Conditions for Demand Curves Whose Curves of Total Revenue, Consumers' Surplus, Total Benefit, and Compromise Benefit are Convex - Kozlik, Adolf (Econometrica, 1940)
IT IS IMPORTANT for problems of storage and of price discrimination to know whether the total-revenue curve is convex or concave. I shall try to state the conditions for the demand curves under which the totalrevenue curve is convex or concave. Let D be the quantity. is the demand curve and the total-revenue curve. The total-revenue curve is convex if d2[DF(D)] /dD2 0. In the first case, d2[DF(D) ]/dD2 O, or DF(D) -2F'(D). If the total-revenue curve is a straight line, DF(D) =2F'(D), and by solving this differential equation we get = a/D + b, where a and b are constants (a>O). These are Marshall's constant-outlay curves, a/D, shifted up or down by any constant quantity b (Figure 2). If the case of a straight-line total-revenue curve, = a+bD, we have F'(D) = -a/D2 (Figures 1, 2), F(D) =2a/D3 (Figure 1), and DF(D) = 2a/D2 (Figure 1). In Figure 2 some demand curves and their marginal curves for straight-line total-revenue curves are represented. If the marginal curve, F'(D) =I'(D), of a given demand curve, = I(D), cuts one of the curves F'(D) =-a/D2 from above at a point P', then its slope at this point is smaller than the slope of the curve F'(D) =-a/D2. Since the slope of the curve F'(D) =I'(D) is F(D) =I(D) and the slope of the curve F'(D) = a/D2 is F(D) = 2a/D3, the relation