ELASTICITIES

The Price Elasticity of Demand

One of the more useful concepts of economics is price elasticity of demand, which measures the responsiveness of the quantity demanded to a change in the price of the good. There are several different ways to measure the price elasticity of demand, but the most precise is simply the inverse of the slope of the demand curve times the quantity demanded divided by the price, all at a particular point of the curve. That is, the price elasticity of demand, denoted here as e_p, is given as e_p = (1/slope)(P/Q). Since the demand curve slopes downward (the slope is negative), this characterization of the price elasticity of demand has a negative sign. Recall also that the slope of a demand curve is defined as the change in price divided by the change in quantity demanded. If we let D indicate change, then e_p=(DQ/DP)x(P/Q).  This, and all formulas, can be written with calculus, just replacing the D with d.  So e_p=(DQ/DP)x(P/Q) becomes e_p=(dQ/dP)x(P/Q) where  dQ/dP is just the derivative of quantity demanded with respect to price -- that is, the derivative of the demand curve.  Rearranging the first specification, we find that e_p=(DQ/Q)/(DP/P). Each term in the parentheses can be interpreted as a percentage change, so in fact the price elasticity of demand is simply the percentage change in quantity divided by the percentage change in price (but keep in mind the negative slope, so price and quantity move in opposite directions).

If the demand curve is linear, the slope is constant, and the price elasticity of demand changes only because P and Q change as we move along the curve. But, as is often the case, the demand curve is nonlinear, the slope changes as we move to different points on the curve. Recall at any point the slope can be measured as the slope of the tangent line to the curve at that point. As a rough measure, however, economists use the arc elasticity of demand measure of elasticity, defined as

e_p = [(Q₂-Q₁)/(Q₂+Q₁)]/[(P₂-P₁)/(P₂+P₁)]

where the subscript 1 indicate initial price and quantity and subscript 2 indicate the new price and quantity. This formulation corrects for the span in price and quantity over the curve, but has the same interpretation.

When e_p=-1, the price elasticity of demand is said to be unit or unitary elastic, and it is a demarcation between the two primary ranges of elasticity. If e_p<-1 (that is, exceeds one in magnitude), demand is said to be elastic, and if 0>e_p>-1 (less than one in magnitude), demand is said to be inelastic.

These demarcations are important, because they indicate what happens to total expenditures if prices change. The total amount spent on a good is just price times quantity. If demand is price elastic, then if price changes, the percentage change in quantity exceeds the percentage change in price. Since price and quantity move in opposite directions on the demand curve, a price increase lowers total expenditure on the good, but a price decrease increases total expenditures. Similarly, since if demand is price inelastic the percentage change in price exceeds the percentage change in quantity, a price increase raises total expenditure, while a price decrease lowers it. If demand is unitary elastic, any change in price leaves total expenditures unchanged.

As an application, suppose the demand for a good is such that e_p=-2. Then a 10% increase in price will cause a 20% decrease in quantity. Total revenue will thus go down. Of more interest is when demand is inelastic. Suppose e_p=.9. Then a 10% increase in price causes only a 9% decrease in quantity, and since the price for all units increases, total expenditure goes up. To see this, let the base be 100 units selling for $10 per unit. Total expenditure would be $1000. But if e_p=.9, a 10% increase in price, to $11 per unit, results in the quantity sold (demanded) falling to 91 units. Total expenditure is now 91x$11 or $1001. Since fewer units are produced (so costs are probably lower) but more revenue is collected, the sellers should make higher profits. So anytime a firm is in an inelastic area of its demand curve, the firm can increase its profits by raising price. Eventually, of course, the firm moves into an elastic area of its demand, raising price lowers both costs and revenues, and what happens to profits then is questionable.

The price elasticity of demand has some important information for policy analysis and implementation. Price elasticities also help businesses in making pricing decisions. Airline companies, for example, can classify customers into two general groups; business customers, who have relatively inelastic demand, and pleasure travelers, with demand that is more elastic.

Figure 2.13 shows each type of flyer's demand curve, business travelers in panel A, and pleasure travelers in panel B. If we start from a base of (P₁,Q₁) for each group, and then raise price the same percent, you can see that total expenditures by business travelers increases, but that for pleasure travelers decreases. We can see that the firms do better by charging business travelers more. We see one reason that full fare tickets, used primarily by business travelers are more susceptible to fare increases than restricted fare tickets, which are used more by pleasure travelers.

Many factors influence the price elasticity of demand. One of the primary factors is the availability of substitutes for the good. The fewer substitutes there are for the good, the more likely it is that demand will be inelastic. Because there are few substitutes for milk, it tends to have a relatively steep (inelastic) demand curve. Gasoline is another example of inelastic demand for shorter time horizons. The only substitutes are other modes of transportation (walking, mass transit) or driving less, not particularly good substitutes. But over time, conservation is also a substitute for gasoline, by more fuel efficient automobiles, and thus the longer range elasticity of demand for gasoline is more elastic than the shorter range. Other factors that affect the price elasticity of demand are how important the good is in the consumers budget (larger share items are more elastic), and whether the good is a luxury (which is relatively elastic) or a necessity (relatively inelastic).

Other Demand Elasticities: Income and Cross Price

Two other demand elasticities of interest are the income elasticity of demand and the cross price elasticity of demand. The income elasticity measures the response of quantity demanded to changes in income, and the cross price elasticity of demand measures the response of quantity demanded to a change in the price of another good. While the price elasticity of demand is based on a movement along the demand curve, these measure the response of the quantity demanded to changes in demand curve shifters. Although they give some indication about how the demand curve shifts with a change in income or the price of another good, the result is dependent on the price of the good in question, and therefore we must be careful not to generalize the result as a measure of the shift in the entire demand curve.

The definitions of these elasticities follow that for the price elasticity of demand. Income elasticity is the percentage change in quantity demanded that results from a percentage change in income. Again let indicate change, so e_I=(DQ/DI)x(I/Q), [ in calculus, e_I=(dQ/dI)x(I/Q)] or rearranging, e_I=(DQ/Q)/(DI/I). If Q and I move in the same direction, so an increase in income causes the quantity demanded to go up, the good is a normal good. If Q and I move in opposite directions, the good is an inferior good. Normal goods that have a more than proportionate change in quantity demanded are called luxury goods. So if income increases by 10%, but the quantity demanded of steak increases by more than 10%, steak would be classified as a luxury good.

Cross price elasticity is defined as the percentage change in the quantity demanded for good A with respect to a percentage change in the price of good B, in notational form e_x=(DQ^A/DP^B)x(P^B/Q^A) [in calculus, e_x=(dQ^A/dP^B)x(P^B/Q^A)], or rearranging, e_x=(DQ^A/Q^A)/(DP^B/P^B). If the two variables move in opposite directions, so an increase in the price of B decreases the quantity demanded of A, the goods are complements. Hot dogs and buns might be such goods, or gasoline and automobile tires. If the variables move in same directions, as the quantity demanded for Coke might do if Pepsi changes its price, the goods are substitutes. Substitutes have positive cross price elasticities, while complements have negative cross price elasticities. The larger the magnitude, the stronger the effect. Coke and Pepsi are very good substitutes, so the cross price elasticity is probably very large and positive. Automobile tire usage depends strongly on the price of gasoline, so in this case the cross price elasticity is large in magnitude but negative.

Income and cross price elasticities tell a lot about pricing for joint goods or, as in the airline example, pricing towards different groups. Together with the price elasticity of demand they provide useful summary information about the demand for a good. A policy analyst can use elasticity measurements to choose between alternative interventions in a market, and to tailor strategies for political gains or improvements in the efficiency of the market. In the next chapter we explore some examples of demand elasticities.

Price Elasticity of Supply

So far we have concentrated on the demand side of the market when looking at elasticities. Analogous to the price elasticity of demand is the price elasticity of supply. The price elasticity of supply is the percentage change in quantity supplied divided by the percentage change in price. Formulas for point and arc measures are the same as those for the price elasticity of demand, except quantity supplied rather than quantity demanded is used. That is, e_s=(DQ^s/DP)x(P/Q^s), [in calculus, e_s=(dQ^s/dP)x(P/Q^s)] or rearranging, e_s=(DQ^s/Q^s)/(DP/P), and the arc formula is e_s = [(Q₂-Q₁)/(Q₂+Q₁)]/[(P₂-P₁)/(P₂+P₁)] where here the Qs are quantities supplied.

Just like the other elasticities, the price elasticity of supply changes from point to point on the supply curve. Since the supply curve is upward sloping, price and quantity will move in the same direction, and the price elasticity of supply has a positive sign. Although not as interesting as the demand elasticities, the price elasticity of supply does provide some useful insights. If we think about the conflict on strategic minerals, discussed above, the price elasticity of supply could be used to summarize what price increase would be needed to bring forth additional supplies. For example, think of the strategic mineral niobium. A 100% increase in price (from $1.60 per pound to $3.20) elicited a five fold increase (500%) increase in quantity supplied, from 1 million pounds to 5 million pounds per year. Thus, the price elasticity of supply measured at approximately 5.