Elasticity

Intuition [Beginner]

Some goods are sensitive to price changes; others are not. If the price of a life-saving medication doubles, people still buy it because they need it. If the price of a luxury brand of chocolate doubles, people switch to cheaper alternatives. Elasticity measures how responsive buyers or sellers are to a change in price or income.

Price elasticity of demand answers the question: when price changes by 1%, by what percentage does quantity demanded change? If a 10% price increase leads to a 20% drop in quantity demanded, demand is elastic (responsive). If the same 10% price increase leads to only a 2% drop, demand is inelastic (unresponsive).

The same logic applies to supply. Price elasticity of supply measures how much the quantity supplied changes when price changes. A farmer who can quickly plant more crops has elastic supply; a gold mine that takes a decade to develop has inelastic supply.

Elasticity matters because it determines who bears the burden of a tax, whether a price increase raises or lowers total revenue, and how much a market responds to shocks. A firm with inelastic demand for its product can raise prices without losing many customers; a firm facing elastic demand cannot.

Visual [Beginner]

Elastic Demand          Inelastic Demand

P |                     P |
  |   \                   |   |
  |    \                  |   |
  |     \                 |   |
  |      \                |   |
  |_______\____ Q         |___|________ Q
  (flat = responsive)     (steep = unresponsive)

Elasticity Type	Formula	Meaning
Price elasticity of demand	% change in Qd / % change in P	How quantity demanded responds to price
Price elasticity of supply	% change in Qs / % change in P	How quantity supplied responds to price
Income elasticity	% change in Qd / % change in income	Normal good if positive, inferior if negative
Cross-price elasticity	% change in Qd of A / % change in P of B	Substitutes positive, complements negative

Magnitude	Term	Meaning
	E	> 1
	E	= 1
	E	< 1
	E	= 0
	E	= infinity

Worked Example [Beginner]

A coffee shop sells 500 cups per day at $3.00each.Afterraisingthepriceto$3.50, sales drop to 400 cups per day.

Step 1: Calculate percentage changes.

Price change: $(3.50-3.00)/3.00\times 100=16.7\%$ increase.

Quantity change: $(400-500)/500\times 100=-20\%$ decrease.

Step 2: Calculate elasticity.

$$E_d=\frac{|\text{\% change in quantity}|}{\text{\% change in price}}=\frac{20\%}{16.7\%}=1.20$$

Demand is elastic ($E_d>1$). The coffee shop's customers are fairly responsive to price.

Step 3: Check revenue.

Old revenue: $500\times \$3.00=\$1,500$.

New revenue: $400\times \$3.50=\$1,400$.

Revenue fell. This confirms elastic demand: when demand is elastic, a price increase reduces total revenue. The shop would have been better off keeping the lower price.

Using the midpoint formula (more accurate for larger changes):

$$E_d=\frac{(Q_2-Q_1)/[(Q_1+Q_2)/2]}{(P_2-P_1)/[(P_1+P_2)/2]}=\frac{(400-500)/450}{(3.50-3.00)/3.25}=\frac{-100/450}{0.50/3.25}=\frac{-0.222}{0.154}=-1.44$$

Using the midpoint method, $|E_d|=1.44$, still elastic.

Check Your Understanding [Beginner]

Formal Definition [Intermediate+]

Price elasticity of demand (point elasticity):

$${\epsilon }_d=\frac{\partial Q_d}{\partial P}\cdot \frac{P}{Q_d}$$

Arc (midpoint) elasticity between points $(P_1,Q_1)$ and $(P_2,Q_2)$:

$${\epsilon }_d^{arc}=\frac{(Q_2-Q_1)/[(Q_1+Q_2)/2]}{(P_2-P_1)/[(P_1+P_2)/2]}$$

Price elasticity of supply:

$${\epsilon }_s=\frac{\partial Q_s}{\partial P}\cdot \frac{P}{Q_s}$$

Income elasticity of demand:

$${\epsilon }_Y=\frac{\partial Q_d}{\partial Y}\cdot \frac{Y}{Q_d}$$

• ${\epsilon }_Y>0$: normal good (demand rises with income)
  • $0<{\epsilon }_Y<1$: necessity
  • ${\epsilon }_Y>1$: luxury good
• ${\epsilon }_Y<0$: inferior good (demand falls with income)

Cross-price elasticity of demand:

$${\epsilon }_{XY}=\frac{\partial Q_d^X}{\partial P_Y}\cdot \frac{P_Y}{Q_d^X}$$

• ${\epsilon }_{XY}>0$: X and Y are substitutes
• ${\epsilon }_{XY}<0$: X and Y are complements
• ${\epsilon }_{XY}=0$: X and Y are unrelated

Elasticity and total revenue:

$$TR=P\times Q$$

$$\frac{d(TR)}{dP}=Q+P\frac{dQ}{dP}=Q\left(1+\frac{P}{Q}\frac{dQ}{dP}\right)=Q(1+{\epsilon }_d)$$

Since ${\epsilon }_d<0$: when $|{\epsilon }_d|>1$, $\frac{d(TR)}{dP}<0$ (price and revenue move in opposite directions). When $|{\epsilon }_d|<1$, $\frac{d(TR)}{dP}>0$ (price and revenue move in the same direction).

Constant elasticity demand function: $Q=A{P}^{\epsilon }$, where $\epsilon$ is constant at all price levels.

Academic Perspectives [Master]

Neoclassical Economics

In the neoclassical framework, elasticity is derived from the underlying utility function. For a consumer with utility $U(x_1,x_2)$ and budget constraint $p_1{x}_1+p_2{x}_2=m$, the price elasticity of demand for good 1 is determined by the Slutsky equation, which decomposes elasticity into substitution and income effects:

$$\frac{\partial x_1}{\partial p_1}=\frac{\partial h_1}{\partial p_1}-x_1\frac{\partial x_1}{\partial m}$$

where $h_1$ is the Hicksian (compensated) demand. The substitution effect is always negative (for own-price); the income effect can be positive (Giffen goods) or negative.

Homogeneous of degree zero in prices and income implies that the sum of own-price, cross-price, and income elasticities equals zero (Cournot aggregation). The sum of expenditure shares times own-price elasticities equals $-1$ (Engel aggregation).

The neoclassical model predicts that all Giffen goods are inferior goods, but not all inferior goods are Giffen. The rarity (or existence) of Giffen goods is debated.

Marshallian Origins

Alfred Marshall (1890) introduced the concept of elasticity in Principles of Economics, defining it precisely with the ratio of percentage changes. Marshall used elasticity to analyze how the burden of taxation is distributed between buyers and sellers (tax incidence). The key result: the more inelastic side of the market bears a larger share of the tax.

Keynesian Economics

Keynesian economists focus on aggregate elasticities. The income elasticity of demand for investment goods is particularly important in Keynesian macro. Keynes argued that investment is highly sensitive to expectations (animal spirits) and interest rates, with a low interest elasticity of investment being a reason monetary policy may be ineffective in a liquidity trap (the IS curve becomes steep).

Austrian School

Austrian economists are skeptical of elasticity as a predictive tool. Following Mises and Hayek, preferences are subjective and constantly changing. Elasticity measured from historical data may not predict future behavior because the economic environment itself changes through entrepreneurial action. Austrians accept elasticity as a descriptive ex post concept but reject it as a basis for central planning or policy prediction.

Behavioral Economics

Behavioral research finds that real-world elasticities are influenced by cognitive biases. Framing effects can alter the perceived price change and thus the response. Loss aversion (Kahneman and Tversky, 1979) implies that demand may be more elastic for price increases than for price decreases of the same magnitude, violating the symmetry assumed in the standard model. Reference price effects show that consumers respond not to the absolute price but to the gap between the current price and an expected reference price.

Marxian Economics

Marxian analysis does not use elasticity as a central concept, but the responsiveness of demand to price is relevant to the theory of realization crises -- the problem of whether sufficient effective demand exists to purchase the output of capitalist production at prices that cover costs and yield profit. A low income elasticity of demand for basic goods combined with unequal income distribution can lead to underconsumption.

Historical Context [Master]

Alfred Marshall introduced the concept of elasticity in Book III, Chapter 4 of Principles of Economics (1890). He defined it as the ratio of the percentage change in quantity demanded to the percentage change in price and used it to analyze tax incidence, monopoly pricing, and international trade.

The concept was refined throughout the 20th century. Hicks and Allen (1934) developed the Slutsky decomposition into substitution and income effects, grounding elasticity in utility theory. The distinction between Marshallian (uncompensated) and Hicksian (compensated) elasticity became standard.

In macroeconomics, elasticity concepts played a central role in the Keynesian-monetarist debate. The interest elasticity of money demand (liquidity preference) and the interest elasticity of investment were key parameters in debates about the effectiveness of fiscal vs. monetary policy.

In international trade, the Marshall-Lerner condition (combining export and import demand elasticities) determines whether a currency devaluation improves the trade balance.

Empirical estimation of demand elasticities became a major research program, pioneered by Stone (1954) with the Linear Expenditure System and extended by Deaton and Muellbauer (1980) with the Almost Ideal Demand System.

Bibliography [Master]

1. Marshall, A. (1890). Principles of Economics. Macmillan. Book III, Chapter 4.
2. Hicks, J. R., & Allen, R. G. D. (1934). "A Reconsideration of the Theory of Value." Economica, 1(1), 52-76.
3. Deaton, A., & Muellbauer, J. (1980). Economics and Consumer Behavior. Cambridge University Press.
4. Kahneman, D., & Tversky, A. (1979). "Prospect Theory: An Analysis of Decision under Risk." Econometrica, 47(2), 263-292.
5. Stone, J. R. N. (1954). "Linear Expenditure Systems and Demand Analysis." Economic Journal, 64(255), 511-527.