Price elasticity of demand and total revenue

The price elasticity of demand is the percentage change in the quantity demanded divided by the percentage change in price. It is a pure number, so the answer is the same whether quantity is in kilos or tonnes and price in rupees or paise.

For an ordinary good it is negative, because the two changes go in opposite directions. Many exam questions quote the size, |e|, and some leave the sign off, so read the question for which it means. The classes are the same either way:

• Elastic: |e| > 1. Quantity responds by a larger percentage than price.
• Unit elastic: |e| = 1.
• Inelastic: |e| < 1.
• Perfectly inelastic: e = 0, a vertical demand curve (UOH 2017). Perfectly elastic: |e| infinite, a horizontal one.

So the range of the price elasticity in absolute terms is from 0 to infinity (UGC NET 2026). Reading a value: if the elasticity of demand for an iPod is −3.5, a 10% rise in price cuts the quantity demanded by 35% (JNU 2012). It says nothing about the steepness of the curve, which depends on the units.

1. Calculus. Differentiate, then multiply by P/Q. For Q = 10 − 2P at P = 2: dQ/dP = −2, Q = 6, so e = (−2)(2/6) = −2/3. That is the question asked by three JNU papers (SIS 2011, 2015 and 2019): find the price at which Mary’s elasticity is −2/3. Since e = −2P/(10 − 2P), setting it equal to −2/3 gives P = 2.
2. Percentages. Given e = −2, a 10% rise in quantity needs a price change of 10 ÷ (−2) = −5%, five per cent lower (UOH 2011).
3. Geometry (for a straight line). |e| equals the length of the line below the point divided by the length above it.
4. Logs. If the demand is written with logarithms on both sides, the coefficient on ln P is the elasticity directly.

More single-point calculations, all the same recipe. Q = 20 − 5P at P = 3: Q = 5, so e = (−5)(3/5) = −3 (UOH 2012). Q = 100 − 2P at P = 25: Q = 50, so e = (−2)(25/50) = −1 (UGC NET 2026). Q = 100 − P − P² at P = 5: Q = 70 and dQ/dP = −1 − 2P = −11, so e = (−11)(5/70) = −0.79, which is inelastic (UGC NET 2025).

For a straight line Q = a − bP, put P = a/(2b) into the formula and you find |e| = 1 exactly in the middle. That is also what CUET 2023 asked in another dress: for P = A − BQ, at Q = A/2B the elasticity has size 1.

Slide the price on the straight line. Above the dashed midpoint demand is elastic and the revenue curve on the right is rising; at the midpoint revenue is at its peak and demand is unit elastic; below it demand is inelastic and revenue is falling. The readouts show all four routes to the elasticity. Then switch to constant elasticity: the number never changes, and revenue is always rising, flat or falling.

A straight-line demand curve has the same slope everywhere, but the elasticity is infinite at the price axis, falls to 1 at the midpoint and reaches 0 where the curve meets the quantity axis. On the way down, the price gets smaller and the quantity bigger, so P/Q shrinks while the slope stays put (UOH 2011: “the price elasticity varies, but the slope remains the same”).

The same fact gives the market’s tests: for the demand Q = 35,000 − 5P, demand is inelastic exactly while |e| = 5P/(35,000 − 5P) < 1, that is for prices from zero up to 3,500 (JNU 2020). The price 3,500 is half of the choke price, 7,000.

Is there a demand curve whose elasticity never changes? Yes: Q = A·P^−b. Take logs: ln Q = ln A − b ln P, a straight line in logs with slope −b. So the elasticity is −b at every price (UOH 2014).

• With b = 1, Q = A/P, the rectangular hyperbola. Spending PQ = A is the same at every price, and the elasticity is −1 everywhere (UOH 2010; JNU 2013 with D = 1/p; CUET 2022). It is the demand of the Cobb–Douglas consumer: for u = X₁X₂ the own-price elasticity is −1 (UOH 2010).
• Written as q^ap^b = c, the size of the elasticity is b/a (JAM 2026). So P²Q⁵ = 392.67 has |e| = 2/5 = 0.4 (JAM 2025).
• In log form, log Q = log A + 0.5 log P gives an elasticity of size 0.5, so demand is inelastic (GATE 2025). Notice that the sign is positive, so this “demand” slopes up; the question only asks about the size.

With more than one variable the same reading works. In Q = A·Y·P_X^β/P_Q^α, the exponent on its own price is −α, so demand is elastic when α > 1. The exponent on the other good’s price, β, is the cross elasticity, positive for a substitute (JNU 2021). Cross elasticities are in Lesson 11.

For a big change in price, the point formula depends on which end you start from. The arc (midpoint) elasticity divides each change by the average of its two values:

If price falls from Rs 12 to Rs 8 and quantity rises from 40 to 60, the quantity change is 20 on a mean of 50, which is 40%, and the price change is 4 on a mean of 10, which is 40%. The arc elasticity has size 1, and indeed revenue is 480 at both points.

What she spends is what the seller receives: total revenue TR = P × Q. When price rises, the price effect raises TR and the quantity effect lowers it. Elasticity tells you which one wins.

The test runs in both directions, and exam questions use both.

• A price cut that lowers the bus company’s revenue means demand is inelastic (CUET 2018). A price rise that leaves revenue unchanged means unit elasticity (UOH 2011). Spending that stays at Rs 15 million a month however the price moves means elasticity 1 (UOH 2011).
• A fall in the price of a good with inelastic demand, such as agricultural products, lowers producers’ revenue (CUET 2022). That is why a bumper harvest leaves farmers worse off (UOH 2016).
• Cell phones with elastic demand: a price cut raises revenue (JNU, four papers). Parking fees with elastic demand: a rise lowers revenue (JNU 2021). Grapes whose spending rises when the price falls from Rs 2 to Rs 1: the elasticity is above 1 (JNU 2017).
• The firm with two drugs, one with close substitutes and one with none, should raise the price of the one with no substitutes: its demand is inelastic (UOH 2018).
• A price rise with inelastic demand raises what consumers spend (UOH 2014 and 2015).

Quantities work the same way. If e = 0.2 in size and price rises by 2.8%, quantity falls by about 0.56%, so spending rises by about 2.8 − 0.56 = 2.2% (JNU 2011).

Since MR is the slope of the revenue curve (drawn against quantity), a revenue curve rises while demand is elastic, peaks where |e| = 1 and falls after that. Hence: if |e| > 1 the total revenue curve has a positive slope (JNU 2021), and if revenue falls as price rises then |e| > 1 (JAM 2026). The lab shows it. For Q = 120 − 4P at P = 20: Q = 40, |e| = 4 × 20/40 = 2 and MR = 20 × (1 − 1/2) = 10.

Trace her best bundle as the price of x falls with income and p_y fixed: the path is the price consumption curve (PCC). Her spending on x is p_xx = m − p_yy. So the PCC tells you what happens to the money spent on x, which is the revenue test again.

• Horizontal PCC (with x on the horizontal axis): y does not change, so spending on x does not change. Demand for x has unit elasticity, and its demand curve is a rectangular hyperbola. This is the Cobb–Douglas case (JNU 2014; UOH 2012 with money on the vertical axis; and UOH 2016, where y is unchanged after a Rs 2 rise in p_x).
• PCC sloping down as the price falls: she buys less y, so spends more on x: demand for x is elastic.
• PCC sloping up: she buys more y, so spends less on x: demand is inelastic.

One more statement from a paper: “Sonia has decided to always spend one-tenth of her income on shoes.” Constant spending whatever the price is unit price elasticity, and constant share of income whatever the income is unit income elasticity. Both are 1 (JNU 2016).

Other things equal, demand is more elastic when:

• the good has close substitutes, or is narrowly defined (Apple Mac against computers in general);
• it is a luxury rather than a necessity (JNU 2015 gave this as the answer);
• it takes a large share of the budget;
• consumers have more time to adjust;
• there is a bandwagon effect (Lesson 9).

Take care with the middle options of questions on this: a broad category is less elastic than a narrow one, and a small share of the budget makes demand less elastic, so those two are wrong in the JNU question.