Elasticity, revenue and who bears a tax

If a quantity rises from 50 to 55, the change is five units and the starting-base percentage increase is five divided by 50, or ten per cent. If an interest rate rises from four per cent to five per cent, it rises by one percentage point but by 25 per cent relative to its initial value. These are different measures. Elasticity uses proportional changes, allowing comparisons across units such as boxes, kilograms or litres. Always identify the relevant base and sign before calculating. A calculator cannot correct an ambiguous definition of the percentage you intend to measure.

An elasticity of minus two means that, locally, a one per cent price increase is associated with about a two per cent reduction in quantity demanded, other relevant conditions unchanged. The absolute magnitude is two. Be explicit about whether you report the signed value or its magnitude. Elastic does not mean that quantity is high; it means quantity responds proportionally more than price. An inelastic response is not necessarily zero. Perfectly inelastic demand has zero quantity response in the specified model and range, a much stronger condition than merely having a magnitude below one.

Step one calculates the changes: price increases by two pounds and quantity decreases by twenty units. Step two calculates the average of each pair of endpoints: eleven pounds and ninety units. Step three divides each change by its own average, giving approximately 0.1818 and negative 0.2222. Step four divides the quantity proportion by the price proportion. The result is negative eleven-ninths, or approximately negative 1.222. Using midpoint bases gives the same magnitude when the direction of comparison is reversed. Do not divide a percentage quantity change by a raw pound change; that would not be an elasticity.

Calculate revenue independently to check the interpretation: ten pounds times one hundred is one thousand pounds; twelve times eighty is nine hundred and sixty. Revenue falls by forty pounds. For positive endpoints, the midpoint elasticity classification is consistent with this finite-change revenue test. If its magnitude is exactly one, the two endpoint revenues are equal. The relationship assumes the observed quantity change is a movement along unchanged demand and that all units sell at the stated price. Revenue is not profit. Costs, returns, discounts and capacity constraints may change whether a pricing decision benefits the seller.

First take the signed changes: quantity rises by four and price falls by two. Divide by the midpoint bases to get about positive 7.692 per cent and negative 10.526 per cent. Dividing those proportions gives negative nineteen-twenty-sixths, approximately negative 0.731. Its magnitude is below one, so demand over this interval is inelastic. Next calculate revenues: 20 times 50 equals 1,000, while 18 times 54 equals 972. The price cut reduces revenue by £28. More customers or more units sold therefore does not necessarily imply more sales revenue, still less more profit.

The same ten-unit response is a larger percentage of a small starting quantity than of a large one. Likewise, a one-pound change is a smaller percentage of a high price than of a low price. That is why a constant slope need not imply constant elasticity. The point formula multiplies the quantity response per pound by price divided by quantity. At five pounds, it gives negative ten times five divided by 150. At fifteen pounds, it gives negative ten times fifteen divided by fifty. Point elasticity describes a small local change; midpoint elasticity summarises an interval between two distinct observations.

A particular café may have many nearby substitutes, while coffee as a broad category has fewer close alternatives for some customers. Buyers may respond to a lasting fuel-price change by changing routes, vehicles or housing, even if tomorrow’s commute barely changes. The relevant time horizon therefore matters. Different groups can also have different responses. An average market elasticity can conceal people who have few short-run alternatives. To use an estimate responsibly, check its product definition, population, period, price range and research design. An elasticity borrowed from a different setting may be a poor guide.

A bakery with unused ovens may increase output quickly, while a fully booked venue cannot create more seats for tonight merely because offers rise. Over a longer period, investment or entry can make supply more responsive. These examples concern conditional responses, not shifts caused by new technology or input-price changes. As with demand, distinguish a local elasticity from an estimate over a large interval. A vertical supply curve means fixed quantity in the model’s relevant range, whereas a horizontal curve represents an extremely responsive supply at the given price. Neither shape is appropriate for every market.

Do not interpret all elasticities as own-price demand elasticity. If bus fares rise and train demand increases, the cross-price response can indicate substitution between travel options. If printer prices rise and demand for ink falls, the goods may be complements. These labels depend on the context and level of aggregation; they are not timeless characteristics of every individual’s preferences. Income elasticity also varies across groups and income ranges. The word inferior has no moral meaning here. As always, an observed correlation is not automatically a causal response when incomes, prices and other conditions move together.

Before introducing the tax, set buyer and seller prices equal to P. Equilibrium is 100 minus 2P equals 10 plus P, giving 90 equals 3P and P equals 30. Supply then gives 40 units. With the tax, using a single price in both original schedules would erase the distinction we need to analyse. Buyers respond to the amount they pay, while sellers respond to what they retain per unit after the tax. We assume a competitive market, compliance, unchanged underlying schedules and no separate administrative costs or other policy effects.

Expand the left side carefully: 100 minus 2Ps minus 12 equals 10 plus Ps. Subtract ten and collect price terms to obtain 78 equals 3Ps. Dividing by three gives 26, then adding the six-pound tax gives the buyer price of 32. Check quantities: demand is 100 minus 64, or 36; supply is 10 plus 26, also 36. Tax revenue is six times 36, or £216. The buyer and seller burdens per traded unit sum to the tax: two plus four equals six. This is a particular model calculation, not a general two-to-one incidence rule.

Use the local elasticity formulas at the original price of 30 and quantity of 40. Supply’s slope is one, so its elasticity is one times 30 divided by 40, or 0.75. Demand’s slope is negative two, giving magnitude two times 30 divided by 40, or 1.5. Sellers are less responsive in this example. This intuition concerns relative ability to adjust quantities. Market power, evasion, price regulation, cross-border options and longer-run changes can alter the result. A positive incidence calculation also does not decide whether a tax is justified; that judgement needs stated goals, rights and other consequences.

Imagine comparing a quiet rainy Tuesday with a crowded sunny Saturday. Weather, visitors, opening hours and stock could differ alongside price. Dividing the observed percentage quantity change by the observed price change gives a descriptive ratio, but not necessarily demand elasticity. A controlled comparison might hold other conditions more nearly constant, while a research design may use a source of price variation unrelated to demand changes. Neither method is automatically flawless. Check selection, spillovers, measurement, compliance and the range of prices studied. Report uncertainty rather than presenting a single estimate as a universal behavioural constant.

A reliable answer has three layers. First comes arithmetic: changes, averages, ratios and units. Second comes interpretation: what the magnitude means for the specified demand or supply relationship. Third comes identification and scope: why the data support a causal statement and where the estimate might fail to transfer. The exercises deliberately include a case where arithmetic is possible but a causal demand interpretation is not established. Written reflections remain ungraded. Use them to explain assumptions and evidence, because checking a number cannot substitute for understanding the model or the way the observations were generated.