Performance Management · Pricing decisions

Profit Maximisation Using Marginal Revenue and Marginal Cost

Updated 11 October 2026 · Fact-checked

Profit is maximised at the output where marginal revenue equals marginal cost (MR = MC). First build the demand equation P = a − bQ. Then MR = a − 2bQ. Set this equal to MC and solve for Q. Put Q back into the demand equation to get the profit-maximising price.

Understand Profit Maximisation Using Marginal Revenue and Marginal Cost

A firm that faces a downward-sloping demand curve must cut its price to sell more units. So selling one extra unit brings in less than its price. The cut applies to every unit, not just the extra one. Marginal revenue (MR) is the extra revenue from selling one more unit. Marginal cost (MC) is the extra cost of making one more unit.

While MR is higher than MC, each extra unit adds to profit, so you keep increasing output. Once MR falls below MC, each extra unit reduces profit. Profit is highest at the point where MR = MC.

In ACCA PM you use a straight-line demand function: P = a − bQ. Here P is the price, Q is the quantity demanded, a is the price at which demand falls to zero (the intercept) and b is how much the price falls for each extra unit sold (the slope). The matching marginal revenue function has the same intercept but twice the slope: MR = a − 2bQ.

In most questions MC is the variable cost per unit and stays constant. Fixed costs do not affect MC, so they do not change the profit-maximising price or output. They only affect the final profit figure.

The answer is a price and a quantity. You find Q first from MR = MC. You then find P from the demand equation, not from the MR equation.

Key rules to remember

Demand function: P = a − bQ; P is price, Q is quantity demanded. This is a straight-line demand curve.
Slope of demand curve (b): b = change in price ÷ change in quantity; Use the price change for a given change in demand. For example, a $5 price change that moves demand by 500 units gives b = 5 ÷ 500 = 0.01.
Intercept of demand curve (a): a = current price + (b × current quantity); This is the price at which demand would be zero. It is the same as the price at Q = 0.
Marginal revenue function: MR = a − 2bQ; Same intercept as the demand function, twice the slope. Derived from total revenue PQ = aQ − bQ² by differentiating.
Profit-maximising condition: MR = MC; MC is usually the constant variable cost per unit.
Profit-maximising quantity: Q = (a − MC) ÷ 2b; Comes from solving a − 2bQ = MC.
Profit-maximising price: P = a − bQ; Substitute the optimal Q into the demand function. Do not use the MR equation.
Maximum profit: Profit = (P − MC) × Q − fixed costs; This holds where MC is the constant variable cost per unit.

How to solve Profit Maximisation Using Marginal Revenue and Marginal Cost questions

Use this order for any question on profit maximisation with MR and MC. It works whether you are given the equations or must build them from data.

1Find b. Divide the price change by the quantity change it causes. Keep the units consistent.
2Find a. Add b × current quantity to the current price. Write the demand equation P = a − bQ.
3Write the marginal revenue equation: MR = a − 2bQ. Double only the b term.
4Identify MC. This is normally the variable cost per unit. Ignore fixed costs here.
5Set MR = MC and solve for Q.
6Substitute Q into the demand equation P = a − bQ to find the price.
7If asked for profit, calculate (P − MC) × Q and deduct fixed costs. Check that your Q and P fit the original data pattern.

Quickest way: Shortcut: Q = (a − MC) ÷ 2b

When to use it: Use this when the demand equation is already given or easily found, and MC is a constant variable cost per unit. It saves the step of writing out MR.

Get a and b from the data. a = current price + b × current quantity.
Compute Q = (a − MC) ÷ (2 × b).
Compute P = a − bQ.
Check: P should be above MC and below a. If not, recheck your arithmetic.
If profit is asked, compute (P − MC) × Q less fixed costs.

Common mistakes in Profit Maximisation Using Marginal Revenue and Marginal Cost

Using MR = a − bQ instead of a − 2bQ.
Students copy the demand equation and forget that MR has twice the slope.
Fix: Always write the demand equation first, then double the b term in MR. Keep a unchanged.
Calculating b upside down, as quantity change ÷ price change.
The data is often given as 'demand rises by 500 units for each $5 cut', and the order gets reversed.
Fix: b is price per unit of quantity: price change ÷ quantity change. Check that b is a small number such as 0.01.
Forgetting to add b × Q when finding a, and using the current price as a.
Students treat the current price as the intercept.
Fix: The current price is a point on the line, not the intercept. Use a = current price + b × current quantity.
Putting Q into the MR equation to find the price.
Students confuse the MR line with the price line.
Fix: MR is used only to find Q. The price always comes from P = a − bQ.
Including fixed costs, or average cost per unit, as MC.
Students use the total cost per unit from a cost card.
Fix: MC is the extra cost of one more unit, normally the variable cost per unit. Deduct fixed costs only when you calculate total profit.
Stating the price but not the quantity, or giving revenue instead of profit.
Students stop once they have solved the equation and do not reread the requirement.
Fix: Underline what is asked: price, output, revenue or profit. Show Q, P and profit clearly in your answer.

Worked examples

Example 1

A company sells a product at $100 and sells 2,000 units a month. Market research shows that each $5 reduction in price increases demand by 500 units. Variable cost is $40 per unit and fixed costs are $90,000 a month. Calculate the profit-maximising price, output and profit.

Show the solution

b = 5 ÷ 500 = 0.01.
a = 100 + (0.01 × 2,000) = 120. Demand equation: P = 120 − 0.01Q.
MR = 120 − 0.02Q.
MC = $40. Set MR = MC: 120 − 0.02Q = 40, so 0.02Q = 80 and Q = 4,000 units.
P = 120 − (0.01 × 4,000) = 120 − 40 = $80.
Contribution = (80 − 40) × 4,000 = $160,000. Profit = 160,000 − 90,000 = $70,000.
Check: a cut from $100 to $80 is four $5 steps, giving 4 × 500 = 2,000 extra units, so demand is 2,000 + 2,000 = 4,000. This matches.

Answer: Profit-maximising output is 4,000 units at a price of $80, giving a monthly profit of $70,000.

Example 2

A firm currently charges $60 and sells 3,000 units a month. Research shows that each $2 increase in price reduces demand by 100 units. Variable cost is $24 per unit and fixed costs are $30,000 a month. Find the profit-maximising price and output and the maximum profit.

Show the solution

b = 2 ÷ 100 = 0.02.
a = 60 + (0.02 × 3,000) = 60 + 60 = 120. Demand equation: P = 120 − 0.02Q.
MR = 120 − 0.04Q.
MC = $24. Set MR = MC: 120 − 0.04Q = 24, so 0.04Q = 96 and Q = 2,400 units.
P = 120 − (0.02 × 2,400) = 120 − 48 = $72.
Contribution = (72 − 24) × 2,400 = 48 × 2,400 = $115,200. Profit = 115,200 − 30,000 = $85,200.
Check: a rise from $60 to $72 is six $2 steps, so demand falls by 6 × 100 = 600 units to 2,400. This matches.

Answer: The profit-maximising price is $72 with output of 2,400 units, giving a monthly profit of $85,200.

Exam tips

In Section C, show the demand equation, the MR equation and the MR = MC line separately. Marks are given for each step, even if the final figure is wrong.
In objective test questions there are no part marks, so check b and a with the original data before you choose an answer. Plugging your Q back into the demand equation takes seconds.
Read the cost data carefully. Use the variable cost per unit as MC, and bring in fixed costs only when asked for profit. Watch for MC that changes at a given volume, which you should state and test.
Expect a short written part on limitations. Mention that the straight-line demand function is an estimate, that other factors such as competitors affect demand, and that MC may not be constant.
Pay attention to units. Prices may be per unit but quantities may be in thousands, so keep the same units in a, b and MC.

Practice questions from Pricing decisions

Profit Maximisation Using Marginal Revenue and Marginal Cost in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Profit Maximisation Using Marginal Revenue and Marginal Cost: frequently asked questions

What is the difference between marginal revenue and marginal cost?

Marginal revenue is the extra revenue from selling one more unit. Marginal cost is the extra cost of producing that unit. Profit rises while MR is above MC and is maximised where they are equal.

How do I derive marginal revenue from the demand equation?

Total revenue is price × quantity, so with P = a − bQ it is aQ − bQ². Differentiating with respect to Q gives MR = a − 2bQ. In the exam you can simply keep a and double b.

How do I calculate a and b in the demand equation?

Find b by dividing the price change by the quantity change it causes. Then find a by adding b × current quantity to the current price. This gives the price at which demand would be zero.

Why do fixed costs not affect the profit-maximising price?

Fixed costs do not change when output changes, so they are not part of marginal cost. They do not alter where MR equals MC. They only reduce the total profit you report at that output.