ACCA F5 Performance Management - 2010 - Study text

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Chapter 4: Pricing decisions

Deriving a demand equation

A demand equation expresses the relationship between sales price and sales

demand at all price levels. A straight-line demand equation has the basic formula.

P = a – bQ

where

P = the sales price

Q = the quantity demanded at that price

a = the sales price when the quantity demanded is 0

b = a constant value, representing the amount of change in sales demand for a given

change in the sales price.

For example, the demand equation for a product might be:

P = 12 – 0.0001Q

This demand equation means that:

 At a selling price of $12, sales demand would be 0

 For every $1 reduction in the sales price, the sales demand would increase by

1/0.0001 = 10,000 units.

If the sales price is $9, sales demand would therefore be 30,000 units, calculated as

follows:

P = 12 – 0.0001Q

9 = 12 – 0.0001Q

0.0001Q = 3

Q = 30,000.

Example

The sales demand equation for a product is straight-line. When the sales price is $40,

sales demand is 0 units. For every $0.50 increase in price, the sales demand falls by

1,000 units.

The demand equation is therefore:

P = 40 – (0.50/1,000)Q

P = 40 – 0.0005Q

Note

For every $0.50 increase in price, sales demand changes by 1,000 units; therefore b in

the formula is 0.50/1,000 = 0.0005.

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3.2 Total cost function

A total cost function is an equation for total costs. When total costs are a

combination of variable costs and fixed costs, a total cost function can be stated as:

C = a + bQ

where

a = total fixed costs

b = variable cost per unit

Q = volume of production and sales

3.3 Maximising profit

The demand equation and total cost function for a product can be used to calculate

the sales price (and sales volume) at which total profits would be maximised.

 The demand equation can be used to calculate marginal revenue (MR).

 The total cost function can be used to calculate marginal cost (MC).

 Profit is maximised at the price and sales volume where marginal revenue

equals marginal cost:

In other words, profit is maximised where MR = MC.

Calculating marginal revenue (MR)

The marginal revenue is the change in total sales revenue that would occur if one

extra unit of the product is sold. If the sales price is constant, the marginal revenue

per unit is its sales price. However, when there is a downward-sloping demand

equation for a product, one extra unit of product can be sold only by reducing the

sales price for all units of the product that are sold.

 Marginal revenue is therefore less than the sales price.

 The marginal revenue will continue to fall as the sales price falls and sales

demand increases.

To obtain a formula for calculating marginal revenue, the demand equation must be

converted into an equation for total revenue.

(1) Total revenue = Price × Quantity

(2) The price is represented by the demand equation, P = a – bQ

Substituting (2) in (1), we get:

(3) Total revenue = (a – bQ) Q, = aQ – bQ

The formula for calculating marginal revenue is obtained by applying differential

calculus to the formula for total revenue. If you are not familiar with differential

calculus, you should learn the following technique:

(1) If total revenue (TR) = aQ – bQ

(2) Marginal revenue (MR) = a – [2b × Q]

Chapter 4: Pricing decisions

Example

Returning to the previous example: The demand curve is P = 40 – 0.0005Q

Therefore the formula for total revenue is:

TR = (40 – 0.0005Q) Q = 40Q – 0.0005Q

Therefore the formula for marginal revenue is

MR = 40 – (2 × 0.0005) Q = 40 – 0.001Q.

Calculating marginal cost (MC)

The marginal cost per unit is the additional cost of making and selling one extra unit

of the product. It can be obtained by applying differential calculus to the total cost

function.

However when the total cost function is a liner cost function TC = a + bQ, the

marginal cost per unit is its variable cost (b).

Profit-maximisation, MR = MC

Profit will be maximised at the sales price (and volume) where MR = MC.

Example

Suppose that the monthly demand equation for a product is P = 40 – 0.0005Q. Fixed

costs per month are $500,000 and the variable cost per unit is $2.

Required

Calculate the sales price and output quantity that maximise the profit each month,

and calculate the amount of that profit.

Answer

The marginal revenue for this demand equation was calculated earlier:

MR = 40 – 0.001Q

MC = 2

Profit is maximised when MR = MC

40 – 0.001Q = 2

0.001Q = 38

Q = 38,000 units.

The sales price when Q = 38,000 is:

P = 40 – 0.0005(38,000) = $21.

Profit is therefore maximised at a sales price of $21.

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Sales (38,000 × $21) 798,000

Variable costs (38,000 × $2) 76,000

Contribution 722,000

Fixed costs 500,000

Profit 222,000

Example

A company is the monopoly producer of Product T. The product is currently sold at

a price of $16 and annual sales demand at this price is 80,000 units. Total annual

fixed costs are $750,000 and variable costs are $5 per unit.

It has been estimated that if the sales price is increased, sales demand would fall by

2,500 units for every $1 increase in price. Similarly, if the sales price is reduced,

demand would increase by 2,500 units for every $1 reduction in the price.

Required

(a) Calculate the annual profit at the current sales price.

(b) Calculate the sales price and output quantity that maximise the annual profit,

and calculate the amount of that profit.

Answer

At current price of $12

Sales (80,000 × $16) 1,280,000

Variable costs (80,000 × $5) 400,000

Contribution 880,000

Fixed costs 750,000

Profit 130,000

We need to derive the demand equation. To do this, we need to calculate the sales

price at which demand Q would be 0 units.

At the moment, demand is 80,000 units when the sales price is $16, and demand will

fall by 2,500 units for every $1 increase in the price. Total sales demand will

therefore be 0 units at a price of:

$18 + $(80,000/2,500) = $18 + $32 = $50.

The demand equation is therefore: P = 50 – (1/2,500)Q

= 50 – 0.0004Q

Total revenue = P × Q

= (50 – 0.0004Q) × Q = 50Q – 0.0004Q

Marginal revenue MR = 50 – (2 × 0.004)

= 50 – 0.0008Q Q

Chapter 4: Pricing decisions

MC = 5

Profit is maximised when MR = MC

50 – 0.0008Q = 5

0.0008Q = 45

Q = 56,250 units.

The sales price when Q = 56,250 is:

P = 50 – 0.0004(56,250) = $27.5.

Profit is therefore maximised at a sales price of $27.50.

Maximum profit

Sales (56,250 × $27.50) 1,546,875

Variable costs (56,250 × $5) 281,250

Contribution 1,265,625

Fixed costs 750,000

Profit 515,625

3.4 Problems using a demand equation

There are several practical difficulties with using a demand equation to establish a

profit-maximising price.

 There may be insufficient reliable data to produce an estimate of the demand

equation.

 The assumption that the demand equation is a straight line may be inaccurate.

 A demand equation normally applies to either total market demand for a

product or to the demand for a product that is made and sold by a company in a

monopoly position in its market. For smaller companies in a competitive market,

selling prices might not be the profit-maximising price, but a price at which the

company can compete successfully against its rivals.

3.5 Incremental revenue and incremental cost

Because of the practical problems with deriving a reliable demand equation,

decisions about changing the selling price for a product are more likely to be

assessed by comparing the incremental revenues and incremental costs that would

arise from the price change.

Reducing the selling price of a product by a specific amount should be expected to

result in higher sales demand. (If it doesn’t, there would be no reason to reduce the

price.) Higher sales demand would be met by increasing the volume of output; and

at higher volumes of output, costs will change.

 The may be no change in variable costs; however variable costs per unit may

change. Labour costs per hour will increase if employees are required to work

overtime to produce the higher volume of output. On the other hand, direct

materials costs per unit might change if volume purchase discounts are available

at the new volume of production.

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 There may be no change in fixed cost expenditure; however fixed cost spending

may be higher at higher volumes of output due to a ‘step’ increase in fixed costs.

To assess the effect of a proposed increase in price on profit, it is necessary to

calculate:

 the incremental sales revenue, which is the increase in total sales revenue that

would occur (or fall in total revenue, if demand is inelastic), and

 the incremental costs, which is the net increase in total costs, both fixed and

variable that would occur.

A reduction in price would be justified if incremental revenue exceeded incremental

costs.

Study the following example carefully.

Example

A company makes and sells a single product. The sales price is $20 per unit. Annual

sales demand is currently 24,000. Annual fixed costs are $250,000, and the variable

cost per unit is $8, made up as follows:

Variable cost per unit

Materials 4

Direct labour 3

Other variable costs 1

Total variable cost 8

The company is considering a proposal to reduce the sales price from $20 to $19. At

the lower price, it is expected that annual sales demand would be 30,000 units. The

following changes in costs would occur if annual production and sales are increased

to 30,000 units:

A favourable volume price discount would be obtained on direct material

purchases, and materials costs would fall by 10% for all units produced by the

company during the year.

The company’s output capacity is 28,000 units during normal working time. At

output volumes above this amount, some overtime would have to be worked. The

overtime premium is 50% on top of normal hourly rates of pay.

Total fixed cost expenditure would increase to $260,000.

Required

(a) Calculate the effect on annual profit of the proposed price reduction and

recommend (on the basis of change in profit) whether the price should be

reduced.

(b) Briefly state any other factors, other than profitability, that might influence the

pricing decision.

Chapter 4: Pricing decisions

Answer

$ $

Sales revenue at new price level: 30,000 × $19

570,000

Sales revenue at current price level: 24,000 × $20

480,000

Incremental revenue 90,000

Total materials cost at new price level: 30,000 × $4 × 90%

108,000

Total materials cost at current price level: 24,000 × $4

96,000

Incremental cost of materials (12,000)

Incremental cost of labour at basic rate of pay: 6,000 units × $3

18,000

Overtime premium for 2,000 units: 2,000 × $3 × 50%

3,000

Total incrmeental cos tof labour (21,000)

Other variable costs: incremental costs: 6,000 units × $1

(6,000)

Incremental fixed cost expenditure $(260,000 – 250,000) (10,000)

Incremental profit 41,000

Conclusion: By reducing the price to $19, annual profits would increase by $41,000.

On the basis of profitability, the sales price should be reduced.

Other factors could affect the decision:

 Profit might be increased even more if the sales price is changed to a different

amount, say $18.

 Management needs to be satisfied that the company has the capacity to produce

the larger volume of output without loss of production efficiency or quality.

 The recommendation that the price should be reduced assumes that the estimate

of sales demand at $19 per unit is reliable. If competitors respond by reducing

their prices by a similar amount, there might be little or no sales volume

increase, and profits might fall rather than increase.

Paper F5: Performance management

Price strategies

 The nature of pricing strategies

 Full cost plus pricing

 Marginal cost plus pricing (mark-up pricing)

 Return on investment (ROI) pricing

 Market skimming prices

 Market penetration prices

 Pricing of complementary products and product line pricing

 Volume discounting

 Price discrimination

 Relevant cost pricing (minimum pricing)

4 Price strategies

4.1 The nature of pricing strategies

Although sales prices in a market are determined largely by factors such as supply

and demand and competition, companies might use a variety of different pricing

strategies, depending on the nature of their business, the nature of the markets in

which they operate and the particular circumstances in which a pricing decision is

made. For example:

For companies in a jobbing industry or contracting industry, each new job or

contract might be different, and this means that a separate price has to be calculated

for each individual job or contract. Some form of cost-plus pricing is therefore often

used in these industries.

When a company brings an entirely new product to the market, can decide whether

to set the price high or low, because there are no rival products on the market.

Several different pricing strategies are described in this section.

4.2 Full cost plus pricing

Full cost plus pricing involves calculating the full cost of an item (such as a job or

contract) – or the expected full cost – and adding a profit margin to arrive at a

selling price.

Profit is expressed as either:

 a percentage of the full cost (a profit ‘mark-up’) or

 a percentage of the sales price (a ‘profit margin’).

In the following example, the profit on full cost is a mark-up of 25% on full cost,

which gives a profit margin of 20% on the sales price.

Chapter 4: Pricing decisions

Variable production costs 600

Other variable costs 200

Absorbed overheads:

Production overheads absorbed 800

Non-production overheads absorbed 300

Full cost 1,900

Profit (added to full cost) 475

Selling price 2,375

Notes on calculating the profit:

 If the mark-up is x% of full cost, the selling price is Full cost + x%. In this

example there is a mark-up of 25% on full cost, and since the full cost is $1,900,

the profit is (25%) $475, giving a sales price of $2,375.

 If the profit margin is y% of the sales price, the sales price is calculated as Full

cost × profit is [y/(100 – y)] × Full cost. In this example, the profit margin is 20%

of the sales price and the full cost is $1,900: therefore the sales price is $1,900 ×

100/(100 – 20) = $2,375.

Advantages of full cost plus pricing

A business entity might have an idea of the percentage profit margin it would like

to earn on the goods or services that it sells. It might therefore decide the average

profit mark-up on cost that it would like to earn from sales, as a general guideline

for its pricing decisions. This can be useful for businesses that carry out a large

amount of contract work or jobbing work, for which individual job or contract

prices must be quoted regularly to prospective customers and there is no obvious

‘fair market’ price.

The percentage mark-up or profit margin does not have to be a fixed percentage

figure. It can be varied to suit the circumstances, such as demand conditions in the

market and what the customer is prepared to pay.

There are also other possible advantages in using full cost plus pricing:

 If the budgeted sales volume is achieved, sales revenue will cover all costs and

there will be a profit.

 It is useful for justifying price rises to customers, when an increase in price

occurs as a consequence of an increase in costs.

Disadvantages of full cost plus pricing

The main disadvantage of cost plus pricing is that it is calculated on the basis of

cost, without any consideration of market conditions, such as competitors’ prices.

 Cost plus pricing fails to allow for the fact that when the sales demand for a

product is affected by its selling price, there is a profit-maximising combination

of price and demand. A cost plus based approach to pricing is unlikely to arrive

at the profit-maximising price for the product.

Paper F5: Performance management

 In most markets, prices must be adjusted to market and demand conditions. The

pricing decision cannot be made on a cost basis only.

There are also other disadvantages:

 The choice of profit margin or mark-up is arbitrary. How is it decided?

 When a company makes and sells different types of products, the calculation of

a full cost becomes a problem due to the weaknesses of absorption costing. The

method of apportioning costs between the different products in absorption

costing is largely subjective. This affects the calculation of full cost and the

selling price. For example, full cost per unit will differ according to whether a

direct labour hour absorption rate or a machine hour absorption rate is used.

Full cost will also differ if activity-based costing is used.

Example

Entity Q makes two products, product X and product Y. These products are both

made by the same work force and in the same department. The budgeted fixed costs

are $900,000. Variable costs per unit are as follows:

Product X

Product Y

Direct costs

Materials

Labour (2 hours)

(3 hours) 18

Expenses (1 machine hour)

(1 machine hour) 6

Budgeted production and sales are 15,000 units of product X and 10,000 units of

product Y.

Required

Calculate the sale prices for each unit of product X and product Y which give a

profit margin of 20% on the full cost, if overheads are absorbed on the following

bases:

(a) on a direct labour hour basis

(b) on a machine hour basis.

Answer

(a) Direct labour hour basis

Budgeted direct labour hours = (15,000 × 2) + (10,000 × 3) = 60,000 hours.

Overhead absorption rate = $900,000/60,000 = $15 per direct labour hour.

ACCA F5 Performance Management - 2010 - Study text - Emile Woolf Publishing

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