Difficulty: Intermediate | Prerequisites: Demand curves, total revenue, basic derivatives.
Price elasticity of demand and marginal revenue are two sides of the same coin: both describe what happens to a firm's revenue when it changes its price. This material bridges the gap between demand analysis (which you covered earlier) and the theory of monopoly pricing (which comes next). You need a solid demand curve and the ability to take a derivative or compute a simple ratio. If you can do those, everything here is mechanical.
Price elasticity of demand measures how sensitive quantity demanded is to a change in price. Marginal revenue is the additional revenue from selling one more unit, which for a downward-sloping demand curve is always less than the price. There is a direct link between the two: MR = P(1 + 1/E_d), and MR = 0 exactly where demand is unit elastic.
Price elasticity of demand (E_d)
The percentage change in quantity demanded divided by the percentage change in price. For a linear demand curve q = a - bP, the point elasticity is E_d = (dq/dP)(P/q) = -b(P/q). It is always negative for a downward-sloping demand curve. In simple terms, it tells you how much buyers pull back when the price goes up.
Elastic demand
Demand where |E_d| > 1. A 1% price increase causes more than a 1% drop in quantity. Total revenue falls when price rises. Think of it as: buyers are very sensitive to price here.
Inelastic demand
Demand where |E_d| < 1. A 1% price increase causes less than a 1% drop in quantity. Total revenue rises when price rises. Buyers are relatively insensitive to price.
Unit elastic demand
Demand where |E_d| = 1. A 1% price increase causes exactly a 1% drop in quantity. Total revenue is at its maximum. Marginal revenue equals zero at this point.
Marginal revenue (MR)
The change in total revenue from selling one additional unit of output. For a linear inverse demand P = a - bq, MR = a - 2bq. In simple terms, it is the extra money you bring in from the last unit sold, after accounting for the fact that you had to lower the price on all units.
The demand curve used in this problem set
q = 24 - 3P
Inverse demand: P = 8 - (1/3)q
Step 1: Find the elasticity at a given price
The slope of the demand curve is dq/dP = -3.
At any price P, quantity is q = 24 - 3P.
Point elasticity: E_d = (dq/dP)(P/q) = -3(P/q) = -3P/(24 - 3P)
Step 2: Compute marginal revenue
Total revenue: TR = P * q = P(24 - 3P) = 24P - 3P^2
MR = dTR/dq. Using the inverse demand P = 8 - (1/3)q: TR = 8q - (1/3)q^2, so MR = 8 - (2/3)q.
Alternatively, MR = P(1 + 1/E_d).
Worked answers from the problem set
P = $2: q = 18, E_d = -3(2)/18 = -1/3 (inelastic), MR = 8 - (2/3)(18) = -4
P = $3: q = 15, E_d = -3(3)/15 = -9/15 = -3/5 (inelastic), MR = 8 - (2/3)(15) = -2
P = $4: q = 12, E_d = -3(4)/12 = -1 (unit elastic), MR = 8 - (2/3)(12) = 0
P = $6: q = 6, E_d = -3(6)/6 = -3 (elastic), MR = 8 - (2/3)(6) = 4
The pattern
As price rises from $2 to $6, demand moves from inelastic to elastic. MR goes from negative to positive, crossing zero at the unit-elastic midpoint (P = $4, q = 12). Revenue is maximised at that midpoint.
Point price elasticity of demand (linear demand q = a - bP)
E_d = -b(P/q) = -bP/(a - bP)
Marginal revenue (from inverse demand P = a - bq)
MR = dTR/dq = a - 2bq
The MR curve has the same vertical intercept as the inverse demand curve but twice the slope.
Relationship between MR and elasticity
MR = P(1 + 1/E_d)
When E_d = -1: MR = 0. When |E_d| > 1 (elastic): MR > 0. When |E_d| < 1 (inelastic): MR < 0.
Revenue-maximising price
Set MR = 0, which happens at the midpoint of a linear demand curve: q = a/2, P = a/(2b) in inverse demand terms.
Firms with market power (monopolists, firms with differentiated products) use the relationship between elasticity and marginal revenue to set prices. A firm operating on the inelastic portion of its demand curve (where MR < 0) could raise its price, sell fewer units, and increase total revenue, so it would never maximise profit there. This is why monopoly pricing always lands on the elastic portion of the demand curve.
Students often confuse the slope of the demand curve with elasticity. The slope (dq/dP) is constant along a linear demand curve, but elasticity varies at every point because it depends on the ratio P/q.
Elasticity is not always negative 1 at the midpoint of any demand curve. That property holds for linear demand curves specifically.
Students sometimes think MR is the same as price. MR equals price only for a perfectly competitive firm (horizontal demand curve). For a downward-sloping demand, MR < P because lowering the price to sell one more unit also lowers the price on all previous units.
A common error is computing elasticity as just the slope (-3 in this problem), forgetting to multiply by P/q.
⚠️ You will likely be given a demand curve and asked to fill in a table of elasticities and marginal revenues at various prices. Know the formulas cold.
⚠️ Expect a question asking you to identify where total revenue is maximised (where MR = 0, where |E_d| = 1).
⚠️ The relationship MR = P(1 + 1/E_d) is a favourite exam formula. Be able to use it both ways: given E_d, find MR; given MR and P, find E_d.
⚠️ Know why a profit-maximising monopolist never operates on the inelastic portion of the demand curve.
True or false: Elasticity is constant along a linear demand curve. False. The slope is constant, but elasticity changes because P/q changes.
Fill in the blank: MR = 0 when demand is ______. Unit elastic (E_d = -1).
True or false: If demand is inelastic and the firm raises its price, total revenue falls. False. Total revenue rises when demand is inelastic and price increases.
Fill in the blank: For a linear inverse demand P = a - bq, the MR curve is MR = ______. a - 2bq.
True or false: Marginal revenue is always less than price for a firm facing a downward-sloping demand curve. True.
Q: A firm faces q = 30 - 5P. What is the price elasticity of demand at P = $3?
A: At P = 3, q = 30 - 15 = 15. E_d = -5(3/15) = -1. Demand is unit elastic at P = $3.
Q: Using the same demand curve, what is marginal revenue at q = 15?
A: Inverse demand: P = 6 - (1/5)q. MR = 6 - (2/5)q = 6 - (2/5)(15) = 6 - 6 = 0. MR = 0 confirms unit elasticity.
Q: A monopolist faces q = 24 - 3P. At what price is total revenue maximised?
A: Set MR = 0. Inverse demand: P = 8 - (1/3)q, so MR = 8 - (2/3)q = 0, giving q = 12 and P = $4. Total revenue is maximised at P = $4.
Q: If E_d = -2 and P = $10, what is MR?
A: MR = P(1 + 1/E_d) = 10(1 + 1/(-2)) = 10(1 - 0.5) = 10(0.5) = $5.
Elasticity and MR feed directly into monopoly pricing: the profit-maximising monopolist sets MR = MC, and the elasticity at that price determines the markup over marginal cost (the Lerner index = -1/E_d). This material also connects to price discrimination, where a firm charges different prices to groups with different elasticities. In later chapters on oligopoly, you will see how the number of firms in the market affects the residual demand elasticity each firm faces.
Price elasticity of demand, point elasticity, arc elasticity, elastic, inelastic, unit elastic, marginal revenue, total revenue, MR curve, inverse demand, linear demand, revenue maximisation, monopoly pricing, Lerner index, markup, demand sensitivity, ECON 302, intermediate microeconomics, Penn State, McLeod