A company launches a new product and has two sets of facts. The market research says that at one price a certain number of units will sell, and that every time the price rises by a fixed amount, a fixed number of buyers drop out. The cost records say what the product costs at two different output levels.
The directors ask the obvious question: what price should we charge, and how many should we make? The last topic showed that elasticity tells you what happens to revenue when the price changes. This topic goes one step further and finds the profit-maximising price and output using a demand function, marginal revenue and marginal cost.
It also shows how the high-low method separates fixed from variable cost, so that the cost side of the problem can be built from two observations. It builds on the earlier topic on price elasticity of demand. The next topic covers cost-based pricing, including cost-plus and relevant cost pricing.
In 30 seconds
A linear demand function is P = a − bQ, where P is the price, Q the quantity, b = change in price ÷ change in quantity (the fall in price needed to sell one more unit) and a = the price at which demand is zero, found by putting one known price and quantity into the equation. Total revenue = P × Q = aQ − bQ², so marginal revenue MR = a − 2bQ.
Marginal cost MC is the variable cost of one more unit, and is constant if variable cost per unit is constant. Profit is greatest where MR = MC. Solve for Q, then put Q into the demand function for the price.
Profit = (P − variable cost) × Q − fixed cost. The high-low method finds variable cost per unit = (cost at the high output − cost at the low output) ÷ (high output − low output), and fixed cost = total cost − variable cost × output. The price found is not necessarily the right launch price: a skimming strategy may start higher. Full marks come from showing each step.
The demand function, marginal revenue and marginal cost
The demand function
Market research often gives two facts: at a given price a given quantity is demanded, and for each stated change in price the quantity changes by a stated amount. From these a straight-line demand function is built:
- P = a − bQ. The price falls as the quantity to be sold rises.
- b. = the change in price ÷ the change in quantity (a positive number).
If each ₦20 rise in price loses 100 units, b = 20 ÷ 100 = 0.2. - a. = the price at which no one buys. Put the known price and quantity into P = a − bQ and solve for a.
If P = ₦900 at Q = 10,000 and b = 0.2, then 900 = a − 2,000 and a = ₦2,900.
Marginal revenue
Total revenue is price times quantity: R = PQ = (a − bQ)Q = aQ − bQ². Marginal revenue is the extra revenue from selling one more unit. For the linear demand function it is MR = a − 2bQ. The marginal revenue is lower than the price, because to sell one more unit the price must fall on all units, not just the extra one.
The examination gives this rule: if P = a − bQ then MR = a − 2bQ.
Marginal cost and the optimum
Marginal cost is the extra cost of one more unit. When variable cost per unit is constant, MC equals it. A company should keep making and selling more units as long as each extra unit adds more revenue than cost, that is, while MR > MC. It stops where MR = MC. At that quantity, profit is at its greatest.
The steps
- Find b and a, and write P = a − bQ.
- Write MR = a − 2bQ.
- Find the variable cost per unit (the MC).
- Put MR = MC and solve for Q.
- Put Q into the demand function for the price.
- Calculate profit: (P − variable cost) × Q − fixed cost.
Finding cost with the high-low method, and judging the answer
Cost data often comes as total costs at several output levels. The high-low method uses the highest and lowest outputs to split total cost into a variable and a fixed element:
- Variable cost per unit = (total cost at high output − total cost at low output) ÷ (high output − low output).
- Fixed cost = total cost at either output − (variable cost per unit × that output).
The method assumes that cost is linear, that the two points are typical, and that the answer holds only within the range of outputs observed. Where cost items are given separately, such as materials, labour and overhead, the method can be applied to each. A cost that varies in steps, such as supervision that needs an extra supervisor above a certain output, is treated separately.
Is the optimum the right price?
The MR = MC price assumes that the demand function is accurate, that costs are linear and that the company can set any price it likes. In practice it should be tested:
- The demand function comes from research on similar but not identical products and may be wrong.
- A skimming strategy may start above the optimum to take high prices from early buyers, if the product is new and different, the life cycle is short, high early cash is needed, entry barriers keep competitors out and demand is uncertain.
- Competitors’ reactions, legal limits on prices, and the effect on the rest of the product range may all alter the price chosen.
- The result holds only for the output range for which cost data exists.
The profit function by differentiation, and when the demand equation fails
The same optimum can be found from the profit function. Revenue is R = aQ − bQ², total cost is TC = fixed cost + variable cost × Q, and profit is R − TC. The slope of the profit function is zero at the maximum: differentiate, set the result to zero and solve for Q; a negative second derivative confirms that it is a maximum, not a minimum.
The answer is the same as MR = MC. Note that the quantity that maximises profit is less than the quantity that maximises revenue, because the last units add less revenue than cost. The demand curve is found from any price at which demand is known and the change in demand per change in price, or from the price at which demand falls to zero, which is the intercept a.
Example 3: a gala dinner. A hotel expects to sell 600 tickets at ₦1,000, and each ₦50 change in price changes demand by 20 tickets. The variable cost is ₦250 a person and 800 places are available. Required: the price and number of tickets for maximum profit.
- Demand curve.
Demand reaches zero after 600 ÷ 20 = 30 steps of ₦50, so a = 1,000 + 1,500 = ₦2,500
b = 50 ÷ 20 = 2.5.
P = 2,500 − 2.5Q. - MR = MC.
2,500 − 5Q = 250, so Q = 450 tickets, within the 800 places. - Price.
2,500 − 2.5 × 450 = ₦1,375.
Contribution = (1,375 − 250) × 450 = ₦506,250, more than at ₦1,000 with 600 tickets (750 × 600 = ₦450,000).
In practice a demand equation has problems:
- there may be too little reliable data to estimate it
- the straight-line assumption may be wrong
- it describes a whole market or a firm with a monopoly position, so a small firm in a competitive market may have to take the market price instead
Because of this, a price change is often judged by the incremental approach: compare the incremental revenue with the incremental costs, including any change in unit variable costs (volume discounts, overtime premiums) and any step in fixed costs.
Example 4: an incremental price decision. A product sells 24,000 units at ₦50. Variable cost is ₦30 (materials ₦16, labour ₦10, other ₦4) and fixed costs are ₦400,000. At ₦47 demand would be 30,000 units. Materials would cost 5% less on all units; normal capacity is 28,000 units and labour above that costs a 50% overtime premium; fixed costs would rise to ₦412,000. Should the price be cut?
- Incremental revenue.
30,000 × 47 = 1,410,000; less 24,000 × 50 = 1,200,000; +₦210,000. - Materials.
30,000 × 16 × 0.95 = 456,000; less 24,000 × 16 = 384,000; ₦72,000. - Labour.
6,000 × 10 = 60,000, plus overtime premium on 2,000 units × 10 × 50% = 10,000; ₦70,000.
Other variable 6,000 × 4 = ₦24,000.
Fixed +₦12,000. - Incremental profit.
210,000 − (72,000 + 70,000 + 24,000 + 12,000) = ₦32,000.
The cut pays, provided the demand estimate holds, there is capacity without loss of quality, rivals do not match the cut, and some other price would not do better.
Layouts for price and output calculations
- b. Change in price ÷ change in quantity. a. Price + b × quantity, from any known pair.
- MR. a − 2bQ.
- Variable cost per unit and fixed cost by the high-low method, if not given.
- Optimum quantity. Q = (a − MC) ÷ 2b.
- Optimum price. P = a − bQ.
- Profit. (P − variable cost) × Q − fixed cost.
Money is in naira and each answer is rounded once, at the end.
Worked examples
Example 1: the optimum from a demand function. A product sells 10,000 units a month at ₦900. Each ₦20 rise in price loses 100 units, and each ₦20 fall gains 100. The variable cost is ₦500 a unit and fixed costs are ₦2,000,000. Required: the optimum price, output and profit.
- b.
20 ÷ 100 = 0.2.
a. 900 + 0.2 × 10,000 = ₦2,900.
So P = 2,900 − 0.2Q. - MR. 2,900 − 0.4Q. MC. 500.
- Optimum.
2,900 − 0.4Q = 500, so 0.4Q = 2,400 and Q = 6,000 units. - Price.
2,900 − 0.2 × 6,000 = ₦1,700. - Profit.
(1,700 − 500) × 6,000 = ₦7,200,000
less fixed costs 2,000,000 = ₦5,200,000. - Comparison.
At ₦900 and 10,000 units: (900 − 500) × 10,000 = 4,000,000 − 2,000,000 = ₦2,000,000.
The optimum earns ₦3,200,000 more, with fewer units sold at a much higher price.
Example 2: costs from the high-low method. Total cost is ₦2,600,000 at 2,000 units and ₦4,200,000 at 6,000 units. Demand: 3,000 units sell at ₦1,500, and each ₦100 rise loses 400 units. Required: the optimum price, output and profit.
- Variable cost.
(4,200,000 − 2,600,000) ÷ (6,000 − 2,000) = 1,600,000 ÷ 4,000 = ₦400 a unit.
Fixed cost. 2,600,000 − 400 × 2,000 = ₦1,800,000. - b.
100 ÷ 400 = 0.25.
a. 1,500 + 0.25 × 3,000 = ₦2,250. - Optimum.
2,250 − 0.5Q = 400, so Q = 3,700 units (inside the observed cost range of 2,000 to 6,000). - Price.
2,250 − 0.25 × 3,700 = ₦1,325. - Profit.
(1,325 − 400) × 3,700 = ₦3,422,500
less 1,800,000 = ₦1,622,500. - Comment. The price is below the ₦1,500 of the research, because the research point was not the optimum. The result depends on the accuracy of the demand estimate.
Expect variants: a demand function given directly, a cost function with fixed and variable parts, a question on the effect of a rise in variable cost, or a discussion of skimming.
Common mistakes
- Using the wrong MR. It is a − 2bQ, not a − bQ.
- Finding b the wrong way round. It is the change in price per unit of quantity.
- Stopping at the quantity. Put Q into the demand function to find the price.
- Forgetting fixed costs when asked for profit.
- Using total variable cost as MC. MC is per unit.
- Applying the cost data outside its range. State the limit.
Quick revision
- P = a − bQ; b = ΔP ÷ ΔQ; a from a known pair. MR = a − 2bQ.
- Profit is maximised where MR = MC; Q = (a − MC) ÷ 2b; price from the demand function.
- High-low: variable cost = Δcost ÷ Δoutput; fixed cost = cost − variable × output.
- Skimming may justify a higher launch price. The estimate may be wrong.
How the marks are earned
- Derive: b, a and the demand function, one mark each.
- Calculate: MR, MC, Q, price and profit in the layout.
- Discuss: limits of the model and skimming.
Practice questions
These are original questions, written in the style of the Skills examination. Work each one before you open the answer.
Recall check: multiple choice
Question 1. Profit is greatest where:
- Revenue is greatest
- Price equals cost
- Volume is greatest
- Marginal revenue equals marginal cost
Show the answer
D. Beyond it each extra unit costs more than it earns.
Question 2. If P = a − bQ, marginal revenue is:
- a − 2bQ
- a − bQ
- 2a − bQ
- a + bQ
Show the answer
A. It falls twice as fast as the price.
Question 3. In the demand function, b equals:
- The price at zero demand
- Fixed cost
- The change in price divided by the change in quantity
- Marginal cost
Show the answer
C. It is the slope.
Question 4. In the demand function, a is:
- The slope
- The price at which demand is zero
- Fixed cost
- Profit
Show the answer
B. It is found from a known price and quantity.
Question 5. When variable cost per unit is constant, marginal cost:
- Rises
- Falls
- Is zero
- Equals variable cost per unit
Show the answer
D. The extra unit costs the same each time.
Question 6. After finding the optimum quantity, the price is found from:
- The demand function
- The fixed cost
- The tax rate
- The high-low method
Show the answer
A. Put Q into P = a − bQ.
Question 7. The high-low variable cost per unit is:
- Total cost divided by units
- Fixed cost divided by units
- The change in cost between high and low outputs divided by the change in output
- The average of all costs
Show the answer
C. It uses only two observations.
Question 8. Fixed cost by the high-low method is:
- Variable cost per unit × output
- Total cost less variable cost per unit × output
- Zero
- Revenue less profit
Show the answer
B. Use either output.
Question 9. Marginal revenue is below price because:
- To sell one more unit the price must fall on all units
- Costs rise
- Tax is paid
- Demand is perfectly elastic
Show the answer
A. The loss on earlier units offsets the gain on the new one.
Question 10. Profit at the optimum is:
- Revenue less variable cost only
- Zero
- Contribution at the optimum less fixed cost
- MR less MC
Show the answer
C. Do not forget fixed cost.
Section B style: written questions
Question 11 (10 marks). (a) Explain why profit is maximised where marginal revenue equals marginal cost. (4 marks) (b) State FOUR limitations of using a demand function to set a price. (4 marks) (c) State the high-low formulae. (2 marks)
Question 12 (15 marks). A product sells 20,000 units at ₦300. Each ₦10 rise in price loses 400 units. The variable cost is ₦200 and fixed costs are ₦1,200,000.
(a) Derive the demand function. (4 marks) (b) State the marginal revenue function. (2 marks) (c) Find the optimum quantity and price. (4 marks) (d) Calculate the profit. (3 marks) (e) Compare it with profit at ₦300. (2 marks)
Question 13 (20 marks). Total cost is ₦3,500,000 at 100,000 units and ₦4,700,000 at 160,000 units. A new product sells 80,000 units at ₦100, and each ₦5 cut in price adds 2,000 units.
(a) Calculate the variable cost per unit and fixed cost. (4 marks) (b) Derive the demand function. (3 marks) (c) Find the optimum price, quantity and profit. (7 marks) (d) Discuss the conditions that would make a skimming price suitable. (6 marks)
Model answers
Show the model answer to Question 11
(a) Each extra unit adds marginal revenue and costs marginal cost. While MR is above MC an extra unit adds to profit; once MR falls below MC an extra unit reduces it. So the profit is greatest where they are equal (4).
(b) Any four:
- the research is on similar but not identical products
- demand may not be linear
- competitors may react
- costs may not be linear or may change
- the price may be limited by law or the product range
- the cost data holds only within its range (1 each)
(c) Variable cost per unit = (high cost − low cost) ÷ (high output − low output); fixed cost = total cost − variable cost × output (2).
Show the model answer to Question 12
(a)
- b = 10 ÷ 400 = 0.025
- a = 300 + 0.025 × 20,000 = 800
- P = 800 − 0.025Q (4).
(b) MR = 800 − 0.05Q (2).
(c)
- 800 − 0.05Q = 200, so Q = 12,000
- P = 800 − 0.025 × 12,000 = ₦500 (4).
(d)
- (500 − 200) × 12,000 = ₦3,600,000
- less 1,200,000 = ₦2,400,000 (3).
(e)
- At ₦300: (300 − 200) × 20,000 = ₦2,000,000, less 1,200,000 = ₦800,000.
- The optimum is ₦1,600,000 better (2).
Show the model answer to Question 13
(a)
- Variable cost (4,700,000 − 3,500,000) ÷ (160,000 − 100,000) = 1,200,000 ÷ 60,000 = ₦20
- fixed cost 3,500,000 − 20 × 100,000 = ₦1,500,000 (4).
(b)
- b = 5 ÷ 2,000 = 0.0025
- a = 100 + 0.0025 × 80,000 = 300
- P = 300 − 0.0025Q (3).
(c)
- MR = 300 − 0.005Q = 20, so Q = 56,000
- P = 300 − 0.0025 × 56,000 = ₦160
- contribution (160 − 20) × 56,000 = ₦7,840,000
- profit 7,840,000 − 1,500,000 = ₦6,340,000 (7).
(d) Any three, with a reason:
- the product is new and different, so buyers will pay to have it early
- a short life cycle needs quick recovery of cost
- high initial cash is needed to recover development cost
- barriers to entry such as patents or high investment keep rivals out
- demand and its sensitivity to price are uncertain (2 each)
Pathfinder on Demand and cost functions
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Demand function and MR=MC launch price with high-low costs; conditions for price skimming.
Revise first: Layouts for price and output calculations
Theme park: premium ticket demand function, MR=MC price, equipment hire benefit and sensitivity.
Revise first: Layouts for price and output calculations
High-low variable and fixed cost; demand function P=a-bx, MR=MC optimum price and profit; skimming conditions.
Revise first: Layouts for price and output calculations
Source: ICAN Pathfinder, Performance Management. Diets covered: March/July Diet, 2020 to May Diet, 2026. All 13 diets in the Pathfinder are mapped.
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