Why Measure Demand Response to Income?
A household’s income can change while the price of a good and other relevant conditions remain fixed. Measuring how quantity demanded responds to that income change gives a way to compare goods and demand responses on a common scale. The measure is an elasticity: it relates a percentage change in quantity demanded to a percentage change in income, rather than comparing changes in dollars with changes in physical units.
The calculation is straightforward when the percentage changes are supplied. When the data give two observations, however, the chosen base matters. The midpoint method uses the average of the two incomes and the average of the two quantities as the respective bases. This makes the calculated response the same whether the observations are read forward or backward.
In this tutorial, \(Q_1\) and \(Q_2\) refer to quantities demanded at incomes \(I_1\) and \(I_2\), respectively. The other determinants of demand, such as the good’s price and consumer preferences, are assumed not to change between observations. If those conditions change too, the observed quantity difference cannot be attributed to income alone without additional analysis.
Income and quantity must be measured consistently within each pair of observations. For example, income can be recorded in dollars or thousands of dollars, provided both income observations use the same unit. Quantity must likewise use the same unit in both observations. The units cancel in the percentage changes, leaving a dimensionless elasticity.
Calculating with Percentage Changes
If a problem directly supplies the percentage changes, divide the quantity-demand percentage change by the income percentage change. Use decimal values for both changes, and preserve their signs: an increase is positive and a decrease is negative. Since both changes are percentages, their ratio does not depend on whether the calculation is written using decimals or percentage numbers, as long as the same convention is used for both.
For example, if income rises by 8% and quantity demanded falls by 2%, the changes as decimals are \(0.08\) and \(-0.02\). The elasticity is \(-0.02/0.08=-0.25\). The result is a ratio, not a percentage-point difference. A frequent source of errors is to divide a number expressed as a percentage by one expressed as a decimal, or to drop a negative sign before forming the ratio.
Worked Example: Calculating from Stated Percentage Changes
A hypothetical household survey estimates that, with the price of a meal-kit subscription and other demand conditions unchanged, a 6% rise in monthly income is associated with a 1.5% increase in quantity demanded. Calculate the income elasticity using the stated percentage changes.
Method. Divide the percentage change in quantity demanded by the percentage change in income. Convert both changes to decimals so the calculation uses one consistent convention.
The quantity-demand change is \(1.5\%=0.015\), and the income change is \(6\%=0.06\). Therefore,
Check. The quantity percentage change is one quarter of the income percentage change, because \(0.25 \times 6\%=1.5\%\). Thus, the ratio is \(0.25\), not \(25\).
Common mistake avoided. Dividing \(1.5\) by \(0.06\) mixes percentage numbers and decimals. The consistent calculation is \(0.015/0.06\), or equivalently \(1.5/6\); both give \(0.25\).
The Midpoint Method for Two Observations
When observations are given as endpoints, calculate each percentage change relative to the average of its two endpoint values. For quantity, the change is \(Q_2-Q_1\) and the base is \((Q_1+Q_2)/2\). For income, the change is \(I_2-I_1\) and the base is \((I_1+I_2)/2\). Then divide the quantity percentage change by the income percentage change.
Identify \(I_1,Q_1\) and \(I_2,Q_2\), making sure each quantity corresponds to the stated income.
Calculate \(Q_2-Q_1\), divide by \((Q_1+Q_2)/2\), and keep the sign.
Calculate \(I_2-I_1\), divide by \((I_1+I_2)/2\), and keep the sign.
Divide the quantity percentage change by the income percentage change. Check that the sign and approximate size agree with the two calculated changes.
The midpoint formula follows directly from the definition: each change is measured as a fraction of its own average endpoint value. Writing out those two fractions gives
This expression also explains why reversing the order of the observations does not change the result. Reversal changes the sign of both differences, \(Q_2-Q_1\) and \(I_2-I_1\), while leaving both averages unchanged. The two sign changes cancel in their ratio. This symmetry is useful when two endpoints describe the same movement but the problem presents them in the opposite order.
The calculation requires a nonzero income change; otherwise the income percentage change is zero and the ratio is undefined. For the usual application, both endpoint incomes are positive. The midpoint method is a convention for calculating an arc elasticity between two observations; it is not the same as choosing one endpoint as the base.
Worked Example: Income and Demand Both Increase
A hypothetical household’s annual income rises from $40,000 to $50,000. Over the same comparison, its annual demand for a particular home-maintenance service rises from 80 to 96 visits. Assume the service price and other relevant demand conditions are unchanged. Calculate the midpoint income elasticity.
Method. Calculate the midpoint percentage changes separately, then divide the quantity change by the income change.
The quantity change is \(96-80=16\) visits. The average quantity is \((80+96)/2=88\) visits. Therefore, the quantity percentage change is
The income change is \(\$50{,}000-\$40{,}000=\$10{,}000\). The average income is \((\$40{,}000+\$50{,}000)/2=\$45{,}000\). Therefore, the income percentage change is
Divide the quantity percentage change by the income percentage change:
Check. The quantity percentage change, about \(0.1818\), is smaller than the income percentage change, about \(0.2222\); their ratio is therefore positive and less than 1. Rounded to four decimal places, the midpoint estimate is \(0.8182\).
Common mistake avoided. Using the initial income and quantity as bases would give different percentage changes. The requested midpoint estimate uses averages of the endpoints for both bases.
Worked Example: Income Falls While Quantity Rises
A hypothetical customer’s annual income falls from $60,000 to $48,000. The customer’s annual demand for a repair plan rises from 30 to 36 units, while price and other relevant demand conditions remain unchanged. Calculate the midpoint income elasticity and check the signs.
Method. Keep the decrease in income negative and the increase in quantity positive. Use the average of each pair as its percentage-change base.
The quantity change is \(36-30=6\) units, and the average quantity is \((30+36)/2=33\) units. Thus, the quantity percentage change is
The income change is \(\$48{,}000-\$60{,}000=-\$12{,}000\), and the average income is \((\$60{,}000+\$48{,}000)/2=\$54{,}000\). Thus, the income percentage change is
The elasticity is
Check. One percentage change is positive and the other is negative, so their ratio must be negative. Rounded to four decimal places, the midpoint estimate is \(-0.8182\).
Common mistake avoided. Replacing the income decrease with a positive change would incorrectly remove the negative sign. The signed change is \(48{,}000-60{,}000=-12{,}000\), not \(12{,}000\).
Why the Base Matters
The midpoint method is especially important when the question explicitly asks for a midpoint or arc calculation. Using the first endpoint as the base answers a different calculation: the percentage changes are then measured relative to the initial observation, and reversing the direction generally changes those percentages. The method should follow the question’s stated convention rather than whichever base is quickest to use.
The following example compares the two calculations using the same data. It illustrates a common trap: using the first observation’s values while labelling the result a midpoint elasticity.
Worked Example: Avoiding the First-Endpoint Trap
A hypothetical household’s annual income rises from $30,000 to $42,000, while its demand for a service rises from 50 to 60 units. Assume other relevant demand conditions are unchanged. Calculate the midpoint elasticity, then compare it with the result from using the first endpoint as the base.
Method. For the midpoint estimate, divide each change by the average of its two endpoints. Calculate the first-endpoint result separately to identify the difference.
The quantity change is \(60-50=10\), and the average quantity is \((50+60)/2=55\). The midpoint quantity percentage change is
The income change is \(\$42{,}000-\$30{,}000=\$12{,}000\), and the average income is \((\$30{,}000+\$42{,}000)/2=\$36{,}000\). The midpoint income percentage change is
The midpoint elasticity is
For comparison, using the first endpoint as the base gives a quantity percentage change of \((60-50)/50=10/50=0.2\), and an income percentage change of \((42{,}000-30{,}000)/30{,}000=12{,}000/30{,}000=0.4\). Their ratio is \(0.2/0.4=0.5000\). That is not the requested midpoint estimate; the midpoint result is \(0.5455\).
Check. The midpoint percentages are approximately \(0.1818\) and \(0.3333\), whose ratio is approximately \(0.5455\). The first-endpoint percentages are \(0.2\) and \(0.4\), whose ratio is \(0.5000\). Each pair of percentages is internally consistent, but only the first pair uses midpoint bases.
Common mistake avoided. Do not calculate \(0.2/0.4\) and report \(0.5000\) as the midpoint elasticity. Those percentages use the initial endpoints, not the averages.
Calculation Checks and Exam Traps
Before finalizing an answer, check the following details:
- Match the observations. Each income must be paired with the quantity demanded observed at that income; do not compare values from different periods or populations without a stated reason.
- Use the requested base. For the midpoint method, divide each change by its own endpoint average, not by the initial value.
- Keep signs until the ratio is formed. A rise is positive, and a fall is negative. A negative result can be correct if the two changes have opposite signs.
- Use consistent units. Income expressed in dollars in one observation and thousands of dollars in the other would make the subtraction invalid. Convert both to the same unit first.
- Check the scale of the result. If the quantity percentage change is \(0.15\) and the income percentage change is \(0.30\), the ratio is \(0.15/0.30=0.5\), not \(50\).
Income elasticity is a numerical measure of the relationship between percentage changes, not a measure of the number of dollars of income or units of demand. The calculation also does not establish that income caused the observed change if other demand conditions changed at the same time. For the midpoint calculation, state the two changes, their bases, and their ratio; this makes both the method and its sign easy to audit.
Check Your Understanding
For each problem, calculate the requested income elasticity using the stated convention. Unless noted otherwise, assume other relevant demand conditions are unchanged.
- A hypothetical customer’s income rises from $24,000 to $30,000, while annual demand for a service rises from 100 to 115 units. Calculate the midpoint income elasticity. (Answer: \(27/43 \approx 0.6279\))
- A hypothetical consumer’s income falls from $80,000 to $72,000, while demand for a good rises from 40 to 44 units. Calculate the midpoint income elasticity, retaining the signs of both changes. (Answer: \(-19/21 \approx -0.9048\))
- Income rises by 12% and quantity demanded rises by 3%, with both percentage changes already calculated on a consistent basis. Calculate income elasticity directly. (Answer: \(0.25\))
- A household’s income rises from $50,000 to $65,000, while its demand for a hypothetical service falls from 200 to 170 units. Calculate the midpoint income elasticity. (Answer: \(-23/37 \approx -0.6216\))
- Income rises from $45,000 to $54,000 and demand rises from 70 to 84 units. Calculate the midpoint elasticity, and compare it with the first-endpoint calculation. (Answer: Midpoint \(1.0000\); first-endpoint calculation \(1.0000\))