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falls to 0.10 Using equation (2.3), we see that the price elasticity of demand for the linear demand curve in Figure 2.16 is given by the formula (2.4) This formula tells us that for a linear demand curve, the price elasticity of demand varies as we move along the curve. Between the choke price (where Q \ufffd 0) and a price of at the midpoint M of the demand curve, the price elasticity of demand is between \ufffdq and \ufffd1. This is known as the elastic region of the demand curve. For prices between and 0, the price elasticity of demand is between \ufffd1 and 0. This is the inelastic region of the demand curve. a\ufffd2b a\ufffd2b a\ufffdb \ufffdQ, P \ufffd ¢Q ¢P P Q \ufffd \ufffdb P Q a\ufffdb P \ufffd a b \ufffd 1 b Q choke price The price at which quantity demanded falls to 0. 10You can verify that quantity demanded falls to 0 at the choke price by substituting P \ufffd into the equation of the demand curve: \ufffd 0 \ufffd a \ufffd a Q \ufffd a \ufffd b aa b b a\ufffdb Pr ic e (do lla rs pe r u nit ) 0 \u3b5Q,P = \u2013\u221e \u3b5Q,P = \u20131 \u3b5Q,P = 0 Q = a \u2013 bP P = \u2013 or a b a a 2 a b Q b a 2b Quantity (units per year) M D Elastic region \u3b5 Q ,P between \u2013 \u221e and \u20131 Inelastic region \u3b5 Q ,P between \u20131 and 0 FIGURE 2.16 Price Elasticity of Demand along a Linear Demand Curve In the region to the northwest of the mid- point M, demand is elastic, with the price elasticity of demand between minus infinity and \ufffd1. In the region to the southeast of the midpoint M, demand is inelastic, with the price elasticity of demand between \ufffd1 and 0. c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 46 2.2 PRICE ELASTICITY OF DEMAND 47 Equation (2.4) highlights the difference between the slope of the demand curve, \ufffdb, and the price elasticity of demand, . The slope measures the absolute change in quantity demanded (in units of quantity) brought about by a one-unit change in price. By contrast, the price elasticity of demand measures the percentage change in quantity demanded brought about by a 1 percent change in price. You might wonder why we do not simply use the slope to measure the sensitivity of quantity to price. The problem is that the slope of a demand curve depends on the units used to measure price and quantity. Thus, comparisons of slope across different goods (whose quantity units would differ) or across different countries (where prices are measured in different currency units) would not be very meaningful. By contrast, the price elasticity of demand expresses changes in prices and quantities in common terms (i.e., percentages). This allows us to compare the sensitivity of quantity demanded to price across different goods or different countries. Constant Elasticity Demand Curves Another commonly used demand curve is the constant elasticity demand curve, given by the general formula: Q \ufffd aP\ufffdb, where a and b are positive constants. For the constant elasticity demand curve, the price elasticity is always equal to the exponent \ufffdb.11 For this reason, economists frequently use the constant elasticity demand curve to estimate price elasticities of demand using statistical techniques. \ufffdb(P\ufffdQ) total revenue Selling price times the quantity of product sold. constant elasticity demand curve A demand curve of the form Q \ufffd aP\ufffdb where a and b are positive constants. The term \ufffdb is the price elas- ticity of demand along this curve. 11We prove this result in the appendix to this chapter. Problem (a) Suppose a constant elasticity demand curve is given by the formula . What is the price elasticity of demand? (b) Suppose a linear demand curve is given by the formula . What is the price elasticity of demand at P \ufffd 30? At P \ufffd 10? Solution (a) Since this is a constant elasticity demand curve, the price elasticity of demand is equal to everywhere along the demand curve. (b) For this linear demand curve, we can find the price elasticity of demand by using equation (2.4): \ufffd1\ufffd2 Q \ufffd 400 \ufffd 10P Q \ufffd 200P\ufffd 1 2 Elasticities along Special Demand Curves Since b \ufffd \ufffd10 and Q \ufffd 400 \ufffd 10P, when P \ufffd 30, and when P \ufffd 10, Note that demand is elastic at P \ufffd 30, but it is inelastic at P \ufffd 10 (in other words, P \ufffd 30 is in the elastic region of the demand curve, while P \ufffd 10 is in the inelastic region). Similar Problems: 2.5, 2.6, 2.13 \ufffdQ, P \ufffd \ufffd10 a 10400 \ufffd 10(10)b \ufffd \ufffd0.33 \ufffdQ,P \ufffd \ufffd10 a 30400 \ufffd 10(30)b \ufffd \ufffd3 \ufffdQ, P \ufffd (\ufffdb)(P\ufffdQ) L E A R N I N G - B Y- D O I N G E X E R C I S E 2 . 6 E S D PRICE ELASTICITY OF DEMAND AND TOTAL REVENUE Businesses, management consultants, and government bodies use price elasticities of demand a lot. To see why a business might care about the price elasticity of demand, let\u2019s consider how an increase in price might affect a business\u2019s total revenue, that is, the selling price times the quantity of product it sells, or PQ. You might think that c02demandandsupplyanalysis.qxd 7/14/10 11:10 AM Page 47 48 CHAPTER 2 DEMAND AND SUPPLY ANALYSIS when the price rises, so will the total revenue, but a higher price will generally reduce the quantity demanded. Thus, the \u201cbenefit\u201d of the higher price is offset by the \u201ccost\u201d due to the reduction in quantity, and businesses must generally consider this trade-off when they think about raising a price. If the demand is elastic (the quantity demanded is relatively sensitive to price), the quantity reduction will outweigh the benefit of the higher price, and total revenue will fall. If the demand is inelastic (the quantity demanded is relatively insensitive to price), the quantity reduction will not be too severe, and total revenue will go up. Thus, knowledge of the price elasticity of demand can help a business predict the revenue impact of a price increase. DETERMINANTS OF THE PRICE ELASTICITY OF DEMAND Price elasticities of demand have been estimated for many products using statistical tech- niques. Table 2.1 presents these estimates for a variety of food, liquor, and tobacco products in the United States, while Table 2.2 presents estimates for various modes of transporta- tion. What determines these elasticities? Consider the estimated elasticity of \ufffd0.107 for cigarettes in Table 2.1, which indicates that a 10 percent increase in the price of cigarettes would result in a 1.07 percent drop in the quantity of cigarettes demanded. This tells us that cigarettes have an inelastic demand: When the prices of all the individual brands of cigarettes go up (perhaps because of an increase in cigarette taxes), overall consumption of cigarettes is not likely to be affected very much. This conclusion makes sense. Even though consumers might want to cut back their consumption when cigarettes become more expensive, most would find it difficult to do so because cigarettes are habit forming. In many circumstances, decision makers do not have precise numerical estimates of price elasticities of demand based on statistical techniques. Consequently, they have to rely on their knowledge of the product and the nature of the market to make edu- cated conjectures about price sensitivity. TABLE 2.1 Estimates of the Price Elasticity of Demand for Selected Food, Tobacco, and Liquor Products Product Estimated Q,P Cigars 0.756 Canned and cured seafood 0.736 Fresh and frozen fish 0.695 Cheese 0.595 Ice cream 0.349 Beer and malt beverages 0.283 Bread and bakery products 0.220 Wine and brandy 0.198 Cookies and crackers 0.188 Roasted coffee 0.120 Cigarettes 0.107 Chewing tobacco 0.105 Pet food 0.061 Breakfast cereal 0.031 Source: Emilio Pagoulatos and Robert Sorensen, \u201cWhat Determines the Elasticity of Industry Demand,\u201d International Journal of Industrial Organization, 4 (1986): 237\u2013250. c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 48 TABLE 2.2 Estimates of the Price Elasticity of Demand for Selected Modes