Professional Scheme
Relevant to Paper 1.2

Questions involving the establishment of an optimal selling price in order to achieve profit maximisation for an organisation have been set in Section B of the last two exams. This article will look in detail at the question set in the June 2004 exam and explain the concepts involved in correctly answering it. The article will then go on to look at other types of question in the area of profit maximisation.

The setting of an optimal short-term profit-maximising selling price is just one aspect of decision-making but it clearly interrelates with other concepts such as
cost-volume-profit analysis. In the pricing decision, not only is the price itself important but also the number of units that will be sold at that price (the quantity or volume), and the costs incurred at that volume. The latter will require a knowledge of cost behaviour - the identification of fixed and variable costs.

Question 3 on the June 2004 exam involved Braithwaite Limited which manufactures and sells a single product. The following information has been extracted from the current year's budget:

The company's production capacity is not being fully utilised and three possible strategies are under consideration. Each strategy involves reducing the unit selling price on all units sold with a consequential effect on the budgeted volume of sales. Details of each strategy are detailed in Table 1. The company does not hold stocks of finished goods.

Required:

1. Calculate for the current year:
   1. the selling price per unit for the product
   2. the weekly sales (in units).
2. Determine, with supporting calculations, which one of the three strategies should be adopted by the company in order to maximise weekly profits.

The selling price per unit is found using the contribution to sales (C:S) ratio and the contribution per unit (CPU). Both of these are important concepts that students should be familiar with. The C:S ratio is the contribution per unit expressed as a percentage of the selling price per unit. The CPU is the selling price per unit less the variable cost per unit. Therefore the selling price is (£8 ÷ 0.40 ) = £20 per unit. Weekly sales give a profit of £22,000 and a contribution of (22,000 + 10,000) = £32,000. At a CPU of £8 this equates with weekly sales of (32,000 ÷ 8) = 4,000 units.

In part (b) it can be seen that the changes in selling price and the resultant volume changes do not represent a straight line relationship. The way to tackle the question is to calculate the total revenue (price x quantity) and the total variable cost for each strategy and hence the total contribution from each strategy. Then select the strategy giving the highest total contribution, as shown in Table 2.

Therefore the strategy giving the highest contribution (and profit) is A. It should be noted that this is not the strategy giving the highest total revenue (C). The effect of the change in volume on total variable costs needs to be taken into account. There is no need to include the total fixed costs in the decision because, in this example, they remain the same whichever strategy is adopted.

Alternative layouts of the calculation are perfectly acceptable. For example, the CPU for each strategy could be calculated and multiplied by the units to give the total contribution.

The question could also be tackled by calculating the marginal (or more correctly the incremental) revenue and marginal (incremental) costs. This is slightly more long-winded but gives a useful insight into the economic concepts involved.

The figures in Table 3 have been derived from the information relating to Braithwaite Limited. From the table it can be seen that by moving from the current position to strategy A, the incremental revenue (£6,240) is greater than the incremental cost (£4,800) but then to move to strategy B would incur an incremental cost greater than the incremental revenue. A similar situation would occur with strategy C. This is an application of the economics concept which states that profits are maximised when marginal revenue equals marginal cost.

Alternatively stated, a firm should continue to produce and sell more units as long as the marginal revenue from doing so exceeds the marginal cost. Finally the net marginal revenue from adopting strategy A is (£6,240 - £4,800) = £1,440. This, of course, is the amount by which the total contribution from adopting strategy A (£33,440 - as before) is above the current total contribution of £32,000.

As stated earlier, the analysis so far has involved a situation where the price/quantity demanded relationship was not linear. Some examination questions are set where that relationship is a continuously linear one which can be represented by an equation. The problem still involves the maximisation of profits. However, it would be very tedious to list in a table all of the possible price/units combinations, so a simpler mathematical approach is recommended.

The following question, which is adapted from the original Pilot Paper for this subject, illustrates the concepts. Lindfield has established that for one of its products the following cost and price functions apply:

Price (per unit in £) = 30 - 0.25 x quantity
Marginal revenue (per unit in £) =
30 - 0.5 x quantity
Total cost = £100 + £5 x quantity

In each case, the quantity is the number of units produced and sold. At what selling price per unit would profits be maximised?

For profit maximisation: marginal revenue (MR) = marginal cost (MC). If Q represents the number of units (produced and sold in a period) then MR = 30 - 0.5Q. An expression for MC now needs to be established. The question gives the total cost (TC) function which can be interpreted as a total fixed cost per period of £100 and a variable cost of £5 per unit. Therefore the MC (per unit) is simply equal to 5 - the extra cost of producing and selling one more unit of product.

Setting MR = MC gives the equation
30 - 0.5Q = 5. Solving this 25 = 0.5Q and Q = 25 ÷ 0.5 = 50. This means that the quantity that needs to be sold in order to maximise profits is 50 units per period. To establish the selling price per unit, which should be set to generate that level of demand, Q = 50 needs to be substituted in the price/demand equation given
(P = 30 - 0.25Q). So P = 30 - 0.25 x 50 and P = 30 - 12.5 = 17.5. The profit maximising selling price is £17.50 per unit. The question could have asked for the value of the maximum profits per period which would be calculated as follows:
£
Total revenue = price x quantity =
17.50 x 50 = 850
Less
Total cost = 100 (total fixed)
+ 5 x 50 (total variable) = (350)
Total profit (maximum) 500

The problem posed in the Lindfield question can be shown diagrammatically. In Figure 1, the linear price/demand, marginal revenue and marginal cost functions are shown. The profit maximising point is where the MR and MC lines intersect. At this point the quantity demanded is 50 units and the corresponding value on the price/demand function is a price of £17.50.

Figure 1: the Lindfield question

In this examination, candidates would be given the equations for the price/demand relationship and for the marginal revenue. They would not be expected to derive these. However, they would be expected to be able to derive the marginal cost function from a straight line total cost function.

In summary, this article has looked at two basic types of pricing question involving profit maximisation. One is where there is discrete information given about the price and quantity demanded at different but specific levels of activity. The recommended approach with this type is to produce a table of revenues and costs for each activity level and to choose the one that gives the highest profit.

The other type of question is where the relationship between price and quantity demanded is linear and represented by an equation. Similarly, the total cost function is linear. The approach here is to use marginal analysis - setting marginal revenue equal to marginal cost to give the quantity to be sold to maximise profits. Then by substitution the optimal selling price can be determined.

David Forster is examiner for Paper 1.2