Chapter 2, Lesson 3

In this lesson students will:

• create a total profit function, given information about production
• use the total profit function to determine profit levels, given information about production
• determine what production level is required for a pre-determined profit level
• determine the break-even point for production
• understand when a company would go into shutdown

The Total Profit Function

Fixed and Variable Costs

For a producer there are typically two main types of costs: fixed costs and variable costs. Fixed costs are costs that are independent of the number of items produced, and may include things like building or machinery purchases. Variable costs are directly dependent on the number of items produced, and may include things like materials required for production.

For the purpose of this lesson, we will create a total cost function, denoted as $TC$, that is composed of both the fixed costs ($FC$) and variable costs ($VC$).

\begin{aligned} TC &= FC + VC \end{aligned}

Since the variable costs are dependent on the number of items produced, we write $VC = Cx$, where $C$ represents the cost to produce each item and $x$ represents how many items are produced. This gives:

\begin{aligned} TC &= FC + Cx. \end{aligned}

Total Revenue

The total revenue ($TR$) represents our gross income, and is also dependent on how many items are produced. Using $S$ to represent our selling price, we can write

\begin{aligned} TR &= Sx. \end{aligned}

Notice, again, that we use the variable $x$ to represent the number of items produced.

Combining Revenue and Costs

Now we are in a position to create the total profit function ($TP$). Total profit can be thought of as our net income, or as total revenue minus total costs. Making a few subsitutions we arrive at the desired formula.

\begin{aligned} TP &= TR - TC \\ &= Sx - (FC + Cx) \\ &= Sx - FC - Cx \\ &= Sx - Cx - FC \\ &= (S - C)x - FC \end{aligned}

In other words, every time we sell an item, we obtain $\(S-C)$ in income (notice that this is exactly the markup we discussed in the last lesson). We must sell a certain amount of goods in order to pay off our fixed costs ($FC$), otherwise our total profit will be negative and we still owe money for the fixed costs. Let’s work through a few examples in the next video.

Break-even Point

Sometimes a company would like to know how many units must be produced and sold in order to start realizing a profit. Recall that if a company has fixed costs, many items might have to be produced and sold before these fixed costs are paid off. If too few items are sold, then the total profit function is negative and we still owe for our fixed costs. The point at which our total profit function is zero dollars is called the break-even point. This is the point at which we have produced and sold enough items to pay off our fixed costs – increasing production past this point allows the company to realize profit. To find the break-even point, we let $TP = \0$ and solve for the unknown $x$.

Shutdown Point

The shutdown point is the point at which a company is better off not producing. From the perspective of the company, it does not make sense to continue selling an item if it is selling at a loss. Since the expression $(S - C)$ in the total profit function controls how much net revenue is earned from selling one item, it makes sense that if $S < C$, or the cost per item is larger than the selling price per item, the company should stop selling this item.

For example, if the cost to produce a watch is $\70$, yet the selling price of the watch is $\50$, then each watch is sold at a loss of twenty dollars since $(S - C) = \50 - \70 = -\20$. There may be instances in which a company will sell at a loss to help recoup lost revenue; however, if the cost to produce an item fluctuates in such a way so that it is higher than the selling price of an item, it typically makes sense for the company to stop producing this item and shut down production.