Price Elasticity

Price elasticity is a dimensionless measure in economics of how the quantity demanded or supplied responds to a change in price, holding other relevant influences constant. It expresses the percentage change in quantity relative to the percentage change in price. Central to microeconomics and supply-and-demand analysis, it distinguishes a large proportional response from a small one, independently of the units used to measure prices and quantities. (openstax.org)

Definition and measurement

Price elasticity of demand concerns buyers’ responses to a good’s own price; price elasticity of supply concerns sellers’ responses. For a differentiable quantity function $Q(P)$, point elasticity at positive price and quantity is

The derivative measures the instantaneous quantity response, while $P/Q$ converts that response into proportional terms. Demand elasticity is ordinarily negative because higher prices reduce quantity demanded. Supply elasticity is ordinarily positive. Demand elasticities are frequently reported as absolute values, so the sign convention must be stated. (open.oregonstate.education)

For a finite movement between two observations, arc elasticity commonly uses the midpoint formula:

Using averages makes the result identical whether the movement is calculated forward or backward. In a hypothetical example, a price increase from 10 to 12 accompanied by a quantity decrease from 100 to 80 gives an arc elasticity of approximately $-1.22$. This describes the interval rather than necessarily the elasticity at either endpoint. (openstax.org)

Elastic, inelastic, and limiting cases

Using the magnitude of elasticity, demand or supply is classified as:

• Elastic: $|\varepsilon|>1$; quantity changes proportionally more than price.
• Inelastic: $0<|\varepsilon|<1$; quantity changes proportionally less than price.
• Unit elastic: $|\varepsilon|=1$; the proportional responses are equal.

These categories describe responsiveness at a point or over a specified interval, not necessarily an entire curve. (openstax.org)

Perfectly inelastic demand or supply has zero elasticity: quantity remains fixed as price changes, producing a vertical curve when price is on the vertical axis. Perfectly elastic demand or supply is represented by a horizontal curve and is conventionally assigned infinite elasticity. These are idealized limiting cases rather than descriptions of most observed markets. (openstax.org)

Elasticity is not the same as slope. A straight downward-sloping demand curve has constant slope but changing elasticity because its price-to-quantity ratio changes. By contrast, a constant-elasticity function such as $Q=AP^\varepsilon$, with $A>0$, has the same point elasticity throughout its positive domain. (openstax.org)

Determinants and time horizons

Demand responsiveness depends partly on the availability of substitutes, consumers’ ability to change consumption, and whether a purchase is regarded as necessary. A narrowly defined product may have alternatives that are unavailable for a broader product category. Consequently, an elasticity estimate must identify the product and market to which it applies. (open.oregonstate.education)

Time also matters. Demand for energy, for example, can be relatively unresponsive immediately because equipment and travel arrangements are fixed. Over longer periods, consumers can replace equipment or change locations. Producers likewise gain opportunities to expand facilities and reorganize production. Both demand and supply are therefore often more elastic in the long run, although this is not a universal rule. (openstax.org)

Revenue and taxation

For a seller facing a fixed demand curve, total revenue is $R=PQ(P)$. Differentiating gives

Thus, a small price increase raises revenue when demand is inelastic and reduces it when demand is elastic. At unit elasticity, its first-order effect on revenue is zero. These statements concern revenue, not profit, which also depends on costs. In particular, maximizing revenue need not maximize profit when marginal cost is positive. (openstax.org)

Relative demand and supply elasticities help determine tax incidence. In a competitive equilibrium, the less elastic side generally bears more of the economic burden of a per-unit tax. Buyers bear more when demand is less elastic than supply; sellers bear more when supply is less elastic than demand. The economic burden is distinct from the obligation to remit the tax. (openstax.org)

Estimation and related measures

Empirical elasticity estimation belongs to econometrics. Observed price and quantity movements do not automatically identify a demand response: both may change because demand and supply shift together. This creates endogeneity, making a simple ordinary least squares relationship potentially misleading. Instrumental variables can help separate relevant price variation, but their validity depends on identification assumptions. Research on electricity demand also shows that serial dependence can distort some instrumental-variable estimates. (arxiv.org)

Own-price elasticity differs from cross-price elasticity, which measures the response of one good’s quantity demanded to another good’s price. Positive cross-price elasticity indicates gross substitutes; negative elasticity indicates gross complements. Income elasticity of demand instead measures responsiveness to income. These measures answer different questions and cannot be substituted for one another. (openstax.org)