{i})
Firm i's profit schedule: profit_{i}(q_{i}) = rev_{i}(q_{i})  cost_{i}(q_{i})
Firm i's marginal demand schedule: md_{i}(q_{i}) = d p_{i}(q_{i}) / d q_{i} = A_{i} / (q_{i} + Q_{i} + B_{i})^{2}
Firm i's marginal revenue schedule: mr_{i}(q_{i}) = d rev_{i}(q_{i}) / d q_{i} = p_{i}(q_{i}) + q_{i} * md_{i}(q_{i})
Firm i's marginal cost schedule: mc_{i}(q_{i}) = d cost_{i}(q_{i}) / d q_{i} = 2 * a_{i} q_{i} + b_{i}
First Order Conditions
Profit Maximization: Li(q_{i}) = d profit_{i}(q_{i}) / d q_{i} = mr_{i}(q_{i})  mc_{i}(q_{i}) = 0
Jacobian
Note: d Q_{i} / d q_{i} = 0; d Q_{i} / d q_{j} = 1, i ≠ j
The Jacobian matrix: for all i and j,
  
d Li(q_{i}) / d q_{j}    
The gradient and Newton methods use the first order equations and the Jacobian matrix to solve for the equilibrium  profit maximizing  values of q_{i}.

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: McGrawHill, 1971.
