Worked Demand Elasticity Calculations

Turning Paired Observations Into Elasticity

A pair of market observations gives two prices and two corresponding quantities demanded. Demand elasticity calculations use the percentage changes between those observations to summarize how quantity demanded changed relative to price. The arithmetic is straightforward, but the result depends on how each percentage change is defined. In particular, the ordinary percentage-change method uses a chosen starting observation as its base, while the midpoint method uses the average of the two observations.

Before calculating, check what the data represent. To interpret a change in quantity as a response to a price change along the same demand relationship, other relevant determinants of demand must be held constant. If income, preferences, or another determinant changes between observations, the quantity difference may reflect more than the price change. In that case, the arithmetic ratio can still be calculated, but it should not automatically be described as the demand response to price alone.

Definition: For two observations, let $P_1$ and $Q_1$ be the starting price and quantity demanded, and let $P_2$ and $Q_2$ be the ending price and quantity demanded. The ordinary percentage-change estimate of price elasticity of demand is the percentage change in quantity demanded divided by the percentage change in price, with both changes measured from the starting observation. The midpoint estimate uses the average of the two prices and the average of the two quantities as the percentage-change bases.

The signed calculations below retain the sign of each change. For the usual case where price rises and quantity demanded falls, the quantity percentage change is negative and the price percentage change is positive, so their ratio is negative. This tutorial focuses on calculating the ratio; interpreting elasticity categories is a separate step.

The Ordinary Percentage-Change Method

For the ordinary method, calculate each change relative to its corresponding starting value. Keep the order consistent: ending value minus starting value in both numerators. Then divide the quantity percentage change by the price percentage change.

\%\Delta Q_d=\frac{Q_2-Q_1}{Q_1}\times 100\%, \qquad \%\Delta P=\frac{P_2-P_1}{P_1}\times 100\%, \qquad \varepsilon_d=\frac{\%\Delta Q_d}{\%\Delta P}.

The factors of $100\%$ cancel in the ratio, so the same result can be obtained by dividing the fractional changes directly. Showing the percentage steps is still useful: it makes the chosen bases visible and helps catch a reversed change or a mismatched denominator.

Worked Example: Ordinary Calculation From a Starting Observation

A hypothetical online supplier records that at a price of $20 per filter, customers demand 600 filters per week. At a price of $30, they demand 400 filters per week. Assume other demand determinants are unchanged. Calculate the ordinary percentage-change estimate, using the $20 observation as the starting point.

Method. Use the starting price and quantity as the two percentage-change bases. The ending observation has $P_2=30$ and $Q_2=400$; the starting observation has $P_1=20$ and $Q_1=600$.

Quantity change. Quantity falls by $400-600=-200$ filters. Relative to the starting quantity of 600, the percentage change is

\%\Delta Q_d=\frac{400-600}{600}\times 100\% =\frac{-200}{600}\times 100\% =-33.333\ldots\%.

Price change. Price rises by $30-20=10$ dollars. Relative to the starting price of $20, the percentage change is

\%\Delta P=\frac{30-20}{20}\times 100\% =\frac{10}{20}\times 100\% =50\%.

Elasticity estimate. Divide the quantity percentage change by the price percentage change:

\varepsilon_d=\frac{-33.333\ldots\%}{50\%} =-0.6666\ldots\approx -0.667.

Rounded to three decimal places, the ordinary estimate is $-0.667$. As a check, using the fractional changes gives $(-200/600)/(10/20)=(-1/3)/(1/2)=-2/3$, the same result. The common mistake here is to use the ending quantity or price as a denominator while claiming to use the starting-value method.

The Midpoint Method

The midpoint method replaces each starting-value denominator with the average of the two observations. For quantity, that base is $(Q_1+Q_2)/2$; for price, it is $(P_1+P_2)/2$. The change in each numerator is still ending value minus starting value. This method is useful when the calculation should give the same result whichever observation is called the starting point.

\%\Delta Q_d^{\,\text{mid}} = \frac{Q_2-Q_1}{(Q_1+Q_2)/2}\times 100\%, \qquad \%\Delta P^{\,\text{mid}} = \frac{P_2-P_1}{(P_1+P_2)/2}\times 100\%, \qquad \varepsilon_d^{\,\text{mid}} = \frac{\%\Delta Q_d^{\,\text{mid}}}{\%\Delta P^{\,\text{mid}}}.

The reason for using averages is symmetry. If the two observations are swapped, each difference changes sign, but each average stays the same. Thus both percentage changes reverse sign together, leaving their ratio unchanged. The ordinary method does not have this property because its starting-value denominators change when the observations are reversed.

\varepsilon_d^{\,\text{mid}} = \frac{(Q_2-Q_1)/((Q_1+Q_2)/2)} {(P_2-P_1)/((P_1+P_2)/2)} = \frac{(Q_2-Q_1)(P_1+P_2)} {(Q_1+Q_2)(P_2-P_1)}.

In the final expression, swapping subscripts 1 and 2 changes the signs of both $Q_2-Q_1$ and $P_2-P_1$, while the sums are unchanged. The two sign changes cancel in the ratio. This establishes why the midpoint estimate is independent of which endpoint is listed first. It does not mean that the method is a point estimate at either observation: it summarizes the finite change between the pair.

Worked Example: Applying Midpoints to Both Changes

A hypothetical transit pass costs $12 in one period and $18 in another. Weekly pass purchases are 720 and 480, respectively. Assume other relevant demand determinants did not change. Calculate the midpoint estimate, showing each midpoint base.

Method. Set $P_1=12,\ P_2=18,\ Q_1=720,$ and $Q_2=480$. Use the average price and average quantity as the percentage-change denominators.

Quantity midpoint and change. The average quantity is $(720+480)/2=600$ passes. The change is $480-720=-240$, so

\%\Delta Q_d^{\,\text{mid}} =\frac{-240}{600}\times 100\% =-40\%.

Price midpoint and change. The average price is $(12+18)/2=15$ dollars. The change is $18-12=6$ dollars, so

\%\Delta P^{\,\text{mid}} =\frac{6}{15}\times 100\% =40\%.

Elasticity estimate. Divide the two midpoint percentage changes:

\varepsilon_d^{\,\text{mid}} =\frac{-40\%}{40\%} =-1.000.

The midpoint estimate is $-1.000$. Reversing the observations would make the quantity change $+240$ and the price change $-6$, while retaining the same average bases; the ratio would still be $-1.000$. The trap to avoid is using the midpoint denominator for quantity but the starting price as the denominator for price. That mixes methods and does not produce the midpoint estimate.

Comparing the Two Calculations Carefully

Both methods use the same observed differences, but they answer the calculation using different bases. Consequently, a pair of observations can produce different numerical estimates. Neither method permits the percentage change in quantity to be divided by the price change in dollars: both numerator and denominator of the elasticity ratio must be percentage changes.

Worked Example: Comparing Methods on One Pair

A hypothetical equipment supplier observes a price increase from $40 to $60 per unit and a quantity decrease from 500 to 350 units per month. Assume the observations reflect the same demand relationship. Calculate the ordinary estimate from the first observation and the midpoint estimate, then compare the results.

Ordinary method. Quantity changes by $350-500=-150$ units. Relative to 500, this is $(-150/500)\times100\%=-30\%$. Price changes by $60-40=20$ dollars. Relative to $40, this is $(20/40)\times100\%=50\%$. Therefore,

\varepsilon_d=\frac{-30\%}{50\%}=-0.600.

Midpoint quantity change. The average quantity is $(500+350)/2=425$. Thus the midpoint quantity percentage change is

\%\Delta Q_d^{\,\text{mid}} =\frac{-150}{425}\times100\% =-35.2941\ldots\%.

Midpoint price change. The average price is $(40+60)/2=50$. Thus

\%\Delta P^{\,\text{mid}} =\frac{20}{50}\times100\% =40\%.

Midpoint estimate. Dividing gives

\varepsilon_d^{\,\text{mid}} =\frac{-35.2941\ldots\%}{40\%} =-0.8823529\ldots\approx -0.882.

Rounded to three decimal places, the ordinary estimate is $-0.600$, and the midpoint estimate is $-0.882$. The difference comes from the percentage-change bases, not from different observed changes. A common error is to report the ordinary result as if it were the midpoint result; always show or identify the denominators so the method is clear.

A Reliable Calculation Routine

For exam-style paired-observation problems, an orderly setup prevents most errors. The following sequence works for either method.

Label the observations.
Write down the starting and ending prices and quantities. Keep the same observation order for both variables.

Choose the base explicitly.
For the ordinary method, use $P_1$ and $Q_1$. For the midpoint method, use $(P_1+P_2)/2$ and $(Q_1+Q_2)/2$.

Calculate both percentage changes.
Use ending minus starting in both numerators. Keep track of the signs and do not substitute dollar or unit changes for percentage changes.

Divide and check.
Divide the quantity percentage change by the price percentage change. Check that the result agrees with the signs and arithmetic in the two percentage changes.

A zero price change makes the percentage-change denominator zero, so this paired-observation ratio is undefined. A zero quantity change, by contrast, produces a zero quantity percentage change when the relevant base is nonzero. These edge cases are reasons to inspect the data before applying the formula mechanically.

Units also deserve a quick check. If price is recorded in dollars and quantity in hundreds of units, the percentage changes still work because each change is divided by a value measured in the same unit. The units cancel within each percentage change. Do not, however, subtract one quantity measured in individual units from another measured in hundreds without first converting them to a consistent unit.

Key takeaway: The ordinary method measures both changes from the selected starting observation. The midpoint method measures both from their respective averages and is unchanged when the observations are reversed. State the method, show each base, calculate both percentage changes, and only then take their ratio.

Check Your Understanding

For each problem, state the method and calculate the signed demand elasticity estimate. Round final estimates to three decimal places where needed.

A hypothetical bakery raises a specialty loaf price from $25 to $30. Weekly quantity demanded falls from 800 to 720 loaves. Find the ordinary estimate using the first observation as the base. (Answer: $-0.500$)
Use the same two bakery observations, but list the $30 price and 720 loaves first, then the $25 price and 800 loaves. Find the ordinary estimate using this listed starting point. (Answer: $-0.667$)
A digital service changes its monthly fee from $10 to $14, while monthly subscriptions change from 900 to 600. Find the midpoint estimate. (Answer: $-1.200$)
A hypothetical repair kit sells for $18 in one period and $24 in another; quantity demanded falls from 560 to 440 kits. Find the midpoint quantity and price percentage changes, then the midpoint estimate. (Answer: quantity change $-24.000\%$, price change $28.571\%$, elasticity $-0.840$)
A supplier records a price rise from $50 to $70 and a quantity decrease from 1,000 to 700. Calculate both the ordinary estimate from the first observation and the midpoint estimate. (Answer: ordinary $-0.750$; midpoint $-1.059$)

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