Price Elasticity

Definition and core intuition

Price elasticity of demand summarizes how much quantity demanded changes when an economic variable (price, other price, or income) changes, measured in percent terms. Because it is a percent change, elasticity is unit-free and comparable across goods and contexts.

There are three standard elasticities:

• •Own-price elasticity (often written $\varepsilon_{p}$ or $\varepsilon_{q,p}$): percent change in quantity demanded for a 1% change in the good's own price.
• •Cross-price elasticity ($\varepsilon_{q,p'}$): percent change in demand for good A for a 1% change in the price of good B.
• •Income elasticity ($\varepsilon_{q,y}$): percent change in demand for a 1% change in income.

Formal (point) definition uses calculus and the derivative — recall from "Derivatives (d2)" how to compute instantaneous rates of change. If demand is $q(q)$ written as a function of price $p$ (a Marshallian demand from "Demand Functions (d3)"), the point own-price elasticity is

Concrete numeric example: Suppose $q(p)=100-2p$. Then $dq/dp=-2$. At $p=10$, $q(10)=100-20=80$. So

Interpretation: a 1% increase in price (~$0.10 increase at$p=10$) reduces quantity by 0.25% (about 0.2 units), so demand is inelastic at that point.

Signs and classification

• •Own-price elasticity is typically negative for normal goods (higher price reduces demand). We often drop the sign and speak of elasticity magnitude: |\varepsilon|. If |\varepsilon|>1, demand is "elastic"; if |\varepsilon|<1, demand is "inelastic".
• •Cross-price elasticity is positive for substitutes, negative for complements. Example numeric: if $q_A=30-2p_A+0.5p_B$ then $\partial q_A/\partial p_B=0.5$, so cross-price elasticity at $p_B=20, q_A=10$ is $0.5\cdot 20/10 =1.0$.
• •Income elasticity >0 indicates a normal good, <0 an inferior good.

Why percent changes? Percent measures are scale-free and align with economic behavior (many responses are multiplicative). For small changes, percent changes are approximated by derivatives, which ties directly to calculus from "Derivatives (d2)".

Elastic vs inelastic regions

Elasticity can vary along a demand curve. For linear demand $q=100-2p$, elasticity depends on $p$ because $dq/dp$ is constant but $q$ changes with $p$. At $p=10$ we found $\varepsilon=-0.25$ (inelastic); at $p=40$, $q=20$ so $\varepsilon=-2\cdot40/20=-4$ (elastic). Thus, the same linear demand is inelastic at low prices (high quantities) and elastic at high prices (low quantities). This matters for pricing decisions: raising price increases revenue when demand is inelastic, but decreases revenue when elastic (numeric demonstration follows in later sections).

Connection to prerequisites

In "Demand Functions (d3)" we derived Marshallian demand functions from utility maximization; those functions are the input to elasticity formulas. In "Derivatives (d2)" we learned how to compute $dq/dp$; that derivative is essential to the point-elasticity formula above. The rest of this lesson builds directly on those tools.

Formulas and step-by-step calculations

We compute point elasticities using partial derivatives of the demand function. Suppose a demand for good i is $q_i(p_i, p_j, y)$ where $p_i$ is its price, $p_j$ other prices, and $y$ income. The standard formulas are:

• •Own-price elasticity:

Numeric example: Let $q(p)=120-3p$. Then $\partial q/\partial p=-3$. At $p=10$, $q=120-30=90$, so $\varepsilon=-3\cdot10/90=-0.333...$ (inelastic).

• •Cross-price elasticity with respect to $p_j$:

Numeric example: If $q_A=50-2p_A+0.4p_B$ and at $(p_A,p_B)=(10,20)$ we have $q_A=50-20+8=38$, then $\partial q_A/\partial p_B=0.4$ and

Interpretation: a 1% increase in $p_B$ raises $q_A$ by 0.21% (they are substitutes).

• •Income elasticity:

Numeric example: If $q=5+0.2y-1.5p$ and $(y,p)=(100,10)$ then $q=5+20-15=10$, $\partial q/\partial y=0.2$, so $\varepsilon_{y}=0.2\cdot100/10=2.0$, a luxury good (income-elastic >1).

Arc elasticity for finite changes

Point elasticities describe infinitesimal changes. For large discrete changes, use arc elasticity (midpoint formula):

Numeric example: Price goes from $p_1=10$ to $p_2=15$ for a demand where $q_1=80,q_2=50$. Then

Sign conventions and intuition

• •A negative own-price elasticity means a price increase reduces demand. We typically report the absolute value for magnitude comparisons.
• •If |\varepsilon|>1 demand is elastic: percent change in quantity exceeds percent change in price.
• •If |\varepsilon|<1 demand is inelastic: quantity responds less than proportionately.

Revenue consequences (numerical rule)

Total revenue $R(p)=p\cdot q(p)$. Take derivative using product rule:

So:

• •If $\varepsilon_{p}<-1$ (elastic), $dR/dp<0$: raising price reduces revenue.
• •If $\varepsilon_{p}>-1$ (inelastic), $dR/dp>0$: raising price increases revenue.

Numeric example: For $q=120-3p$ at $p=10$ we found $\varepsilon=-0.333...$. Then $dR/dp=q(1+\varepsilon)=90(0.6667)=60>0$, so raising price increases revenue locally.

Connection to "Derivatives (d2)" and "Demand Functions (d3)"

All point elasticity formulas rely on computing partial derivatives from the demand function (as in "Derivatives (d2)"), and the demand function itself may be derived from utility maximization (Marshallian demand) as studied in "Demand Functions (d3)". If you have a Marshallian demand, apply these derivative formulas to get the economic elasticities.

Motivation and algebra

Empirically and theoretically, it is common to model demand in "log–log" form because coefficients directly equal elasticities. A log–log model writes demand as:

where $\beta$ is the own-price elasticity, $\gamma$ the income elasticity, and each $\delta_k$ is a cross-price elasticity. If the error term $\varepsilon$ is small or mean-zero, point estimates of $\beta$ give constant elasticities across observations.

Equivalently, exponential form (constant-elasticity functional form) is:

where $A=e^{\alpha}$.

Concrete numeric example: Suppose empirical estimates give $\alpha=3.912, \beta=-1.5, \gamma=0.8$. Then the model implies

Compute a concrete number: take $p=10, y=100$. Then $e^{3.912}\approx 49.98$ and

Interpretation: a 1% increase in $p$ reduces $q$ by 1.5% exactly, at all points.

Why constant elasticity is useful

• •Interpretability: coefficients equal elasticities directly without multiplication by p/q.
• •Estimation: regressions of $\ln q$ on $\ln p$ yield coefficient estimates that are standard errors and hypothesis-testable.
• •Multiplicative effects: percentage impacts add when variables change multiplicatively.

From log–log to numerical elasticity

If you estimate $\hat\beta=-1.5$, then at any observed point the own-price elasticity is $-1.5$. Numeric check: if $p$ rises by 2%, predicted $q$ falls by 3%.

Revenue and optimization in constant-elasticity form

If demand is $q=A p^{\beta}$ (ignore income for simplicity) then revenue is

Revenue-maximizing price occurs when $dR/dp=0$, but monotonicity depends on $\beta+1$. If $\beta<-1$, revenue decreases with price; if $\beta>-1$, revenue increases. However, for monopoly with constant marginal cost $c$ the profit-maximizing price uses the Lerner formula (requires elasticity):

Numeric example: If $\varepsilon_{p}=-2$ and $c=10$, then $(p-10)/p=0.5$, so $p=20$.

Estimation notes and small-change approximation

For small price changes $\Delta p/p$ the percent change in quantity is approximately $\varepsilon_{p}\cdot \Delta p/p$. Numeric example: with $\varepsilon=-1.5$, a 2% price rise implies a 3% fall in quantity.

Every formula must link to a numeric example: the algebra above applied $\alpha,\beta,\gamma$ to $p=10,y=100$ to compute $q\approx99.7$.

Limitations and edge cases

• •Log–log requires strictly positive variables (prices, quantities, income). If zeros occur, you must use alternative specifications or add small constants (with caveats).
• •Constant elasticity is an approximation; real demand may vary with income and prices in non-constant ways. Compare constant-elasticity prediction to a linear demand numeric example: linear $q=100-2p$ at $p=10$ gave elasticity -0.25, but no single constant matches the whole curve.

Connection to prerequisites

You use "Derivatives (d2)" implicitly when differentiating $\ln q$ to show the elasticity equals the slope: $$\frac{d\ln q}{d\ln p}=\beta.$$Also, if$q(p)$ stems from a Marshallian demand in "Demand Functions (d3)", you can log-transform that demand and interpret estimated coefficients as elasticities if the functional form fits.

Practical uses for elasticities

1) Pricing strategy and revenue forecasting

Firms use price elasticity to decide whether to raise or lower price. The rule (numeric) is: if local |\varepsilon|<1 (inelastic), raising price increases revenue; if |\varepsilon|>1 (elastic), lowering price increases revenue. Example numeric: demand $q=120-3p$ at $p=10$ had |\varepsilon|=0.333 so a price hike increases revenue.

2) Monopoly pricing and markups (Lerner index)

Monopolists set price using elasticity: $$\frac{p-MC}{p}=-\frac{1}{\varepsilon}.$$

Numeric example: with constant marginal cost $MC=5$ and elasticity $\varepsilon=-2$, the markup fraction is 0.5, so $p=10$.

3) Tax incidence and welfare analysis

Elasticity governs how much of a per-unit tax is passed to consumers vs producers. If demand is perfectly inelastic (|\varepsilon|=0), consumers bear full tax; if perfectly elastic (|\varepsilon|=\infty), producers bear it. Numeric illustration: small tax $t$ with linear demand and supply changes quantity; compute pass-through using elasticities.

4) Cross-price effects and product relationships

Cross-price elasticities tell whether goods are substitutes or complements and by how much. Example numeric: a cross-elasticity of 0.8 between two beverages implies a 10% price increase for B raises A’s demand by 8%.

5) Income elasticity for forecasting and product classification

Income elasticity distinguishes luxuries (elasticity>1), necessities (0<elasticity<1), and inferior goods (elasticity<0). Example numeric: gasoline income elasticity 0.2 implies a 10% income rise increases demand by 2%.

Empirical estimation and data issues

• •When estimating elasticities from data, log–log regressions give constant elasticity estimates. Example numerical regression output: $\ln q = 3.9 -1.5\ln p + 0.8\ln y$ implies own-price elasticity -1.5.
• •Beware of endogeneity: price may be correlated with unobserved demand shocks. Instrumental variables are often required.
• •For bounded or zero quantities, log specifications are problematic; consider linear or Poisson pseudo-maximum-likelihood variants.

Downstream connections (what this enables)

Elasticities are inputs to:

• •Welfare analysis (consumer surplus change): small-change approximation uses $\Delta CS \approx -\int p dq$, which simplifies using elasticity in constant-elasticity models.
• •General equilibrium and policy analysis: elasticities feed into CGE models to simulate tax or subsidy effects.
• •Industrial organization: demand elasticities are central to merger simulation, mark-up models, and demand estimation.

Numeric policy example: If demand for cigarettes has price elasticity -0.4, a 10% tax increase (raising price by 10%) reduces demand by 4% and increases tax revenue roughly by 6% (because demand is inelastic).

Summary

Price elasticities translate derivative information from "Derivatives (d2)" into economically meaningful percent changes. They rely on demand functions derived in "Demand Functions (d3)" and feed into pricing, tax policy, welfare calculations, and empirical demand estimation. Constant-elasticity (log–log) forms make interpretation straightforward and are widely used in applied work, but always check fit and consider endogeneity and zero observations.