  We consider a Cournot oligopoly model where multiple suppliers (oligopolists)
compete by choosing quantities. We compare the social welfare achieved at a
Cournot equilibrium to the maximum possible, for the case where the inverse
market demand function is convex. We establish a lower bound on the efficiency
of Cournot equilibria in terms of a scalar parameter derived from the inverse
demand function, namely, the ratio of the slope of the inverse demand function
at the Cournot equilibrium to the average slope of the inverse demand function
between the Cournot equilibrium and a social optimum. Also, for the case of a
single, monopolistic, profit maximizing supplier, or of multiple suppliers who
collude to maximize their total profit, we establish a similar but tighter
lower bound on the efficiency of the resulting output. Our results provide
nontrivial quantitative bounds on the loss of social welfare for several convex
inverse demand functions that appear in the economics literature.
