Calculation of optimum selling price using differential calculus

By Jackie, Researcher

Topic: Education

Area of discussion: Management and Cost Accounting

Chapter: Pricing decisions and profitability analysis

The primary objective of this posting is to demonstrate the computation of optimal selling prices using differential calculus. Ideally, the theoretical solution to pricing decisions is derived from economic theory, which explains how the optimal selling price is determined. Interestingly, it is possible to derive simultaneously the optimum output level and selling price using differential calculus, if the demand and cost schedules are known. For discussion purpose, I have taken a past year question from ICAEW Management Accounting; this is an advanced question and it focuses a lot on “equation”.

Suggested answers with workings:

Solution for a:

Fundamentally, profit maximization is achieved when dTC/dx = dTR/dx (or when MC = MR). Therefore, it is advisable to calculate the marginal cost (MC) and the fixed costs (FC) first, so that we can use them to form the total cost (TC) function later. On the other hand, total revenue (TR) function can be found by multiplying selling price (SP) per unit with the output quantity. To solve ‘question a’, we need to calculate what is the maximum profit when the new machine is not leased and when the new machine is leased. Decision on whether to lease or not to lease the machine will depend on the option that can give a higher profit.

Note: The cost of materials per unit (i.e. £2/unit) could be found by dividing £400,000 with 200,000 units. However, do not apply this method (i.e. £90,000 ÷ 200,000 units = £0.45/unit) to compute piecework rate because the computed figure is a sunk cost and hence, it is irrelevant for decision making purposes as the price of the piecework rate will increase to £0.50 per unit in the coming quarter.

[Cont'd]

Note: If the new machine is leased, then the cost of materials will now be £1 (i.e. £2/unit x 50%) as the quality control problems will be eliminated, resulting in a halving of the usage of materials. Besides, the cost of leasing the new machine per quarter (i.e. £115,000) will now be an additional fixed cost element.

Solution for b(i):

Again, this is quite similar with ‘question a’. Profit maximization is achieved when dTC/dx = dTR/dx (or when MR = MC).

Solution of b(ii):

Please bear in mind that, profit maximization ≠ sales maximization. Total revenue will be maximized when MR=0.

Additional readings, related links and references:

For more questions on pricing decisions and profitability analysis, you may go to this link; there are a lot of past year questions taken directly from CIMA, ACCA and ICAEW.

http://www.tlint.com/emea/drury_6e/student/instreviewques/11questions.pdf

Product costing/Pricing strategy: Emphasize more on economics. Computations of profit maximization are included and well-explained. Ample graphs and tables are provided for illustration purposes.

http://www2.accaglobal.com/archive/sa_oldarticles/43958

For students who are interested to learn this in a greater level of details, you may download extra notes, print out, and keep a copy for your own references/studies purposes.

http://download.nos.org/srsec311new/L.No.41.pdf

Optimization problems & solutions in calculus. This is a very short article. It explains the formula: f'(x) = a(x^(a-1))

http://www.ehow.com/info_8062745_optimization-problems-solutions-calculus.html

Calculus review and minor short exercises.

http://www.sfu.ca/~zhent/files/Calculus%20Review.pdf