# 2.3 Contribution Margin

## LEARNING OBJECTIVES

• Solve cost-volume-profit analysis problems using the contribution margin approach.

Suppose you run a small business that produces custom-ordered t-shirts.  You have fixed costs of $2,700 per month. Each t-shirt has a variable cost of$25.  You sell each t-shirt for $40. What happens to the$40 you receive for each t-shirt?  Because each t-shirt you sell has a $25 variable cost, you need to use$25 of the $40 selling price to cover the variable costs of each t-shirt. That leaves you with$15 from each t-shirt sold.  Does this mean that you make $15 in profit for every t-shirt you sell? Not exactly. Do not forget that you also have to cover the$2,700 fixed costs.  Before you can make any profit, the fixed costs must be paid.  Until all of the fixed costs are paid, the leftover $15 from the$40 selling price is applied to the fixed costs.  Once you have sold enough t-shirts to pay off the fixed costs, the $15 becomes profit. This happens at the break-even point. In other words, you have to use the$15 to pay the fixed costs until you reach the break-even point, and after the break-even point the $15 becomes profit. The$15 is called the contribution margin.

## Contribution Margin

The contribution margin ($CM$) is the difference between the selling price per unit and the variable cost per unit.

$\displaystyle{CM=S-VC}$

where

• $CM$ is the contribution margin.  This is the amount of money that remains available to pay for fixed costs once the unit variable cost is removed from the selling price of the product.
• $S$ is the selling price per unit.  The unit selling price of the product.
• $VC$ is the variable cost per unit.  The unit variable cost is the typical or average variable cost associated with an individual unit of output.

A jewelry store sells watches for $120 each. The fixed costs are$6,800 per month and the variable cost per watch is $70. What is the contribution margin? What does the contribution margin represent? Solution: Step 1: The given information is $\begin{eqnarray*} S & = & \120 \\ VC & = & \70 \\ FC & = & \6,800 \end{eqnarray*}$ Step 2: Calculate the contribution margin. $\begin{eqnarray*} CM & = & S-VC \\ & = & 120-70 \\ & = & \50 \end{eqnarray*}$ Step 3: Interpret the contribution margin. For every watch sold below the break-even point,$50 is applied to pay off the fixed costs.  For every watch sold above the break-even point, a $50 profit is made. ## Net Income using Contribution Margin In accounting and marketing, the contribution margin is the amount that each unit sold adds to the net income of the business. This approach allows you to understand the impact on net income of each unit sold. The contribution margin determines, on a per-unit basis, how much money is left over after unit variable costs are removed from the price of the product. This leftover money is then available to pay for the fixed costs. Ultimately, when all fixed costs have been paid for, the leftover money becomes the profits of the business. If not enough money is left over to pay for the fixed costs, then the business has a negative net income and loses money. Recall from the previous section that net income is the different between the total revenue and total costs. $\displaystyle{NI=S \times x-(FC+VC \times x)}$ where • $NI$ is the net income. The amount of money left over after all costs have been paid is the net income. If the number is positive, then the business is profitable. If the number is negative, then the business suffers a loss. • $S$ is the selling price per unit. The unit selling price of the product. • $x$ is the number of units produced. This is the number of units produced or sold or the total output that incurred the total variable costs. • $FC$ is the fixed costs. These are the costs that do not change regardless of the level of production. • $VC$ is the variable cost per unit. The unit variable cost is the typical or average variable cost associated with an individual unit of output. Rearranging this formula, we get $\displaystyle{NI=(S-VC) \times x-FC}$ Because $CM=S-VC$, we can express the net income in terms of the contribution margin $\displaystyle{NI=CM \times x-FC}$ where • $NI$ is the net income. The amount of money left over after all costs have been paid is the net income. If the number is positive, then the business is profitable. If the number is negative, then the business suffers a loss. • $CM$ is the contribution margin. This is the amount of money that remains available to pay for fixed costs once the unit variable cost is removed from the selling price of the product. • $x$ is the number of units produced. This is the number of units produced or sold or the total output that incurred the total variable costs. • $FC$ is the fixed costs. These are the costs that do not change regardless of the level of production. If you have no units sold, there is no offsetting revenue to pay for the fixed costs. Consequently, your net income is negative and equal to the total fixed costs associated with your business. With each unit sold, the contribution margin is available to pay off the fixed costs. Once the fixed costs are paid, the contribution margin becomes profit. ## EXAMPLE A manufacturer makes winter boots. The manufacturer sells the boots for$135 a pair.  The fixed costs are $5,400 per month and the variable cost per pair of boots is$60.

1. What is the contribution margin?
2. What is the net income when 100 pairs of boots are sold a month?
3. What is the net income when 65 pairs of boots are sold a month?

Solution:

Step 1:  The given information is

$\begin{eqnarray*} S & = & \135 \\ VC & = & \60 \\ FC & = & \5,400 \end{eqnarray*}$

Step 2:  Calculate the contribution margin.

$\begin{eqnarray*} CM & = & S-VC \\ & = & 135-60 \\ & = & \75 \end{eqnarray*}$

Step 3:  Calculate the net income for $x=100$ units.

$\begin{eqnarray*} NI & = & CM \times x-FC \\ & = & 75 \times 100-5,400 \\ & = & \2,100 \end{eqnarray*}$

At 100 units, the net income is $2,100. Because the net income is positive, the manufacturer made a profit at 100 units. Step 4: Calculate the net income for $x=65$ units. $\begin{eqnarray*} NI & = & CM \times x-FC \\ & = & 75 \times 65-5,400 \\ & = & -\525 \end{eqnarray*}$ At 65 units, the net income is -$525.  Because the net income is negative, the manufacturer incurred a loss at 65 units.

In the net income formula show below, $x$ is the number of units produced and sold.

$\displaystyle{NI=CM \times x-FC}$

Rearranging this formula to solve for $x$, we can express the number of units sold, $x$, in terms of the net income, contribution margin, and fixed costs.

$\displaystyle{\mbox{Number of Units Sold}=\frac{FC+NI}{CM}}$

• $NI$ is the net income. The amount of money left over after all costs have been paid is the net income. If the number is positive, then the business is profitable. If the number is negative, then the business suffers a loss.
• $CM$ is the contribution margin. This is the amount of money that remains available to pay for fixed costs once the unit variable cost is removed from the selling price of the product.
• $FC$ is the fixed costs.  These are the costs that do not change regardless of the level of production.

## EXAMPLE

A manufacturer makes winter boots.  The manufacturer sells the boots for $135 a pair. The fixed costs are$5,400 per month and the variable cost per pair of boots is $60. From the previous example, the contribution margin is$75.

1. How many pairs of boots must be sold in a month to produce a profit of $3,600 ? 2. How many pairs of boots must be sold in a month to incur a loss of$1,275?

Solution:

Step 1:  The given information is

$\begin{eqnarray*} S & = & \135 \\ VC & = & \60 \\ FC & = & \5,400 \\ CM & = & \75 \end{eqnarray*}$

Step 2:  Calculate the number of units for $NI=3,600$.  Because this is a profit, the net income is positive.

$\begin{eqnarray*} \mbox{Number of Units Sold} & = & \frac{FC+NI}{CM} \\ & = & \frac{5,400+3,600}{75} \\ & = & 120 \end{eqnarray*}$

A profit of $3,600 occurs when 120 pairs of boots are sold. Step 3: Calculate the number of units for $NI=-1,275$. Because this is a loss, the net income is negative. $\begin{eqnarray*} \mbox{Number of Units Sold} & = & \frac{FC+NI}{CM} \\ & = & \frac{5,400-1,275}{75} \\ & = & 55 \end{eqnarray*}$ A loss of$1,275 occurs when 55 pairs of boots are sold.

## TRY IT

You run a small business making leather backpacks.  You sell each backpack for $150. Your monthly fixed costs are$7,200 and the variable cost per backpack is $60. 1. What is the contribution margin? 2. What is your net income when you sell 100 backpacks a month? 3. How many backpacks are sold in a month when you incur a loss of$1,350?
Click to see Solution

1. Contribution margin.

$\begin{eqnarray*} CM & = & 150-60 \\ & = & \90 \end{eqnarray*}$

2. Net income for 100 backpacks.

$\begin{eqnarray*} NI & = & CM \times x-FC \\ & = & 90 \times 100-7,200 \\ & = & \1,800 \end{eqnarray*}$

3. Units sold for loss of $1,350. $\begin{eqnarray*} \mbox{Number of Units Sold} & = & \frac{FC+NI}{CM} \\ & = & \frac{7,200-1,350}{90} \\ & = & 65 \end{eqnarray*}$ ## Break-Even Analysis using Contribution Margin Recall the t-shirt business example at the beginning of this section where the fixed costs are$2,700, the variable cost is $25 per unit and the selling price is$40 per unit.  The contribution margin of $15 is first used to pay the fixed costs. At the break-even point, all of the fixed costs are covered and the$15 contribution margin becomes profit after the break-even point.  How many shirts, $x$, do you need to sell to cover the fixed costs?  That is, what is the value of $x$ so that $15\times x=2,700$?  Solving for $x$ yields 180 shirts, which is the break-even point.  This means that for every t-shirt you sell above the break-even point of 180 shirts, you make $15 in profit and for every t-shirt you sell below the break-even point of 180 shirts, you make no profit and the$15 contribution margin goes to paying the fixed costs.

The break-even point is the number of units sold so that the net income is zero.  From the above formula,

$\displaystyle{\mbox{Number of Units Sold}=\frac{FC+NI}{CM}}$

To find the break-even point, we set $NI=0$ in this formula.  Then,

$\displaystyle{\mbox{Break-Even Point}=\frac{FC}{CM}}$

where

• $CM$ is the contribution margin. This is the amount of money that remains available to pay for fixed costs once the unit variable cost is removed from the selling price of the product.
• $FC$ is the fixed costs.  These are the costs that do not change regardless of the level of production.

## EXAMPLE

An electronics manufacturer produces an electronic toy.  The manufacturer sells the toy for $145 each. The fixed costs per month are$15,200.  The variable cost to make each toy is $65. 1. What is the contribution margin? 2. How many toys does the manufacturer need to produce and sell each month to break-even? 3. What is the revenue at the break-even point? 4. What is the net income if the manufacturer sells 75 units above the break-even point? 5. Suppose the fixed costs decrease by$2,000 with no changes to the variable costs or selling price.  What is the new break-even point?
6. Suppose the variable costs increase by $4 with no changes to the fixed costs or selling price. What is the new break-even point? Solution: Step 1: The given information is $\begin{eqnarray*} S & = & \145 \\ VC & = & \65 \\ FC & = & \15,200 \end{eqnarray*}$ Step 2: Calculate the contribution margin. $\begin{eqnarray*} CM & = & S-VC \\ & = & 145-65 \\ & = & \80 \end{eqnarray*}$ Step 3: Calculate the break-even point. $\begin{eqnarray*}\mbox{Break-Even Point} & = & \frac{FC}{CM} \\ & = & \frac{15,200}{80} \\ & = & 190 \end{eqnarray*}$ The manufacturer must produce and sell 190 toys a month to break-even. Step 4: Calculate the revenue for $x=190$ toys. $\begin{eqnarray*} TR & = & S \times x \\ & = & 145 \times 190 \\ & = & \27,550 \end{eqnarray*}$ Step 5: Calculate the net income for $x=265$ units (75 units above the break-even point of 190). $\begin{eqnarray*} NI & = & CM \times x-FC \\ & = & 80 \times 265-15,200 \\ & = & \6,000 \end{eqnarray*}$ At 75 units above the break-even point of 190 units, the net income is$6,000.  Because the net income is positive, this is a profit.

Step 6:  Calculate the break-even point when $FC=15,200-2,000=13,200$.  Because the selling price and variables costs do not change, the contribution margin is still $80. $\begin{eqnarray*}\mbox{Break-Even Point} & = & \frac{FC}{CM} \\ & = & \frac{13,200}{80} \\ & = & 165 \end{eqnarray*}$ If the fixed costs decrease by$2,000, the manufacturer must produce and sell 165 toys a month to break-even.

Step 7:  Calculate the break-even point when $VC=65+4=69$.  Because the variable costs changed, the contribution margin will also change.  The contribution margin must be recalculated before finding the new break-even point.

$\begin{eqnarray*}\mbox{New } CM & = & S-VC \\ & = & 145-69 \\ & = & \76 \\ \\ \mbox{Break-Even Point} & = & \frac{FC}{CM} \\ & = & \frac{15,200}{76} \\ & = & 200 \end{eqnarray*}$

If the variable costs increase by $4, the manufacturer must produce and sell 200 toys a month to break-even. ### NOTE If either the selling price or variable costs (or possibly both) change, the contribution margin will also change, and so needs to be recalculated in these situations. In the previous example, because the contribution margin is not affected by the fixed costs, we did not have to recalculate the contribution margin in the part of the question where just the fixed costs changed. However, we did have to recalculate the contribution margin when the variable costs changed. ## TRY IT You run a business that produces and sells bicycles. The fixed costs per month are$35,000 and the variable cost per bike is $375. You sell each bike for$500.

1. What is the contribution margin.
2. What is the break-even point?
3. Suppose the fixed costs increase by 20% with no changes to the selling price or variable costs.  What is the new break-even point?
4. Suppose the selling price decreases by $25 with no changes to the fixed costs or variable costs. What is the new break-even point? Click to see Solution 1. Contribution margin. $\begin{eqnarray*} CM & = & 500-375 \\ & = & \125 \end{eqnarray*}$ 2. Break-even point. $\begin{eqnarray*} \mbox{Break-Even Point} & = & \frac{FC}{CM} \\ & = & \frac{35,000}{125} \\ & = & 280 \end{eqnarray*}$ 3. New break-even point for $FC=\42,000$. $\begin{eqnarray*} \mbox{Break-Even Point} & = & \frac{FC}{CM} \\ & = & \frac{42,000}{125} \\ & = & 336 \end{eqnarray*}$ 4. New break-even point for $S=\475$. $\begin{eqnarray*} CM & = & S-VC \\ & = & 475-375 \\ & = & \100 \\ \\ \mbox{Break-Even Point} & = & \frac{FC}{CM} \\ & = & \frac{35,000}{100} \\ & = & 350 \end{eqnarray*}$ ## EXAMPLE A sporting goods manufacturer makes baseball caps. The baseball caps sell for$28 each.  The contribution margin per baseball cap is $12.35. To break-even each month, the manufacturer needs a total revenue of$12,600.

1. What is the break-even point?
2. What are the monthly fixed costs?
3. What is the net income if the number of baseball caps sold is 80 units below the break-even point?

Solution:

Step 1:  The given information is

$\begin{eqnarray*} S & = & \28 \\ CM & = & \12.35 \\ \mbox{Break-even revenue} & = & \12,600 \end{eqnarray*}$

Step 2:  Calculate the break-even point.

$\begin{eqnarray*}\mbox{Break-Even Revenue} & = & S \times x \\ 12,600& = & 28 \times x \\ \frac{12,600}{28}& = & x \\ 450 & = & x \end{eqnarray*}$

The manufacturer must produce and sell 450 baseball caps a month to break-even.

Step 3:  Calculate the fixed costs.

$\begin{eqnarray*} \mbox{Break-Even Point} & = & \frac{FC}{CM} \\450 & = & \frac{FC}{12.35} \\ 450 \times 12.35 & = & FC \\ 5,557.50 & = & FC \end{eqnarray*}$

Step 4:  Calculate the net income for $x=370$ units (80 units below the break-even point of 450).

$\begin{eqnarray*} NI & = & CM \times x-FC \\ & = & 12.35 \times 370-5,557.50 \\ & = &- \980 \end{eqnarray*}$

At 80 units below the break-even point of 450 units, the net income is -$980. Because the net income is negative, this is a loss. ## Exercises Note: There are other ways to solve most of these questions, but use the contribution margin to answer the following questions. 1. In the current period, Blue Mountain Packers in Salmon Arm, British Columbia, had fixed costs of$228,000 and a total cost of $900,000 while maintaining a level of output of 6,720 units. Next period sales are projected to rise by 20%. What total cost should Blue Mountain Packers project? Click to see Answer$1,034,000

2. Fred runs a designer candle-making business out of his basement. He sells the candles for $15 each, and every candle costs him$6 to manufacture. If his fixed costs are $2,300 per month, what is his projected net income or loss next month, for which he forecasts sales of 225 units? Click to see Answer -$275

3. Gayle is thinking of starting her own business. Total fixed costs are $19,000 per month and unit variable costs are estimated at$37.50. From some preliminary studies that she completed, she forecasts sales of 1,400 units at $50 each, 1,850 units at$48 each, 2,500 units at $46 each, and 2,750 units at$44 each. What price would you recommend Gayle set for her products?

$46 4. What level of output would generate a net income of$15,000 if a product sells for $24.99, has unit variable costs of$9.99, and total fixed costs of $55,005? Click to see Answer 4,667 5. What is the unit contribution margin on a product line that has fixed costs of$1,800,000 with a break-even point of 360,000 units?

$5 6. Franklin has started an ink-jet print cartridge refill business. He has invested$2,500 in equipment and machinery. The cost of refilling a cartridge including labor, ink, and all other materials is $4. He charges$14.95 for his services. How many cartridges does he need to refill to break even?

229

7. Hasbro manufactures a line of children’s pet toys. If it sells the toy to distributors for $2.30 each while variable costs are$0.75 toy, how many toys does it need to sell to recover the fixed cost investment in these toys of $510,000? What total revenue would this represent? Click to see Answer 329,033,$756,775.90

8. If the break-even point is 15,000 units, the selling price is $95, and the unit variable cost is$75, what are the company’s total fixed costs?

$300,000 9. Louisa runs a secretarial business part time in the evenings. She takes dictation or handwritten minutes and converts them into printed word-processed documents. She charges$5 per page for her services. Including labor, paper, toner, and all other supplies, her unit variable cost is $2.50 per page. She invested$3,000 worth of software and equipment to start her business. How many pages will she need to output to break even?

1200

10. Ashley rebuilds old laptops as a home hobby business. Her variable costs are $125 per laptop and she sells them for$200. She has determined that her break-even point is 50 units per month. Determine her net income for a month in which she sells 60 units.

$750 11. Shardae is starting a deluxe candy apple business. The cost of producing one candy apple is$4.50. She has total fixed costs of $5,000. She is thinking of selling her deluxe apples for$9.95 each.
1. Determine her unit break-even point at her selling price of $9.95. 2. Shardae thinks her price might be set too high and lowers her price to$8.95. Determine her new break-even point.
3. An advertising agency approaches Shardae and says people would be willing to pay the $9.95 if she ran some “upscale” local ads. They would charge her$1,000. Determine her break-even point.
4. If she wanted to maintain the same break-even units as determined in a., what would the price have to be to pay for the advertising?

a. 918; b. 1,124; c. 1,101; d. $11.04 12. Robert is planning a wedding social for one of his close friends. Costs involve$865 for the hall rental, $135 for a liquor license,$500 for the band, and refreshments and food from the caterer cost $10 per person. If he needs to raise$3,000 to help his friend with the costs of his wedding, what price should he charge per ticket if he thinks he can fill the social hall to its capacity of 300 people?

$25 13. École Van Belleghem is trying to raise funds to replace its old playground equipment with a modern, child-safe structure. The Blue Imp playground equipment company has quoted the school a cost of$49,833 for its 20m by 15m megastructure. To raise the funds, the school wants to sell Show ‘n’ Save books. These books retail for $15.00 each and cost$8.50 to purchase. How many books must the school sell to raise funds for the new playground?

7,667

14. Calculate the following:
1.  By what percentage does the unrounded unit break-even point change if the unit contribution margin increases by 1% while all other numbers remain the same?
2. By what percentage does the unrounded unit break-even point change if total fixed costs are reduced by 1% while all other numbers remain the same?
3. What do the solutions to the above questions illustrate?

a. -0.99%; b. -1%; c. lowering fixed costs is better.

15. Suppose the selling price per unit is $100, the variable costs per unit is$60 and the fixed costs are $250,000 1. What is the contribution margin? 2. Calculate the break-even point in both units and revenue. 3. Suppose the fixed costs are reduced by 15% and the unit variable costs rise by 5%. What is the new break-even point and break-even revenue? 4. Suppose the unit variable costs are reduced by 10% and the fixed costs rise by 5%. What is the new break-even point and break-even revenue? 5. Suppose the total fixed costs are reduced by 20% and the unit variable costs rise by 10%. What is the new break-even point and break-even revenue? 6. Suppose the unit variable costs are reduced by 15% and the fixed costs rise by 15%. What is the new break-even point and break-even revenue? Click to see Answer a.$61.73; b. 6,250, $625,000; c. 5,744,$574,400; d. 5,707, $570,700; e. 5,883,$588,300; f. 5,868, \$586,800