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Elasticity practice problems with answers

Ten exam-style elasticity problems with complete worked solutions — price elasticity of demand by the midpoint and point methods, the total-revenue test, income and cross-price elasticity, price elasticity of supply, and how elasticity decides who bears a tax. Work each problem on paper before opening the solution; the calculation is easy to follow and hard to reproduce under exam pressure unless you have done it yourself. Free to use, no signup.

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Problem 1

The price of a streaming subscription rises from 10 euros to 12 euros per month, and the number of subscribers falls from 200,000 to 170,000. Calculate the price elasticity of demand using the midpoint method and classify demand over this range.
Show worked solution
Midpoint percentage change in quantity: 170−200(170+200)/2=−30185≈−16.2%\frac{170 - 200}{(170 + 200)/2} = \frac{-30}{185} \approx -16.2\%. Midpoint percentage change in price: 12−10(12+10)/2=211≈18.2%\frac{12 - 10}{(12 + 10)/2} = \frac{2}{11} \approx 18.2\%. So ϵD=−16.2%18.2%≈−0.89\epsilon_D = \frac{-16.2\%}{18.2\%} \approx -0.89. Because ∣ϵD∣<1|\epsilon_D| < 1, demand is inelastic over this range: the percentage fall in quantity is smaller than the percentage rise in price, so total revenue rises (from 2.0 million to 2.04 million euros per month).

Problem 2

Demand for a good is QD=300−5PQ_D = 300 - 5P. Calculate the point price elasticity of demand at P=20P = 20 and interpret the result.
Show worked solution
At P=20P = 20: Q=300−5(20)=200Q = 300 - 5(20) = 200. Point elasticity is ϵD=dQdP⋅PQ=−5×20200=−0.5\epsilon_D = \frac{dQ}{dP} \cdot \frac{P}{Q} = -5 \times \frac{20}{200} = -0.5. A 1 percent price increase reduces quantity demanded by about 0.5 percent — demand is inelastic at this point, so a price rise here would raise total revenue.

Problem 3

A firm estimates the price elasticity of demand for its product at −2.5-2.5. It is considering a 5 percent price cut. What happens to quantity demanded and to total revenue?
Show worked solution
Quantity demanded rises by approximately 2.5×5%=12.5%2.5 \times 5\% = 12.5\%. Because demand is elastic, the percentage gain in quantity exceeds the percentage loss in price, so total revenue rises: new revenue ≈0.95×1.125=1.069\approx 0.95 \times 1.125 = 1.069 times the old level, a gain of about 6.9 percent. This is the total-revenue test: when demand is elastic, price and total revenue move in opposite directions.

Problem 4

For the linear demand curve QD=300−5PQ_D = 300 - 5P, find the price and quantity at which demand is unit elastic, and state where along the curve demand is elastic and where it is inelastic.
Show worked solution
Set ∣ϵD∣=1|\epsilon_D| = 1: 5⋅P300−5P=15 \cdot \frac{P}{300 - 5P} = 1, so 5P=300−5P5P = 300 - 5P and P=30P = 30, Q=150Q = 150. This is the midpoint of the demand curve (the choke price is 60 and the maximum quantity is 300). Above P=30P = 30 demand is elastic (∣ϵD∣>1|\epsilon_D| > 1); below it demand is inelastic. Elasticity changes along a straight line even though the slope is constant, because the ratio P/QP/Q changes.

Problem 5

Consumer incomes rise by 8 percent and the quantity demanded of restaurant meals rises by 12 percent, with prices unchanged. Calculate the income elasticity of demand and classify the good.
Show worked solution
ϵI=%ΔQ%ΔI=12%8%=1.5\epsilon_I = \frac{\%\Delta Q}{\%\Delta I} = \frac{12\%}{8\%} = 1.5. The elasticity is positive, so restaurant meals are a normal good; because it exceeds 1, they are a luxury (income-elastic) good — spending on them rises more than proportionally with income. In a recession, demand for such goods falls sharply.

Problem 6

The price of printers rises by 10 percent and, holding everything else constant, the quantity demanded of ink cartridges falls by 4 percent. Calculate the cross-price elasticity of demand and describe the relationship between the goods.
Show worked solution
ϵXY=%ΔQX%ΔPY=−4%10%=−0.4\epsilon_{XY} = \frac{\%\Delta Q_X}{\%\Delta P_Y} = \frac{-4\%}{10\%} = -0.4. The negative sign means the goods are complements: a dearer printer reduces demand for cartridges because the two are used together. A positive value would have indicated substitutes.

Problem 7

When the market price of wheat rises from 8 to 10 dollars per bushel, quantity supplied rises from 400 to 440 million bushels within the season. Calculate the price elasticity of supply using the midpoint method and explain why it is likely to be larger over a longer horizon.
Show worked solution
Midpoint percentage change in quantity: 440−400420≈9.5%\frac{440 - 400}{420} \approx 9.5\%. Midpoint percentage change in price: 10−89≈22.2%\frac{10 - 8}{9} \approx 22.2\%. So ϵS≈9.5%22.2%≈0.43\epsilon_S \approx \frac{9.5\%}{22.2\%} \approx 0.43 — inelastic. Within one season, farmers have already planted; only marginal adjustments (more fertiliser, less waste) are possible. Over several seasons they can shift land from other crops, so the quantity response to the same price change is larger and supply becomes more elastic.

Problem 8

In a market, the price elasticity of demand is −0.4-0.4 and the price elasticity of supply is 1.21.2. A per-unit tax is introduced. Approximately what share of the tax is borne by buyers, and why?
Show worked solution
The buyers' share of a tax is approximately ϵSϵS+∣ϵD∣=1.21.2+0.4=0.75\frac{\epsilon_S}{\epsilon_S + |\epsilon_D|} = \frac{1.2}{1.2 + 0.4} = 0.75, so buyers bear about 75 percent and sellers about 25 percent. The side of the market that responds less to price (here demand, which is inelastic) cannot easily avoid the tax and therefore bears more of it. Who the tax is legally levied on makes no difference to this split.

Problem 9

Explain why the price elasticity of demand for petrol is much lower in the week after a price rise than two years after it. Name the determinant of elasticity involved and give two concrete adjustments consumers make over time.
Show worked solution
The determinant is the time horizon (adjustment time). In the short run, consumers' habits and capital are fixed: they own a particular car, live a particular distance from work, and have committed journeys, so they can only trim driving at the margin. Over two years they can buy a more fuel-efficient or electric car, move closer to work, switch to public transport or car-pooling, or change jobs. Because more substitutes become available the longer the horizon, the same price rise produces a larger percentage fall in quantity — demand becomes more elastic over time.

Problem 10

A monopolist faces the demand curve QD=300−5PQ_D = 300 - 5P and has zero marginal cost. Which price maximises total revenue, and what is the maximum revenue? Relate your answer to elasticity.
Show worked solution
Total revenue is TR=P⋅Q=P(300−5P)=300P−5P2TR = P \cdot Q = P(300 - 5P) = 300P - 5P^2. Setting dTRdP=300−10P=0\frac{dTR}{dP} = 300 - 10P = 0 gives P=30P = 30, Q=150Q = 150, and TR=30×150=4,500TR = 30 \times 150 = 4{,}500. This is exactly the unit-elastic point found in Problem 4: revenue is maximised where ∣ϵD∣=1|\epsilon_D| = 1, because raising price into the elastic region loses more in quantity than it gains in price, and lowering price into the inelastic region does the reverse. With zero marginal cost, revenue maximisation is also profit maximisation.

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