San José State University
Department of Economics

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 The Social Cost of a Oligopolistic Pricing

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 = pmax − (1/b)q and thus the demand function can be expressed as q = b(pmax−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 popt. The optimal market price is then equal to popt. Thus the socially optimal level of production qoptis then

#### qopt = b(pmax−popt)

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

#### poli = (1/(n+1))pmax + (n/(n+1))popt

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

#### qoli = b(pmax−poli) which reduces to qoli = b(pmax−(1/(n+1))pmax − (n/(n+1))poptand still further to qoli = (n/(n+1))b(pmax−popt)

Since b(pmax−p)=qopt

## 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 qoli to qopt. This is the area of a trapezoid. Thus the social cost of the oligopoly Soli is

#### Soli = [qopt−qoli][poli+popt]/2

From the previous expression for qoli it follows that

#### qopt−qoli = (1/(n+1))qopt = (1/n)qoli

The average of poli and popt works out to be

## The Revenue of the Oligopolists

The revenus Roli is simply

## The Ratio of Social Cost to Oligopolist Revenues

The ratio is

#### Soli/Roli = (1/n)qoli[poli+popt]/2/(qolipoli) which reduces to Soli/Roli = (1/2n)[1 + popt/poli] which can be expressed as Soli/Roli = (1/2n)[1+(2n+1)(popt/pmax)]/[1+n(popt/pmax)]

Let the ratio Soli/Roli be denoted as σoli and the ratio popt/pmax as γ. Thus

#### σoli = (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,

Thus

#### qoli = b(pmax−poli) = b(1-w)b(pmax-popt)

Since qopt=b(pmax-popt) the above relation reduces to:

Furthermore

#### qopt−qoli = wqoptand thus qoli−qoli = (w/(1-w))qoli

The social cost of oligopolistic pricing is then

#### Soli = (w/(1-w))qoli[poli+popt]/2 and since the olipolists' revenue is Roli = qolipoli σoli = ½(w/(1-w))[wpmax+(2-w)popt]/[wpmax+(1-w)popt] which reduces to σoli = 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).