## 3GP.1 Contribution Margin

### 3GP.1.E1

A firm has $2,000,000 in revenue and $1,000,000 in total costs for a given period. 60% of the firm’s total costs are variable costs. The firm produces and sells 6,000 units during the period.

**Required**

**(A)** What is the firm’s *total contribution margin*?

**(B)** What is the firm’s *unit contribution margin* (round to cents if necessary)?

**(C)** What is the firm’s *contribution margin ratio*?

#### Answer (A)

The firm’s total contribution margin is total revenue minus total variable costs. Total variable costs is 60% of $1,000,000, which is $600,000.

Total contribution margin is $2,000,000 total revenue – $600,000 total variable costs = $1,400,000.

#### Answer (B)

(I) The unit contribution margin is simply the total contribution margin per unit.

$1,400,000 total contribution margin / 6,000 units = $233.33 unit CM

(II) Alternatively, one could calculate revenue and variable cost into unit values, then find unit CM as the

$2,000,000 total revenue / 6,000 units = $333.33 revenue per unit

$600,000 total variable costs / 6,000 units = $100 variable cost per unit

333.33 – 100 = 233.33 unit CM

(III) Or, one could calculate the contribution margin ratio from total revenue and total variable cost and then multiply this by the revenue per unit.

($2,000,000 – 600,000) / 2,000,000 = 0.7 CM ratio

0.7 CM ratio * 333.33 revenue per unit = $233.33 unit CM

#### Answer (C)

The firm’s contribution margin ratio is either total contribution divided by total revenue or unit CM divided by unit revenue (the unit quantity cancels out in the division process leaving the same CM ratio in both calculations).

$1,400,000 / $2,000,000 = 0.7 CM ratio

Or

$233.33 / $333.33 = 0.7 CM ratio

If you have forgotten which number belongs in the numerator in the CM ratio equation, remember the CM ratio is the percent of each dollar of revenue that is left over after variable costs are paid. Thus

### 3GP.1.M1

A firm sells a product for $16 per unit (for a total of $100,000 in total revenue). Each unit costs $3.50 in variable costs (for a total of $21,875 in total variable costs).

**Required**

**(A)** What is the product’s *total contribution margin*?

**(B)** What is the firm’s *contribution margin ratio*** **?

**(C)** What is the firm’s *unit contribution margin*** **(round to cents if necessary)?

#### Answer (A)

Total contribution margin is $100,000 total revenue – $21,875 total variable costs = $78,125.

#### Answer (B)

The firm’s contribution margin ratio is either total contribution divided by total revenue or unit CM divided by unit revenue.

$78,125 / $100,000 = 0.78125

Or

($16 revenue per unit – $3.50 variable cost per unit) / $16 = 0.78125 CM ratio

#### Answer (C)

(I) Unit contribution margin is CM ratio times revenue per unit.

0.78125 CM ratio * 16 revenue per unit = $12.50 unit CM

(II) Alternatively, one could calculate the units from the data available, then divide

$100,000 total revenue $/ 16 revenue per unit = 6,250 units

$78,125 total contribution margin / 6,250 units = $12.50 unit CM

## 3GP.2 Single-product CVP

## 3GP.2.E1

Last year the firm sold 18,000 units for total revenue of $450,000. Each unit incurred $20 in variable costs. The firm incurred $50,000 of fixed costs. The firm’s fixed costs and CM ratio will remain the same this year.

**Required**

**(A)** What was the firm’s margin of safety last year in sales units?

**(B)** What was the firm’s contribution margin ratio and degree of operating leverage last year?

**(C)** How many unit sales and total revenue does the firm need to earn to reach a target profit of $150,000 (assume the firm’s cost structure remains the same)?

**(D)** How much total revenue did the firm need to earn to breakeven last year?

#### Answer (A)

Margin of safety is the number of the sales units or sales dollars that are earned above and beyond the breakeven point. This question is specifically asking about margin of safety in terms of sales units. First, we must calculate the breakeven point in units.

Breakeven point in units is simply the fixed costs divided by unit CM. Unit CM is $5 (450,000 total revenue / 18,000 sales units – $20 variable cost per unit).

$50,000 fixed costs / $5 unit CM = 10,000 units sold at breakeven.

The margin of safety is simply the units sold (i.e. 18,000) minus the breakeven point (i.e. 10,000). Margin of safety is 8,000 units. These are the number of units the firm sold above and beyond the units that had to be sold to breakeven.

#### Answer (B)

Contribution margin ratio is simply contribution margin divided by revenue. In part A we already calculated unit CM (i.e. $5), so we will divide that $5 by unit revenue of $25 ($450,000 total revenue / 18,000 units sold = $25).

5 / 25 = 0.2 CM ratio.

Degree of operating leverage is simply total contribution margin divided by total profit. Total contribution margin is $90,000 (i.e. $5 unit CM * 18,000 units sold), and total profit is $40,000 (i.e. 90,000 total contribution margin – 50,000 fixed costs).

$90,000 / $40,000 = 2.25

This represent how many dollars of contribution margin were per dollar of profit earned. A higher number indicates higher fixed costs.

#### Answer (C)

Revenue required for target profit is simply the quotient of (in the numerator) the sum of target profit and fixed costs and (in the denominator) CM ratio. Sales units required for target profit is the same equation except the denominator is unit CM.

($150,000 + $50,000) / 0.2 = $1,000,000 sales revenue required for target profit.

(150,000 + 50,000) / $5 = 40,000 units sold at target profit.

#### Answer (D)

In part A we already determined the breakeven point in unit sales (i.e. 10,000 unit sales). We can simply multiply this number by the price per unit to arrive at breakeven in sales revenue (i.e. 10,000 * $25 = $250,000 sales revenue required to breakeven).

Alternatively, we can calculate sales revenue using one of the formulas from part C (specifically the formula with CM ratio in the denominator), but setting target profit equal to zero.

($0 target profit + $50,000 fixed costs) / 0.2 CM ratio = $250,000 sales revenue at breakeven.

## 3GP.3 Multi-product CVP

### 3GP.3.E1

A firm has two products, product 1 and product 2, which it sells in the ratio of 2:3. Product 1 sells for a price of $120 per unit and has $100 of variable cost per unit. Product 2 sells for a price of $55 per unit and has variable costs of $30. The firm’s fixed costs are $70,000.

**Required**

**(A)** How many units of Product 2 does the firm have to sell if it aims to earn $200,000 in profit?

**(B) **How many dollars of Product 1 revenue does the firm need to sell to breakeven?

#### Answer (A)

In order to calculate the unit sales or revenue at a target profit tor a multi-product firm, we must calculate the composite unit CM, that is, the unit CM of each bundle.

The unit CM per unit of Product 1 is $20 (i.e. $120 sales price – $100 variable cost per unit), and the firm sells two units of Product 1 per bundle. The unit CM of Product 2 is $25 (i.e. $55 sales price – $30 variable costs per unit), and the firm sells three units of Product 2 per bundle.

$20 * 2 + $25 * 3 = $115 unit CM per bundle.

Now we can calculate the number of bundles that must be sold to reach the target profit, using the same equations as we used in a single-product firm.

($200,000 target profit + $70,000 fixed costs) / $115 unit CM per bundle = 2347.826

For simplicity we can round up to 2348 bundles sold at the target profit. But we’re not done yet. This gives us the number of bundles that must be sold, but not the units of Product 2 that must be sold. Since each bundle includes three units of Product 2 the answer to this problem is simply the propduct of 2348 bundles and 3 Product 2 units per bundle, i.e. 7,044 units of Product 2 sold at target profit.

#### Answer (B)

The simplest way to answer this question is to first figure out how many units of Product 1 must be sold and then scale up that up by Product 1’s sales price to calculate the total Product 1 revenue.

In Part A we calculated the number of bundles sold at a target profit of $200,000. We can use that same equation but set the target profit to zero.

($0 target profit + $70,000 fixed costs) / $115 unit CM per bundle = 608.696

We can round this to 609 bundles sold at breakeven. Since each bundle includes two units of Product 1, the number of units of Product 1 sold at breakeven is 1,218.

Each unit of Product 1 is sold for $120, so the total Product 1 revenue at breakeven is $146,160.

### 3GP.3.M1

A firm earned $2,750,000 in revenue last year and had a degree of operating leverage of 2. The firm also incurred $250,000 in fixed costs. The firm sells three products in a 1:5:3 ratio. (Assume the firm’s cost structure and sales mix remain the same this year.)

**Required**

**(A) **If there is enough information provided above, what is the total revenue required for the firm to reach a target profit of $1,000,000 this year? If there is not enough information provided, what information is missing?

**(B)** What was the firm’s margin of safety last year (in sales dollars)?

#### Answer (A)

Yes, there is enough information to complete this problem, even though we do not have information on the individual product line’s unit CMs. We have enough information to calculate the firm’s overall CM ratio, which reflects the firm’s sales mix. As long as that sales mix stays the same we can bypass the creation of bundles and so forth.

Even though the problem does not explicitly indicate total contribution margin (which is the numerator in calculating CM ratio), we know the firm’s fixed costs and degree of operating leverage, which allows us to algebraically tease out the firm’s variable costs.

2 = total CM / (total CM – 250,000)

2 * total CM – $500,000 = total CM

$500,000 = total CM

CM ratio = $500,000 total CM / $2,750,000 total revenue = 0.182

Now, applying the equation for total revenue required at a target profit level, we have the following.

($1,000,000 target profit + $250,000 fixed costs) / 0.182 CM ratio = $6,868,131.86 total revenue required at target profit

#### Answer (B)

The breakeven point last year was calculated as follows.

($0 + 250,000) / 0.182 = $1,373,626.37

The firm earned 2,750,000 in revenue, which is $1,376,373.62 more than the breakeven point. That last figure is the margin of safety for last year.

## 3GP.4 Special-Order

### 3GP.4.E1

A firm is considering a special order for 500 units of product TY365. TY365 sells on the external market for $100 and costs $50 in variable costs.

**Required**

**(A)** Assuming the firm has plenty of excess capacity and no extra costs are required to complete the special order, what is the minimum price for the special order?

**(B)** Still assuming plenty of excess capacity but now assuming $10,000 of extra costs are required to complete the special order, what is the minimum price for the special order?

**(C)** Still assuming plenty of excess capacity but now assuming $27,500 of extra costs are required to complete the special order, what is the minimum price for the special order?

#### Answer (A)

The minimum price for a special order where there is plenty of excess capacity and no extra costs to complete the order is simply the variable cost per unit, i.e. $50.

#### Answer (B)

Using the equation for minimum price, this $10,000 must be spread across the units of the special order and then added to the variable cost per unit.

$10,000 extra special order costs / 500 units in order + $50 variable costs per unit = $70 minimum price

#### Answer (C)

This problem is like part B, using the same equation.

$27,500 / 500 + $50 = $105 minimum price

While this price is higher than the general price, the buyer might still accept it, assuming the extra costs the producer is taking upon itself will add sufficient value to special-order units. However, if so, this suggests the seller is taking substantial risk and will likely look for contractual or real guarantees to mitigate this risk. The seller might require a more-binding contract, securitization, or prepayment.

### 3GP.4.M1

A firm is considering a special order for 1,000 units of product MG721. MG721 sells on the external market for $100 and costs $60 in variable costs.

**(A)** Assuming the firm has plenty of excess capacity and no extra costs are required to complete the special order, what is the minimum price for the special order?

**(B)** Still assuming no extra costs but now assuming the firm only has excess capacity for 250 units, what is the minimum price for the special order?

**(C)** If the firm only has excess capacity for 250 units and must incur a cost of $12,000 extra costs to complete the special order, what is the minimum price for the special order?

#### Answer (A)

The minimum price in this special case (plenty of capacity, no extra costs) is simply the variable cost per unit, i.e. $60.

#### Answer (B)

Using the equation for

1,000 special order units – 250 excess unit capacity = 750 lost unit sales.

750 lost unit sales * ($100 revenue per unit – $60 variable costs per unit) / 1,000 special order units + $60 variable costs per unit = $90 minimum price

#### Answer (C)

We use the same equation as in part B, with the addition of the $12,000 extra costs.

1,000 – 250 = 750 lost unit sales.

($12,000 + 750 * $40) / 1,000 + $60 = $102 minimum price

This price is high, but may still be economically feasible if providing the firm’s excess capacity to the buyer and if incurring the extra costs for the special order are of sufficient value to the buyer.

## 3GP.5 Constrained Resource Decisions

### 3GP.5.E1

A firm produces two products. Both products require inspection from a small number of certified inspection personnel in the firm. In total, these employees can provide just 100 inspection hours in a given period.

Product 1 is sold for $150 per unit, and each unit incurs $25 of variable cost. Product 2 sells for $75 per unit, and each unit incurs $25 of variable cost. However, each unit of Product 1 takes 0.5 inspection hours, while each unit of Product 2 takes 0.1 inspection hour.

Demand for the two products is effectively fixed at 150 units of Product 1 and 600 units of Product 2.

**Required**

How many units of each product will the firm optimally produce?

#### Answer

This problem requires prioritizing the two product lines based on which of the two gives the firm the most contribution margin for its time on the constraint. This requires us to divide unit CM for each product by those products’ respective use of the constraint: inspection hours in this case.

The unit CM for Product 1 is $125 (i.e. 150 – 25 = 125) and for Product 2 is $50 (i.e. 75 – 25 = 50). Now, by dividing these by the products’ time on the constraint, we get the following.

$125 unit CM / 0.5 inspection hours per unit = $250 CM per inspection hour

$50 unit CM / 0.1 inspection hours = $500 CM per inspection hour

Product 2 should be prioritized and only after all its demand is met should Product 1 be produced from the remaining inspection hours. Product 2 demand is such that the firm will fill all demand before running out of inspection hours. Specifically, 600 units of demand * 0.1 inspection hours per unit = 60 inspection hours. We will produce all 600 units of demand for Product 2.

There are 40 inspection hours left for Product 1 (i.e. 100 total inspection hours – 60 Product 2 inspection hours = 40 Product 1 inspection hours). This means the firm will produce 80 units of Product 1 (i.e. 40 inspection hours / 0.5 inspection hours per unit = 80 units).

Therefore, at the optimal production arrangement, the firm will produce 600 units of Product 2 and 80 units of Product 1.