22-Mec-B4 Integrated Manufacturing Systems · December 2019
Question 6 of 7: Break-Even Analysis and the Profit Effect of a Cost Trade
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Examinations, December 2019 —
16-Mec-B4 Integrated Manufacturing Systems. Three hours, OPEN BOOK, any
non-communicating calculator permitted. Seven questions are printed and
any five constitute a complete paper; all questions are of equal value,
so each is worth 20 marks of the 100 available. Note 1 of the paper invites the
candidate to state any assumption made where a question is open to
interpretation, and this solution uses that licence wherever the source withholds
a datum. Every one of the seven questions is worked below, because the set is a
study resource rather than a three-hour sitting.
variable cost down 20 per cent, fixed cost up 10 per cent
(b)
Sales growth considered
up 5 per cent
Find. (a) the break-even volume before and after the cost
reduction, and the price that would restore the original break-even volume;
(b) the change in annual profit at constant sales, and the profit if sales also
rise 5 per cent.
Approach. Both parts are the same linear cost model,
$\pi = (p-v)Q - F$, used in two directions. In (a) the volume is the unknown and
then the price is the unknown; in (b) the variable cost is not given directly and
must first be recovered by difference from the income statement.
Part (a) — state the break-even condition. Break-even is
the volume at which contribution exactly covers fixed cost,
$$Q_{BE} = \frac{F}{p - v}$$
where $p - v$ is the contribution margin per unit. Nothing else in the model
matters at that point.
Compute the present break-even volume. The present margin is
$20.00 - 12.50 = 7.50$ CAD per unit, so
$$Q_{BE} = \frac{15{,}000}{7.50}$$
$$\boxed{\ Q_{BE} = 2{,}000\ \text{units per year}\ }$$
Recompute it at the lower manufacturing cost. The margin rises
to $20.00 - 11.80 = 8.20$ CAD per unit, so
$$Q_{BE}' = \frac{15{,}000}{8.20} = 1{,}829.3$$
$$\boxed{\ Q_{BE}' = 1{,}830\ \text{units per year}\ }$$
a fall of 8.5 per cent. Put the other way, at the old volume of 2,000 units the
70-cent saving is now worth $8.20(2{,}000) - 15{,}000 = 1{,}400$ CAD of profit
where before there was none.
Invert the relation to find the price that holds the old break-even
point. Requiring $Q_{BE} = 2{,}000$ with the new variable cost gives
$$p' = v' + \frac{F}{Q_{BE}} = 11.80 + \frac{15{,}000}{2{,}000} = 11.80 + 7.50$$
$$\boxed{\ p' = 19.30\ \text{CAD per unit}\ }$$
The whole 70-cent cost saving is passed to the customer, which is exactly what
holding the break-even volume constant means: the contribution margin has to stay
at 7.50 CAD per unit. Rebuilding the income statement at 2,000 units confirms it,
$19.30(2{,}000) - [15{,}000 + 11.80(2{,}000)] = 38{,}600 - 38{,}600 = 0$.
Part (b) — recover the variable cost from the income
statement. The question gives revenue, profit and fixed cost but not
variable cost, so take it by difference,
$$V = S - F - \pi = 1{,}000{,}000 - 250{,}000 - 150{,}000 = 600{,}000\ \text{CAD}$$
which is a variable-cost ratio of 0.60 and a contribution ratio of 0.40. This step
is the one the question is really testing.
Apply the two changes the new system brings. The variable cost
falls by 20 per cent and the fixed cost rises by 10 per cent,
$$V' = 0.80(600{,}000) = 480{,}000, \qquad F' = 1.10(250{,}000) = 275{,}000\ \text{CAD}$$
so at unchanged sales
$$\pi' = S - V' - F' = 1{,}000{,}000 - 480{,}000 - 275{,}000$$
$$\boxed{\ \pi' = 245{,}000\ \text{CAD, an increase of } 95{,}000\ \text{CAD}\ }$$
that is 63.3 per cent above the present profit. The change can be read directly as
the variable saving less the fixed increase,
$120{,}000 - 25{,}000 = 95{,}000$ CAD, which is the check worth writing down.
Add the 5 per cent sales increase. Variable cost moves with
volume while fixed cost does not, so
$$S'' = 1.05(1{,}000{,}000) = 1{,}050{,}000, \qquad V'' = 1.05(480{,}000) = 504{,}000$$
$$\pi'' = 1{,}050{,}000 - 504{,}000 - 275{,}000$$
$$\boxed{\ \pi'' = 271{,}000\ \text{CAD}\ }$$
The extra 26,000 CAD is simply the contribution on the extra sales,
$50{,}000 \times (1 - 0.48) = 26{,}000$ CAD, and total profit is 80.7 per cent
above the original 150,000 CAD.
Note what the trade has done to risk. Break-even sales in
dollars are $F/(1 - V/S)$, which moves from $250{,}000/0.40 = 625{,}000$ CAD to
$275{,}000/0.52 = 528{,}800$ CAD. So this particular trade improves profit
and lowers the break-even point, because the variable saving outweighs the
fixed increase. That is not automatic — substituting fixed cost for variable
cost usually raises operating leverage and therefore raises break-even — and
it is worth stating explicitly in the recommendation.
