Source: Besanko & Braeutigam, Microeconomics 5th ed.
Difficulty: Intermediate | Prerequisites: Part 1 study notes (supply/demand curve shifts, choke price), basic algebra (solving two equations in two unknowns).
Once you know how to shift curves, the next question is: by how much does quantity respond to a price change? That is what price elasticity of demand measures. This part of Chapter 2 also introduces equilibrium solving with multiple variables (exogenous factors like rainfall or income), which is the algebraic foundation for comparative statics throughout the rest of the course. If you are comfortable with setting Qd = Qs and solving, the rest of intermediate micro becomes far more manageable.
Price elasticity of demand tells you the percentage change in quantity demanded for a 1% change in price. When |elasticity| > 1, demand is elastic; when |elasticity| < 1, it is inelastic. For a linear demand curve, elasticity varies along the curve, changing from inelastic at low prices to elastic at high prices, with unit elasticity at the midpoint.
Price elasticity of demand (εQ,P)
The percentage change in quantity demanded divided by the percentage change in price. Formally: εQ,P = (dQ/dP) x (P/Q). In simple terms, it tells you how sensitive buyers are to price changes. A large absolute value means buyers react strongly; a small one means they barely notice.
Elastic demand
When |εQ,P| > 1. A 1% price increase causes quantity demanded to fall by more than 1%. Think of luxury goods or items with many substitutes: people walk away quickly when the price rises.
Inelastic demand
When |εQ,P| < 1. A 1% price increase causes quantity demanded to fall by less than 1%. Think of necessities like petrol or insulin: people keep buying even when the price goes up.
Unit elastic demand
When |εQ,P| = 1. The percentage change in quantity exactly equals the percentage change in price. For a linear demand curve, this occurs at the midpoint.
Exogenous variable
A variable determined outside the model. In the corn market example, rainfall (R) and disposable income (I) are exogenous: they affect supply and demand but are not determined by the market for corn. In simple terms, these are the "given" values you plug in.
Comparative statics
The method of comparing one equilibrium to another after an exogenous variable changes. You solve for equilibrium before the change and after, then compare prices and quantities.
For Qd = a - bP, the derivative dQ/dP = -b. So:
εQ,P = -b x (P / Q)
This means elasticity is not constant along a linear demand curve. It depends on where you are (which P and Q values you plug in).
Given: Qd = 300 - 100P. Here, b = 100, so dQ/dP = -100.
Computing the table:
P | Qd = 300 - 100P | εQ,P = -100 x (P/Q) |
|---|---|---|
0.10 | 290 | -100 x (0.10/290) = -0.034 |
0.45 | 255 | -100 x (0.45/255) = -0.176 |
0.50 | 250 | -100 x (0.50/250) = -0.200 |
0.55 | 245 | -100 x (0.55/245) = -0.224 |
2.50 | 50 | -100 x (2.50/50) = -5.000 |
Unit elasticity occurs where |εQ,P| = 1. Set |-100 x (P/Q)| = 1:
100 x P / (300 - 100P) = 1
100P = 300 - 100P
200P = 300
P = 1.50
Inelastic range: Prices below $1.50 (where |εQ,P| < 1). At low prices, quantity demanded is large relative to price, so the ratio P/Q is small and elasticity is low in absolute value.
Elastic range: Prices above $1.50 (where |εQ,P| > 1). At high prices, quantity demanded is small relative to price, so the ratio P/Q is large and elasticity is high in absolute value.
The slope (dQ/dP = -100) is constant, but the ratio P/Q is not. Near the bottom of the curve (low P, high Q), P/Q is tiny, so elasticity is close to zero. Near the top (high P, low Q), P/Q is large, so elasticity is large in absolute value. The midpoint of the curve is where elasticity equals -1.
Given:
Supply: Qs = 20R + 100P
Demand: Qd = 4000 - 100P + 10I
where R = rainfall, I = disposable income, P = price of corn.
Set Qs = Qd: 20R + 100P = 4000 - 100P + 10I
Collect P terms: 200P = 4000 + 10I - 20R
Solve for P: P = (4000 + 10I - 20R) / 200 = 20 + 0.05I - 0.1R
Substitute back into the supply equation to find Q: Q = 20R + 100(20 + 0.05I - 0.1R) = 20R + 2000 + 5I - 10R = 2000 + 10R + 5I
So the equilibrium expressions are:
P* = 20 + 0.05I - 0.1R
Q* = 2000 + 10R + 5I
Higher R increases supply (Qs shifts right) while demand is unaffected.
From the equilibrium expressions: as R rises, P* falls (the -0.1R term) and Q* rises (the +10R term). This matches the graph: the supply curve shifts right along a fixed demand curve, lowering the equilibrium price and raising the equilibrium quantity.
Lower I reduces demand (Qd shifts left) while supply is unaffected.
From the equilibrium expressions: as I falls, P* falls (the +0.05I term shrinks) and Q* falls (the +5I term shrinks). This matches the graph: the demand curve shifts left along a fixed supply curve, lowering both the equilibrium price and the equilibrium quantity.
Supply curve vertical intercept: Set Qs = 0, solve for P: P = -20R/100 = -R/5 (negative, so the supply curve starts from the origin area).
Demand curve vertical intercept (choke price on demand side): Set Qd = 0, solve for P: P = (4000 + 10I) / 100 = 40 + 0.1I.
Price elasticity of demand (point elasticity)
εQ,P = (dQ/dP) x (P/Q)
For a linear demand curve Q = a - bP: εQ,P = -b x (P/Q)
Unit elasticity price (linear demand Q = a - bP)
Set |εQ,P| = 1 and solve: P = a / (2b). This is always the midpoint of the demand curve.
Equilibrium condition
Qs(P, exogenous variables) = Qd(P, exogenous variables). Solve for P*, then substitute back for Q*.
Comparative statics shortcut
Once you have P* and Q* as functions of the exogenous variables, you can read the effect of any change directly from the signs and magnitudes of the coefficients. A positive coefficient on R in Q* means more rainfall raises equilibrium quantity.
Elasticity matters for pricing strategy. A firm selling a product with inelastic demand (say, insulin) can raise prices without losing many customers, while a firm selling an elastic product (say, a particular brand of bottled water) will see sales collapse if it tries to charge much more than competitors. Governments use elasticity estimates to predict how much revenue a tax will generate: taxing an inelastic good raises more revenue because consumers do not cut back much.
Students often think elasticity is the same as slope. It is not. Slope (dQ/dP) is constant for a linear demand curve, but elasticity varies along the curve because it also depends on the ratio P/Q.
Students sometimes believe that a "steep" demand curve is always inelastic. Steepness depends on the units of measurement on the axes, so visual steepness can be misleading. Use the formula.
When solving for equilibrium, students occasionally forget that Qs and Qd are both functions of P and set P equal to something on only one side. Always set the full Qs expression equal to the full Qd expression.
A negative elasticity value does not mean "bad." Demand curves are (almost) always downward-sloping, so εQ,P is negative by default. What matters is the absolute value.
⚠️ Elasticity calculation questions are very common. You will likely need to compute εQ,P at a specific price point using the formula, not just know the concept.
⚠️ Expect to be asked where on a linear demand curve demand is elastic vs. inelastic, and why. The answer is always: above the midpoint price = elastic, below = inelastic.
⚠️ Equilibrium problems with exogenous variables (like Question 4) test whether you can do algebra under pressure. Practice the steps until they are automatic: set Qs = Qd, collect terms, solve for P, substitute back.
⚠️ Comparative statics questions ask: "What happens to P* and Q* when R increases?" Read the answer directly from your solved expressions.
True or False: Elasticity is constant along a linear demand curve. (False. Slope is constant; elasticity varies.)
Fill in the blank: For Q = 300 - 100P, demand is unit elastic at P = ______. (P = 1.50.)
True or False: If |εQ,P| = 3, a 1% price increase leads to a 3% decrease in quantity demanded. (True.)
Fill in the blank: To find equilibrium, set ______ = ______. (Qs = Qd.)
True or False: In the corn model, an increase in rainfall raises the equilibrium price. (False. It lowers the equilibrium price.)
Q: Given Qd = 500 - 50P, calculate the price elasticity of demand at P = 4.
A: First find Q at P = 4: Q = 500 - 50(4) = 300. Then εQ,P = -50 x (4/300) = -0.667. Since |0.667| < 1, demand is inelastic at this price.
Q: For the same demand curve Qd = 500 - 50P, at what price is demand unit elastic?
A: Use P = a/(2b) = 500/(2 x 50) = 5.00. Demand is unit elastic at P = $5.00.
Q: In the corn market with Qs = 20R + 100P and Qd = 4000 - 100P + 10I, suppose R = 10 and I = 100. Find the equilibrium price and quantity.
A: Set Qs = Qd: 20(10) + 100P = 4000 - 100P + 10(100). So 200 + 100P = 4000 - 100P + 1000. Then 200P = 4800. P* = 24. Q* = 200 + 100(24) = 2600.
Q: In the same corn market, if rainfall increases from R = 10 to R = 15 (holding I = 100), what happens to equilibrium price and quantity?
A: Using P* = 20 + 0.05(100) - 0.1R: at R = 10, P* = 24; at R = 15, P* = 23.50. Price falls by $0.50. Using Q* = 2000 + 10R + 5(100): at R = 10, Q* = 2600; at R = 15, Q* = 2650. Quantity rises by 50 units. More rain shifts supply right, lowering price and raising quantity.
Price elasticity of demand connects directly to total revenue analysis (Chapter 2/3): when demand is elastic, a price cut increases total revenue; when inelastic, a price cut decreases it. The comparative statics method used in the corn market problem is the same technique you will use for tax incidence analysis, where you solve for equilibrium with and without a tax. Elasticity also feeds into monopoly pricing (later chapters), where a profit-maximising firm sets marginal revenue equal to marginal cost, and marginal revenue depends on elasticity.
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