Tags: marginal analysis, marginal revenue, marginal cost, marginal profit, MR equals MC, profit maximisation, monopolist pricing, demand function derivation, total revenue, microeconomic theory, Texas A&M
Marginal analysis is the core tool for managerial decision-making. The central rule: keep increasing output as long as the additional (marginal) profit from one more unit is positive. At the optimum, marginal profit equals zero, which is equivalent to setting marginal revenue equal to marginal cost. This section covers deriving demand functions, computing MR from a demand curve, and solving for the profit-maximising price and quantity.
Marginal revenue (MR)
The amount of additional revenue the firm earns from selling one more unit of output. Formally, the derivative of total revenue with respect to quantity.
Marginal cost (MC)
The additional cost of producing one more unit of output. Formally, the derivative of total cost with respect to quantity.
Marginal profit (Mπ)
The change in profit from producing and selling one more unit. Equal to MR minus MC. At the profit-maximising output, marginal profit equals zero.
MR = MC rule
The profit-maximising condition. A firm should expand output until the revenue gained from the last unit exactly equals the cost of producing it.
The MR "trick" for linear inverse demand
If the inverse demand curve is P = a – bQ, then total revenue is R = aQ – bQ², and marginal revenue is MR = a – 2bQ. The MR curve has the same intercept as the demand curve but twice the slope.
Demand function
An equation relating quantity demanded to price (and potentially other variables). Can be written as Q = f(P) or in inverse form as P = f(Q).
Monopolist
A single seller in a market. The monopolist faces the entire market demand curve and chooses price and quantity to maximise profit, subject to the demand constraint.
This comes up in applied problems (e.g. Lam's Garden restaurant).
You need two observations: (P₁, Q₁) and (P₂, Q₂)
Calculate the slope: m = (P₂ – P₁) / (Q₂ – Q₁)
Plug one point into P = mQ + b to solve for the intercept b
Example: Sweet and Sour Chicken was $3.85 at Q = 125, then $4.29 at Q = 100
Slope = (4.29 – 3.85) / (100 – 125) = 0.44 / (–25) = –0.0176
4.29 = –0.0176(100) + b → b = 6.05
Inverse demand: P = –0.0176Qd + 6.05
Demand function: Qd = –56.818P + 343.75
Worked example: Conigan Box Company
Demand: P = 100 – Q
MC = $10 (constant), so MC = AC
Total revenue: R = PQ = 100Q – Q²
MR = 100 – 2Q (the "trick": double the slope coefficient)
Set MR = MC: 100 – 2Q = 10 → Q = 45
Price: P = 100 – 45 = 55
Profit: TR – TC = (55 × 45) – (10 × 45) = 2,475 – 450 = 2,025
Worked example: another monopolist
MR = 45 – 0.05Q, MC = 30
Set MR = MC: 45 – 0.05Q = 30 → Q = 300
P = 45 – 0.025(300) = 37.5
Profit = (37.5)(300) – 30(300) = 11,250 – 9,000 = 2,250
You can also write out the profit function π = TR – TC, take its derivative, and set Mπ = 0.
Worked example:
P = 100 – 2Q, C = 200 + 4Q
π = (100 – 2Q)Q – (200 + 4Q) = 100Q – 2Q² – 200 – 4Q = –2Q² + 96Q – 200
Mπ = –4Q + 96 = 0 → Q* = 24
P* = 100 – 2(24) = 52
Verify via MR = MC:
MR = 100 – 4Q, MC = 4
100 – 4Q = 4 → Q = 24 ✓
π = –200 + 80Q – 0.2Q²
Mπ = 80 – 0.4Q
Set to zero: 80 – 0.4Q = 0 → Q* = 200
π = –150 + 360Q – 36Q²
Mπ = 360 – 72Q
Set to zero: 360 – 72Q = 0 → Q* = 5
When given a table of P, Q, TR, and TC:
Profit at any Q = TR – TC
Marginal profit of the nth unit = Profit(n) – Profit(n–1)
Example from the table: Profit at Q = 3 is 39 – 12 = 27. Marginal profit of the 3rd unit is 27 – 21 = 6.
If Mπ > 0 at current output, the firm should increase output (because MR > MC).
Keep increasing until Mπ = 0.
Do not increase to "full capacity" unless MR > MC all the way there.
A competitive firm sells at a fixed market price, so MR = P (constant).
If MR > MC at all output levels, the firm should keep expanding until it hits its capacity limit.
Example: P = $12, C = 200 + 4Q → MC = $4. Since MR ($12) always exceeds MC ($4), produce as much as possible.
A change in fixed cost does not affect marginal revenue or marginal cost (since neither depends on fixed cost).
Therefore, the profit-maximising output and price do not change when fixed costs change. Only total profit shifts.
Revenue is maximised where MR = 0, not where MR = MC.
Example: P = 2,500 – 10Q → R = 2,500Q – 10Q², MR = 2,500 – 20Q
Set MR = 0: Q = 125
MR trick for linear demand: If P = a – bQ, then MR = a – 2bQ
Profit-maximising condition: MR = MC, or equivalently Mπ = dπ/dQ = 0
Marginal profit from a table: Mπ(n) = [TR(n) – TC(n)] – [TR(n–1) – TC(n–1)]
Revenue-maximising condition: MR = 0
⚠️ The MR "trick" only works for linear inverse demand (P = a – bQ). MR has the same intercept but twice the slope: MR = a – 2bQ.
⚠️ "Marginal revenue" is the additional revenue from one more unit. The exam answer uses exactly this phrasing: "the amount of additional revenue from a unit increase in output."
⚠️ If marginal profit is positive, increase output. If marginal profit is negative, decrease output. Stop where Mπ = 0.
⚠️ When monopoly MC increases, price increases and quantity decreases.
⚠️ A decrease in fixed cost does not change MR or MC, so the optimal Q and P stay the same.
⚠️ Revenue maximisation (MR = 0) gives a different answer from profit maximisation (MR = MC). Do not confuse them.
⚠️ For a competitive firm with constant MR > MC everywhere, the answer is to produce at capacity.
Q: Conigan Box has demand P = 100 – Q and constant MC = $10. What are the profit-maximising price and quantity?
A: Set MR = MC. MR = 100 – 2Q = 10 → Q = 45, P = 55. Profit = 45 × 45 = 2,025.
Q: What is marginal revenue?
A: The amount of additional revenue from a unit increase in output.
Q: If a monopolist's MC rises, what happens to price and quantity?
A: Price increases and quantity decreases.
Q: A firm's profit is π = –200 + 80Q – 0.2Q². What is the profit-maximising output?
A: Mπ = 80 – 0.4Q = 0 → Q = 200.
Q: A firm's profit is π = –150 + 360Q – 36Q². What is the optimal output?
A: Mπ = 360 – 72Q = 0 → Q = 5.
Q: A firm is producing where marginal profit is positive. What should it do?
A: Increase output because MR is greater than MC. Continue until marginal profit equals zero.
Q: A firm produces at its profit-maximising output. Fixed costs fall. What happens to MR and MC?
A: Neither marginal revenue nor marginal cost changes. Fixed costs do not affect marginal values.
Q: If P = 2,500 – 10Q, what quantity maximises total revenue?
A: MR = 2,500 – 20Q = 0 → Q = 125.
Q: Using the revenue-cost table, what is profit at Q = 3 and what is the marginal profit of the 3rd unit?
A: Profit at Q = 3 is TR – TC = 39 – 12 = 27. Marginal profit of the 3rd unit is 27 – 21 = 6.
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