Elasticity

BIBLIOGRAPHY

The quantity of a commodity that people want to purchase (demand) per unit of time or want to sell (supply) per unit of time depends in part upon the price of the commodity. Economists have sought ways to describe and measure the relationship between changes in the rate of quantity demanded or supplied and changes in price.

Alfred Marshall (1885, p. 260) first used the term elasticity to refer to the percentage change in quantity demanded (?q/q) divided by the percentage change in price (?p/p), and indicated that elasticity at C, on the demand curve ABCDE (see Figure 1), is CG divided by CF. He elaborated these ideas in his Principles (1890, book 3, chapter 3) and the concept has since been adopted by the profession.

Prior to Marshall, there had been no generally accepted terminology or measure of the relationship between price and quantity, although concepts similar to elasticity had been suggested. For example, Mill (1848, book 3, chapter 2, sec. 4) compared percentage changes in price and in quantity, but he did not divide the two to get what we now call elasticity. Cournot (1838, chapter 4, sec. 24) showed the relationship between elasticity and total revenue (see below) by comparing the ratio of the change in quantity demanded to the change in price (?q/?p) and the ratio of quantity demanded to price (q/p), but he did not suggest dividing the two ratios to measure elasticity.

Elasticity of demand. Elasticity of demand is defined as the percentage change in the quantity demanded, divided by the percentage change in price. It is derived from the demand curve, which shows the absolute quantity demanded as a function of price. Since the rate of demand rises if the price falls, and vice versa, the elasticity of demand is actually negative, except that it is common to ignore the algebraic sign and use the absolute value. By definition, demand is inelastic if the elasticity is less than one and is elastic if the elasticity is greater than one. An elasticity of one is called unit, or unitary, elasticity.

Elastic demand. If demand is elastic, a given percentage increase in price causes a larger percentage decline in the quantity demanded, or a given percentage reduction in price causes a larger percentage increase in quantity demanded. In the limiting case of perfectly (infinitely) elastic demand, which is illustrated by the horizontal line segment DE in Figure 1 (if it is visualized as extended horizontally so as to constitute the entire demand curve), any increase in price, no matter how small, will cause the quantity demanded to decline to zero; and any reduction in price, again no matter how small, will cause the quantity demanded to increase without limit.

Demand is usually elastic for commodities for which there are good substitutes, so that the de-

mand facing a firm is more elastic than the demand facing an industry, since a buyer may substitute the products of other firms in the industry. It is part of the definition of perfect competition that there must be so many firms in the industry producing such excellent substitutes that the demand facing the firm is perfectly (infinitely) elastic, even though the demand facing the industry may be inelastic. [SeeCompetition.]

Inelastic demand. Conversely, if demand is inelastic the percentage change in quantity is smaller than the percentage change in price. An increase in price therefore causes a proportionately smaller decline in the quantity demanded, and a reduction in price causes a proportionately smaller increase in the quantity demanded. In the limiting case of zero elasticity (perfectly inelastic demand), which is illustrated by the vertical line segment AB in Figure 1 (if it is visualized as extended vertically so as to constitute the entire demand curve), neither an increase nor a reduction in price has any effect upon the quantity demanded.

Products that have no good substitutes, like tobacco, or that require a very small proportion of the consumer’s income, like salt, frequently have inelastic demands.

Relation to total revenue. Elasticity of demand also shows how the total dollar value of sales of the commodity responds to a change in its price. For example, if the price of a commodity with inelastic demand is increased, the total revenue of the seller will increase even though the quantity demanded declines somewhat, since the increase in price more than offsets the reduction in volume. This creates a presumption that a seller faced with an inelastic demand will raise his price and will continue to do so until demand becomes elastic. Similar reasoning shows that a reduction in price reduces total revenue if demand is inelastic but increases total revenue if demand is elastic, since the increase in volume more than offsets the reduction in price. Finally, an increase in price reduces total revenue if demand is elastic and does not change total revenue if demand has unit elasticity.

Relation to marginal revenue. There is also a logical relationship between the elasticity of demand and marginal revenue. Marginal revenue is defined as the change in the total revenue per unit change in quantity sold, if the relationship between the price and the quantity sold is given by the demand curve. Positive marginal revenue therefore implies that total revenue increases if the quantity sold increases, or declines if the quantity sold declines. Since the relation between price and quantity is given by the demand curve, an increase in sales volume must be accompanied by a decline in price. Thus, a positive marginal revenue means that total revenue increases as volume increases and price declines. Since a decline in price increases total revenue only if demand is elastic, positive marginal revenue must be associated with elastic demand. Furthermore, since profit maximization requires that marginal cost equal marginal revenue, and marginal cost is always positive, it follows that the marginal revenue of the profit-maximizing firm must be positive and the demand elastic.

Similar reasoning shows that negative marginal revenue is associated with inelastic demand: if demand is inelastic, an increase in price reduces volume and increases total revenue, so that marginal revenue is negative. The intermediate case of unit elasticity of demand implies zero marginal revenue, since any change in price is just balanced by a change in volume, and total revenue is the same at any price or at any volume.

Relation to price. Elasticity of demand usually changes with the price of the product. Thus, it is incorrect to say that the demand for cigarettes is inelastic or that the demand for butter is elastic. The correct statement is that these demands are elastic or inelastic at the prices that are currently charged. At higher prices the demand for cigarettes might well become elastic, while the demand for butter might well become inelastic if its price fell far enough below the price of oleomargarine.

In the particular case of the straight-line demand curve, which is so frequently used for illustrative purposes in economics, the demand is elastic and marginal revenue is positive throughout the upper half of the possible price range, while demand is inelastic and marginal revenue negative throughout the lower half of the price range.

Elasticity of supply. Defined as the percentage change in the quantity offered for sale divided by the percentage change in price, elasticity of supply is derived from the supply curve, which shows the absolute amount producers wish to sell as a function of price. Since sellers wish to sell more at a higher price, the elasticity of supply is positive. If the supply curve is inelastic, a relatively larger increase in price is required to produce a given (percentage) increase in quantity supplied. If the supply curve is elastic, the sellers are more responsive to a change in price, and the increase in the quantity supplied is larger, on a percentage basis, than the increase in price.

Other elasticities. Marshall originally introduced the concept of elasticity in connection with demand and supply curves, but economists soon started to use the ratio of the percentage changes to describe the relationship between any two variables, provided only that the variables are functionally related. By 1913, Johnson (p. 503) was able to write, “This form of expression [the ratio of percentage changes] corresponds to the general notion of elasticity.”

The most important elasticities, other than demand and supply, are income elasticity of demand and cross-elasticity of demand.

Income elasticity of demand. The demand for a commodity depends upon its price, consumers’ incomes, the prices of other commodities on which consumers might spend their incomes, and perhaps on other variables. Quantity demanded is therefore functionally related to price, if incomes and other prices do not change; and the (price) elasticity of demand is a property of this function. Quantity demanded is also functionally related to income, if the price of the commodity and other prices do not change. The income elasticity of demand is a property of this function and is defined as the percentage change in the quantity demanded divided by the percentage change in income.

If income elasticity of demand is negative, consumers purchase less of the commodity as their incomes rise, and the commodity is, by definition, an inferior good. If income elasticity is positive, but less than one, consumers buy more of the product as their incomes rise (if prices do not change), but spend a smaller proportion of their incomes on the commodity. For commodities with income elasticities greater than one, consumers spend a higher proportion of a higher income on the commodity, and the quantity demanded grows at a faster (percentage) rate than income grows.

Cross-elasticity of demand. Quantity demanded is also functionally related to the prices of other commodities, if the commodity’s own price and consumers’ incomes do not change. Cross-elasticity of demand is a property of this function and is defined as the percentage change in the quantity demand of commodity A divided by the percentage change in the price of commodity B.

The commodities A and B are, by definition, sub stitutes if the cross-elasticity is positive, since an increase in the price of B then causes a reduction in the demand for B and an increase in the demand for A. Conversely, if cross-elasticity is negative, the commodities are complements, since an increase in the price of B causes consumers to demand less of both commodities.

Mathematical definitions. Point elasticity. If the variables q, p, and y are related by the function q = f (p, y ), the p elasticity of q, or the elasticity of q with respect to p, is customarily defined as ?_qp = (?q/?p)(P/q), no matter what variables the letters represent.

If, as is customary, q represents the quantity demanded, p the price, and y the consumers’ incomes, then ?_qp is the (price) elasticity of demand and is negative. Marginal revenue, M, is defined as M = ?(pq)/?q, and it can be shown that M = p( l + l/?_qp).

Arc elasticity. Sometimes the functional relationship between p and q is unknown, but two points, B and D, with coordinates (q₁,p₁) and (q₂, p₂) as in Figure 1, are known; and the arc elasticity between the two points is desired. The usual practice is to assume that the demand curve passing through these points is a straight line, HBIDJ, and to calculate the point elasticity at I, the point midway between B and D. The formula for the arc elasticity, so defined, is

While this procedure is arbitrary, it has two advantages over calculating the elasticity at either B or D. First, the elasticity has the same value whether the change is from B to D or from D to B. More important, however, is the fact that the relationships between elasticity, total revenue, and marginal revenue, which were discussed above, are valid if elasticity is calculated at I, but not if it is calculated at either B or D.

Dlran Bodenhorn

[See alsoDemand and supply; and the biographies ofCournot; Marshall; Mill.]

BIBLIOGRAPHY

Cournot, Antoine Augustin (1838) 1960 Researches Into the Mathematical Principles of the Theory of Wealth. New York: Kelley. ? First published in French.

Johnson, W. E. 1913 The Pure Theory of Utility Curves.Economic Journal 23:483–513.

Marshall, Alfred 1885 On the Graphic Method of Statistics. Pages 251–260 in Royal Statistical Society, London, Jubilee Volume of the Statistical Society. London: Stanford.

Marshall, Alfred (1890) 1961 Principles of Economics. 9th ed., 2 vols. New York and London: Macmillan. ? A variorum edition. The eighth edition is more convenient for normal use.

Mill, John Stuart (1848) 1961 Principles of Political Economy, With Some of Their Applications to Social Philosophy. 7th ed. Edited by W. J. Ashley. New York: Kelley.