Measure responsiveness with a named convention

Measure responsiveness with a named convention

Start with the question

Can a price rise lower total revenue?

Inspect the teaching record

A fictional ticket seller raises price from 10 to 12 credits and observes quantity falling from 100 to 80. For this worked exercise, assume the same customers and no other demand changes.

This is fictional teaching evidence. It is not an observation about an actual community or economy.

Explain the mechanism

Price elasticity compares proportional quantity response with proportional price change. The midpoint method uses the average of the old and new values, making reversal consistent. Observational changes are not automatically causal estimates. Revenue equals price times quantity, not profit.

Follow the worked example

Midpoint quantity change = −20/90; price change = 2/11. Elasticity = (−20/90)/(2/11) ≈ −1.22. Revenue falls from 1000 to 960. The absolute responsiveness exceeds one.

Challenge the first explanation

A second event sells 95 tickets at 12. It has a different performer, so treating the difference as a pure price response would confound the comparison.

Make and revise your evidence

Reproduce the midpoint calculation and revenue comparison. Rewrite the second-event claim to identify the missing comparison design.

Revise your answer to “Can a price rise lower total revenue?” using both source IDs. Separate the checked calculation from a judgment, and identify evidence that could reverse your judgment.

Name the idea

Elasticity: A proportional response divided by a proportional change in its proposed driver.

Assumption: A condition accepted for a model or argument; it needs justification before applying the result elsewhere.

Extend the investigation

Independently extend the investigation “Measure responsiveness with a named convention” with a documented real dataset or scholarly source. Preserve definitions, recreate one calculation or evidence table, compare a rival explanation and report why the teaching example cannot establish the real-world conclusion.

Use one calculation convention for different questions

The ticket example asks how purchases respond to that ticket’s own price. A producer, a competing product and a household income change ask different questions. Before calculating, write down which quantity responds and which variable changes. Keep all other relevant conditions fixed in each teaching comparison.

For positive starting and ending values, calculate a midpoint proportional change by dividing the change by the average of the two values. If quantity goes from Q₁ to Q₂, its proportional change is (Q₂−Q₁) ÷ ((Q₁+Q₂)/2). Subscripts 1 and 2 mean before and after. Do the same for the driver X, then divide the quantity change by the driver change. Both changes are proportions, so elasticity has no units. Using percentages for both gives the same ratio; do not mix a percentage with an unscaled proportion.

A value of 2 does not mean “two more items.” It means the proportional quantity response is twice the proportional driver change over the stated interval. Reversing before and after changes both signs, leaving the midpoint ratio unchanged. Two observed points do not establish that only the named driver caused the change.

How much can sellers adjust?

Two hypothetical production plans begin at price 10 credits and supply 20 units per week. Under a price of 12, the short-horizon plan can supply 22; the longer-horizon plan can supply 30 after adjusting capacity. These are separate assumed response schedules for the same initial conditions, not observations of a real producer or a forecast that every industry behaves this way.

Predict which schedule has the greater supply elasticity. Calculate the common price-change denominator first, then the two quantity numerators. Plot quantity horizontally and price vertically, on identical scales. Explain why changing capacity over a longer horizon can permit a larger response without guaranteeing it.

Check the two supply responses

Price change is (12−10)/11=2/11, approximately 18.18%. Short-horizon quantity change is (22−20)/21=2/21, approximately 9.52%. Its supply elasticity is (2/21)/(2/11)=11/21≈0.524: a less-than-proportional, inelastic response over this interval.

Longer-horizon quantity change is (30−20)/25=10/25=40%. Its elasticity is (10/25)/(2/11)=2.2: an elastic response. Both are positive because the stated supply quantity rises with price. These are supply responses, not the ticket-demand elasticity above. Storage, available inputs and the time required to change capacity can matter; the invented values do not estimate their real effects.

The graph shows only the stated before/after segments. Reconstruct both endpoints in your own table before inspecting it. No claim about quantities outside those segments follows from these observations.

On narrow screens, focus the graph region and use arrow keys to scroll. Text above and the point labels give equivalent numerical evidence.

Do not substitute slope for elasticity

Two separate hypothetical supply schedules are Q=P and Q=P−8, using positive quantities in the interval P=10 to P=12. Both add one unit for each extra credit: the same slope in these units. Calculate the midpoint elasticity for each and explain why equal slopes need not imply equal elasticities.

Check the same-slope comparison

The first schedule changes Q from 10 to 12: quantity and price have the same proportional change, so elasticity is 1. The second changes Q from 2 to 4: quantity change is 2/3 while price change remains 2/11, so elasticity is 11/3≈3.667. Equal absolute changes are different proportions of the starting levels. Re-expressing quantity in dozens changes a graph’s numerical slope; it does not change the proportional elasticity.

A fixed quantity of 20 while price changes from 10 to 12 has supply elasticity 0, represented by a vertical line with quantity on the horizontal axis. A horizontal ideal supply line represents an arbitrarily large response at a fixed price. The finite midpoint formula cannot divide by a zero price change: report it as undefined for that pair, rather than silently dividing by zero. “Perfectly elastic” describes the ideal horizontal relationship or limiting behavior. It is not a finite elasticity calculated from two equal prices. Draw these two limiting relationships on separate, labeled axes and explain the assumptions each makes.

Identify which other variable changed

Use the new responsiveness source record in the notebook. In each case, hold the responding product’s own price and all unlisted drivers fixed. The following values are fictional and quantities refer to the same weekly period.

• Case S: the price of product A rises from 5 to 6. Demand for product B rises from 40 to 44.
• Case C: A’s price rises from 5 to 6. In a different market, B’s demand falls from 40 to 32.
• Case N: household income rises from 100 to 120. Demand for a product rises from 10 to 13.
• Case I: income rises from 100 to 120. Demand for a different product falls from 10 to 9.

Build a four-row ledger: responding quantity, driver, midpoint numerator, midpoint denominator, signed elasticity and interpretation. Predict each sign before doing the arithmetic. For S and C, the driver is another product’s price; for N and I it is income. These are shifts of the responding product’s demand at its fixed own price, not movements along its unchanged demand curve.

Check signs, arithmetic and classifications

For S, the denominator is 1/5.5=2/11 and the numerator is 4/42=2/21. Cross-price elasticity is 11/21≈0.524, positive: the products act as substitutes under the stated conditions. For C, the numerator is −8/36=−2/9, so cross-price elasticity is −11/9≈−1.222, negative: they act as complements.

For N and I, the common income change is 20/110=2/11. N’s numerator is 3/11.5=6/23, giving income elasticity 33/23≈1.435. Demand rises with income: a normal good, with a more-than-proportional response here. I’s numerator is −1/9.5=−2/19, giving −11/19≈−0.579. Demand falls with income: an inferior good in this range. “Inferior” names the response to income, not poor quality or a judgment about its buyers.

Do not take absolute values before interpreting cross-price or income elasticity; their signs carry the classification. For ordinary downward-sloping own-price demand, textbooks often report the magnitude of the negative response. Always label which convention you use.

Apply to a new record before opening feedback

In an unfamiliar fictional market, price rises from 8 to 12 and supply rises from 12 to 18. Separately, income rises from 80 to 100 and a product’s demand falls from 12 to 10, holding its own price fixed. Compute both midpoint elasticities, label each driver and write one sentence interpreting each result. Then critique a report that compares the two raw quantity changes and calls the first response “three times as elastic.” Save your ledger in the source notebook and record whether you used feedback.

Review the unfamiliar calculation

Supply quantity changes by 6/15=0.4; price changes by 4/10=0.4. Supply elasticity is 1. In the income case, quantity changes by −2/11 while income changes by 20/90=2/9. Income elasticity is −9/11≈−0.818, an inferior-good response over that interval. The raw changes 6 and −2 use different bases and drivers. Their ratio is not the ratio of elasticities, and it obscures the economically meaningful sign.

This panel turns the task into reviewed practice; it does not preserve an independent first attempt. A stronger causal claim would additionally need evidence that the other conditions were actually held fixed. Merely naming an elasticity does not supply that evidence.

Return to the tax-incidence investigation and identify whose response would need measuring before assigning a tax burden. A legal obligation to remit a tax is not a measurement of elasticity. The separate subsidy investigation shows why a demand shift and a price wedge require different diagrams.