Source: Besanko & Braeutigam, Microeconomics 5th ed., Ch. 2
Tags: choke price, price elasticity of demand, elastic demand, inelastic demand, linear demand curve, multi-variable equilibrium, supply and demand equations, rainfall, income, intermediate micro
Difficulty: Intermediate Prerequisites: You should be comfortable with supply and demand curve shifts (see the companion study notes on Curve Shifts). You also need basic algebra, including solving linear equations for an unknown variable.
This set of notes picks up where the curve-shift notes left off. Instead of just knowing which direction a curve moves, you now learn to measure how responsive quantity is to price (elasticity), find the price that kills off demand entirely (choke price), and work with supply and demand as algebraic equations that include variables like rainfall and income. These tools let you solve for exact equilibrium values and predict how those values change when external conditions shift. Most exam problems in intermediate micro require this level of precision.
The choke price is the price at which quantity demanded falls to zero. Price elasticity of demand measures how sensitive buyers are to price changes, on a percentage basis. Along a linear demand curve, demand is elastic at high prices and inelastic at low prices. When supply and demand equations include extra variables (like income or weather), changing those variables shifts one curve and produces a new algebraic equilibrium.
Choke price
The price at which quantity demanded drops to zero. For a linear demand curve Q = a - bP, you find it by setting Q = 0 and solving for P. It is the vertical intercept of the demand curve (when price is on the vertical axis).
Think of it as the price so high that nobody buys the good at all.
Price elasticity of demand (εQ,P)
A measure of how sensitive quantity demanded is to a change in price. Formally: εQ,P = (% change in Qd) / (% change in P), or equivalently εQ,P = (dQ/dP) × (P/Q). The result is typically negative (price up, quantity down), and we often refer to its absolute value.
In simple terms, it tells you whether a price change causes a big or small reaction in buying behaviour.
Elastic demand
Demand is elastic when |εQ,P| > 1. A 1% price increase causes more than a 1% drop in quantity demanded. Buyers are quite responsive to price.
Inelastic demand
Demand is inelastic when |εQ,P| < 1. A 1% price increase causes less than a 1% drop in quantity demanded. Buyers are relatively unresponsive to price.
Unit elastic demand
Demand is unit elastic when |εQ,P| = 1. A 1% price increase causes exactly a 1% drop in quantity demanded. For a linear demand curve Q = a - bP, this occurs at the midpoint, where P = a/(2b).
Linear demand curve
A demand curve of the form Q = a - bP, which graphs as a straight line. Along a linear demand curve, the slope (dQ/dP = -b) is constant, but elasticity varies: elastic at high prices, unit elastic at the midpoint, and inelastic at low prices.
Multi-variable supply and demand
Supply and demand equations that include variables beyond the good's own price, such as income (I), rainfall (R), or the prices of other goods. For example, Qs = 20R + 100P and Qd = 4000 - 100P + 10I. Changing one of these extra variables shifts the relevant curve and produces a new equilibrium.
Given a linear demand curve Q = a - bP, the choke price is the value of P that makes Q = 0.
Worked example (Besanko Q2):
Demand: Q = 50 - 100P
Set Q = 0: 0 = 50 - 100P
Solve: 100P = 50, so P = 0.50
The choke price is $0.50. At any price above $0.50, nobody buys.
Graphically, this is where the demand curve hits the price axis (the vertical axis when price is plotted vertically).
For a linear demand curve Q = a - bP, the point elasticity formula is:
εQ,P = (dQ/dP) × (P/Q) = -b × (P/Q)
Because the slope (-b) is constant but P/Q changes along the curve, elasticity varies from point to point.
Worked example (Besanko Q3): Bagel demand, Qd = 300 - 100P
Here b = 100 (so dQ/dP = -100). The elasticity at any price is εQ,P = -100 × (P/Q).
P | Qd | εQ,P | Elastic or inelastic? |
|---|---|---|---|
$0.10 | 290 | -0.035 | Inelastic |
$0.45 | 255 | -0.176 | Inelastic |
$0.50 | 250 | -0.200 | Inelastic |
$0.55 | 245 | -0.225 | Inelastic |
$1.50 | 150 | -1.000 | Unit elastic |
$2.50 | 50 | -5.000 | Elastic |
Key results from this example:
At prices below $1.50, demand is inelastic (|ε| < 1).
At P = $1.50, demand is unit elastic (|ε| = 1). This is the midpoint of the demand curve: P = a/(2b) = 300/(2 × 100) = 1.50.
At prices above $1.50, demand is elastic (|ε| > 1).
This pattern holds for every linear demand curve: elastic in the upper half, inelastic in the lower half, unit elastic at the midpoint.
When supply or demand depends on variables beyond the good's own price, changing one of those variables shifts the relevant curve.
Worked example (Besanko Q4): Corn market
Supply: Qs = 20R + 100P (R = rainfall)
Demand: Qd = 4000 - 100P + 10I (I = disposable income)
Effect of an increase in rainfall (R rises):
R enters the supply equation with a positive coefficient (+20). More rainfall means more corn supplied at every price.
The supply curve shifts rightward (S to S₂).
Demand is unchanged.
Result: equilibrium price falls (P to P₂) and equilibrium quantity rises (Q to Q₂).
Intuition: better growing conditions increase supply, which pushes price down and quantity up.
Effect of a decrease in disposable income (I falls):
I enters the demand equation with a positive coefficient (+10). Lower income means less corn demanded at every price (corn is a normal good in this model).
The demand curve shifts leftward (D to D₂).
Supply is unchanged.
Result: equilibrium price falls (P to P₂) and equilibrium quantity falls (Q₂ < Q).
Intuition: poorer consumers buy less corn, reducing both the market-clearing price and quantity.
Finding equilibrium algebraically:
Set Qs = Qd and solve for P. For example, if R = 10 and I = 100:
20(10) + 100P = 4000 - 100P + 10(100)
200 + 100P = 5000 - 100P
200P = 4800
P = 24
Q = 200 + 100(24) = 2600
Choke price (linear demand Q = a - bP):
Set Q = 0 and solve: P_choke = a / b
Point price elasticity of demand:
εQ,P = (dQ/dP) × (P/Q)
For a linear curve Q = a - bP, this simplifies to: εQ,P = -b × (P/Q)
Midpoint (unit elastic point) of a linear demand curve:
P_midpoint = a / (2b), and Q_midpoint = a / 2
Finding equilibrium with multi-variable equations:
Set Qs = Qd, substitute known values of the extra variables (R, I, etc.), and solve for P. Then plug P back into either equation to find Q.
Elasticity is central to pricing decisions. A firm selling a product with inelastic demand (petrol, insulin) can raise prices and see revenue increase, because the quantity drop is proportionally small. A firm facing elastic demand (a particular brand of trainers when close substitutes exist) would lose more revenue than it gains from a price hike.
The choke price concept appears in energy markets. Regulators and analysts estimate the choke price for oil, at which electric vehicles and renewables fully displace petroleum demand.
Students often assume elasticity is the same as slope. It is not. Slope (dQ/dP) is constant along a linear demand curve, but elasticity changes because the P/Q ratio changes. Two curves can have the same slope but different elasticities at a given price.
Students sometimes think demand is either elastic or inelastic everywhere. Along a linear demand curve, it is both, depending on where you are. The upper half is elastic; the lower half is inelastic.
When computing the choke price, students occasionally set P = 0 instead of Q = 0. The choke price is the price that chokes off demand (Q = 0), not the quantity at a price of zero.
In multi-variable problems, students sometimes forget to hold the extra variables constant when drawing a single supply or demand curve. Each curve is drawn for a specific value of R, I, etc. Changing that value shifts the curve.
⚠️ Elasticity calculations are a near-certainty on any intermediate micro exam. Know the formula and be able to apply it at specific price-quantity pairs.
⚠️ The choke price is a quick calculation worth easy marks. Do not confuse it with the x-intercept (which is the quantity when P = 0).
⚠️ For multi-variable equilibrium problems, exams typically ask you to solve for equilibrium P and Q given specific values of R, I, etc., and then ask what happens when one of those values changes. Be comfortable doing both the algebra and the graphical analysis.
Fill in the blank: For the demand curve Q = 50 - 100P, the choke price is ______.
$0.50
True or False: Along a linear demand curve, elasticity is constant.
False. Slope is constant, but elasticity varies because the P/Q ratio changes.
Fill in the blank: Demand is elastic when |εQ,P| is ______ than 1.
Greater.
True or False: If rainfall enters a supply equation with a positive coefficient, an increase in rainfall shifts supply to the left.
False. A positive coefficient means more rainfall increases supply, shifting it rightward.
Fill in the blank: The midpoint of a linear demand curve Q = 300 - 100P occurs at P = ______ and Q = ______.
P = 1.50, Q = 150.
Q: A linear demand curve is given by Q = 200 - 50P. What is the choke price? At what price is demand unit elastic?
A: Choke price: set Q = 0, so 50P = 200, P = $4.00. Unit elastic at the midpoint: P = 200/(2 × 50) = $2.00, Q = 100.
Q: Given the demand curve Qd = 300 - 100P, calculate the price elasticity of demand at P = $0.50. Is demand elastic or inelastic at this price?
A: At P = 0.50, Qd = 300 - 100(0.50) = 250. εQ,P = -100 × (0.50/250) = -0.20. Since |ε| = 0.20 < 1, demand is inelastic.
Q: Suppose Qs = 20R + 100P and Qd = 4000 - 100P + 10I. If R = 10 and I = 200, find the equilibrium price and quantity.
A: Set Qs = Qd: 20(10) + 100P = 4000 - 100P + 10(200). This gives 200 + 100P = 6000 - 100P, so 200P = 5800, P = 29. Q = 200 + 100(29) = 3100.
Q: In the corn market above, what happens to the equilibrium if disposable income (I) increases? Which curve shifts, and in which direction?
A: I has a positive coefficient in the demand equation, so higher income raises Qd at every price. The demand curve shifts rightward. Supply stays put. The new equilibrium has a higher price and a higher quantity.
Q: Why does a linear demand curve have both elastic and inelastic regions, even though its slope is constant?
A: Elasticity is not the same as slope. It equals the slope times (P/Q). Near the top of the curve, P is large and Q is small, making the P/Q ratio large and |ε| > 1 (elastic). Near the bottom, P is small and Q is large, making P/Q small and |ε| < 1 (inelastic).
Elasticity is the bridge between Chapter 2 and nearly everything that follows. In consumer theory (Ch. 4-5), you will see how elasticity relates to the shape of indifference curves and budget constraints. In producer theory and market structure chapters, elasticity determines how firms set prices: a monopolist, for instance, always prices in the elastic portion of the demand curve because total revenue falls in the inelastic region.
The multi-variable equilibrium technique from Q4 is the same tool you will use for comparative statics throughout the course. Any time an exam gives you supply and demand equations with an extra parameter and asks "what happens if this parameter changes," you are doing exactly what you practised here.
The choke price reappears in welfare analysis. Consumer surplus is the area of the triangle between the demand curve and the equilibrium price, and the choke price gives you the top vertex of that triangle.
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