Selling Price and Breakeven for the Green Manufacturing Company: 22-Mec-B4 Integrated Manufacturing Systems, December 2013, Q6

Question 6: Selling Price and Breakeven for the Green Manufacturing Company (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 one-year cost and market study for a leased-facility product line, with selling expenses stated as a percentage of sales revenue rather than as a fixed sum, and a profit requirement stated per unit.

Given data
Item	Amount
Estimated annual sales	24,000 units
Materials	$96,000
Direct labour	$14,400
Overhead	$24,000
Administrative expenses	$28,000
Selling expenses	15 per cent of sales
Required profit	$1.02 per unit

Find. (a) the unit selling price that recovers all costs and delivers the required profit; and (b) the breakeven volume in units and in sales dollars, treating overhead and administrative expense as fixed and everything else as variable.

Approach. Because selling expense depends on the price being solved for, write a single revenue equation in which price appears on both sides, solve it algebraically, then reclassify the same costs into fixed and variable pools to obtain the contribution margin and hence the breakeven point.

Figure 6.1 — Cost–volume–profit chart at the price computed below. Total revenue rises at $9.1608 per unit and total cost at $5.9741 per unit from a fixed base of $52,000; the two lines cross at 16,318 units and $149,486 of sales. The planned volume of 24,000 units lies well inside the profit region.

Part (a) — total the costs that are independent of price. Materials, direct labour, overhead and administrative expense are all quoted as annual sums and none of them depends on the selling price: $$\text{Cost}_{\text{fixed in }P}=96{,}000+14{,}400+24{,}000+28{,}000=\$162{,}400$$ Selling expense is deliberately excluded here, because it is quoted as a percentage of sales and therefore changes with the very price being sought.
Write the profit equation with price as the unknown. With $N=24{,}000$ units and $P$ the unit price, revenue is $NP$, selling expense is $0.15NP$, and the required profit is $1.02 per unit, so $$NP=\text{Cost}_{\text{fixed in }P}+0.15NP+1.02N$$ Collecting the price terms, $$NP(1-0.15)=162{,}400+1.02(24{,}000)=162{,}400+24{,}480=\$186{,}880$$
Solve for the selling price. Dividing by $N(1-0.15)=24{,}000\times 0.85=20{,}400$, $$P=\frac{186{,}880}{20{,}400}=9.1608 \qquad\Rightarrow\qquad \boxed{\ P=\$9.16\ \text{per unit}\ }$$
Check the answer by rebuilding the income statement. At $9.1608 per unit, annual sales are $24{,}000\times 9.1608=219{,}858.82$ dollars; selling expense is fifteen per cent of that, $0.15\times 219{,}858.82=32{,}978.82$ dollars; adding the $162,400 of other costs gives a total cost of $195,378.82, which leaves $219{,}858.82-195{,}378.82=24{,}480.00$ dollars of profit — exactly $1.02 per unit on 24,000 units. The price is confirmed.
Part (b) — reclassify the costs as fixed or variable. The question states that overhead and administrative expenses are fixed and all other costs vary with volume: $$F=24{,}000+28{,}000=\$52{,}000\ \text{per year}$$ On a per-unit basis the variable costs are $$v_{\text{mat}}=\frac{96{,}000}{24{,}000}=\$4.00,\qquad v_{\text{lab}}=\frac{14{,}400}{24{,}000}=\$0.60,\qquad v_{\text{sell}}=0.15P=0.15(9.1608)=\$1.3741$$ $$v=4.00+0.60+1.3741=\$5.9741\ \text{per unit}$$ Selling expense is variable by its own definition: it is a fixed proportion of each sales dollar, so it is a fixed amount per unit sold at a given price.
Form the contribution margin. Each unit sold contributes $$\mathrm{CM}=P-v=9.1608-5.9741=\$3.1867\ \text{per unit}$$ towards the fixed costs, and the contribution-margin ratio is $\mathrm{CM}/P=3.1867/9.1608=0.34786$, that is 34.79 per cent of every sales dollar. Note that the ratio can also be obtained directly as $0.85-5.10/9.1608$, since 85 per cent of revenue survives the selling expense and $5.10 per unit is the material and labour content.
Divide the fixed cost by the contribution margin. The breakeven volume is the point at which accumulated contribution exactly covers the fixed cost: $$Q_{\mathrm{BE}}=\frac{F}{\mathrm{CM}}=\frac{52{,}000}{3.1867}=16{,}318 \qquad\Rightarrow\qquad \boxed{\ Q_{\mathrm{BE}}=16{,}318\ \text{units per year}\ }$$
Express the same point in sales dollars. Either multiply the breakeven units by the price, or divide the fixed cost by the contribution-margin ratio — the two must agree: $$S_{\mathrm{BE}}=Q_{\mathrm{BE}}P=16{,}318\times 9.1608=\$149{,}486 \qquad\text{and}\qquad S_{\mathrm{BE}}=\frac{F}{\mathrm{CM}/P}=\frac{52{,}000}{0.34786}=\$149{,}486$$ $$\boxed{\ S_{\mathrm{BE}}=\$149{,}486\ \text{of sales per year}\ }$$ At the planned volume of 24,000 units the firm therefore operates $24{,}000-16{,}318=7{,}682$ units above breakeven, a margin of safety of 32 per cent of planned sales — a comfortable but not generous cushion for a new product.

Final results
Quantity	Symbol	Value
Costs independent of price	—	$162,400 per year
Required profit	1.02N	$24,480 per year
Selling price	P	$9.16 per unit
Annual sales at that price	NP	$219,859
Fixed cost (overhead + administrative)	F	$52,000 per year
Variable cost per unit	v	$5.9741
Contribution margin per unit	CM	$3.1867
Contribution-margin ratio	CM/P	34.79 per cent
Breakeven volume	Q_BE	16,318 units per year
Breakeven sales	S_BE	$149,486 per year
Margin of safety at 24,000 units	—	7,682 units (32 per cent)

Topic. Cost–volume–profit analysis: pricing to a required unit profit when one cost element is a percentage of sales, and breakeven analysis by contribution margin.

Key relations. $NP=\text{other costs}+sNP+\pi N$ giving $P=\dfrac{\text{other costs}+\pi N}{N(1-s)}$, where $s$ is the selling-expense fraction and $\pi$ the required profit per unit; $\mathrm{CM}=P-v$; $Q_{\mathrm{BE}}=F/\mathrm{CM}$; $S_{\mathrm{BE}}=F/(\mathrm{CM}/P)=Q_{\mathrm{BE}}P$; margin of safety $=(Q_{\text{plan}}-Q_{\mathrm{BE}})/Q_{\text{plan}}$.

Why this works. The difficulty of part (a) is entirely circular: the selling expense cannot be computed until the price is known, and the price cannot be computed until the selling expense is known. Writing the relationship as one equation with $P$ on both sides and collecting terms resolves the circularity in a single line, and the factor $(1-s)$ that emerges has a clean interpretation — only 85 cents of every sales dollar is available to cover costs and profit. Part (b) then rests on the observation that fixed costs are the only costs a unit does not have to carry itself, so each unit's surplus over its own variable cost is available to retire them; dividing one by the other counts how many units that takes. The two routes to breakeven sales must agree because $Q_{\mathrm{BE}}P$ and $F/(\mathrm{CM}/P)$ are algebraically identical, which makes them a free arithmetic check.

Common pitfall. Including the selling expense in the cost total before solving, using its value at some assumed price — this produces a price that is wrong and then self-consistently wrong at every later step. A second common error in part (b) is forgetting that the selling expense is variable and loading it into the fixed pool: doing so raises the fixed cost to $84,979 and inflates the contribution margin to $4.56, giving a breakeven of 18,635 units — roughly 14 per cent too high. A third is rounding the price to $9.16 before computing the contribution margin, which shifts the breakeven by a few units; carry the unrounded value through and round only the reported answers.

Source. Chase, Jacobs & Aquilano, Operations and Supply Chain Management, 16th ed., Ch. 2 (Strategy) and Appendix A (Financial Analysis — breakeven analysis); Nahmias & Olsen, Production and Operations Analysis, 7th ed., Ch. 1; Fraser & Jewkes, Global Engineering Economics, 6th ed., Ch. 1 for the Canadian cost-classification conventions.

Question 6 of 7: Selling Price and Breakeven for the Green Manufacturing Company

Question 6: Selling Price and Breakeven for the Green Manufacturing Company (20 marks)