A previous work found that the gross social cost of monopoly pricing can be related to the revenue received by the monopolist. That study found that the social cost of a monopoly price greater than the optimal price is between 50 and 100 percent of the monopolist's revenue. This study extends that analysis to a protected oligopoly operating according to conjectural variations models such as the Cournot and von Stakelberg models. For background on the theory of oligopolies see Oligopoly Theory.

Let n be the number of firms in the market and no other firms are permitted in this market. The oligopolists are somehow protected from potential competition. The products of the firms are assumed to be identical. The demand function for the product is linear and given by

q = a − bp

The inverse demand function is then

p = (a/b) − (1/b)q
which can expressed as
p = p_max − (1/b)q
and thus the demand function
can be expressed as
q = b(p_max−p)

In another study it is shown that the marginal cost relevant for efficiency pricing is the minimum average cost of the marginal plant. Let this marginal cost be denoted as p_opt. The optimal market price is then equal to p_opt. Thus the socially optimal level of production q_optis then

q_opt = b(p_max−p_opt)

As shown in oligopoly theory the price p_oli established by n firms acting independently and presuming no reaction on the part of the other firms (Cournot oligopolists) is

p_oli = (1/(n+1))p_max + (n/(n+1))p_opt

This means that the total quantity marketed by the oligopolists q_oli is

q_oli = b(p_max−p_oli)
which reduces to
q_oli = b(p_max−(1/(n+1))p_max − (n/(n+1))p_opt
and still further to
q_oli = (n/(n+1))b(p_max−p_opt)

Since b(p_max−p)=q_opt

q_oli = (n/(n+1))q_opt

Social Cost of Oligopolistic Pricing

The gross social cost of the oligopoly price being above the socially optimal price is the area under the inverse demand curve from q_oli to q_opt. This is the area of a trapezoid. Thus the social cost of the oligopoly S_oli is

S_oli = [q_opt−q_oli][p_oli+p_opt]/2

From the previous expression for q_oli it follows that

q_opt−q_oli = (1/(n+1))q_opt = (1/n)q_oli

The average of p_oli and p_opt works out to be

[p_oli+p_opt]/2 = (1/(2n(n+1))[p_max+(2n+1)p_opt]/2

The Revenue of the Oligopolists

The revenus R_oli is simply

R_oli = q_olip_oli

The Ratio of Social Cost to Oligopolist Revenues

The ratio is

S_oli/R_oli = (1/n)q_oli[p_oli+p_opt]/2/(q_olip_oli)
which reduces to
S_oli/R_oli = (1/2n)[1 + p_opt/p_oli]
which can be expressed as
S_oli/R_oli = (1/2n)[1+(2n+1)(p_opt/p_max)]/[1+n(p_opt/p_max)]

Let the ratio S_oli/R_oli be denoted as σ_oli and the ratio p_opt/p_max as γ. Thus

σ_oli = (1/2n)[1+(2n+1)γ]/[1+nγ]

Thus the ratio of social costs to the revenue of the oligopolists is a function only of the ratio of the marginal cost to the maximum price for the product. This ratio, γ, can only be in the range of zero to one. For γ=0 σ is equal to (1/2n). For γ=1 σ is equal to (1/n).

Conclusion

For a market of n Cournot-type oligopolists the ratio of the social cost of the oligopoly pricing to the revenue of the oligopolists has to be at least (1/2n) and can be no more than (1/n).

Generalization

For more general models of oligopoly the weights are not just based upon the number of firms. Cournot-type firms have unit weight but von Stakelberg leader firms get double weight. In general,

p_oli = wp + (1-w)p_opt

Thus

q_oli = b(p_max−p_oli) = b(1-w)b(p_max-p_opt)

Since q_opt=b(p_max-p_opt) the above relation reduces to:

q_oli = (1-w)q_opt

Furthermore

q_opt−q_oli = wq_opt
and thus
q_oli−q_oli = (w/(1-w))q_oli

The social cost of oligopolistic pricing is then

S_oli = (w/(1-w))q_oli[p_oli+p_opt]/2
and since the olipolists' revenue is
R_oli = q_olip_oli
 
σ_oli = ½(w/(1-w))[wp_max+(2-w)p_opt]/[wp_max+(1-w)p_opt]
which reduces to
σ_oli = ½(w/(1-w))[1 + ((2-w)/w)γ]/[1 + ((1-w)/w)[_opt]

For γ=0, σ=½(w/(1-w)) and for γ=1, σ=w/(1-w).