Source: Lecture notes / textbook Ch. 2 (Texas A&M University)
Tags: MR equals MC, marginal revenue, marginal cost, profit maximisation condition, sensitivity analysis, what-if analysis, pure selling, revenue maximisation, franchiser-franchisee conflict, multiproduct pricing, substitutes, complements, interdependent demand, exchange rates, t-shirt problem, word problem
The MR = MC rule is the cleanest route to optimal price and quantity: keep producing as long as the revenue from one more unit exceeds its cost, and stop when they are equal. This framework extends to sensitivity analysis (fixed costs do not change the optimum; variable cost changes do), pure selling problems where MC is roughly zero, multiproduct firms with interdependent demand, and international pricing affected by exchange-rate shifts.
Marginal revenue (MR)
The increase in total revenue from producing one additional (infinitesimally small) unit of output. Mathematically: MR = dR/dQ.
Marginal cost (MC)
The increase in total cost from producing one additional unit of output. Mathematically: MC = dC/dQ.
MR = MC rule (profit maximisation condition)
At the profit-maximising output, marginal revenue equals marginal cost. If MR > MC, the firm should produce more. If MR < MC, the firm should produce less. This is equivalent to setting marginal profit equal to zero.
Pure selling (revenue maximisation)
A special case where marginal cost is negligible (close to zero), so the profit maximisation condition simplifies to MR = 0. Common in industries with high fixed costs and near-zero variable costs, such as airlines and stadiums.
Sensitivity analysis (what-if analysis)
Examining how the optimal decision changes when underlying parameters shift, e.g. overhead costs rising or raw material prices increasing.
Franchiser (franchisor)
The parent company that owns the brand, supplies, know-how, and national advertising. Earns revenue as a percentage of the franchisee's sales.
Franchisee
The operator who runs a local outlet under the parent brand. Pays a fixed fee and a percentage of revenue to the franchisor.
Interdependent demand
When the demand for one of a firm's products depends on the quantity sold of another. Products can be substitutes (selling more of A reduces demand for B) or complements (selling more of A increases demand for B).
Cannibalization
Lost sales of an existing product caused by the introduction of a new product from the same firm.
This is an easier and equivalent alternative to setting marginal profit equal to zero.
Using the microchip manufacturer example:
Inverse demand: P = 170 - 20Q
Revenue: R = 170Q - 20Q²
Cost: C = 100 + 38Q
Derive marginal revenue and marginal cost:
MR = dR/dQ = 170 - 40Q
MC = dC/dQ = 38
Set MR = MC and solve:
170 - 40Q = 38, so 40Q = 132, so Q = 3.3 lots
Then P = 170 - 20(3.3) = $104,000 and profit = $117,800. Identical to the calculus approach from Part 1.
MR shortcut for linear demand: if the inverse demand is P = a - bQ, then MR = a - 2bQ. The MR curve has the same intercept but twice the slope of the inverse demand line.
The stop-go intuition:
MR > MC: the last unit brings in more than it costs. Produce more.
MR < MC: the last unit costs more than it earns. Pull back.
MR = MC: the sweet spot. No further gain from adjusting output in either direction.
Overhead (fixed cost) increases:
Fixed costs do not appear in MR or MC. A rise in rent or plant overhead does not change the profit-maximising P and Q. It lowers total profit, but the optimal operating point stays the same.
In the profit function Pi = -20Q² + 132Q - 100, the fixed cost is the constant term (-100). When you differentiate, it drops out. The derivative that determines optimal Q is unaffected.
This is a powerful implication: once you have decided to produce, sunk and fixed costs are irrelevant to how much you produce or what price you set.
In the long run, persistent losses from high fixed costs may cause the firm to exit the market entirely, but that is a different decision from the short-run P and Q choice.
Raw material (variable cost) increases:
Variable costs do affect MC. If the cost per lot rises, MC shifts up, the MR = MC intersection moves to a lower Q and a higher P. You must recalculate the optimum.
A small business normally charges $8 per shirt. Total cost per shirt is $6 ($3 variable + $3 fixed). Normal profit margin: $2 per shirt. The business is in a slump.
An order comes in offering $5 per shirt for 600 shirts.
Per-unit (marginal) analysis:
MR = $5 (the price the buyer is offering)
MC = $3 (only the variable portion; the $3 fixed cost is paid regardless)
Since MR ($5) > MC ($3), accept the order.
A rent increase would not change this decision. Rent is overhead, not a marginal cost.
Total profit analysis confirms it:
Accept: Pi = 600 x (5 - 6) = -$600
Reject (assuming no other orders): Pi = -Fixed costs = -600 x $3 = -$1,800
Accepting cuts the loss by $1,200.
The key insight: when comparing options, only marginal (variable) costs matter. Fixed costs are the same either way.
The franchiser earns a percentage of revenue, so its objective is to maximise revenue. The franchisee keeps the residual after costs, so its objective is to maximise profit. These are different goals.
What the franchisee gives the parent company: a fixed fee plus a monthly percentage of revenue.
What the parent company provides: branded supplies, operational know-how, national advertising, reputation, established business model.
The conflict plays out in specific decisions:
Remodelling premises: franchiser favours it (boosts revenue), franchisee resists (raises costs)
Raising prices: franchiser may resist (could lower revenue), franchisee might favour (could raise profit margin)
Expanding promotional discounts: franchiser favours (drives volume and revenue), franchisee weighs cost
Longer store hours and multiple express lines: franchiser favours (more sales), franchisee bears the staffing cost
Why not profit sharing? Because profit = revenue - costs, and costs are very difficult to monitor. Revenue can be checked from cash registers, but costs can be exaggerated. Revenue-based contracts are simpler to enforce.
Some businesses have near-zero marginal costs. Airlines (once the plane, pilot, and fuel are paid for, one more passenger costs almost nothing) and stadiums (one more spectator is essentially free) are classic examples.
When MC is approximately 0, the MR = MC condition becomes simply MR = 0. The firm maximises revenue rather than profit, because with no variable costs, revenue and profit move together.
Most firms sell more than one product. The pricing approach depends on the relationship between products:
Independent products: no interaction. Apply MR = MC to each product separately.
Interdependent demand (substitutes): selling more of product A reduces demand for product B. Example: Panera introducing a new sandwich that steals sales from an existing sandwich. You must account for the lost revenue on B when pricing A.
Interdependent demand (complements): selling more of A increases demand for B. Example: introducing a healthy salad that pairs well with an existing soup. Pricing A can be more aggressive because it lifts sales of B.
Worked example (Frontega sandwich F and Chicken Salad S):
MC(F) = 80 cents, MC(S) = 40 cents
Demand for F: Q(F) = 140 - P(F)/2
Demand for S: Q(S) = 180 - P(S) - 2Q(F)
The -2Q(F) term in S's demand captures cannibalization: each additional Frontega sold reduces Chicken Salad demand by 2 units.
Solve by setting MR(F) = MC(F) and MR(S) = MC(S) simultaneously (two equations, two unknowns). Solution: Q(F) = 30, Q(S) = 40.
This framework also informs whether to introduce a new product at all, not just how to price it.
Exchange-rate shifts change the relative price of domestic vs. foreign goods, which shifts demand curves and therefore MR.
Example: U.S. steel vs. Japanese steel in the U.S. market.
Starting exchange rate: 2 yen / $1. U.S. price $10 = 20 yen. Prices equivalent.
New exchange rate: 1 yen / $1. The yen has appreciated (become more expensive in dollar terms).
At the old yen price (20 yen), the dollar equivalent is now $20, not $10.
Japanese steel is now more expensive in dollar terms. U.S. consumers switch to domestic producers.
Demand for U.S. steel rises; demand for Japanese steel falls.
Since demand changes, the MR function changes, and the firm must recalculate its optimal P and Q using MR = MC with the new demand curve.
MR (linear inverse demand shortcut): if P = a - bQ, then MR = a - 2bQ
Profit maximisation condition: MR = MC (equivalently, dPi/dQ = 0)
Pure selling condition: MR = 0 (when MC is approximately 0)
Microchip example solved via MR = MC: 170 - 40Q = 38, Q = 3.3
⚠️ MR = MC is the single most important condition in this chapter. Be able to derive it, solve it, explain the intuition, and draw it on a graph.
⚠️ Fixed costs do not affect the profit-maximising P and Q. This is a commonly tested sensitivity-analysis point. If overhead rises, the optimal output stays the same; only total profit changes.
⚠️ Variable cost changes do alter the optimum. A rise in raw material costs shifts MC up and changes the MR = MC solution.
⚠️ For the t-shirt word problem, remember that marginal cost is the variable cost only. Do not include fixed costs when deciding whether to accept an order.
⚠️ Know the franchiser-franchisee conflict: the parent maximises revenue, the operator maximises profit. Be ready to explain why each party prefers different operational decisions.
⚠️ Pure selling: when MC is approximately 0, the condition simplifies to MR = 0. Know the airline/stadium examples.
⚠️ With substitutes, introducing a new product cannibalises the old one. The firm must account for lost revenue on existing products. With complements, the new product boosts existing sales.
⚠️ Exchange-rate changes shift demand, which changes MR, which changes the optimal P and Q.
Q: State the profit maximisation condition and explain its intuition in plain language.
A: MR = MC. If the revenue from one more unit exceeds its cost (MR > MC), produce it. If the cost exceeds the revenue (MR < MC), do not. At the optimum, the last unit just breaks even on the margin.
Q: A firm's fixed costs double. Does the profit-maximising quantity change? Why or why not?
A: No. Fixed costs do not appear in MR or MC. The optimal Q (and P) remain the same. Only total profit falls.
Q: In the t-shirt problem, the order offers $5 per shirt when total cost is $6. Why should the firm still accept the order?
A: Because marginal cost is only $3 (the variable portion). The $3 in fixed cost is paid regardless. Since MR ($5) > MC ($3), each shirt contributes $2 toward covering fixed costs. Rejecting the order means absorbing the full fixed-cost loss.
Q: What is pure selling, and when does it apply?
A: Pure selling is when marginal cost is approximately zero, so the firm simply sets MR = 0 to maximise profit (which is effectively revenue maximisation). It applies in industries like airlines and stadiums where costs are overwhelmingly fixed.
Q: Why does the franchiser want to maximise revenue rather than profit?
A: Because the franchiser's income is a percentage of the franchisee's revenue, not profit. Revenue is also easier to verify from cash registers, whereas costs can be exaggerated.
Q: Two products are substitutes. How does introducing product A affect the optimal pricing of existing product B?
A: Selling more of A reduces demand for B, lowering B's MR. The firm must account for this cross-product effect when pricing A. Ignoring cannibalization leads to overproduction of A and underpricing relative to the true optimum.
Q: An exchange-rate change makes foreign goods more expensive in the domestic market. What happens to the domestic firm's optimal output?
A: Demand for the domestic product rises, shifting its demand curve (and MR curve) outward. The new MR = MC intersection yields a higher optimal Q and potentially a different optimal P. The firm must recalculate.
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