Elasticity: calculating the number and reading its meaning

Overview

Calculating price, income and cross elasticities, and what the sign and the size of each number tell you.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   A bus operator raises the fare from £2.50 to £2.75 and weekly passenger journeys fall from 40,000 to 34,000. Calculate the price elasticity of demand and the change in revenue, and say whether the fare rise was a good commercial decision.

   Model answer

   The price rose by 0.25 divided by 2.50, which is 10 per cent, and journeys fell by 6,000 divided by 40,000, which is 15 per cent. Price elasticity of demand is -15 divided by 10, which is -1.5, so demand is elastic. Revenue was 2.50 multiplied by 40,000, which is £100,000, and is now 2.75 multiplied by 34,000, which is £93,500: a fall of £6,500, or 6.5 per cent. Because demand is elastic the proportional loss of passengers outweighs the proportional rise in the fare, so on revenue grounds the rise was a mistake. Buses run to a timetable, so almost none of the cost disappears with the lost passengers and profit falls too.

2. 2

   A water company raises its price from £1.20 to £1.32 per cubic metre and demand falls from 500,000 to 480,000 cubic metres. What is the price elasticity of demand, what happens to revenue, and what does the result show?

   Model answer

   The price rose 0.12 divided by 1.20, which is 10 per cent, and quantity fell 20,000 divided by 500,000, which is 4 per cent. Elasticity is -4 divided by 10, which is -0.4, so demand is inelastic. Revenue rises from 1.20 multiplied by 500,000, which is £600,000, to 1.32 multiplied by 480,000, which is £633,600, an increase of £33,600 or 5.6 per cent. A supplier of an essential good with no substitutes can raise price and raise revenue, which is precisely why such firms are subject to price regulation.

3. 3

   Why is price elasticity of demand normally negative, and what do economists mean when they call a value of -0.4 inelastic?

   Model answer

   The law of demand means price and quantity demanded move in opposite directions, so one of the two percentage changes in the ratio is negative and the elasticity is negative for almost every good. Inelastic means the response is proportionally smaller than the price change, that is, the value is between zero and one in absolute size. A figure of -0.4 says a 10 per cent price rise cuts the quantity demanded by only 4 per cent. Some writers drop the minus sign and compare absolute values; either way the comparison that matters is with one, not with zero.

4. 4

   State the relationship between price elasticity of demand and the effect of a price rise on total revenue, and explain why it holds.

   Model answer

   If demand is price elastic, meaning the absolute value exceeds one, a price rise reduces total revenue and a price cut raises it. If demand is inelastic, the reverse is true. Total revenue is price multiplied by quantity, and a price change moves one of these up and the other down, so the outcome depends on which proportional movement is larger. Elasticity is exactly the ratio of those two proportional movements, so it settles the question directly.

5. 5

   A firm facing unit elastic demand raises its price by 5%. What happens to total revenue, and is it right to say revenue is unchanged?

   Model answer

   Unit elasticity means a 5 per cent price rise brings a 5 per cent fall in quantity, so revenue changes by a factor of 1.05 multiplied by 0.95, which is 0.9975, a fall of 0.25 per cent. Revenue is exactly unchanged only for an infinitesimally small price change; for any discrete change the product of the two opposing movements leaves a small shortfall. The claim is a good approximation for small changes rather than an exact identity, and the shortfall grows as the price change gets larger.

6. 6

   Why does a straight-line demand curve not have the same price elasticity of demand at every point along it?

   Model answer

   The slope is constant, but elasticity compares proportional changes, and the same absolute price change is a small proportion of a high price and a large proportion of a low one. Near the top of the curve price is high and quantity low, so a small proportional price change meets a large proportional quantity response and demand is elastic. Near the bottom the reverse holds and demand is inelastic. At the midpoint elasticity is exactly one, and that is where total revenue reaches its maximum.

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