Rothbard on V shaped average and total cost curves

Abstract

R othbard (1993, pp. 638-45) refuted the important economic fallacy that excess capacity is a normal consequence of profit maximizing behavior by businesses in some industries when they are in long-run equilibrium. And, in so doing provided a manifest example of misuse of mathematics in modern economics. According to standard theory, given a U-shaped, average-cost curve (ACC), in equilibrium, a firm whose demand is perfectly competitive will operate at the point where its horizontal demand curve is just tangent to the ACC; i.e., at the point where average cost (AC) is at its minimum. Alternatively, a firm in an industry characterized by monopolistic competition will face a downward-sloping demand curve. In that case, again in equilibrium, the firm will operate where the demand curve is just tangent to the U-shaped ACC. However, in that case, the point of tangency will occur at lesser quantity than that at which AC is at its minimum. Rothbard, however, puts an end to this notion, despite its vast popularity within the profession. He notes that a necessary condition for the above conclusion is that the ACC be smooth. As proof, he offers his famous diagram (1993, p. 644, fig. 72), the essential burden of which is that though the ACC curve slopes downward continuously until it reaches its minimum and then slopes upward continuously, and is in fact a graph of a continuous function, nevertheless the function is not differentiable at critical points, including especially, at its minimum. Our figure 1 accentuates this even more, drawing the ACC not in a quasi-U shape, but in a V shape. Note that in either of these cases, the downward sloping demand curve can, in contravention of the standard theory, touch 1 the ACC curve at the minimum point of the latter.