Price elasticity

For two comparable endpoints, denote price by P₀ and P₁ and quantity by Q₀ and Q₁. Midpoint elasticity is [(Q₁−Q₀)/((Q₁+Q₀)/2)] divided by [(P₁−P₀)/((P₁+P₀)/2)]. The averages make the interval comparison symmetric. They do not make it causal. Comparable price and quantity observations are required to evaluate the expression.

A subscription comparison needs the same eligible population or a justified exposure adjustment, the same renewal opportunity and a defined service period. Count the billed plan separately from subscriber accounts included in it. Record upgrades, downgrades and cancellations: a customer leaving a version may remain with the service. Quantity cannot be made comparable merely by giving two totals the same label.

Revenue is P×Q, whereas contribution depends on the costs that change with the offer. If comparable per-unit variable cost v is identified, the simple contribution comparison is (P₁−v)Q₁ versus (P₀−v)Q₀. Fixed commitments, acquisition, delayed churn and quality effects require additional evidence. When costs or credible demand comparisons are unavailable, the decision remains unresolved rather than becoming an assumed profit forecast.

Spotify’s reported annual monthly ARPU bridge is the exhibit. It separates management-attributed price, mix and currency effects on an aggregate revenue-per-account measure. It provides real figures for a measurement boundary; it supplies neither matched quantity exposure nor a controlled price response, so no elasticity is computed.

The observation window can also miss delayed behavior. A customer may keep a plan through a billing cycle and cancel later, or try a substitute only when renewal arrives. Record exposure, choice and service dates; inspect cohorts over the horizon the decision affects. Absence of immediate cancellation does not establish long-run sensitivity.

The denominator is often the first failure. A household subscription with several accounts is not equivalent to several separately billed subscriptions. Upgrading, downgrading and cancelling are different responses. If a Family plan attracts people who would otherwise buy Individual plans, customer count and revenue can move in opposite directions without any contradiction. Estimate own-plan response and substitution between plans when those choices matter.

The paired [Spotify case](/en/breakdowns/spotify-versioning-prices) provides an actual measurement boundary. Its filing reports annual monthly Premium ARPU rising from €4.39 to €4.69 in 2024 and attributes part of the movement to price increases, with mix and currency offsets. ARPU is an aggregate realized measure; it is not the US posted price or the treatment price faced by a stable cohort. The disclosed bridge cannot identify a demand curve. [2024 Form 20-F, Premium ARPU](https://www.sec.gov/Archives/edgar/data/1639920/000163992025000003/ck0001639920-20241231.htm).

A test can still mislead if the control group receives different features, if customers influence one another, or if acquisition offers change during the window. A market comparison needs evidence about those differences. If a credible comparison is unavailable, report observed retention and revenue as observations and use a range of response assumptions for the decision. Do not label the range as measured elasticity.

A higher price can reduce quantity and still improve current contribution, yet damage access, future adoption or service trust. Those trade-offs should be decided openly rather than concealed inside an elasticity number. The calculation informs the choice; it does not choose the firm’s objective.