Project: Elasticity in economics
Topics and skills: Derivatives
Economists apply the term elasticity to supply, demand, income, capital, labor, and many other variables in systems with input and output. In a few words, elasticity describes how changes in the input to a system are related to changes in the output. Because elasticity involves change, it also involves derivatives.
In this project we investigate the idea of elasticity as it applies to demand functions. It's a common experience that as the price of an item increases; the number of sales of that item generally decreases. This relationship is expressed in a demand function.
1. Suppose a gas station has the linear demand function D(p) = 1200 - 200p (Figure 1). According to this function, how many gallons of gas can the gas station owners expect to sell per month if the price is set at $4 per gallon?
2. Evaluate D'(p) and show that the demand function is decreasing. Explain why demand functions are usually decreasing functions.
3. Suppose the price of a gallon of gasoline (Steps 1 and 2) increases from $3.50 to $4.00 per gallon; call this change Δp. What is the resulting change in the number of gallons sold; call it D? (Note that the change is a decrease, so it should be negative.)
4. Now express the answer to Step 3 in terms of percentages: What is the percent change in price, Δp/p, when it increases from $3.50 to $4.00 per gallon? What is the resulting percent change in the number of gallons sold ΔD/D?
5. The elasticity in the demand is the ratio of the percent change in demand to the percent change in price; that is, (ΔD/D)/(Δp/p). Compute the elasticity for the changes in Steps 3 and 4 (it should be negative).
6. The elasticity is simplified by considering small changes in p and D. In this case we use the definition of the derivative and write
E = limΔp→0(ΔD/D)/(Δp/p) = limΔp→0 (ΔD/ΔP)(p/D) = (dD/dp)(p/D).
Now the elasticity is a function of p. Show that for the gasoline demand function the elasticity is
E(p) = -(p/6-p).
7. The elasticity may be interpreted as the percent change in the demand that results for every one percent change in the price. For example if E(p) = -2, a one-percent increase in price produces a two-percent decrease in demand. If the price of gasoline is p = $4.50 and there is a 3.5% increase in the price, what is the elasticity and the corresponding percent change in the number of gallons sold?
8. Graph the gasoline demand elasticity function for 0 ≤ p < 6.
9. When -∞ < E < -1, the demand is said to be elastic. When -1 < E < 0, the demand is said to be inelastic. When E = -∞, the demand is perfectly elastic and when E = 0 the demand is perfectly inelastic. Essential goods such as basic foods tend to have inelastic demands; discretionary items, such as electronic equipment have elastic demands. Explain the meaning of these terms in this context.
10. For what prices is the gasoline demand function elastic and inelastic?
11. The demand for processed pork in Canada is described by the function D(p) = 286 - 20p1. Graph the demand function, compute the elasticity, and graph the elasticity. For what prices is the demand function elastic and inelastic?
12. Show that the general linear demand function D(p) = a - bp, where a and b are positive real numbers, has a decreasing elasticity for 0 ≤ p < a/b. Show that for the general linear demand function, E = -1 when p = a/2b.
13. Not all demand functions are linear. Compute the elasticity for the exponential demand function D(p) =ae-bp, where a and b are positive real numbers.
14. Show that the demand function D(p) =a/pb, where a and b are positive real numbers, has a constant elasticity for all positive prices.