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In a log-log demand model, what does a price coefficient of -1.4 tell you?

level: middleimportance: must knowfreq 58%

answer

  1. both sides are on the same scale
  2. the answer is a ratio, not units
  3. percent per percent
  4. compare the magnitude against 1
  5. d log q over d log p

basics

~20 s

It is a price elasticity: a 1% price increase is associated with roughly a 1.4% fall in quantity, holding the other predictors fixed. Because the size exceeds 1, demand is elastic, so raising price lowers revenue.

solid answer

~50 s

When both the outcome and the predictor are logged, the slope is a percent-per-percent ratio, which economists call an elasticity: `d log(q) / d log(p) = -1.4`, so a 1% price rise goes with about a 1.4% drop in quantity demanded. The number is unit-free, so it does not change if you switch currency or measure quantity in cases rather than units, which is what makes elasticities comparable across products. The magnitude relative to 1 is the interesting threshold: above 1 in absolute value demand is elastic and a price rise shrinks revenue, since revenue changes by roughly `1 + elasticity` percent per 1% price move - here about -0.4%. Two caveats I would state: the constant-elasticity form assumes the same percent response at every price level, and the coefficient is causal only if price is exogenous in this data.

go deeper

for a junior

Know the headline: with both sides logged the coefficient is a percent change per percent change, and a negative price coefficient means quantity falls as price rises.

for a middle

Explain where the percent-per-percent reading comes from, that the number is unit-free, and what magnitude above one implies for revenue. Expect to derive it from d log q over d log p.

for a senior

Show judgment about the constant-elasticity assumption, the observed price range, zero or missing quantities, and whether price moved for reasons unrelated to demand before calling the estimate an elasticity.

for a principal

Own how elasticity estimates get used in pricing decisions: the risk of extrapolating one pooled number across segments, and how much identification evidence you require before letting a model set prices.

## The algebra of a log-log slope A log-log, or constant-elasticity, model is `log(q) = b0 + b1 * log(p) + ... + error` Differentiating, `b1 = d log(q) / d log(p)`, and since `d log(x) = dx / x` is a relative change, `b1` is `(percent change in q) / (percent change in p)` That ratio is the definition of an elasticity. With `b1 = -1.4`, a 1% price increase corresponds to roughly a 1.4% decrease in quantity. Exponentiating shows the same thing multiplicatively: the model is `q = exp(b0) * p^(-1.4) * ...`, a power law, so the elasticity is constant at every price level rather than varying along the curve. ## Why the unit-free property matters A level-level price slope is measured in units-per-currency, so it changes if you move from dollars to cents or from bottles to cases, and it cannot be compared across products with different price ranges. An elasticity survives all of those rescalings, because both numerator and denominator are ratios. This is why demand teams report elasticities: -1.4 for one product and -0.6 for another is an immediately meaningful comparison even when one sells for 3 and the other for 300. ## Elastic, inelastic, and revenue The threshold is 1 in absolute value. - `|elasticity| > 1` - elastic. Quantity responds proportionally more than price moves. - `|elasticity| < 1` - inelastic. Quantity barely reacts. - `|elasticity| = 1` - unit elastic. Revenue is flat to small price changes. Revenue is `p * q`, and in logs `log(revenue) = log(p) + log(q)`, so the percent change in revenue for a 1% price rise is approximately `1 + b1`. At -1.4 that is -0.4%: raising price by 1% loses about 0.4% of revenue, because the quantity loss more than eats the per-unit gain. At -0.6 the same 1% price rise would add about 0.4% of revenue. This one line is often the whole point of estimating the model, and interviewers ask for it directly. ## Sign and magnitude discipline Downward-sloping demand means the coefficient should be negative; a positive price coefficient usually signals that something else is moving with price - quality tiers, seasonality, or price changes that follow demand rather than lead it. Reporting the elasticity as "1.4" without a sign, or as "1.4 fewer units", both misread the model. And the percent change is a percent of the current quantity, so 1.4% of a base of 10,000 units is 140 units at that point, not everywhere. ## The whole family of log forms It helps to hold the four cases together, because interviewers often walk across them: - level-level: b units of y per unit of x. - log-level (outcome logged only): roughly `100 * b` percent change in y per one-unit change in x. - level-log (predictor logged only): roughly `b / 100` units of y per 1% change in x. - log-log: b percent change in y per 1% change in x - the elasticity. Mixing these up is the single most common way candidates lose the question. A quick check: whatever side is logged is the side measured in percent. ## Assumptions worth naming The constant-elasticity form is a modelling choice, not a fact. It imposes the same percentage response at a price of 2 and at a price of 200; if the true response steepens at high prices, the fitted single number is an average over the observed range and should not be extrapolated past it. Logging also requires strictly positive values, so zero-quantity periods have to be handled deliberately rather than dropped by accident. Finally, the estimate is descriptive: calling it "the" price elasticity implies that price moved for reasons unrelated to demand, which needs either an experiment or an identification argument, not just a good fit. ## What interviewers listen for The percent-per-percent phrasing, the sign, the comparison against 1, and the revenue implication. A candidate who adds that the number is unit-free and that constant elasticity is an assumption is answering at a level above the screen.

  • What does an elasticity below -1 imply for revenue if you raise the price?
    Revenue falls. Since `log(revenue) = log(p) + log(q)`, the percent change in revenue per 1% price rise is about `1 + elasticity`, which is `1 - 1.4 = -0.4`. So a 1% price increase costs roughly 0.4% of revenue: the extra margin per unit does not cover the units lost. If instead the elasticity were -0.6, demand would be inelastic and the same price rise would add about 0.4% of revenue.
  • How does the interpretation change if only the predictor is logged, not the outcome?
    You get a semi-elasticity in the other direction. In `y = b0 + b1 * log(x)`, a 1% increase in x raises y by about `b1 / 100` units of y, because a 1% change adds roughly 0.01 to `log(x)`. So the outcome stays in its original units and only the predictor is read in percent. The rule that keeps this straight: whichever side is logged is the side measured in percent.
  • What does a constant-elasticity model assume that might not hold?
    That the same percentage response applies at every price level, since the fitted form is a power law with one exponent. Real demand often steepens near the top of the price range or flattens at the bottom, and the single estimate is then an average over the observed data. I would treat it as valid inside the price range in the sample, avoid extrapolating, and check whether splitting by price band or adding curvature changes the story.

saying these in an interview costs you the question

  • Reads -1.4 as 1.4 fewer units sold
  • Says a 1% price rise cuts quantity by 140%
  • Drops the sign and reports the elasticity as 1.4
  • Calls demand inelastic when the magnitude exceeds 1
  • Assumes the elasticity is causal with no exogeneity argument

context

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