Lumpy Demand and the Diagrammatics of Aggregation

Abstract

Economists are used to sketching demand curves that do not reach the vertical axis. What happens to the demand curve at low quantities is often ignored, which is understandable if the relevant cases for analysis all have prices that intersect the demand curve at quantities measurably far from zero. Nevertheless, in textbook treatments of the summation of individual demand curves (horizontal for rival goods or vertical for nonrival goodst), the individual curves are pictured as nicely intersecting the vertical axis. The point of intersection is sometimes referred to as the choke price. In textbook treatments of aggregation, the demand curve is continuous, but not continuously differentiable, through the choke price. A hidden implication of this is that the inverse demand curve, which, abstracting from income effects, is the marginal valuation schedule, is represented by the same diagram. Adding the demand curves horizontally or the marginal valuation curves vertically poses no particular problem. In distinction to the usual treatment, we were motivated by the real possibility that small amounts of a good are worthless to the consumer. For example, a pint of gasoline would not allow one to operate an automobile in a useful way, but several tankfuls could be a valuable input toward vacation travel. Wheels for one's automobile are generally worthless until you have at least four; 20 feet of fencing may be worthless if your property line is 30 feet; a bridge spanning 40