# 2.1 Cost-Volume-Profit Analysis Terminology

## LEARNING OBJECTIVES

- Understand the terms used in cost-volume-profit analysis and their relationships to each other.

The end of the month is approaching, and bills are coming due. As you sit at your kitchen table trying to figure out your budget for next month, you wonder whether you will be able to afford that concert you had been planning on attending. Some of your costs remain unchanged from month to month, such as your rent, internet service, gym membership, and insurance. Other costs tend to fluctuate with your usage, such as your utilities, cellphone bill, vehicle fuel, and the amount of money spent on recreational activities. Together, these regular and irregular costs total to next month’s costs.

Examining a few recent paycheque stubs, you calculate the average monthly net income you bring home from your hourly cashier position at Sobeys. The exact amount of each paycheque, of course, depends on how many hours you work. Besides your short-term costs, you need to start saving for next year’s tuition. Therefore, your budget needs to include regular deposits into your savings account to meet that goal. Once you have put your bills, paycheques, and goals together, you hope that your budget will balance. If there is a shortfall, you will have to miss out on those concert tickets.

Budgeting at work is no different in principle from your home budget. Businesses also need to recognize the different types of costs they incur each month, some of which remain the same and some of which fluctuate. Businesses need to pay for these costs by generating revenues, which correspond to your paycheque. As with your education goals, businesses also require profits to grow. A business needs to understand all of these numbers so it can plan its activities realistically.

A business needs to understand how to relate its total costs to its total revenue in order to determine its total profitability levels. This process is called **break-even analysis, **which is a technique that allows a business to determine the level of sales required to cover its costs and how to set targets to achieve profitability.

## Costs

A **cost** is an outlay of money required to produce, acquire, or maintain a product, which includes both physical goods and services. In business, examples of costs include rent, employees salaries, taxes, utilities, raw materials, and maintenance, to name a few. Costs can be split into two distinct categories: fixed costs and variable costs.

**Fixed costs** ([latex]FC[/latex]) are costs that do not change with the level of production or sales (call this “output” for short). In other words, whether the business outputs nothing or outputs 10,000 units, these costs remain the same. Some examples of fixed costs include rent, insurance, property taxes, salaries unrelated to production (such as management), production equipment, office furniture, and much more. **Total fixed costs** are the sum of all fixed costs that a business incurs.

A **variable cost** is a cost that changes with the number of units produced. In other words, if the business produces nothing there are no variable cost. However, if the business produces just one unit (or more) then a variable cost appears. Some examples of variable costs include material costs of products, production labor (hourly or piecework wages), sales commissions, repairs, maintenance, and more. **Total variable costs** ([latex]TVC[/latex]) are the sum of all variable costs that a business incurs at a particular level of output.

In calculating business costs, fixed costs are commonly calculated on a total basis only because the business incurs these costs regardless of any production. However, variable costs are commonly calculated both on a total and per-unit basis to reveal the overall cost along with the cost associated with any particular unit of output. When these variable costs are assigned on an individual basis it is called a **variable cost per unit **([latex]VC[/latex]). The calculation of the per unit variable cost has a further benefit because it allows managers to explore how the total business costs vary at different levels of output.

The variable cost per unit is

[latex]\displaystyle{VC=\frac{TVC}{x}}[/latex]

where

**[latex]VC[/latex] is the variable cost per unit.**The unit variable cost is the typical or average variable cost associated with an individual unit of output. Being a dollar cost, the unit variable cost is rounded to two decimals.**[latex]TVC[/latex] is the total variable cost.**The total variable costs is the sum of all the variable costs that were incurred at a particular level of output.**[latex]x[/latex] 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.

Rearranging the variable cost per unit formula, the total variable costs is

[latex]\displaystyle{TVC=VC \times x}[/latex]

## EXAMPLE

Assume a company produces 10,000 units and wants to know its unit variable cost. It incurs production labour costs of $3,000, material costs of $1,875, and other variable costs totaling $1,625.

**Solution:**

**Step 1: ** In this case, all of the costs (production labour and material costs) are variable costs. The total variable costs are

[latex]\begin{eqnarray*} TVC & = & 3,000+1,875+1,625 \\ & = & \$6,500 \end{eqnarray*}[/latex]

**Step 2: ** The variable cost for unit is

[latex]\begin{eqnarray*} VC & = & \frac{TVC}{x} \\ & = & \frac{6,500}{10,000} \\ & = & \$0.65 \end{eqnarray*}[/latex]

The variable cost associated with one unit of production is $0.65.

### NOTE

The definitions of variable and fixed costs along with their typical associated examples represent a simplified view of how the real world operates. The complexities involved in real-world business costs complicate the fundamentals of managing a business. Here is how this textbook addresses the complexities of atypical operations, changing fixed costs, and decreasing unit variable costs:

**Atypical Operations.**Although there are “normal” ways that businesses operate, there are also businesses that have atypical operations. What is a fixed cost to one business may be a variable cost to another. For example, rent is usually a fixed cost. However, some rental agreements include a fixed base cost plus a commission on the operational output. This textbook does not venture into any of these atypical costs and instead focuses on common cost categorizations.**Changing Fixed Costs**. In real-world applications, fixed costs do not remain flat at all levels of output. As output increases, fixed costs tend to move upwards in steps. For example, at a low level of output only one manager (on salary) may be needed. As output increases, eventually another manager needs to be hired, perhaps one for every 20,000 units produced. In other words, up to 20,000 units the fixed costs would be constant, but at 20,001 units the fixed costs take a step upwards as another manager is added. The model presented in this textbook does not address these upward steps and treats fixed costs as a constant at all levels of output.**Decreasing Unit Variable Costs**. Production tends to realize efficiencies as the level of output rises, resulting in the unit variable cost dropping. This is commonly known as achieving**economies of scale**. As a consumer, you often see a similar concept in your retail shopping. If you purchase one can of soup, it may cost $1. However, if you purchase a bulk tray of 12 cans of soup it may cost only $9, which works out to $0.75 per can. This price is lower partly because the retailer incurs lower costs, such as fewer cashiers to sell 12 cans to one person than to sell one can each to 12 different people. Now apply this analogy to production. Producing one can of soup costs $0.75. However, a larger production run of 12 soup cans may incur a cost of only $6 instead of $9 because workers and machines can multitask. This means the unit variable cost would decrease by $0.25 per can. However, the model in this textbook assumes that unit variable costs always remain constant at any given level of output.

The **total costs** ([latex]TC[/latex]) is the sum of the fixed costs and total variable costs.

[latex]\displaystyle{TC=FC+TVC}[/latex]

where

**[latex]TC[/latex] is the total costs.**The total costs is the sum of all of costs incurred by the business.**[latex]FC[/latex] is the fixed costs.**These are the costs that do not change regardless of the level of production.**[latex]TVC[/latex] is the total variable costs.**The total variable costs is the sum of all the variable costs that were incurred at a particular level of output.

## EXAMPLE

You are considering starting your own home-based internet business. After a lot of research, you have gathered the following financial information.

Dell computer | $214.48 monthly lease payments |

Office furniture | $186.67 monthly rental |

High-speed internet connection | $166.88 per month |

Wages | $30 per hour |

Utilities (fixed) | $13 per month |

Utilities (variable) | $0.20 per hour |

Software plus upgrades | $20 per month |

Licences and permits | $27 per month |

Google click-through rate | $10 per month |

Generating and fulfilling sales of 430 units involves 80 hours of work per month. Based on industry response rates, your research also shows that to achieve your sales you require a traffic volume of 34,890 Google clicks.

On a monthly basis, calculate the total fixed cost, total variable cost, and unit variable cost.

**Solution:**

**Step 1: ** Sort the costs into fixed and variable costs.

Fixed Costs |
Variable Costs |
||

Dell computer | $214.48 | Wages | $30 per hour |

Office Furniture | $186.67 | Utilities | $0.20 per hour |

Internet | $166.88 | ||

Utilities | $13 | ||

Software | $20 | ||

Licences and permits | $27 | ||

Google click-through rate | $10 |

**Step 2:** Add up the fixed costs.

[latex]\begin{eqnarray*}FC & = & 214.48+186.67+166.88+13+20+27+10 \\ & = & \$638.03 \end{eqnarray*}[/latex]

**Step 3:** Calculate the total variable costs based on the 80 hours.

[latex]\begin{eqnarray*} TVC & = & 30 \times 80+0.2 \times 80 \\ & = & $2,416 \end{eqnarray*}[/latex]

**Step 4: ** Calculate the variable cost per unit based on the 430 units sold.

[latex]\begin{eqnarray*} VC & = & \frac{2,416}{430} \\ & = & \$5.62 \end{eqnarray*}[/latex]

## Revenue and Selling Price

The **selling price per unit** ([latex]S[/latex]) is the amount the business charges its customers to purchase one unit of an item produced.

**Revenue** is how much money or gross income the sale of the product at a certain output level brings into the business. **Total revenue** ([latex]TR[/latex]) is the entire amount of money received by a company for selling its product calculated by multiplying the quantity sold by the selling price ([latex]S[/latex]).

[latex]\displaystyle{TR=S \times x}[/latex]

where

**[latex]TR[/latex] is the total revenue.**This is how much money or gross income the sale of the product at a certain output level brings into the organization.**[latex]S[/latex] is the selling price per unit.**The unit selling price of the product.**[latex]x[/latex] 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.

## EXAMPLE

A business sells a product for $10 each. How much revenue does the business generate if it produces and sell 5,000 units?

**Solution:**

**Step 1:** The given information is

[latex]\begin{eqnarray*} S & = & 10 \\ x & = & 5,000 \end{eqnarray*}[/latex]

**Step 2: ** Calculate the total revenue.

[latex]\begin{eqnarray*} TR & = & S \times x \\ & = & 10 \times 5,000 \\ & = & \$50,000 \end{eqnarray*}[/latex]

## Net Income and the Break-Even Point

The **net income** ([latex]NI[/latex]) is the difference between the total revenue and the total costs.

[latex]\displaystyle{NI=TR-TC}[/latex]

Note that many companies use the terms net earnings or net profit instead of the term net income. Net income is based on a certain level of output. Here, we assume that the number of units produced or purchased exactly matches the number of units that are output or sold by the company. In other words, there is no inventory, and its associated costs, to consider.

There are three scenarios to consider.

- Total revenue is greater than total costs ([latex]TR \gt TC[/latex]). In this case the net income is positive ([latex]NI \gt 0[/latex]) and the business makes a
**profit**. - Total revenue is less than total costs ([latex]TR \lt TC[/latex]). In this case the net income is negative ([latex]NI \lt 0[/latex]) and the business incurs a
**loss**. - Total revenue equals total cost ([latex]TR=TC[/latex]). In this case the net income is zero ([latex]NI=0[/latex]). This is the
**break-even point**.

If you are starting your own business and head to the bank to initiate a start-up loan, one of the first questions the banker will ask you is your break-even point. You calculate this number through **break-even analysis**, which is the analysis of the relationship between costs, revenues, and net income with the sole purpose of determining the point at which total revenue equals total cost. This **break-even point** is the level of output (in units or dollars) at which all costs are paid but no profits are earned, resulting in a net income equal to zero.

## EXAMPLE

A business sells an item for $75 each. The variable costs are $58 per item and the fixed costs are $4,250 per month.

- What is the net income if the business sells 375 units?
- What is the net income if the business sells 200 units?
- What is the net income if the business sells 250 units?

**Solution:**

**Step 1: ** The given information is

[latex]\begin{eqnarray*} S & = & \$75 \\ VC & = & \$58 \\ FC & = & \$4,250 \end{eqnarray*}[/latex]

The total revenue function is [latex]TR=75 \times x[/latex]. The total cost function is [latex]TC=4,250 +58 \times x[/latex].

**Step 2:** Calculate the net income for 375 units.

[latex]\begin{eqnarray*} NI & = & TR-TC \\ & = & 75 \times 375-(4,250+58 \times 375) \\ & = & 28,125-26,000 \\ & = & \$2,125 \end{eqnarray*}[/latex]

The net income at 375 units is $2,125. Because the net income is positive, at 375 units the business generates a profit.

**Step 3:** Calculate the net income for 200 units.

[latex]\begin{eqnarray*} NI & = & TR-TC \\ & = & 75 \times 200-(4,250+58 \times 200) \\ & = & 15,000-15,850 \\ & = & -\$850 \end{eqnarray*}[/latex]

The net income at 200 units is -$850. Because the net income is negative, at 200 units the business incurs a loss.

**Step 4:** Calculate the net income for 250 units.

[latex]\begin{eqnarray*} NI & = & TR-TC \\ & = & 75 \times 250-(4,250+58 \times 250) \\ & = & 18,750-18,750 \\ & = & \$0 \end{eqnarray*}[/latex]

The net income at 250 units is $0. Because the net income is zero, 250 units is the break-even point.

#### Attribution

“4.6: Cost-Revenue-Net Income Analysis” from Introduction to Business Math by Margaret Dancy is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

“5.1: Cost-Revenue-Net Income Analysis” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.