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

Product-mix decisions when capacity constraints exist

By Jackie, Researcher

Topic: Education

Area of discussion: Management and Cost Accounting

Chapter: Measuring relevant costs and revenues for decision-making

The primary objective of this posting is to critically describe the key concept that should be applied for presenting information for product-mix decisions when capacity constraints apply. In this discussion, I have taken a past year question from CIMA Cost Accounting 2 and prepared sample solutions with detailed workings to aid illustration purposes. Personally, I like this question because it is very tricky as ‘duo ranking system’ applies due to the nature of allotment and the presence of ‘market commitment’ in this question.  

Suggested answers with workings:

Solution for a(i):

Note: Variable costs are quoted per acre, but selling prices are quoted per tonne. Therefore, it is necessary to calculate the planned sales revenue per acre. 

Solution for a(ii):

Ordinarily, the allocation of scarce resources will strictly follow the ascending order of rankings (in term of contribution amount); starting with the product that gives the highest contribution per limiting factor and then, leaving behind the balance of scarce resources to be taken up by other products that give lesser contribution per limiting factor. However, ‘market commitment’ exists in this case. So, the ‘market commitment’ will be the first one to get allocated. For profit maximization purposes, ‘market commitment’ has to be filled up with the product that gives the lowest contribution per limiting factor and the balancing figure to be taken up by other products that give a higher contribution per limiting factor. Also, due to the ‘market commitment’, the nature of allocation of scarce resource (i.e. land space) has been splitted into two.

Solution for b(i):

‘No market commitment and the land could be cultivated in such a way that any of the above crops could be produced’ simply means that the clause (i.e. the land that is being used for the production of carrots and parsnips can be used for either crop, but not for potatoes or turnips. The land being used for potatoes and turnips can be used for either crop, but not for carrots or parsnips. In order to provide an adequate market service, the gardener must produce each year at least 40 tonnes each of potatoes and turnips and 36 tonnes each of parsnips and carrots.) was not in effect anymore.

Solution for b(ii):

Note: To get 100 acres, sum up all the area occupied (i.e. 25 + 20 + 30 +25); while the contribution of £960/acre is taken from solution for a(i).

Solution for b(iii):

There are two ways to calculate this. The first one is to calculate the BEP in acres by dividing the fixed costs with the contribution per acre, and then used the computed figure to further multiply with its sales revenue per acre. (i.e £54,000 ÷ £960/acre = 56.25 acres, then 56.25 acres x £1,620/acre = £91,125). The second one is shown below by dividing the fixed costs with the contribution/sales ratio. Both methods are well explained in Cost-Volume-Profit analysis’ chapter.

Additional readings, related links and references:

Throughput accounting and TOC, one step beyond limiting factor analysis. The time frame is so short until all operating expenses are treated as fixed including direct labour; only direct materials are treated as variable. http://www.accaglobal.com/content/dam/acca/global/pdf/sa_nov11_throughput2.pdf

Reframing the Product Mix Problem using the Theory of Constraints
http://www.orsnz.org.nz/conf34/PDFs/Mabin.pdf

Product Mix: Determining My Winners and Losers

http://aggie-horticulture.tamu.edu/ornamental/economic-fact-sheets/product-mix-determining-my-winners-and-losers/

Limiting factors analysis

http://www.accaglobal.com/content/dam/acca/global/PDF-students/2012s/sa_jan10_coulthurst.pdf

Short-term Decisions Overview and Product Mix Decisions 

http://www.youtube.com/watch?v=gUKXLihYxzw

Fair Value Vs Historical Cost

By Jackie, Researcher

Topic: Education

Area of discussion: Financial Accounting and Reporting

Chapter: Accounting Concepts

  

The objectives of this posting are: to critically compare and contrast the usefulness of fair value and historical cost (in different conditions); as well as to identify, explain, and analyse the benefits and limitations of both fair value and historical cost.

Fair Value

Fair value is also known as “market value” or “current value”; it is a rational and unbiased estimate of the potential market price of a good, service, or asset.

Arguments in support of fair value concept

1. It is recommended to use fair value concept during inflation period or when the currency is unstable or fluctuate.
2. It is also suitable to be used when there is an active and observable market; such market usually exists in popular and densely-populated area where there will be a lot of buyers and sellers trading daily.
3. Depending on the nature of the assets, this valuation method is suitable for assets that undergo appreciation (i.e. value goes up) instead of depreciation (i.e. value goes down). For example: land and buildings, livestock and raw materials. Assets that follow this concept are required to undergo revaluation process after a specific time (if needed) and also test for impairment (if any) before the assets values are to be displayed in the financial statements (inside the annual reports).
4. It upholds the accounting qualitative characteristic: relevance.
5. Information about the assets values is updated. Hence, it is useful for decision making purposes (especially for potential buyers) 

Arguments against fair value concept

1. Valuing assets using current market price is very subjective. It involves a lot of predictions, assumptions and careful analysis in order to get an estimated figure. However, not everyone can agree with that figure as the figure is usually open to dispute or debate as it involves a lot of judgements, thoughts and opinions which can be either personal or professional. Furthermore, the amount has not been “realised” yet (i.e. not incurred yet). Thus, it cannot be fully trusted.    
2. Unlike historical cost, those estimated figures are usually given in a “range of values” instead of one clear-cut value for each asset.
3. In reality, active and observable market is highly dependent on location. Therefore, if the assets (usually properties) are situated in an isolated location, it is very hard for the property agents to estimate the price of the properties because there is insufficient data to be used as comparison and analysis purposes.
4. Some experts believe that as compared to historical cost, fair value concept is easier to be manipulated by unethical corporate directors and managers.
5. It will be quite troublesome sometimes, because the preparers of financial statements are required to make some changes to the subsequent recording and readjust the depreciation amount. Thus, a lot of paperwork and additional disclosures are involved.

Historical Cost

According to the historical cost concept, assets are shown at the original cost price (i.e. the figures shown during the initial recording of those particular assets at the date of acquisition). 

Arguments in support of historical cost concept

1. It is suitable to be used when the currency is stable.
2. Depending on the nature of the assets, historical cost is suitable to be applied to any assets that undergo depreciation instead of appreciation. For example: plant and machinery, motor vehicles and office equipment. Assets that follow this type of valuation method are required to be depreciated accordingly usually per annum by using appropriate depreciation methods such as straight-line, reducing balance or sum-of-digits methods. Assets values that are displayed in financial statements are also known as Net Book Values (NBVs). NBVs are computed by using initial cost less accumulated depreciation.
3. Valuing assets at their original cost is very objective. Everyone can accept and agree on it because it is based on a factual occurrence. Not only that, it also has a very concrete support in term of nature of evidence such as printed receipts, legal documents and purchase agreements.  
4. It upholds the accounting qualitative characteristic: reliability.
5. It gives one clear-cut and precise value for each asset.
6. It is convenient and easy to use. It does not required active and observable market nor additional disclosure simply because there is no real change. 

Arguments against historical cost concept

1. Information about the assets values is outdated. Accountants called it as “sunk cost”. Thus, it will not be helpful in decision making processes anymore.
2. During the period of rising prices, using historical cost accounting might poses adverse effects on financial statements as well as some financial ratios. For example, in the Statement of Comprehensive Income (i.e. the Income Statement), the understated depreciation will cause the profit to be overstated. Meanwhile, in the Statement of Financial Position (i.e. the Balance Sheet), the cost of assets, accumulated depreciation of the assets and net book values of the assets will be all understated. On the other hand, financial ratios like ROCE (Return On Capital Employed) and ROA (Return On Assets) will be overstated. This is because the numerator tends to be overstated and the denominator tends to be understated. Ultimately, it makes the figures to be “unpresentable” and “unreflective” to the current situation. Not only that, it also poses risk and danger to the users of financial statements (especially for the shareholders and potential investors) as the information is misleading.

Additional readings, related links and references:

“Causes and consequences of choosing historical cost versus fair value”; this journal’s discussion is mainly on real estate. (IAS 40 – Investment Property) 

http://www3.nd.edu/~carecob/May2008Conference/Papers/RiedlMRS03062008.pdf

Historical cost vs. market (fair) value. Emphasize more on theories. http://commerceducation.blogspot.com/2011/04/historical-cost-vs-market-fair-value.html

Does fair value accounting for non-financial assets pass the market test? http://faculty.chicagobooth.edu/valeri.nikolaev/PDF/FairvaluePaper_RAST_Conference.pdf

Historical cost vs. fair value. Examples are given with calculations too.

http://business-accounting-guides.com/historical-cost/

The fair move: The trend towards fair value accounting http://www.usc.edu/org/InsightBusiness/ib/articles/articlescontent/08_4_Nishan%20Perera.html