For a number of reasons, economists are interested in the amount by which the quantity demanded of a good changes in response to a change in the price of the good. Economists call this concept the price elasticity of demand. In precise terms, price elasticity of demand is defined as the percentage change in quantity demanded divided by the percentage change in price.
Price elasticity of demand can be measured in two ways. Your textbook outlines the midpoint method of determining elasticity. This web note will review that method and will present an alternative, often simpler, method for deriving the point elasticity of demand.
Consider a particular demand curve. Suppose we want to calculate the price elasticity of demand for a price change from $5 per unit to $10 per unit. The percentage change in price is 100%, because the change in price is $5, and the original price is also $5. As P rises, quantity demanded falls from 100 units to 60 units, a decline of 40 percent (40 divided by 100). Thus, the price elasticity of demand is 40% divided by 100%, which equals - 0.4. (We will typically ignore the minus sign and just say that the price elasticity of demand is 0.4.)
Suppose we had begun at P = $10 and reduced the price by $5. What would the estimated elasticity be? A $5 price reduction, starting from P = $10 amounts to a 50% change in the price. Quantity demanded increases from 60 to 100; but 40 divided by 60 is 2/3. Our elasticity measure thus is 2/3 divided by 1/2, which equals 4/3, or 1.33. Whoa! If we start from the lower point, our elasticity measure is 0.4. If we start from the upper point, our elasticity is 1.33. Something seems amiss.
The problem is that elasticity changes continuously along the demand curve. We can see this by developing the elasticity formula a bit further. The percentage change in quantity demanded is DQ/Q, while the percentage change in price is DP/P. Thus, elasticity is - [(DQ/Q)/( DP/P)]. This can be rewritten as
The first term, DQ/DP, is simply the inverse of the slope. If the demand curve is linear (a straight line), as ours is, the slope is constant. So DQ/DP doesn't change as we move along the curve. However, the second term, the ratio of P to Q, changes continuously along the demand curve. The slope of the demand curve shown above is - 1/8 (= 5/-40). But at P = $5, the P/Q ratio is 0.05, while at P = $10, the P/Q ratio is 0.167. Note that the P/Q ratio increases in size as we move up to the left along the demand curve. Thus, the elasticity of demand increases as we move up to the left along the demand curve. Because of this, the elasticity measure over a range of the demand curve differs according to the starting point used.
The midpoint approach averages the prices and quantities demanded, thus arriving at an average elasticity estimate for the range of values covered on the demand curve. In our example, the average price is $7.50, the average quantity 80 units. The elasticity is the inverse of the slope (in this case, 8) times the average price divided by the average quantity. That is, ed = 8 x (7.50/80) = 0.75. Note that this value lies between our previous estimates of 0.4 and 1.33.
Quick, what is the elasticity of demand for a tiny movement in price away from P = $7.50? Before you reply, "How should I know?", note that we just answered that question. For $7.50 is the midpoint price in the preceding example. Since our demand curve is linear, the slope, and hence its inverse, is a constant. We can find P/Q for any point on the demand curve. Thus, we can derive the point elasticity of demand for any P,Q combination on the demand curve.
What's the elasticity of demand at P = $5, Q = 100? We know the slope is 1/8, so the elasticity is 8 x (5/100) = 0.4. At P = $10, Q = 60, the elasticity is 8 x (10/60) = 1.33. At P = $8.75, ed = 8 x (8.75/70) = 1.0. That is, at P = $8.75, the demand curve is unit elastic. At any price below $8.75, we are on the inelastic portion of the demand curve. At any price above $8.75, we are on the elastic portion of the demand curve.
In our classroom exercises and examples, I will use the point elasticity formula. Given the importance of knowing the exact price at which the demand curve is unit elastic, using the point method makes more sense than using the midpoint method. If we used a nonlinear (curved) demand curve, deriving point elasticity would be more difficult (since the slope of the demand curve would change along the curve). But we don't, so we'll stick with point elasticity in our work.