MBA Demand Elasticity Concept Card Template

Demand Elasticity Concept Card: [Product, market or course]

A concept card on demand elasticity for the MBA managerial economics core: own-price, cross-price and income elasticity, arc (midpoint) versus point elasticity, estimating elasticity from sales data with a log-log regression, and the link between elasticity, revenue and the profit-maximising price. It is for MBA students preparing for class, problem sets and exams, and for managers who must judge a price change with real data. It gives the idea in plain language, the formulas with a notation table, one fully solved example, one half solved, one exercise, demand and revenue charts, common mistakes and a self-test with an answer key. Source basis: Marshall (1890) for the concept of elasticity, Lerner (1934) for the markup rule, and standard econometric practice for log-log demand estimation, written up in original words with fictional data. Replace every [bracketed] value and every row marked Example.

Document control

Contents

• When to use it
• The idea in plain language
• Formulas and notation
• Assumptions and limits
• How managers use it
• Decision flow
• Worked example A (fully solved): estimating and using an elasticity
• Revenue and profit on a linear demand curve
• Worked example B (half solved): cross-price and income elasticity
• Exercise C (unsolved)
• Connections to other cards
• Common mistakes and misconceptions
• Self-test questions
• My notes and next steps
• Sources and further reading
• Answer key
• Revision history

When to use it

Use elasticity whenever a decision depends on how much buyers respond to a change in price, in a rival's price or in income: setting or changing a price, forecasting volume after a promotion, judging a rival's price cut, deciding whether a product will grow with the economy, or checking whether a current price is anywhere near optimal.

The idea in plain language

Elasticity answers one question: if this driver changes by 1%, by what percentage does the quantity demanded change? Because both changes are percentages, the units cancel, so you can compare a USD 2 coffee with a USD 2,000 laptop. Marshall (1890) introduced the concept for price.

• Own-price elasticity is normally negative: a higher price means fewer units. Demand is elastic when the absolute value is above 1 (quantity moves more than price, in percentage terms), inelastic when it is below 1, and unit elastic at exactly 1.
• Cross-price elasticity measures the response to another product's price. Positive means substitutes (a rival's backpack gets cheaper, ours sells less); negative means complements (tents get cheaper, sleeping mats sell more).
• Income elasticity measures the response to buyers' income. Above 1: a luxury that grows faster than income; between 0 and 1: a necessity; below 0: an inferior good that buyers leave as they get richer.

Elasticity depends on the availability of substitutes (more substitutes, more elastic), on how narrowly the product is defined (one brand is more elastic than the whole category), on the share of the budget it takes, and on time (demand is more elastic in the long run, when buyers can adjust). The elasticity your firm faces is almost always more elastic than the market's, because your customers can switch to rivals.

The link to money is simple and essential: when demand is elastic, a price cut raises revenue; when it is inelastic, a price cut lowers revenue. Revenue is at its maximum where elasticity is exactly -1. Profit is a different matter, because costs change too; profit is maximised where demand is elastic, not at unit elasticity.

Formulas and notation

Point elasticity (at one point on a known demand curve):

$\varepsilon = \frac{dQ}{dP}\cdot\frac{P}{Q}, \qquad \varepsilon_{xy} = \frac{\partial Q_x}{\partial P_y}\cdot\frac{P_y}{Q_x}, \qquad \varepsilon_I = \frac{\partial Q}{\partial I}\cdot\frac{I}{Q}$

Arc (midpoint) elasticity (between two observations), which gives the same answer whichever direction you move:

$\varepsilon_{\text{arc}} = \frac{(Q_1 - Q_0)/\bar{Q}}{(P_1 - P_0)/\bar{P}}$

Log-log (constant-elasticity) demand, estimated by ordinary least squares on many observations:

$\ln Q = \beta_0 + \beta_1 \ln P + \beta_2 \ln P_y + \beta_3 \ln I + u$

Here $\beta_1$ is the own-price elasticity, $\beta_2$ the cross-price elasticity and $\beta_3$ the income elasticity, each holding the other variables constant. The model assumes the elasticity is the same at every price.

Elasticity, revenue and price. Marginal revenue can be written as

$MR = P\left(1 + \frac{1}{\varepsilon}\right)$

so MR is positive only when $|\varepsilon| > 1$. Setting $MR = MC$ gives the markup rule of Lerner (1934):

$\frac{P - MC}{P} = -\frac{1}{\varepsilon} \qquad \Longleftrightarrow \qquad P^* = MC \cdot \frac{\varepsilon}{1 + \varepsilon}$

Linear demand $Q = a - bP$ has a different elasticity at every point, $\varepsilon = -bP/Q$: inelastic at low prices, elastic at high prices, and unit elastic at the revenue-maximising price $P = a/(2b)$.

Assumptions and limits

How managers use it

1. Price changes. Estimate the volume effect of a proposed change, then compute the contribution effect. A price rise in inelastic demand raises both revenue and profit; in elastic demand it lowers revenue but can still raise profit if the margin gain outweighs the volume loss.
2. Checking a current price. Compare the actual margin $(P - MC)/P$ with $1/|\varepsilon|$. A margin far below $1/|\varepsilon|$ suggests the price is too low, far above suggests it is too high.
3. Promotion planning. Promotional elasticities are usually much larger than regular-price elasticities because of stocking-up and brand switching; do not use them to set the everyday price.
4. Competitive response. Cross-price elasticities show which rivals' moves matter most and whether to match a rival's cut.
5. Portfolio and cannibalisation. A positive cross-price elasticity between two of your own products means a price cut on one takes sales from the other.
6. Forecasting across the cycle. Income elasticity tells you how volume will move with the economy; luxury lines fall faster in a downturn (connect with mba/mba-macro-indicators-card).
7. Taxes, tariffs and cost pass-through. The more elastic the demand, the less of a cost increase a firm can pass on to buyers without losing volume.

Decision flow

The diagram shows how to go from data to a price decision. The two most common short-cuts, using a promotional response as a regular elasticity and stopping at revenue instead of contribution, are the boxes marked as checks.

Worked example A (fully solved): estimating and using an elasticity

Situation. Northfield Outdoor Co. (fictional) sells the Ridgeline 35 L hiking backpack. Over 12 weeks in spring, the merchandising team varied the price in planned steps, with no advertising change and no stock-outs. The backpack costs USD 30 a unit (marginal cost). What is the own-price elasticity, and is any of the tested prices close to the profit-maximising price?

Excel: =LN(B2) and =LN(C2) filled down for columns D and E (data in rows 2 to 13).

Step 1. Arc elasticity from two weeks. Take week 4 (USD 99, 300 units) and week 7 (USD 75, 441 units).

The two answers differ a lot, because the simple ratio depends on which week you call the start. The midpoint method fixes that, but any two-point estimate also picks up week-specific noise: it uses 2 of 12 observations.

Step 2. Log-log regression on all 12 weeks.

The estimate is -1.59: a 1% price rise reduces units by about 1.6%, so demand for the Ridgeline is elastic. The 95% interval excludes -1, so we can be fairly confident demand is elastic in this price range.

Step 3. Profit-maximising price. With constant elasticity $\varepsilon = -1.5884$ and $MC = 30$:

$P^* = 30 \times \frac{-1.5884}{1 - 1.5884} = 80.99$

Excel: =30*-1.5884/(1-1.5884). Check with the markup rule: $(80.99 - 30)/80.99 = 0.6296 = 1/1.5884$. Predicted weekly units at USD 80.99 are $e^{12.9666} \times 80.99^{-1.5884} = 398.1$ (=EXP(12.9666)*80.99^-1.5884), and weekly contribution is about USD 20,300.

Excel for each row, with the unrounded step 2 results in cells named Intercept and Slope: units =EXP(Intercept)*B2^Slope, revenue =B2*C2, contribution =(B2-30)*C2. Typing the rounded coefficients (12.9666 and -1.5884) instead moves revenue and contribution by up to USD 3 a week.

Interpretation. Revenue keeps rising as price falls (with constant elastic demand there is no interior revenue maximum), but contribution is almost flat between USD 79 and USD 89 and peaks near USD 81. The most common everyday price in the test (USD 89) gives up only about USD 80 a week; the main value of the analysis is to show that a deeper cut would not pay and that a rise to USD 99 would cost about USD 330 a week. The chart shows the observed weeks and the fitted constant-elasticity curve.

Revenue and profit on a linear demand curve

Situation. Northfield's two-person tent has weekly demand $Q = 900 - 6P$ (Example) across the chain, and a marginal cost of USD 40.

The profit-maximising price (USD 95) is above the revenue-maximising price (USD 75), in the elastic part of the curve. Moving from USD 75 to USD 95 lowers revenue by USD 2,400 (33,750 to 31,350) but raises contribution by USD 2,400 (15,750 to 18,150), because 120 fewer tents are made at USD 40 each, saving USD 4,800 of cost. The table and chart use USD 10 steps; the contribution peak at USD 95 lies between the USD 90 and USD 100 rows (both USD 18,000).

Excel with price in column A: units =900-6*A2, revenue =A2*B2, contribution =(A2-40)*B2, elasticity =-6*A2/B2.

Worked example B (half solved): cross-price and income elasticity

Situation. Northfield sells the Ridgeline in 16 regional store groups (fictional data). For one season the merchandising team recorded each region's average Ridgeline price, the average price of the main rival backpack and the region's median household income (USD thousand), with weekly units per store. A log-log regression of $\ln Q$ on $\ln P$, $\ln P_{rival}$ and $\ln I$ gives:

R squared 0.9609, n = 16. Excel: with ln Q in column E and ln own price, ln rival price and ln income in columns B to D, =LINEST(E2:E17,B2:D17,TRUE,TRUE) returns the coefficients in reverse order (income first, intercept last) in its first row and the standard errors in the second.

Tasks:

1. Own-price elasticity: [ ]. Is demand elastic? [ ]
2. Cross-price elasticity: 0.736. The rival backpack is a [substitute / complement] because [ ].
3. The rival raises its price by 5%. Approximate change in Ridgeline units: $0.736 \times 5\% = 3.68\%$. Exact change: $1.05^{0.736} - 1 =$ [ ] (=1.05^0.736-1).
4. Income elasticity: [ ]. The Ridgeline behaves like a [luxury / necessity / inferior good] in this sample. A 10% higher income is associated with [ ]% more units (exact, =1.10^1.4822-1).
5. Northfield raises its own price by 4%: units change by [ ]% (exact).
6. The own-price elasticity here (-2.34) differs from Worked example A (-1.59). Give two reasons.
7. Using -2.34 and MC = USD 30, the markup rule gives $P^* = 30 \times 2.3405/1.3405 = 52.38$. Why should you not recommend USD 52.38 from this regression? [ ]

Exercise C (unsolved)

Northfield's insulated water bottle has weekly chain demand $Q = 2400 - 40P$ (Example) and a marginal cost of USD 18.

1. Compute the point elasticity at USD 30 and at USD 33.
2. Compute the arc (midpoint) elasticity between USD 30 and USD 33. Why does it differ from both point values?
3. Find the revenue-maximising price, quantity and revenue.
4. Find the profit-maximising quantity and price, and the weekly contribution.
5. Verify the Lerner condition at the profit-maximising price.
6. The team proposes USD 30 "because it maximises sales revenue". Write two sentences to the team explaining what they are giving up.

Connections to other cards

• Pricing (mba/mba-pricing-card): the markup rule, price discrimination by segment elasticity, bundling and dynamic pricing all start from the elasticities estimated here.
• Market structure (mba/mba-market-structure-card): the elasticity a firm faces depends on the number of rivals and product differentiation; under perfect competition it is infinite.
• Game theory (mba/mba-game-theory-card): cross-price elasticities give the payoffs in a pricing game.
• Regression (mba/mba-regression-for-managers-card, academic/stats-linear-regression): estimation, confidence intervals, omitted-variable bias and diagnostics.
• Marketing mix (mba/mba-marketing-mix-card): advertising and promotion elasticities follow the same logic.
• Macro indicators (mba/mba-macro-indicators-card): income elasticity connects firm volume to GDP growth.
• Course guide (mba/mba-microeconomics-guide): where this card sits in the course and the integrative mini-case.

Common mistakes and misconceptions

1. Stopping at revenue. Elastic demand means a price cut raises revenue, not profit. Always compute contribution.
2. Using the start value in one direction and the end value in the other. Use the midpoint method for two points, or a regression for many.
3. Confusing slope with elasticity. The slope $dQ/dP$ has units (units per dollar); elasticity is unit-free and changes along a linear demand curve.
4. Using LOG10 in one column and LN in another. The log-log slope is the same with either base only if both variables use the same base; mixing them gives a wrong number.
5. Reading the cross-price sign backwards. Positive means substitutes, negative means complements.
6. Using a promotional response as the regular elasticity. Stocking-up and brand switching during promotions inflate the short-run response.
7. Extrapolating outside the observed price range. A log-log model fitted on USD 75 to 99 says little about USD 50.
8. Ignoring endogenous prices. Prices raised in strong weeks bias the estimate toward zero; record why prices changed.
9. Using the market elasticity for one brand. A single brand faces a much more elastic demand than the category.
10. Believing a firm should price where demand is inelastic. At an inelastic point MR is negative; a profit maximiser would raise the price.

Self-test questions

1. A 5% price rise cuts units by 2%. What is the elasticity, and what happens to revenue?
2. Why does the midpoint method give the same elasticity for a move from USD 99 to USD 75 as for USD 75 to USD 99?
3. On $Q = 900 - 6P$, at what price is demand unit elastic, and why is that the revenue-maximising price?
4. A log-log regression gives a slope of -0.8 for your brand. What does the markup rule imply, and what should you suspect about the estimate or the current price?
5. The cross-price elasticity of tents with respect to sleeping mats is -0.4. Interpret it.
6. Income elasticity is -0.3. What kind of good is it, and how will volume move in a recession?
7. With elasticity -4 and MC = USD 60, what is the profit-maximising price and the Lerner index?
8. In Worked example A, why is the 95% confidence interval of the slope important for the pricing decision?
9. Give two reasons why an elasticity estimated from promotion weeks is larger in absolute value than the regular one.
10. Write MR in terms of P and elasticity, and use it to show why MR is zero at unit elasticity.

My notes and next steps

Sources and further reading

• Marshall, A. (1890). Principles of Economics. Macmillan. (Book III, the elasticity of wants.)
• Lerner, A. P. (1934). The concept of monopoly and the measurement of monopoly power. Review of Economic Studies, 1(3), 157–175.
• Microsoft Support, documentation of the Excel functions SLOPE, INTERCEPT, RSQ, LINEST and T.INV.2T.
• Seabold, S., and Perktold, J. (2010). statsmodels: Econometric and statistical modeling with Python. Proceedings of the 9th Python in Science Conference, 92–96.
• Textbooks (further reading, by name only): Pindyck and Rubinfeld, Microeconomics; Besanko, Dranove, Shanley and Schaefer, Economics of Strategy; Baye and Prince, Managerial Economics and Business Strategy; Wooldridge, Introductory Econometrics: A Modern Approach (for log-log models and endogeneity); Nagle and Müller, The Strategy and Tactics of Pricing.
• Open resource: OpenStax, Principles of Economics (3rd edition), chapter on elasticity, CC BY-NC-SA 4.0, openstax.org.

Answer key

Worked example B.

1. -2.34; elastic ($|\varepsilon| > 1$), and the 95% interval (-3.06 to -1.62) lies entirely below -1.
2. Substitute: the coefficient is positive, so a higher rival price raises Ridgeline sales.
3. $1.05^{0.736} - 1 = 3.66\%$ (the approximation 3.68% is close for small changes).
4. 1.48; a luxury in this sample (above 1). $1.10^{1.4822} - 1 = 15.17\%$ more units.
5. $1.04^{-2.3405} - 1 = -8.77\%$ (=1.04^-2.3405-1).
6. Possible reasons: regional price differences may reflect local competition (stronger rivals where prices are lower), so the cross-section picks up more than the pure price response; the regional sample spans a different price range and time; the weekly test in Example A has different noise; the regression controls for rival price and income, which Example A did not; both are estimates with confidence intervals that overlap only partly.
7. USD 52.38 is far below the observed price range (USD 78 to 96), so it extrapolates the constant-elasticity form where there is no data; the cross-sectional estimate may also be biased by regional differences in competition. Test a modest price change first.

Exercise C.

1. At USD 30: $Q = 1200$, $\varepsilon = -40 \times 30/1200 = -1.000$. At USD 33: $Q = 1080$, $\varepsilon = -40 \times 33/1080 = -1.222$.
2. Midpoint: $(-120/1140)/(3/31.5) = -1.105$ (=((1080-1200)/AVERAGE(1200,1080))/((33-30)/AVERAGE(30,33))). It is an average over the arc, so it lies between the two point values.
3. $P = 2400/80 = 30$, $Q = 1200$, revenue USD 36,000.
4. Inverse demand $P = 60 - Q/40$, $MR = 60 - Q/20 = 18$, so $Q = 840$, $P = 39$, contribution $(39 - 18) \times 840 = 17{,}640$ USD a week (=(39-18)*840).
5. $\varepsilon = -40 \times 39/840 = -1.857$; $(39 - 18)/39 = 0.5385 = 1/1.857$.
6. Example: "USD 30 maximises revenue, but at that price each extra unit costs USD 18 and brings in nothing extra, so contribution is only USD 14,400 a week. At USD 39 we sell 360 fewer bottles and earn USD 17,640, USD 3,240 a week more." (At USD 30: $(30 - 18) \times 1200 = 14{,}400$.)

Self-test.

1. $\varepsilon = -2\%/5\% = -0.4$: inelastic, revenue rises (by about 2.9%: $1.05 \times 0.98 - 1 = 0.029$).
2. It divides each change by the average of the two values, so the base is the same in both directions.
3. At $P = 75$, where $\varepsilon = -1$. Since $MR = P(1 + 1/\varepsilon)$, MR = 0 there: revenue can no longer be raised by moving price.
4. The markup rule has no solution for $|\varepsilon| < 1$ (it would need a margin above 100%). Either the current price is too low (raise it), or the estimate is biased toward zero (prices set in response to demand, promotions, stock-outs).
5. Complements: a 1% rise in the price of sleeping mats is associated with 0.4% fewer tents sold.
6. An inferior good; volume rises in a recession as buyers trade down.
7. $P^* = 60 \times 4/3 = 80$; Lerner index $(80 - 60)/80 = 0.25 = 1/4$.
8. The interval (-1.85 to -1.33) excludes -1, so we are confident demand is elastic and prices should not be raised as if demand were inelastic; its width also shows how uncertain the optimal price is (re-run the markup rule at both ends: about USD 65.5 and USD 120.6).
9. Stocking-up (buyers bring purchases forward, so later weeks fall) and brand switching or store switching that reverses when the promotion ends; promotions also come with displays and advertising.
10. $MR = P(1 + 1/\varepsilon)$. At $\varepsilon = -1$, $1 + 1/\varepsilon = 0$, so MR = 0.

Revision history

Standard and licence note

Based on the standard treatment of demand elasticity in managerial economics (Marshall 1890, Lerner 1934) and on standard log-log regression practice. The text, the data, the examples and the questions are original; no textbook text is copied or paraphrased, and Northfield Outdoor Co. is fictional. The template is independent and not affiliated with or endorsed by any business school or software vendor. Check your course's notation and conventions before you rely on it. Template licence: MIT.