Break-even for a multiple product firm ________.

Question

BlackTom AI · Accepted Answer

A multi-product firm's break-even must be expressed in terms of a composite (or “average”) unit because products differ in price, variable cost and sales mix. The composite unit’s contribution margin (CM) is the weighted average CM of the individual products; dividing total fixed costs by that CM yields the break-even quantity of composite units.

1) Relevant concepts:
- Contribution margin per unit: CMi = Pi − Vi (price minus variable cost).
- Sales mix (weights): proportion of each product in total units sold.
- Composite-unit CM: weighted average CM across products.
- Break-even (units): Fixed Costs / CM per unit.

2) Reasoning step-by-step:
- Compute each product’s CM.
- Multiply each CM by its sales-mix weight (or by number of units in a composite bundle).
- Sum these weighted CMs to get CM of one composite unit.
- Divide total fixed costs by that composite CM to find break-even composite units.
- If needed, multiply composite units by each product’s mix to get break-even units by product.

3) Intermediate-step example:
Assume two products, A and B, sales mix 3:2 (i.e., composite = 3 A + 2 B), PA = $10, VA = $6, PB = $8, VB = $4.
- CMA = 10 − 6 = $4; CMB = 8 − 4 = $4.
- Composite CM = 3·4 + 2·4 = 12 + 8 = $20 per composite (5 units total), or $20 per bundle.
- CM per composite unit (bundle) = $20.
- If Fixed Costs = $10,000, break-even = 10,000 / 20 = 500 bundles (or 500·3 = 1500 A and 500·2 = 1000 B).

4) Why this answer is correct:
Using a weighted (composite) CM accounts for differences in prices, variable costs and the sales mix; dividing fixed costs by this composite CM gives the correct break-even quantity expressed in the consistent unit (composite bundle).

5) Why other approaches are incorrect:
- Dividing fixed costs by an average price ignores variable costs and misstates CM.
- Calculating separate single-product break-evens without enforcing the sales mix ignores the firm’s product mix and yields inconsistent totals.
- Using an unweighted aggregate CM without considering mix will misallocate contributions.

6) Conclusion:
Thus, break-even for a multiple-product firm can be calculated by dividing total fixed costs by the contribution margin of a composite unit, which matches the provided correct answer.

Break-even for a multiple product firm ________.单项选择题

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