A first look at elasticity

Studies collected by OpenStax put the price elasticity of demand for restaurant meals at 2.27 and for housing at 0.12: a 10% price rise cuts restaurant meals by 22.7% and housing by 1.2%.

How strongly do people respond?

The law of demand says that quantity falls when price rises. It does not say by how much. If the price of electricity rises by 10%, most households still run the fridge and the lights. If the price of restaurant meals rises by 10%, many people cook at home instead. Economists measure this sensitivity with the price elasticity of demand: the percentage change in quantity demanded divided by the percentage change in price.

Because price and quantity demanded move in opposite directions, the result is negative, and by convention it is reported without the minus sign. If the number is above 1, demand is elastic: quantity reacts more than price. Below 1 it is inelastic. Exactly 1 is called unitary. The same idea applies to supply: the percentage change in quantity supplied divided by the percentage change in price.

The midpoint method

A percentage change depends on the starting point: going from 100 to 120 is +20%, but going back from 120 to 100 is −16.7%. To get the same answer in both directions, economists divide each change by the average of the two values. In OpenStax's example the price falls from $70 to $60 and the quantity rises from 2,800 to 3,000. The quantity changes by 200 ÷ 2,900 = 6.9%, the price by −10 ÷ 65 = −15.4%, and the elasticity is 6.9 ÷ 15.4 = 0.45. Demand is inelastic there.

Higher up the same straight-line demand curve, between $120 and $130, the quantity falls from 1,800 to 1,600. Now the quantity changes by 200 ÷ 1,700 = 11.76%, the price by 10 ÷ 125 = 8.0%, and the elasticity is 1.47. The slope of the line is the same everywhere, yet the elasticity changes. At the top of a demand curve the quantity is small, so a change of 200 is a large percentage; at the bottom the quantity is large and the same 200 is a small percentage. Slope and elasticity are different things.

The table is a selection from OpenStax Table 5.2, estimates drawn from many studies. Necessities with few substitutes, like housing and electricity, are inelastic. Things people can easily do without or replace, like eating out, are elastic.

Who really pays a tax?

Elasticity also decides who bears a tax. Put a tax of $2 per unit on sellers. In the model the supply curve moves up by $2, and the market settles where buyers pay $6 and sellers keep $4 after paying the tax. The price paid by buyers rose by $1 and the price kept by sellers fell by $1, so each side bears half. That is because, at the starting point, demand and supply in this market are equally elastic.

Now make demand steeper by setting its flatness to 0.25. Buyers barely react to price, so sellers can pass most of the tax on: buyers pay $6.60, sellers keep $4.60, and buyers carry 80% of the tax. Set flatness to 3 and the opposite happens: buyers pay $5.50, sellers keep $3.50, and buyers carry only 25%. OpenStax states the rule: the tax falls mostly on whichever side of the market is less elastic. Cigarette demand among adult smokers has an elasticity of about 0.3, which is why cigarette taxes mostly show up as higher prices rather than much less smoking.

The green rectangle in the model is the government's revenue: the tax times the quantity still sold, $2 × 40 = $80 in the starting case. The red triangle is the deadweight loss, the gains from the 20 trades that no longer happen, ½ × 20 × $2 = $20. The more elastic the two curves, the more trades disappear and the smaller the revenue.