## Elasticities of Supply and Demand

We have seen that the demand for a good depends on its price, as well as on consumer income and on the prices of other goods. Similarly, supply depends on price, as well as on variables that affect production cost. For example, if the price of coffee increases, the quantity demanded will fall, and the quantity supplied will rise. Often, however, we want to know how much supply or demand will rise or fall. How sensitive is the demand for coffee to its price? If price increases by 10 percent, how much will demand change? How much will demand change if income rises by 5 percent? We use elasticities to answer questions like these.

An elasticity is a measure of the sensitivity of one variable to another. Specifically, it is a number that tells us the percentage change that will occur in one variable in response to a 1 percent change in another variable. For example, the price elasticity of demand measures the sensitivity of quantity demanded to price changes. It tells us what the percentage change in the quantity demanded for a good will be following a 1 percent increase in the price of that good.

Lets look at this in more detail. Denoting quantity and price by Q and P, we write the price elasticity of demand as where %AQ simply means "percentage change in Q" and %AP means "percentage change in P."4 But the percentage change in a variable is just the absolute change in the variable divided by the original level of the variable. (If the Consumer Price Index were 200 al the beginning of the year and increased to 204 by the end of the year, the percentage change-or annual rate of inflation-would be 4/200 = .02, or 2 percent.) So we can also write the price elasticity of demand as5

The price elasticity of demand is usually a negative number. When the price of a good increases, the quantity demanded usually falls, so AQ/AP (the change in quantity for a change in price) is negative, and therefore EP is negative.

When the price elasticity is greater than 1 in magnitude, we say that demand is price elastic because the percentage decline in quantity demanded is greater than the percentage increase in price. If the price elasticity is less than 1 in magnitude, demand is said to be price inelastic. In general, the elasticity of demand for a good depends on the availability of other goods that can be substituted for it. When there are close substitutes, a price increase will cause the consumer to buy less of the good and more of the substitute. Demand will then be highly price elastic. When there are no close substitutes, demand will tend to be price inelastic.

Equation (2.1) says that the price elasticity of demand is the change in quantity associated with a change in price (AQ/AP) times the ratio of price to quantity (P/Q). But as we move down the demand curve, AQ/AP may change, and the price and quantity will always change. Therefore, the price elasticity of demand must be measured at a particular point on the demand curve and will generally change as we move along the curve.

This is easiest to see for a linear demand curve, that is, a demand curve of the form

4 The symbolA is the Greek capital letter delta; it means "the change in," SoAX means "the change in the variable X," say, from one year to the next.

In terms of infinitesimal changes (letting theAP become very small), E = (P/Q)(dQ/dP).

As an example, consider the demand curve

For this curve, AQ/AP is constant and equal to -2 (a APof 1 results in a AQ of -2). However, the curve does not have a constant elasticity. Observe from Figure 2.10 that as we move down the curve, the ratio P/Q falls, and therefore the elasticity decreases in magnitude. Near the intersection of the curve with the price axis, Q is very small, so Ep = -2(P/Q) is large in magnitude. When P = 2 and Q = 4, Ep = -1. And at the intersection with the quantity axis, P = 0 so Ep — 0.

Because we draw demand (and supply) curves with price on the vertical axis and quantity on the horizontal axis, AO/AP = (I/slope of curve). As a result, tor any price and quantity combination, the steeper the slope of the curve, the less elastic demand is. Figures 2.1 la and b show two special cases. Figure 2.11a shows a demand curve that is infinitely elastic. There is only a single price P* at which consumers will buy the good; for even the smallest increase in price above this level, quantity demanded drops to zero, and for any decrease in price, quantity demanded increases without limit. The demand curve in Figure 2.11b, on the other hand, is completely inelastic. Consumers will buy a fixed quantity Q*, no matter what the price.

We will also be interested in elasticities of demand with respect to other variables besides price. For example, demand for most goods usually rises when FIGURE 2.10 Linear Demand Curve. The price elasticity of demand depends not only on the slope of the demand curve, but also on the price and quantity. The elasticity therefore varies along the curve as price and quantity change. Slope is constant for this linear demand curve. Near the top, price is high and quantity is small, so the elasticity is large in magnitude. The elasticity becomes smaller as we move down the curve.

Price

Price

Price Q* Quantity

Quantity

Q* Quantity

FIGURE 2,11a Infinitely Elastic Demand. For a horizontal demand curve, AQ/AP is infinite. (A tiny change in price leads to an enormous change in demand.) The elasticity of demand is therefore infinite.

FIGURE 2.11b Completely Inelastic Demand. For a vertical demand curve, AQ/AP is zero. The quantity demanded is the same no matter what the price, so the elasticity of demand is zero.

aggregate income rises. The income elasticity of demand is the percentage change in the quantity demanded Q resultingfrom a 1 percent increase in income /:

The demand for some goods is also affected by the prices of other goods. For example, because butter and margarine can easily be substituted for each other, the demand for each depends on the price of the other. A cross-price elasticity of demand refers to the percentage change in the quantity demanded for a good that results from a 1 percent increase in the price of another good. So the elasticity of demand for butter with respect to the price of margarine would be written as r AQfc/Qfr p,„ A Qb ,„v bPm ~ &PJPm " Qb APm <Zd)

where Qh is the quantity of butter and Pm is the price of margarine.

In this example of butter and margarine, the cross-price elasticities will be positive because the goods are substitutes-they compete in the market, so a rise in the price of margarine, which makes butter cheaper relative to margarine than it was before, leads to an increase in the demand for butter. (The demand curve for butter will shift to the right, so its price will rise.) But this is not always the case. Some goods are complements; they tend to be used to-

gether, so that an increase in the price of one tends to push down the consumption of the other. Gasoline and motor oil are an example. If the price of gasoline goes up, the quantity of gasoline demanded falls-motorists will drive less. But the demand for motor oil also falls. (The entire demand curve for motor oil shifts to the left.) Thus, the cross-price elasticity of motor oil with respect to gasoline is negative.

Elasticities of supply are defined in a similar manner. The price elasticity of supply is the percentage change in the quantity supplied resulting from a 1 percent increase in price. This elasticity is usually positive because a higher price gives producers an incentive to increase output.

We can also refer to elasticities of supply with respect to such variables as interest rates, wage rates, and the prices of raw materials and other intermediate goods used to manufacture the product in question. For example, for most manufactured goods, the elasticities of supply with respect to the prices of raw materials are negative. An increase in the price of a raw material inpu means higher costs for the firm, so other things being equal, the quantity supplied will fall. 