There are two general types of theories for oligopoly:

• 1. Conjectural Variation Models
• 2. Limit Pricing Models.

In conjectural variation models the firms in the industry are taken as given and each firm makes certain assumptions about what the others reactions will be to its own actions. For example, in the Cournot model each firm assumes there will be no reaction on the part of the other firms. In the limit price models one firm chooses its action taking into account the possible entry or exit of competitors to or from the market.

Conjectural Variation Models of Oligopoly

Undifferentiated Product Market

(1) p = p₀ - b*Q,

where Q is the total production of all the firms in the market. Let the cost function for the i-th firm be

(2) C_i = C_i0 + C_i1*q_i,

where q_i is the output of the i-th firm. The profit of the i-th firm, U_i, is then

(3) U_i = p*q_i - C_i = (p₀ - b*Q)*q_i - C_i0 - C_i1*q_i.

The first order condition for maximizing U_i with respect to q_i is:

(4) ∂U_i/∂q_i = (p₀ - b*Q) -b*(∂Q/∂q_i)*q_i - C_i1 = 0.

The Cournot Model

In the Cournot Model each firm presumes no reaction on the part of the other firms to a change in its output. Thus, ∂Q/∂q_i = 1. Therefore the first order condition for a maximum profit of the i-th firm is:

(5) p₀ - b*(Q_oi + 2q_i) = C_i1,

where Q_oi is the output of the firms other than the i-th. When this is solved for q_i the result is:

(6) q_i = (p₀ - C_i1)/2b - Q_oi/2.

However it is more convenient to represent the first order condition and its solution as:

(7) p₀ - b*(Q + q_i) = C_i1
and
q_i = (p₀ - C_i1)/b - Q.

Now we can sum the above equation over the n firms. The result is:

(8) Q = n(p₀/b) - C₁/b - n*Q,

where C₁ is the sum of the C_i1. The solution for Q is:

(9) Q = [n/(n+1)](p₀/b) - [1/(n+1)]C₁/b.

When this output is substituted into the inverse demand function the result is:

(10) p = [1/(n+1)]p₀ + [1/(n+1)]C₁,

or if we let c₁=C₁/n:

(11) p = [1/(n+1)]p₀ + [n/(n+1)]c₁,

where c₁ represents the average of the marginal costs of the n firms. We see from (11) that as the number of firms increase without bound the market price approaches c₁.

If one follows through with this model one would have to take in consideration that the firms with above average marginal cost could be making a loss on variable costs and would cease production.

The von Stackelberg Leader-Follower Model

(12) ∂U_L/∂q_L = (p₀ - b*Q) -b*(1/2)*q_L - C_L1 = 0.

Thus

(13) q_L = (p₀ - C_L1)(2/3b) - 2Q_oL/3.

Carrying through with the analysis as shown below indicates that the market price will be:

(14) p = [1/(n+2)]p₀ + [(n+1)/(n+2)]c₁,

where c₁ is now the weighted average of the marginal costs of the firm with all of the follower firms given an equal weight and the leader firm given a weight of twice that of the follower firms. The leader firm has the effect on the industry of two follower firms. Otherwise the result is the same as in the case of the Cournot oligopoly.

The General Case

(15)(∂Q/∂q_i)*q_i = (p₀ - C_i1)/b - Q,
or
q_i = W_i(p₀/b) - W_i(C_i1/b) - W>i*Q,

where W_i = 1/(∂Q/∂q_i). Summing over i gives:

(16) Q = N(p₀/b) - Nc₁/b - NQ,

where N is the sum of the weights W_i and c₁ is the weighted average of the marginal costs C_i1. Thus,

(17) Q = [n/(n+1)]*(p₀-c₁)/b.

This result when substituted into the inverse demand function gives:

(18) p = [1/(n+1)]p₀ + [n/(n+1)]c₁.

This is the same as the Cournot solution with the number of firms replaced by the effective number of competitors, the sum of the reciprocals of (∂Q/∂q_i). From (18) we also have that the change in oligopoly price is a weighted average of the shifts in the marginal costs and shifts in the demand function as given by the parameter p₀.

Differentiated Products

For this case it is convenient and also necessary to use a matrix formulation of the problem. It is assumed that each firm produces a different product. Let P and Q be the column vectors of prices and outputs. The inverse demand function is taken to be linear and of the form:

(19) P = P₀ - BQ.

The cost function for each firm is of the form C_i = C_0i + C_1i*q_i. The first order condition for a maximum profit with respect to q_i is:

(20) p_i0 - Σ_j[b_ij*q_j] + q_i*[-Σ_j[b_ij*(∂q_j/∂q_i)] - C_i1 = 0,

where Σ_j[] denotes the summation with respect to the index j. The set of these first order conditions for i=1,...,n can be represented in matrix form as:

(21)P - BQ - DQ - C₁ = 0,

where D is a diagonal matrix whose diagonal elements are:

(22) d_ii = Σ_j[b_ij*(∂q_j /∂q_i).

It is the diagonal matrix created from the diagonal elements of the matrix BJ, where J = [(∂q_j/∂q_i)].

The solution for Q is

(23) Q = G(P₀ - C₁),

where G is the inverse of (B+D). The vector of market prices is:

(24) P = (I-BG)P₀ + BGC₁.

Thus the oligopoly prices given as P are weighted averages of the P₀ parameters and the marginal costs C₁.

Limit Pricing Models of Oligopoly

An abreviated version of a limit pricing model of oligopoly is given at Limit.

The economic welfare implications of limit pricing oligopoly are pursued at Oligopoly and Welfare.

(To be continued.)