After this page you can
Calculate price elasticity of demand by both the original-quantity and the average-quantity method, and use it to say whether a price rise will raise or lower revenue
What elasticity is, before any symbols
Put the price of something up and you sell less of it. Everybody knows that much. The useful question is how much less, and that is the only thing price elasticity of demand measures: the proportion by which sales fall, divided by the proportion by which the price rose.
It is a ratio of two percentages, so it has no units. That is deliberate. A number of rupees per unit would be useless for comparison, because it would depend on whether you were selling cement by the bag or by the tonne. A ratio of percentages lets you compare a motor premium with a cup of tea.
Why the question matters: revenue pulls in two directions
Revenue is price multiplied by quantity. Raise the price and one of those two numbers goes up while the other goes down. Which effect wins is not a matter of opinion and not a matter of the product being a luxury or a staple. It is settled entirely by the ratio.
Write PED for the elasticity. Because quantity moves the opposite way to price, PED is negative for an ordinary good, so economists talk about its size and quote the sign separately.
Three cases, and each one is a statement about revenue:
Inelastic, |PED| < 1. Quantity moves proportionally less than price. The price gain beats the quantity loss, so a price rise raises revenue.
Elastic, |PED| > 1. Quantity moves proportionally more than price. The quantity loss beats the price gain, so a price rise lowers revenue.
Unit elastic, |PED| = 1. The two exactly cancel, and revenue is at its maximum. This is not a coincidence or a curiosity; it is the definition of the top of the revenue hill.
That last point is the mechanism to carry away. A seller who does not know their elasticity does not know which side of the hill they are standing on, and so does not know whether to raise the price or cut it.
The mechanism students miss: elasticity changes along one curve
A straight-line demand curve has a constant slope. It does not have a constant elasticity, and confusing the two is the single most common error in this topic.
The reason is in the formula. Written as a point measure,
The first factor is the slope, and on a straight line it never changes. The second factor is the ratio of price to quantity, and that changes at every point on the line. High up the curve price is large and quantity small, so P/Q is large and demand is elastic. Low down, price is small and quantity large, so P/Q is small and demand is inelastic.
So on any straight-line demand curve the top half is elastic, the bottom half is inelastic, and elasticity passes through exactly −1 at the midpoint — which is where revenue peaks. The worked example below builds the table and shows this happening.
Two ways to calculate it, and why they disagree
Over a finite stretch of the curve there is a choice of denominator, and the two choices give different answers.
Original quantities
Divide each change by the value it started from. Simple, and it has an unpleasant property: the answer depends on which end you start at. Measuring the same stretch upwards and downwards gives two different elasticities, because the starting values differ.
Average quantities
Divide each change by the average of the two endpoint values — the midpoint of the stretch. Because the average is the same whichever direction you travel, this method gives one answer per stretch instead of two. It is the honest measure of a stretch, and the only extra work is one addition and one halving.
Both appear in exam questions, and a question will tell you which it wants. If it does not, use average quantities and say so in one line: a stated method is never marked wrong, an unstated one sometimes is.
What makes demand elastic
Four things do most of the work, and each is a mechanism rather than a list item.
Substitutes. The more close alternatives a buyer has, the more of them leave when the price rises. This is why elasticity depends on how narrowly you draw the market: demand for one insurer's motor cover is far more elastic than demand for motor cover.
Share of the budget. A price rise on something that takes a tiny share of spending is barely noticed, so quantity hardly moves. A price rise on a large item provokes a search.
Necessity. If going without is not an option, quantity cannot fall much however the price moves. Cover that a law or a lender requires you to hold is the clearest case.
Time. Demand is always more elastic in the long run, because switching takes time — finding another supplier, changing a habit, replacing equipment. A price rise that looks harmless after a month can look expensive after two years.
The same ratio with a different denominator gives the other two elasticities on the syllabus. Divide by the percentage change in income and you have income elasticity, which is positive for a normal good and negative for an inferior one. Divide by the percentage change in another good's price and you have cross elasticity, positive for substitutes and negative for complements. Nothing new has to be learned; only the denominator changes.
Why an actuary is made to learn this
An insurer that raises premiums does not lose a random sample of its policyholders. It loses the ones with the most alternatives and the least need for cover, and those tend to be the better risks. So the quantity loss is worse than it looks, because the claims cost of the policies that remain is higher than the claims cost of the ones that left. Elasticity measures the size of the exit; adverse selection explains who walked out of the door.
Questions students ask
Can elasticity be zero or infinite?
Yes, and both are useful limiting cases. A vertical demand curve has PED of zero: quantity does not respond to price at all. A horizontal one is infinitely elastic: at that price buyers will take any amount, and a paisa above it they take none. The second is exactly the situation a single firm in perfect competition faces, which is why that market structure behaves the way it does.
Why would anyone use the original-quantities method if it gives two answers?
Because for a very small change the two methods agree to any accuracy you need, and over a small change the original-quantities version is the one that corresponds to the calculus definition. The disagreement only becomes embarrassing over a large stretch, which is precisely when the average-quantities version should be used.
Does a firm want elastic or inelastic demand?
Inelastic, and it will spend money to get it. That is the economic purpose of a brand, a loyalty programme, a long contract and a switching cost: each one removes an alternative from the buyer's reach, steepens the demand curve the firm faces, and makes a price rise survivable. It is also why competition authorities take an interest in all four.
Put the price up and you may take less money, not more
A worked example
An invented example, built for this page. A general insurer in Jaipur sells a personal-accident add-on cover. Its own pricing study gives the monthly demand schedule
where P is the annual premium in rupees and Q is the number of policies sold in a month. Every figure here was chosen to make the arithmetic clean; none of it describes a real product or a real market.
The schedule and the revenue it brings in
Premium P Policies Q Revenue P × Q Point PED = −2P/Q ₹100 1,000 ₹1,00,000 −0.20 ₹200 800 ₹1,60,000 −0.50 ₹300 600 ₹1,80,000 −1.00 ₹400 400 ₹1,60,000 −2.00 ₹500 200 ₹1,00,000 −5.00 ₹600 0 ₹0 infinite The slope is the same everywhere on this schedule: every ₹100 on the premium costs 200 policies. The elasticity is not the same anywhere. It grows from 0.20 to 5.00 as you climb, because
P/Qgrows, and it passes through 1.00 at ₹300 — which is exactly where revenue peaks at ₹1,80,000.Check the revenue peak a second way
Revenue as a function of price is
TR = P(1,200 − 2P) = 1,200P − 2P². Differentiate and set to zero:dTR/dP = 1,200 − 4P = 0 → P = 300At P = 300, Q = 600 and TR = ₹1,80,000, and the point elasticity is
−2 × 300 / 600 = −1.00. The calculus and the table agree, and they agree on both the price and the elasticity. That is the revenue test and the elasticity test giving the same answer, which they always must.The same stretch, three ways: ₹200 to ₹300
Original quantities, going up. Start at P = 200, Q = 800.
%ΔQ = (600 − 800) / 800 = −25.0000%
%ΔP = (300 − 200) / 200 = +50.0000%
PED = −25.0000 / 50.0000 = −0.5000Original quantities, going down. Same stretch, now starting at P = 300, Q = 600.
%ΔQ = (800 − 600) / 600 = +33.3333%
%ΔP = (200 − 300) / 300 = −33.3333%
PED = 33.3333 / −33.3333 = −1.0000−0.5000 one way and −1.0000 the other, for the same piece of the same curve. Nothing has been calculated wrongly. The method simply has no single answer for a stretch, and that is its defect.
Average quantities. Average premium
(200 + 300)/2 = 250; average policies(800 + 600)/2 = 700.%ΔQ = −200 / 700 = −28.5714%
%ΔP = +100 / 250 = +40.0000%
PED = −28.5714 / 40.0000 = −0.7143Travel the stretch the other way and ΔQ becomes +200 while ΔP becomes −100. The averages are unchanged, so the answer is −0.7143 again. One stretch, one elasticity.
Now answer the insurer's question
The premium is ₹200 and the proposal on the table is ₹300.
Elasticity over the stretch is −0.7143, and 0.7143 is less than 1, so demand is inelastic here and the price rise should raise revenue. The table confirms it: ₹1,60,000 becomes ₹1,80,000, a gain of ₹20,000 a month. Two independent routes, one answer.
Push on to ₹400 and the picture reverses. Average quantities over ₹300 to ₹400: average P = 350, average Q = 500, so
%ΔQ = −200/500 = −40.0000%and%ΔP = 100/350 = +28.5714%, giving PED = −1.4000. Elastic, so revenue should fall — and it does, from ₹1,80,000 back to ₹1,60,000.The sanity checks
Elasticity rose steadily as the premium rose, as it must on a straight line. It crossed 1.00 exactly where revenue turned. The two arc elasticities bracket that crossing — 0.7143 below it, 1.4000 above it. And the midpoint-method answers are, in both cases, between the two original-quantity answers for the same stretch, which is what averaging a denominator does. If any one of those four things had failed, there would be an error in the working.
One last thing worth noticing, because it is the whole commercial point. Moving the premium from ₹200 to ₹300 earns ₹20,000 a month more and sheds 200 policies. Whether that is a good trade depends on what those 200 policies were costing in claims — and on whether the ones who left were the cheap risks or the dear ones. Elasticity tells you how many go. It does not tell you who.
Where this fits in CB2
This page explains one idea from CB2 Business Economics. The full CB2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.
Other CB2 topics explained
- Income and substitution effects: splitting one price change in two
- Profit maximisation and the shutdown decision, from one cost table
- Cournot and Bertrand: why the same two firms give two different answers
- Adverse selection and moral hazard, and what an insurer does about them
- National income and the multiplier: where the rest of the spending comes from
- The IS-LM model and crowding out, solved with numbers
- How monetary policy reaches the economy: the transmission mechanism
Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.