Source: Practice MCQs for Exam 1, Texas A&M University
Tags: price elasticity of demand, income elasticity, cross-price elasticity, elastic supply, inelastic demand, unit elastic, total revenue, point elasticity, linear demand curve elasticity, vertical demand curve, Engel curve
Elasticity measures how responsive one variable is to a percentage change in another. Price elasticity of demand, income elasticity, and cross-price elasticity each tell you something different about a good. Along a linear demand curve, the slope is constant but elasticity varies. The relationship between elasticity and total revenue is a staple exam topic: when demand is inelastic, a price increase raises total revenue.
Price elasticity of demand
The percentage change in quantity demanded divided by the percentage change in price. It is typically negative for a downward-sloping demand curve.
Income elasticity of demand
The percentage change in quantity demanded resulting from a one percent increase in income. Positive for normal goods, negative for inferior goods.
Cross-price elasticity of demand
The percentage change in quantity demanded of one good resulting from a one percent increase in the price of another good. Positive for substitutes, negative for complements.
Elastic demand
Demand where the absolute value of price elasticity is greater than 1. A 1% price change causes a more-than-1% change in quantity demanded.
Inelastic demand
Demand where the absolute value of price elasticity is less than 1. A 1% price change causes a less-than-1% change in quantity demanded.
Unit elastic demand
Price elasticity of exactly -1 (absolute value 1). Total revenue is at its maximum at the unit-elastic point.
Elastic supply
A 1% change in price causes a larger percentage change in quantity supplied. The absolute value of supply elasticity exceeds 1.
Engel curve
A curve showing the relationship between income and the quantity demanded of a good, holding prices constant. For normal goods it slopes upward; for inferior goods it slopes downward. You can identify a normal good by inspecting its Engel curve.
Price elasticity of demand = (% change in quantity demanded) / (% change in price).
This is the definition. It is not the slope of the demand curve, and it is not the change in quantity divided by the change in price (that would be the slope, missing the percentage conversion).
For a demand function Q = f(P), the point price elasticity at a specific (P, Q) pair is:
E = (dQ/dP) × (P/Q)
Where dQ/dP is the derivative of Q with respect to P (the slope coefficient on P in a linear demand equation).
Along any downward-sloping straight-line demand curve:
The slope is constant (it is a straight line).
The price elasticity varies from point to point.
Near the top of the curve (high P, low Q), demand is elastic.
Near the bottom (low P, high Q), demand is inelastic.
At the midpoint, demand is unit elastic.
This is a commonly tested distinction. Constant slope does not mean constant elasticity.
A vertical demand curve means quantity demanded does not change regardless of price. The price elasticity is zero (perfectly inelastic).
Normal good: income elasticity is positive (quantity demanded rises with income).
Inferior good: income elasticity is negative (quantity demanded falls with income).
You identify a normal good by inspecting its Engel curve. An upward-sloping Engel curve indicates a normal good.
Positive cross-price elasticity: the goods are substitutes. A rise in the price of good 1 increases the quantity demanded of good 2.
Negative cross-price elasticity: the goods are complements. A rise in the price of good 1 decreases the quantity demanded of good 2.
If demand is inelastic (|E| < 1): a price increase raises total revenue. Quantity falls proportionally less than price rises, so revenue goes up.
If demand is elastic (|E| > 1): a price increase decreases total revenue.
If demand is unit elastic (|E| = 1): total revenue does not change with a small price change.
The total revenue curve derived from a linear downward-sloping demand curve is inverted U-shaped. Revenue rises as price increases from zero, peaks at the unit-elastic midpoint, then falls.
When a good is price inelastic, consumer expenditure on that good increases when price increases. This follows directly from the total revenue relationship: if quantity falls by a smaller percentage than price rises, total spending (P × Q) goes up.
Price elasticity of demand:
E_p = (% ΔQ) / (% ΔP) = (dQ/dP) × (P/Q)
Income elasticity of demand:
E_I = (% ΔQ) / (% ΔI) = (dQ/dI) × (I/Q)
Cross-price elasticity:
E_xy = (% ΔQ_x) / (% ΔP_y) = (dQ_x/dP_y) × (P_y/Q_x)
Demand: Q = 240 - 4P_e + 2I + P_b + A
Given: Q = 240, P_e = 10, P_b = 10, A = 2.
Step 1: Find I.
240 = 240 - 4(10) + 2I + 10 + 2
240 = 240 - 40 + 2I + 12
240 = 212 + 2I
2I = 28
I = 14
Step 2: Point price elasticity at P_e = 10, Q = 240.
dQ/dP_e = -4 (the coefficient on P_e).
E = (-4) × (10/240) = -40/240 = -1/6.
Step 3: Is demand elastic or inelastic?
|E| = 1/6 < 1, so demand is inelastic.
Step 4: Effect of a small price increase on total revenue.
Since demand is inelastic, a price increase raises total revenue. (Quantity falls proportionally less than price rises.)
Step 5: Is the good normal or inferior?
The coefficient on I is +2 (positive). As income rises, quantity demanded rises. The good is normal.
⚠️ Along a linear demand curve, the slope is constant but elasticity varies. This is tested almost every time.
⚠️ A vertical demand curve has zero elasticity, not infinite.
⚠️ Cross-price elasticity negative = complements. If a rise in the price of good A shifts the demand for good B to the left, A and B are complements.
⚠️ When demand is inelastic, raising the price increases total revenue (and consumer expenditure). This is the single most common elasticity-revenue link on exams.
⚠️ The total revenue curve from a linear demand curve is inverted U-shaped.
⚠️ To recognise a normal good, check the Engel curve (upward slope) or the sign of income elasticity (positive).
Q: The price elasticity of demand equals what?
A: The percentage change in quantity demanded divided by the percentage change in price.
Q: Along a downward-sloping straight-line demand curve, what stays constant and what varies?
A: The slope is constant, but the price elasticity varies from elastic (top) to inelastic (bottom).
Q: A good with a vertical demand curve has what kind of elasticity?
A: Zero elasticity (perfectly inelastic). Quantity does not respond to price at all.
Q: If a rise in the price of good 1 decreases the quantity demanded of good 2, what is the sign of the cross-price elasticity?
A: Negative. The goods are complements.
Q: Demand is inelastic. Price increases. What happens to total revenue?
A: Total revenue increases. The percentage drop in quantity is smaller than the percentage rise in price.
Q: In the Scenario 1 demand function Q = 240 - 4P_e + 2I + P_b + A, with the given values, what is the point price elasticity?
A: -1/6. Calculated as (-4) × (10/240).
Q: In Scenario 1, are erasers a normal or inferior good?
A: Normal. The coefficient on income (I) is +2, which is positive.
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