Question 2 of 6: Minimum Selling Price for a 20% Return on Investment

Question 2: Minimum Selling Price for a 20% Return on Investment (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A single equipment train (total capital investment $150,000, installed) with a 5-year life; the base case makes 100 kg/yr on one 40 h/week shift. The annual cost blocks and the 20% target return are:

Find. The minimum selling price (in $/kg) that yields a 20% ROI for (a) 100 kg/yr with three day-shift employees, and (b) 200 kg/yr with a day shift plus a night shift (three employees each, night pay 25% higher).

Approach. Write ROI as net annual profit divided by capital invested; the net profit is revenue minus the total product cost, where the total product cost is the operating costs plus straight-line depreciation (the 5-year life is what makes depreciation a genuine annual cost). Set the profit equal to 20% of the capital and back out the price.

1. Depreciation and required profit (common to both parts). With straight-line depreciation over the 5-year life and no salvage, $$d=\frac{\text{TCI}}{n}=\frac{150{,}000}{5}=30{,}000\ \tfrac{\$}{\text{yr}}, \qquad \text{profit}_{\text{req}}=0.20\times150{,}000=\boxed{30{,}000\ \tfrac{\$}{\text{yr}}}.$$ The pricing identity for every case is therefore revenue = fixed + owner + labour + variable + depreciation + required profit, and the price is that revenue divided by the annual production.
2. Part (a): 100 kg/yr, one shift, three employees. The operating cost sums the four blocks: $$C_{\text{op}}=\underbrace{40{,}000}_{\text{fixed}}+\underbrace{120{,}000}_{\text{owner}}+\underbrace{3(55{,}000)}_{=165{,}000\ \text{labour}}+\underbrace{400(100)}_{=40{,}000\ \text{variable}}=365{,}000\ \tfrac{\$}{\text{yr}}.$$ Adding depreciation gives the total product cost $C_{\text{TPC}}=365{,}000+30{,}000=395{,}000$, and the revenue needed for the 20% return is $395{,}000+30{,}000=425{,}000\ \$/\text{yr}$. Dividing by 100 kg: $$P_a=\frac{425{,}000}{100}=\boxed{4{,}250\ \tfrac{\$}{\text{kg}}}.$$
3. Part (b): 200 kg/yr, two shifts. Running a second (night) shift doubles output on the same equipment, so capital and depreciation are unchanged. The night-shift employees are paid 25% above the $55,000 day rate: $$c_{\text{night}}=1.25\times55{,}000=68{,}750\ \tfrac{\$}{\text{yr per employee}}.$$ Fixed costs and the owner's salary do not change; only labour (now 3 day + 3 night) and the variable cost (now 200 kg) scale: $$C_{\text{op}}=40{,}000+120{,}000+3(55{,}000)+3(68{,}750)+400(200)=611{,}250\ \tfrac{\$}{\text{yr}}.$$ Total product cost $=611{,}250+30{,}000=641{,}250$; revenue for the 20% return $=641{,}250+30{,}000=671{,}250\ \$/\text{yr}$. Over 200 kg: $$P_b=\frac{671{,}250}{200}=\boxed{3{,}356.25\ \tfrac{\$}{\text{kg}}}.$$
4. Interpretation — economy of scale. Doubling output cuts the minimum price by about 21% (from $4,250 to $3,356/kg) even though a wage premium was paid, because the fixed costs, the owner's salary and the depreciation are now spread over twice as many kilograms. This is the classic dilution of fixed charges that drives producers toward higher capacity utilisation.