Elasticity Price elasticity of demand is a measure used in economics to show the responsiveness, or elasticity, of the quantity demanded of a good or service to a change in its price. Due to the inverse relationship between price and quantity demanded, economists often refer to price elasticity of demand as a positive value (i.e., in absolute value terms). Q2 − Q1 Q1 %ΔQ = 1. Point Elasticity Formula: ε p = P2 − P1 %ΔP P1 If we know the equation of the linear demand curve, the point elasticity formula is given by εP = %ΔQ 1 P = × %ΔP slope Q 2. Arc Elasticity Formula (= Mid-Point method): Q2 − Q1 ⎛ Q1 + Q2 ⎞ ⎜ ⎟ %ΔQ 2 ⎠ εp = = ⎝ P2 − P1 %ΔP ⎛ P1 + P2 ⎞ ⎜ ⎟ ⎝ 2 ⎠ Exercise 1: Consider the market for bicycles. The demand is given by P= -1.5Qd+ 60. a) Suppose that the price changes from $15 to $30. Using the standard percentage change formula (Point method), what is the price elasticity of demand? Using the demand equation, we can get Qd =30 when P=$15, and Qd =20 when P=$30, so ED = |(20-30)/30 / (30-15)/15| = 1/3 b) Suppose that the price changes from $15 to $30. Using the midpoint method, what is the price elasticity of demand? Arc-elasticity = |(20-30)/(20+30)/(30-15)/(30+15)| = 3/5 c) When P=15, what is the price elasticity of demand using the point method? Is it elastic or inelastic? At P=15, if the price goes up, does the total revenue increase, decrease, or stay the same? We can use this formula ED = (1/|slope|)x(P/Q) Using the demand equation, we can get Qd =30, when P=$15. ED = (1/|-1.5|) x (15/30) = 1/3 So ED = 1/3 < 1. Then the demand is inelastic. Therefore, when the price goes up, the total revenue increases. 1 d) When Qd =10, what is the price elasticity of demand using the point method? Is it elastic or inelastic? At Qd =10, if the price goes up, does the total revenue increase, decrease, or stay the same? Using the demand equation, we can get P=$45, when Qd =10. ED = (1/|-1.5|) x (45/10) = 1/3 So ED = 3 > 1. Then the demand is elastic. When the price goes up, the total revenue decreases. e) At what price is the price elasticity of demand, using the point method, equal to 1? The unit elastic point occurs at the midpoint of the demand curve if the demand curve is linear and intersects both the y-axis and x-axis. When we draw the demand curve, the y-intercept is 60, and the x-intercept is 40. Therefore, the midpoint of the demand curve is (20, 30), which is the unit-elastic point. The answer is P=$30. f) Suppose you are a producer of bicycles. To maximize total revenue, at what price should you sell bicycles? How much is the maximized total revenue? At the midpoint, which is the unit elastic point, the total revenue will be maximized. Therefore, P=$30, and Total Revenue= $30*20 = $600. Income elasticity of demand is the ratio of the percentage change in quantity demanded to the percentage change in income. If the income elasticity of demand is negative, then good A is an inferior good. When the income elasticity is positive, good A is a normal good. Income Elasticity of Demand: ΔQ % change in quanitity of good A demanded %ΔQ Q εI = = = ΔY % change in income %ΔY Y Cross-price elasticity of demand is the ratio of the percentage change in quantity demanded of good X to the percentage change in price of good Y. When this measure is positive goods X and Y are substitutes. When this measure is negative goods X and Y are complements. Cross-Price Elasticity of Demand: ε XY ΔQ X Q % change in quanitity of X demanded %ΔQ X = = = X ΔPY % change in price of good Y %ΔPY PY Exercise 2: a) The cross-price elasticity of demand for doughnuts and coffee is equal to -1.5. So, what is the relationship between doughnuts and coffee? (Complements / Substitutes) Suppose that the price of coffee increases by 10%. What is the percentage change in the quantity demanded of doughnuts? 2 The cross-price elasticity of demand is defined as the percentage change in the quantity demanded of good A divided by the percentage change in the price of good B. Thus, the percentage change in the quantity demanded of doughnuts must be -15%. Since the cross-price elasticity of demand is negative, then doughnuts and coffee are complements. b) The income elasticity of demand for doughnuts is equal to 1.2, you read in the newspaper that the economy has entered a recession and you are interested in how this recession will affect your doughnut business. Holding everything else constant, what do you anticipate will happen to the demand for doughnuts? Are doughnuts a normal good given this information? The income elasticity of demand for doughnuts is equal to the percentage change in the quantity demanded of doughnuts divided by the percentage change in income. Since a recession is a slowdown in total production in the economy and therefore is typically accompanied by higher rates of unemployment, we should anticipate that income will fall. Since the income elasticity of demand is positive, doughnuts are a normal good. Thus, the demand for doughnuts at every price will also fall: the demand curve for doughnuts will shift to the left holding everything else constant. 3