Problem:

Suppose a firm sells 20,000 units when the price is $16, but sells 30,000 units when the price falls to $14.

1. Calculate the percentage change in the quantity sold over this price range using the midpoint formula.
2. Calculate the percentage change in the price using the midpoint formula.
3. Find the price elasticity of demand over this range of prices. State whether demand is elastic or inelastic over this range.
4. Suppose the firm's elasticity of demand is constant over a large range of prices, equal to the value found in part c. If the price were to fall another 4%, what should the firm predict will happen to its quantity sold?

Answer:

1. The midpoint formula uses the average of the two quantities as the reference point for computing the percentage change. In this example, the percentage change is (30,000 – 20,000)/25,000 = 0.40, or 40%.
2. The percentage change is (16 – 14)/15 = 0.133, or 13.3%.
3. The price elasticity of demand is the ratio of the percentage change in quantity to the percentage change in price. In this example, E_d = 40/13.3 = 3. Since E_d is bigger than one, demand is elastic.
4. The elasticity of demand equals the percentage change in quantity divided by the percentage change in price. Rearranging this relationship, the percentage change in quantity is equal to the elasticity of demand times the percentage change in price. In this example, E_d = 3 and the price change is 4%, so quantity sold will increase by 12%. 12% = 3 x 4%.

Problem:

Suppose a firm sells 70 units when the price is $6, but sells 80 units when the price falls to $4.

1. Calculate the firm's revenue at each of the prices.
2. Use the total-revenue test to determine whether demand is elastic or inelastic over this range.
3. Verify your previous answer by calculating the elasticity of demand using the midpoint formula.

Answer:

1. Revenue equals price times quantity sold. At P = $6, revenue equals $420. $420 = $6 x 70. At P = $4, revenue = $4 x 80 = $320.
2. Revenue falls when the price falls, suggesting demand is inelastic over this range.
3. E_d = [(80 – 70)/75] / [(6 – 4)/5] = .133/.40 = .33, or 1/3. This is less than one, verifying that demand is inelastic.