Worked example 1 — Price elasticity of demand, calculated four ways

The law of demand tells you the direction in which quantity moves when price moves. It never tells you the size of the movement, and the size is the only thing a finance minister, a monopolist or a competition regulator actually needs. Elasticity supplies the size. This example takes an individual demand schedule, computes elasticity on it by each of the recognised methods, and then shows what each answer is good for.

| Price (Rs.) | Quantity demanded | |---|---| | 1 | 5 | | 2 | 4 | | 3 | 3 | | 4 | 2 |

Two things to notice before touching the arithmetic. This is an individual schedule, not a market one: a market demand schedule is the quantities all consumers will buy at all possible prices at a given moment of time, and the market curve is the horizontal summation of the individual curves. And the schedule is drawn ceteris paribus — the prices of related goods, the income of the consumer and their tastes and preferences are all held constant, because that is what the law of demand assumes.

There are four methods of measuring price elasticity: the percentage or proportionate method, the total outlay or total expenditure method, the point elasticity or geometric method, and the arc elasticity method.

The percentage method divides the proportionate change in quantity by the proportionate change in price. Take the move from Rs. 2 to Rs. 3.

• Change in quantity: 4 to 3, a fall of 1 on a base of 4 — a 25 per cent fall.
• Change in price: Rs. 2 to Rs. 3, a rise of 1 on a base of 2 — a 50 per cent rise.
• Elasticity = 25 ÷ 50 = 0.5.

Quantity moved by a smaller percentage than price, so the coefficient is less than one. On the five degrees of elasticity, this is inelastic: quantity changes by a smaller percentage than price, so total expenditure falls when price falls and rises when price rises.

The trap to avoid. Run the same calculation backwards, from Rs. 3 to Rs. 2:

• Quantity 3 to 4 — a rise of 1 on a base of 3, 33.3 per cent.
• Price Rs. 3 to Rs. 2 — a fall of 1 on a base of 3, 33.3 per cent.
• Elasticity = 1.0.

The same stretch of the same curve gives 0.5 one way and 1.0 the other, because the base of each percentage changed. That is not a mistake in either calculation; it is the reason the arc method exists. The point elasticity method measures elasticity at a given point on the demand curve using derivatives rather than finite changes, while arc elasticity is a measure of average responsiveness over some finite stretch of the curve. When an examiner gives two price-quantity pairs and asks for the elasticity, say which base you have used and why — that is the mark.

The total outlay or total expenditure method measures elasticity by watching what happens to total spending. It gives only the nature of elasticity, not an exact numerical value. Demand is elastic when total expenditure on the good increases as its price falls; unitary elastic when total outlay stays the same though more units are bought; and inelastic when total expenditure falls as the price falls.

Compute total outlay at each row — price multiplied by quantity:

| Price (Rs.) | Quantity | Total outlay (Rs.) | Reading, as price falls from the row above | |---|---|---|---| | 4 | 2 | 8 | — | | 3 | 3 | 9 | outlay rose from 8 to 9 → elastic over this stretch | | 2 | 4 | 8 | outlay fell from 9 to 8 → inelastic over this stretch | | 1 | 5 | 5 | outlay fell from 8 to 5 → inelastic |

Read the column and the whole shape of the curve appears without a single coefficient being computed. Outlay rises, peaks at Rs. 3, then falls. Where outlay is at its maximum, elasticity is passing through unity: above that price demand is elastic, below it inelastic. That is the standard result about a straight-line demand curve, and this four-row schedule reproduces it in miniature.

Notice too that the total-outlay reading of the Rs. 3-to-Rs. 2 stretch (outlay falls as price falls, therefore inelastic) sits alongside a percentage coefficient of 1.0 computed on the Rs. 3 base. The two methods are answering slightly different questions — one about the whole stretch, one about a point — which is exactly the ambiguity the arc method was invented to manage.

The five degrees, and where our numbers land:

| Degree | What happens | Value | Our schedule | |---|---|---|---| | Perfectly elastic | Demand is infinite at the prevailing price; the slightest rise in price makes quantity demanded fall to zero | ∞ | not present | | Perfectly inelastic | Even substantial changes in price leave demand unaffected | 0 | not present | | Unitary elastic | Quantity demanded changes by exactly the same percentage as price; total expenditure stays the same | 1 | around Rs. 3, where outlay peaks | | Elastic | Quantity changes by a larger percentage than price; total expenditure rises when price falls | > 1 | above Rs. 3 | | Inelastic | Quantity changes by a smaller percentage than price; total expenditure falls when price falls | < 1 | below Rs. 3 |

A coefficient is a number; the examiner wants to know why it took that value. The determinants follow, and an answer that pairs the arithmetic with two or three of them is worth far more than one that stops at 0.5.

• Nature of the commodity. Necessaries — salt, kerosene, matchboxes, textbooks, seasonal vegetables — have less than unitary elastic demand. Luxuries — an air conditioner, costly furniture, fashionable garments — are greater than unitary elastic. Comforts sit between.
• Substitutes. Goods with substitutes are relatively more elastic; goods with none, such as cigarettes or liquor, are inelastic.
• Proportion of income spent. Toothpaste and needles take a very small share of income and are inelastic; cloth takes a large share and is elastic.
• Postponability. Building a house can be postponed, so demand for bricks, cement and sand is elastic.
• Time. Demand is inelastic in the short period and elastic in the long, because habits can be changed more conveniently over time.
• Price level. Elasticity is high at a high price level and low at a low one — which is precisely the pattern the outlay column above displays.

There are five uses of elasticity. Three of them are legal, and each converts this arithmetic into a proposition about a statute.

1. Price determination by a firm with market power. A monopolist fixes a higher price in markets where demand for his product is inelastic and a lower price in segments where it is elastic, discriminating in price to maximise profit. That is not merely a pricing tactic; it is the conduct that competition law regulates as discriminatory pricing, and the elasticity difference between segments is the second of the five conditions that must hold before third-degree price discrimination can work at all.
2. Tax design. Government fixes higher tax rates on goods having inelastic demand and lower rates on goods having elastic demand. Our schedule says the commodity is inelastic below Rs. 3 — so a tax imposed at a low prevailing price will be borne mostly by the buyer and will not much reduce the quantity sold. That is the whole economic logic of excise and GST rate structures.
3. Market definition. Cross elasticity, the sister measure, decides the relevant product market: where a rise in the price of X sends buyers to Y, the two belong in the same market. Worked example 4 takes that up.

It does not give any Indian elasticity estimate for any real commodity. The schedule above is a teaching schedule and the coefficients are computed from it; treat both as illustrative arithmetic, not as data. Where an examiner asks for elasticity in India, the correct answer describes the method and the determinants, names the policy uses, and says candidly that the coefficient depends on the commodity and the period.

1. Elasticity measures the responsiveness of the quantity demanded of a good to a change in its quantitative determinant; the three kinds are price, income and cross.
2. Percentage method: proportionate change in quantity ÷ proportionate change in price. On the schedule, Rs. 2 → Rs. 3 gives 0.5.
3. The answer changes with the direction of travel because the percentage base changes; hence point and arc methods.
4. Total outlay method: outlay rises as price falls → elastic; unchanged → unitary; falls → inelastic. On the schedule outlay peaks at Rs. 3.
5. Uses: monopoly pricing and price discrimination, tax rate design, terms of trade, forecasting, and devaluation decisions.