[Hill03]

I apologize if this issue is address in the lectures but I have yet to make the time to listen to the class in its entirety. My question concerns the best way to determine demand elasticity in a situation – whether it be elastic, inelastic, or unit elastic. My economics textbook (ECONOMICS: PRINCIPLES & PRACTICES, Gary Clayton, pages 102-104) offers two distinct but not always compatible definitions for each of the categories of elasticity (elastic, inelastic, and unit elastic). The first definition involves the relative percent changes in price and quantity demanded while the second, which the book calls the “total expenditures test,” involves the increase or decrease in price compared to the increase or decrease in total expenditures. So for unit elastic the book makes the following statements:
“Sometimes demand for a product or service falls midway between elastic and inelastic. When this happens, demand is unit elastic, meaning that a given changes in prices causes a proportional change in quantity demanded. In other words, when demand is unit elastic, the percent change in quantity roughly equals the percent change in price. For example, a five percent drop in price would cause a five percent increase in quantity demanded.”
“In each case, the change in expenditures depends on the elasticity of the demand curve. … If there is no change in expenditure, demand is unit elastic.”
But in the following case, the two definitions contradict. At $3, 2 units are demanded; and at $2, 3 units are demanded. Total expenditures remains the same: $6. But the percent decrease in price is not proportional to the percent increase in units demanded. Price decreased by 33.3%, while quantity demanded increased by 50%.
What am I missing?

[jmherbener]

Your computation uses discrete units and the range of your computation is large relative to the entire demand curve. If you graph the entire demand curve implied by your calculation, it would have a y-axis intercept at $5 and 0 units and an x-axis intercept at ) dollars and 5 units. So your computation uses half of the demand curve. But the demand curve, technically, is only unit elastic at its mid-point.

If you shifted the demand curve parallel outward and to the right, the disparity in your elasticity computations would decrease. For example the demand curve with the y-intercept at $10 and the x-intercept at 10 units would have equal expenditures at at price of $6 and quantity of 5 and at a price of $5 and a quantity of 6. The elasticity computations would be 6/5 for lowering the price from $6 to $5 and 5/6 for raising it from $5 to $6, which are both closer to one. Technically, a linear demand curve is only unit elastic at its mid-point.

The way to reconcile the two computations when using discrete numbers is to use the average of the two numbers for calculating the base in the percent computation. This is a standard technique for such computations when using discrete numbers. For example, the percent change in quantity in your example would be 2-3 divided by the average of the numbers, that is 2+3 divided by 2. So the percent change in quantity is 0.4 (ignoring the minus sign) and the percent change in price would be 3-2 divided by 3+2/2, which is also 0.4.

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