San José State University
Department of Economics 

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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
The inverse demand function is then
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_{opt}is then
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
This means that the total quantity marketed by the oligopolists q_{oli} is
Since b(p_{max}−p_{})=q_{opt}
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
From the previous expression for q_{oli} it follows that
The average of p_{oli} and p_{opt} works out to be
The revenus R_{oli} is simply
The ratio is
Let the ratio S_{oli}/R_{oli} be denoted as σ_{oli} and the ratio p_{opt}/p_{max} as γ. Thus
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).
For a market of n Cournottype 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).
For more general models of oligopoly the weights are not just based upon the number of firms. Cournottype firms have unit weight but von Stakelberg leader firms get double weight. In general,
Thus
Since q_{opt}=b(p_{max}p_{opt}) the above relation reduces to:
Furthermore
The social cost of oligopolistic pricing is then
For γ=0, σ=½(w/(1w)) and for γ=1, σ=w/(1w).
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