Daniel l. Rubinfeld

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iJ 118 Part 2 Producers, Consumers, and Competitive Markets vVhen demand is inelastic (i.e., Ep is less than 1 in magnitude), the quantity demanded is relatively unresponsive to changes in price. As a result, total expenditure on the product increases 'when the price increases. Suppose, for example, that a family currently uses 1000 gallons of gasoline a year 'when the price is 51 per gallon; suppose also that our family's price elasticity of demand for gasoline is -05. If the price of gasoline increases to $1.10 (a 10-percent increase), the consumption of gasoline falls to 950 gallons (a 5-percent decrease). Total expenditure on gasoline, however, will increase from 51000 (1000 gallons X Sl per gallon) to 51045 (950 gallons x 51.10 per gallon). If PRICE INCREASES, If PRICE DECREASES, DEMAND EXPENDITURES EXPENDITURES Inelastic Increase Decrease Unit elastic Are unchanged Are unchanged Elastic Decrease Increase Chapter 4 Individual and Market Demand 9 isoelastic demand curve Demand curve with a constant price elasticity. In §2.3, we show that when the demand curve is linear, demand becomes more elastic as the price of the product increases. In contrast, when demand is elastic (Ep is greater than 1 in magnitude), total expenditure on the product decreases as the price goes up. Suppose that a family buys 100 pounds of chicken per year, at a price of S2 per pound; the price elasticity of demand for chicken is -1.5. If the price of chicken increases to 52.20 (a 10-percent increase), our family's consumption of chicken falls to 85 pounds a year (a 15-percent decrease). Total expenditure on chicken will also fall, from S200 (100 pounds x $2 per pound) to 5187 (85 pounds x 52.20 per pound). When the price elasticity of demand is constant all along the demand curve, we say that the curve is isoelastic. Figure 4.11 shows an isoelastic demand curve. Note how this demand curve is bmved inward. In contrast, recall from Section 2.3 what happens to the price elasticity of demand as we move along a li11ear demand curve. Although the slope of the linear curve is constant, the price elasticity of demand is not. It is zero when the price is zero, and it increases in magnitude until it becomes infinite when the price is sufficiently high for the quantity demanded to become zero. Price of Movie Tickets 9 - - .. (5) A special case of this isoelastic curve is the unit-elastic dema11d curve: a demand curve with price elasticity always equal to -1, as is the case for the curve in Figure 4.11. In this case, total expenditure remains the same after a price change. A price increase, for instance, leads to a decrease in the quantity demanded that leaves the total expenditure on the good unchanged. Suppose, for example, that the total expenditure on first-I'm) movies in Berkeley, California, is $5.4 million per year, regardless of the price of a movie ticket. For all points along the demand curve, the price times the quantity will be $5.4 million. If the price is $6, the quantity will be 900,000 tickets; if the price increases to $9, the quantity will drop to 600,000 tickets, as shown in Figure 4.11. Table 4.3 summarizes the relationship between elasticity and expenditure. It is useful to review this table from the perspective of the seller of the good rather than the buyer. (VVhat the sellers perceive as total revenue, the consumers view as total expenditures.) VVhen demand is inelastic, a price increase leads only to a small decrease in quantity demanded; thus, the seller's total revenue increases. But when demand is elastic, a price increase leads to a large decline in quantity demanded and total revenue falls. When calculating demand elasticities, we must be careful about the price change or quantity change in question. For a large price change (say, 20 percent), the value of the elasticity will depend on the precise point at which we measure the price and quantity along the demand curve. For this reason, it is useful to distinguish between a point elasticity of demand and an arc elasticity of demand. 6 ----1--, I I I I 3 ----:--1------ 600 900 1,800 Thousands of Movie Tickets elasticity of demand is -1.0 at every price, the total expenditure is the demand curve D. o The point elasticity of demand is defined as the price elasticity at a particular point all the dema11d curve. Note that this is the concept of elasticity that we used throughout Chapter 2. It is calculated by substituting for tlP/tlQ in the elasticity formula the magnitude of the slope of the dema11d curve at that poi1lt. (tlP/tlQ is the slope for small tlP because price is measured on the vertical axis and quantity demanded on the horizontal axis.) As a result, equation (4.1) becomes Point elasticity: Ep = (P/Q)(1/s10pe) (4.2) There are times when we want to calculate a price elasticity over some portion of the demand curve rather than at a single point. Suppose, for example, that vve are contemplating an increase in the price of a product from $8 to $10 and expect the quantity demanded to fall from 6 units to 4. How should we cal ~ulate the price elasticity of demand Is the price increase 25 percent (a $2 l11crease divided by the original price of $8), or is it 20 percent (a $2 increase diVided by the new price of $10) Is the percentage decrease in quantity demanded 33 1/3 percent (2/6) or 50 percent (2/4) point elasticity of demand Price elasticity at a particular point on the demand curve.
120 Part 2 Producers, Consumers, and Competitive Markets arc elasticity of demand Price elasticity calculated over a range of prices. There is no correct answer to such questions. We could calculate the price elasticity using the original price and quantity. If so, we 'would find that Ep = (-331/3 percent/25 percent) = -L33. Or \eve could use the new price and quantity, in which case \'\'e would find that Ep = (-50 percent/ 20 percent) = -25. The difference between these two calculated elasticities is large, and neither seems preferable to the other. We can resolve this problem by using the arc elasticity of demand: the elnsticity cnlcllinted over n mnge of prices. Rather than choose either the initial or the final price, we use an average of the two, P; for the quantity demanded, we use Q. Thus the arc elasticity of demand is given by - "'M J& k!§i§ iBM 16 15 H 13 .\......_______---, Total Demand 12 11 Individual and Market Demand Arc elasticity: Ep (:lQ/:::'P)(P/Q) (4.3) In our example, the average price is $9 and the average quantity 5 units. Thus the arc elasticity is Ep (-2/$2)($9/5) -L8 The arc elasticity will always lie somewhere (but not necessarily halfway) between the point elasticities calculated at the lower and the higher prices. Although the arc elasticity of demand is sometimes useful, economists generally use the word "elasticity" to refer to a point elasticity Throughout the rest of this book, we will do the same, unless noted otherwise. 4 3 Export Demand 2 D B O~--------~L_ __________ L-__________ L_~ ______ ~ 1000 2000 3000 4000 \"lheat (million bushels per year) The total world demand for wheat is the horizontal sum of the domestic demand AB and the export demand CD. Even though each individual demand curve is linear, the market demand curve is kinked, reflecting the fact that there is no export demand when the of wheat is than about $9 bushel. I n Chapter 2 (Example 2.4), we explained that the demand for US wheat has two components: domestic demand (by US consumers) and export demand (by foreign consumers). Let's see how the total demand for 'wheat during 1998 can be obtained by aggregating the domestic and foreign demands. Domestic demand for wheat is given by the equation Qoo = 1700 - 107P an elasticity of - OA Export demand is more elastic than domestic demand because poorer cOlmh'ies that import US wheat tum to other grains and foodstuffs if wheat prices rise. 3 To obtain the world demand for wheat, we set the left side of each demand equation equal to the quantity of wheat (the variable on the horizontal axis). We then add the right side of the equations, obtaining where Qoo is the number of bushels (in millions) demanded domestically, and P is the price in dollars per busheL Export demand is given by Qoo + QOE = (1700 107P) + (1544 - 176P) = 3244 - 283P QOE = 1544 176P (Note that this is the same total demand equation for 1998 given in Example 2.4.) This generates the line segment EF in Figure 4.12. where QOE is the number of bushels (in millions) demanded from abroad As shown in Figure 4.12, domestic demand, given by AB, is relatively price inelastic (Statistical studies have shown that price elasticity of domestic demand is about - 0.2.) However~ export demand, given by CO, is more price elastic, with 3 ,For a survey of statistical studies of demand and supply elasticities and an analysis of the US wheat market, see Larry Salathe and Sudchada Langley, "An Empirical Analysis of Altemativ~ Export Subsidy Programs for US Wheat," AgriclIltllral Ecollomics Research 38, No.1 (Winter 1986)
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Daniel l. Rubinfeld

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