1.3 Linear Functions and Models
Linear Cost, Revenue, and Profit

Next tutorial: Linear Demand, Supply, and Time-Change Models

(This topic is also in Section 1.3 in Finite Mathematics, Applied Calculus and Finite Mathematics and Applied Calculus)

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To use this tutorial, you should be familiar with methods for finding the equation of a straight line with specified data. See the preceding tutorial to learn about some methods.

Using linear functions to describe or approximate relationships in the real world is called linear modeling. Here, we study several kinds of linear model:

Linear Cost Functions

A cost function gives us the cost as a function of the number of items. Here, we are interested in linear cost functions.

Linear Cost Function

A linear cost function expresses cost as a linear function of the number of items. In other words,

    C = mx + b

Here, C is the total cost, and x is the number of items. In this context, the slope m is called the marginal cost and b is called the fixed cost.

Example

The daily cost to print x paperback sci-fi novels is

    C(x) = 3.5x + 1000 dollars.

Note that C is measured in dollars, and x is measured in books (paperback sci-fi novels, to be precise).

The marginal cost is m = 3.5, and the fixed cost is b = 1000.

To understand what these quantities tell us, let us compute some costs:

Q The daily cost if you print no books is $  

This allows us to interpret the fixed cost, $1000, as that part of the cost that is not effected by the number of books printed. Included in the fixed cost are, for instance, morgage payments, salaries, and insurance.
Q The daily cost to print 100 books is $  
Q The daily cost to print 101 books is $  
Q The daily cost to print each additional book is $  

This allows us to interpret the marginal cost, m = $3.50, as the cost to print each additional books.In general, the marginal cost is the cost per additional item.

Units of Measurement.

As we saw above, x is measured in items (books in the above example) and C is measured in units of currency (dollars in the above example). Also:

  • The fixed cost b is measured in units of C (dollars in the last example, since it is the "y-intercept," and C is playing the role of y here.)
  • The marginal cost m is measured in units of C per unit of x (dollars per book in the last example, since it is the slope, measuring units of y per unit of x.)

Q The cost to rent DVDs at the local BlockBuilder outlet is $7 for the first, and $4 per additional DVD.

The cost equation is:
    C =

Q Anmother BlockBuilder oulet charges $15 for three DVDEs and $24 for 6.

Its cost function is:
    C =

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Linear Revenue and Profit Functions

Revenue results from the sale of items, and profit is the excess of revenue over costs. Both the revenue and profit depend on the number of items, x, we buy and sell, and so, like the cost function, they too are functions of x.

Linear Revenue and Profit Functions

Revenue
The revenue resulting from one or more business transactions is the total payment received, sometimes called the gross proceeds. If R(x) is the revenue from selling x items at a price of m each, then R will be the linear function R(x) = mx and the selling price m can also be called the marginal revenue.

Example
Your publishing company sells sci-fi paperbacks to a large retailer for $4.75 per book. The revenue function is then

    R(x) = 4.75x dollars.

The marginal revenue is m = $4.75 per book. Notice again that the units of measurement of the slope are units of y (dollars) per unit of x (books).

Profit
The profitis the net proceeds, or what remains of the revenue when costs are subtracted. If the profit depends linearly on the number of items, the slope m is called the marginal profit. Profit, revenue, and cost are related by the following formula.

    Profit = Revenue - Cost
    P = R - C.

If the profit is negative, say -$500, we refer to a loss (of $500 in this case). To break even means to make neither a profit nor a loss. Thus, break-even occurs when P = 0, or

    R = C       Break Even

The break-even point is the number of items x at which break-even occurs.

Example
Going back to the sci-fi books, we already have the cost and revenue functions:
    C(x) = 3.5x + 1000Daily cost to make x books
    R(x) = 4.75x Revenue from sale of x books
Q Profit = P(x) = dollars.  
Q For break-even, x = books.  

Thus, you should sell 800 books per day to break even, and more than that to make a profit of $1.25 per additional book. ($1.25 is the marginal profit.)

You are the manager of Sassy Surf Creations, the new trend-setting clothing manufacturer. Unfortunately, your accounting practices are somewhat haphazard, and you do not keep track of all your costs. Last week, you noticed that a batch of 200 very exclusive Tai Kwon Do Dragon T shirts cost the company a total of $2050 to make, while this week a batch of 100 shirts cost the company $1300. Sassy Surf Creations sells the shirts for $15 each.

Q The cost function is:
    C(x) =
 

Q The revenue function is:
    R(x) =
 

Q The profit function is:
    P(x) =
 

Q To break even, Sassy should make and sell at least:
    x = shirts per day
 

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Now go on to Part B: Linear Demand, Supply, and Time-Change Models to see other important applications of linear models.

Alternatively, try some of the exercises in Section 1.3 of the textbook, or press "Review Exercises" on the sidebar to see a collection of exercises that covers the whole of Chapter 1.

Last Updated: March, 2006
Copyright © 2001, 2007 Stefan Waner

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