Difficulty: Intermediate | Prerequisites: Units 1A–1B (math skills, supply and demand)
Elasticity measures how responsive quantity is to a change in price, income or the price of another good. It is the bridge between the qualitative logic of supply and demand ("demand falls when price rises") and the quantitative question that matters in practice ("by how much?"). This unit introduces four types of elasticity, each with its own formula, sign convention and interpretation.
Elasticity tells you the percentage change in one variable for a one-percent change in another. Price elasticity of demand and supply measure responsiveness to own-price changes. Income elasticity and cross elasticity reveal whether goods are normal, inferior, substitutes or complements. You need to be comfortable with the basic formula, the midpoint formula and the calculus-based approach.
Price elasticity of demand (Ed)
The percentage change in quantity demanded divided by the percentage change in price. It measures how sensitive buyers are to price changes. In simple terms, it answers: "If the price goes up by 1%, how much does the quantity demanded change?"
Elastic demand
|Ed| > 1. Quantity demanded is highly responsive to price changes. A small price increase leads to a proportionally larger drop in quantity demanded. The demand curve looks relatively flat.
Inelastic demand
|Ed| < 1. Quantity demanded is not very responsive to price changes. Even a large price increase leads to a proportionally smaller drop in quantity demanded. The demand curve looks relatively steep.
Unit elastic demand
|Ed| = 1. The percentage change in quantity demanded exactly matches the percentage change in price.
Perfectly elastic demand
|Ed| = infinity. The demand curve is horizontal. Any price increase above the given level reduces quantity demanded to zero.
Perfectly inelastic demand
|Ed| = 0. The demand curve is vertical. Quantity demanded does not respond to price at all.
Midpoint formula (arc elasticity)
A method of calculating elasticity that uses the average of the two prices and the average of the two quantities as the base. This avoids the problem of getting a different answer depending on which point you start from.
Income elasticity of demand (Ei)
The percentage change in quantity demanded divided by the percentage change in income. The sign tells you whether the good is normal (positive) or inferior (negative). The size tells you whether a normal good is a necessity (0 < Ei < 1) or a luxury (Ei > 1).
Normal good
A good for which demand increases when income rises. Income elasticity is positive.
Inferior good
A good for which demand decreases when income rises. Income elasticity is negative. Think of it as the budget option you stop buying once you can afford something better.
Cross elasticity of demand (Exy)
The percentage change in quantity demanded of good X divided by the percentage change in the price of good Y. The sign reveals the relationship between the two goods.
Substitutes
Goods with a positive cross elasticity. When the price of good Y rises, demand for good X rises (consumers switch).
Complements
Goods with a negative cross elasticity. When the price of good Y rises, demand for good X falls (the goods are used together).
The basic formula
Ed = (%change in Qd) / (%change in P) = (dQ/Q) / (dP/P).
Sign of the calculation
Because of the law of demand, Ed is always negative (price up, quantity down, or vice versa). In practice, economists often report the absolute value |Ed| so that comparisons are easier.
Magnitude and demand curve shapes
|Ed| > 1 (elastic): demand curve is relatively flat. Revenue falls when price rises.
|Ed| < 1 (inelastic): demand curve is relatively steep. Revenue rises when price rises.
|Ed| = 1 (unit elastic): revenue is unchanged when price changes.
|Ed| = 0 (perfectly inelastic): vertical demand curve.
|Ed| = infinity (perfectly elastic): horizontal demand curve.
The midpoint formula (arc elasticity)
Used when you have two discrete points. Instead of choosing one point as the base, you average the two:
Ed = [(Q2 - Q1) / ((Q2 + Q1)/2)] / [(P2 - P1) / ((P2 + P1)/2)]
This gives the same answer regardless of which direction you calculate.
Solving using averages
Plug in the two quantities and two prices directly into the midpoint formula. Compute the numerator (percentage change in Q using the average Q) and the denominator (percentage change in P using the average P) separately, then divide.
Solving using calculus
Point elasticity uses the derivative: Ed = (dQ/dP) * (P/Q). Take the derivative of the demand function with respect to P, then multiply by the ratio P/Q at the specific point. This is exact at a single point rather than an approximation over an arc.
The basic formula
Es = (%change in Qs) / (%change in P) = (dQs/Qs) / (dP/P).
Sign of the calculation
Because of the law of supply, Es is always positive (price and quantity supplied move in the same direction).
The midpoint formula
Same structure as for demand:
Es = [(Q2 - Q1) / ((Q2 + Q1)/2)] / [(P2 - P1) / ((P2 + P1)/2)]
Solving using calculus
Es = (dQs/dP) * (P/Qs). Take the derivative of the supply function with respect to P, then multiply by the ratio P/Qs at the point of interest.
The basic formula
Ei = (%change in Qd) / (%change in income).
What the sign tells you
Ei > 0: the good is a normal good (demand rises with income).
Ei < 0: the good is an inferior good (demand falls as income rises).
What the size tells you (for normal goods)
0 < Ei < 1: the good is a necessity. Demand rises with income, but less than proportionally.
Ei > 1: the good is a luxury. Demand rises faster than income.
The midpoint formula
Ei = [(Q2 - Q1) / ((Q2 + Q1)/2)] / [(M2 - M1) / ((M2 + M1)/2)], where M is income.
Solving using calculus
Ei = (dQ/dM) * (M/Q). Take the derivative of the demand function with respect to income, then multiply by the ratio M/Q at the point of interest.
The basic formula
Exy = (%change in Qd of good X) / (%change in price of good Y).
What the sign tells you
Exy > 0: goods X and Y are substitutes. When Y gets more expensive, consumers switch to X.
Exy < 0: goods X and Y are complements. When Y gets more expensive, consumers buy less of both.
Exy = 0 (or close to it): the goods are unrelated.
The midpoint formula
Exy = [(Qx2 - Qx1) / ((Qx2 + Qx1)/2)] / [(Py2 - Py1) / ((Py2 + Py1)/2)]
Solving using calculus
Exy = (dQx/dPy) * (Py/Qx). Take the partial derivative of the demand function for X with respect to the price of Y, then multiply by the ratio Py/Qx.
E_d = \frac{\%\Delta Q_d}{\%\Delta P} = \frac{dQ}{dP} \cdot \frac{P}{Q}E_s = \frac{\%\Delta Q_s}{\%\Delta P} = \frac{dQ_s}{dP} \cdot \frac{P}{Q_s}E_i = \frac{\%\Delta Q_d}{\%\Delta M} = \frac{dQ}{dM} \cdot \frac{M}{Q}E_{xy} = \frac{\%\Delta Q_x}{\%\Delta P_y} = \frac{dQ_x}{dP_y} \cdot \frac{P_y}{Q_x}\text{Midpoint (general):} \quad E = \frac{(X_2 - X_1) / \left(\frac{X_2 + X_1}{2}\right)}{(Y_2 - Y_1) / \left(\frac{Y_2 + Y_1}{2}\right)}Students often confuse "elastic" with "steep." A steep demand curve is inelastic, not elastic. A flat curve is elastic. The steeper the curve, the less responsive quantity is to price changes.
Students sometimes forget to take the absolute value when comparing demand elasticities. Ed is always negative, so you must compare |Ed| values to determine whether demand is elastic or inelastic.
Confusing the sign of income elasticity with the sign of price elasticity is common. Price elasticity of demand is negative (law of demand). Income elasticity can be positive or negative, and the sign carries real information (normal vs. inferior).
Students sometimes assume that cross elasticity of zero means the goods are complements. It means they are unrelated. Complements have negative cross elasticity.
Expect to calculate elasticity using both the midpoint formula and the calculus (point elasticity) method. Know when to use which.
The sign interpretation of income and cross elasticity is heavily tested. Be ready to classify goods as normal, inferior, luxury, substitute or complement based on a calculated value.
Questions often ask whether demand is elastic or inelastic at a given point, then follow up with a question about what happens to total revenue if price changes. Know the link: elastic demand means a price increase reduces revenue.
The midpoint formula is the default for discrete data (two price-quantity pairs). The calculus method is the default when you are given a demand function and asked about a specific point.
True or false: A perfectly inelastic demand curve is horizontal. (False. It is vertical.)
Fill in the blank: If income elasticity is -0.4, the good is a(n) ___ good. (Inferior.)
True or false: If cross elasticity between goods A and B is positive, they are complements. (False. They are substitutes.)
Fill in the blank: If |Ed| = 0.6, demand is ___. (Inelastic.)
True or false: The midpoint formula gives the same elasticity value regardless of which point you start from. (True. That is its advantage over the basic percentage-change formula.)
Q: Suppose the price of a good rises from $10 to $14 and quantity demanded falls from 100 to 80. Calculate the price elasticity of demand using the midpoint formula.
A: %change in Q = (80 - 100) / ((80 + 100)/2) = -20/90 = -0.222. %change in P = (14 - 10) / ((14 + 10)/2) = 4/12 = 0.333. Ed = -0.222 / 0.333 = -0.667. |Ed| = 0.667, so demand is inelastic over this range.
Q: Given Qd = 50 - 5P, calculate the price elasticity of demand at P = 4.
A: dQ/dP = -5. At P = 4, Q = 50 - 5(4) = 30. Ed = (-5)(4/30) = -0.667. |Ed| = 0.667, inelastic at this point.
Q: Income rises from $30,000 to $40,000 and quantity demanded of a good rises from 200 to 280. Calculate income elasticity using the midpoint formula and classify the good.
A: %change in Q = (280 - 200)/((280 + 200)/2) = 80/240 = 0.333. %change in M = (40,000 - 30,000)/((40,000 + 30,000)/2) = 10,000/35,000 = 0.286. Ei = 0.333/0.286 = 1.165. Positive, so normal good. Greater than 1, so it is a luxury.
Q: The price of good Y rises from $5 to $7, and the quantity demanded of good X rises from 150 to 180. What is the cross elasticity and what does it tell you?
A: %change in Qx = (180 - 150)/((180 + 150)/2) = 30/165 = 0.182. %change in Py = (7 - 5)/((7 + 5)/2) = 2/6 = 0.333. Exy = 0.182/0.333 = 0.545. Positive, so goods X and Y are substitutes.
Elasticity builds directly on the supply-and-demand framework from Unit 1B. The derivative you learned in Unit 1A is the engine of point elasticity. In Unit 1D, elasticity determines how the burden of a tax is shared between buyers and sellers: the more inelastic side of the market bears the larger share.
Income elasticity connects to consumer theory later in the course, where you will study how consumers allocate budgets across goods as income changes. Cross elasticity connects to market structure: firms in monopolistic competition sell products with high but finite cross elasticity.
Related Terms / Search Tags
ECON 207, elasticity, price elasticity of demand, price elasticity of supply, income elasticity, cross elasticity, midpoint formula, arc elasticity, point elasticity, elastic, inelastic, unit elastic, normal good, inferior good, luxury good, necessity, substitutes, complements, total revenue test, Penn State, PSU microeconomics