2.3: Modeling Revenue, Costs, and Profit (2024)

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In the last section we looked at the economic model for supply and demand. We were particularly interested in the point of market equilibrium. In this section we will look at the model for revenue, cost and profit. As with the previous section we will begin with assumptions that make as many things as possible linear.

Revenue and a review of demand price.

The simple model for revenue is

\[ \text{revenue} = \text{quantity}*\text{price}\text{.} \nonumber \]

However, in the previous section we worked with two price functions, the supply price and the demand price. Since we can only make a sale if the consumer is willing to buy, we typically use the demand price in computing revenue. Our model is now

\[ \text{revenue} = \text{quantity}*\text{demand price(quantity)}\text{.} \nonumber \]

If the demand price is a linear function, then revenue is a quadratic function.

Example 2.2.1: Finding Revenue From Linear Demand Price.

Figure $2.2.2.$ Video presentation of this example

We have determined that the demand price function for widgets is

\[ \text{demand price}(q)=10-q/1000\text{,} \nonumber \]

if the quantity is between 2000 and 8000. Find the revenue function and graph it over the region where it is defined.

Solution

We set up a chart in Excel with revenue defined as $\text{supply price} * \text{quantity}\text{.}$

When we graph we note that the scales are quite different for price and revenue. Thus we want to use secondary axes to capture the scale of both price and revenue. We can also put different labels on the two vertical axes.

Cost.

Once again we will start with a simplified model for cost.

For our (simplified) model we will break costs into fixed costs and variable costs.
Fixed costs include the costs of being in business. They might include license fees, rent for a store or plant, and the cost of furnishings and equipment.
Variable costs are tied to the amount you produce or sell. They might include raw material for a manufacturer or the cost of goods for someone in sales.
For our simplified model we assume that variable costs are proportional to quantity. This makes our cost function linear.
For our simplified model variable costs= unit costs*quantity.
Thus costs= fixed costs + unit costs*quantity.

Example 2.2.3: Finding Linear Cost.

We can set up a small gizmo manufacturing shop for $6,000. The raw materials for producing gizmos cost $14 per unit. Find the cost function for gizmo production. Find the cost of producing 2500 gizmos.

Solution

The fixed costs are the $y$ value of the $y$-intercept of the cost function. The per unit material cost is the slope of the function. We have

\[ \text{cost}=6000+14*\text{quantity}\text{.} \nonumber \]

If we substitute 2500 for the quantity, our costs are

\[ \text{cost}(2500)=6000+14*2500=41000\text{.} \nonumber \]

Profit.

For the third piece of the model, we look at profit. We have the simple formula

\[ \text{profit} = \text{revenue} - \text{cost}\text{.} \nonumber \]

For our simple examples where cost is linear and revenue is quadratic, we expect the profit function to also be quadratic, and facing down. We will obviously be interested in the spots where the profit function either crosses the axis or reaches a maximum.

Figure $2.2.4.$ Video presentation of the next two examples

Example 2.2.3: Finding Profit.

Break-Even Point.

The last example illustrates a reality of manufacturing and retail. If a business has a fixed cost or startup expense, it will have a loss if it does not sell enough.

The point at which revenues equal expenses (cost) is called the break-even point.

This is important in preparing a business proposal, because the bank will want to know if the break even point is a reasonable amount before it lends any money.

Example 2.2.6: Find Break-Even Points.

Find break-even points for previous example. Explain what those points mean in practical terms.

Solution

We look at the chart from the previous example.

We can find break-even points by using Goal Seek and setting profit to 0 while changing quantity. In this case, we see that we have break-even points when the quantity is 2000 or 5000, since those numbers were already on our chart.

The first break even point tells us that we need to lower our price to no more than $13.00 to attract enough customers to be able to turn a profit. The second break even point says that is we bring our price down below $10, we will not be able to bring in enough customers to make a profit.

Example 2.2.7: Repeat, Starting With Data.

Figure $2.2.8.$ Video presentation of this example

We have the following data from the gizmo market, with quantity and costs measured in millions.

Quantity	7.81	10.07	11.99	13.84	15.80
Demand Price	$12.07	$9.05	$7.60	$6.64	$5.64
Cost	$60.05	$70.09	$79.98	$89.90	$99.83

Assuming that price and cost are well modeled by linear equations, find the break-even points and explain what they mean with units included in the explanation.

To find the break-even point when we are given data instead of an equation, we usually follow this procedure: Find the best fitting equations for price and cost. From those equations, produce formulas for revenue and profit. Use the formulas to find the break-even points using either algebra or Excel.

Solution

We put the data into Excel and ask for best fitting lines.

This produces the desired cost and price functions.

\begin{align*} \text{demand price} \amp =-0.7796 q+17.478\\ \text{cost} \amp = 5.00251 q+20.162\text{.} \end{align*}

We enter these functions in new columns in the spreadsheet and then compute projective revenues and profit. We then use Goal Seek to find places where the projected profit is 0. The first break-even point tells us that we expect to break even if we sell 1.83 million units. We can do that by setting the price at $16.05. The second break-even point is at 14.15 million units. We reach that sales volume by lowering the price to $6.45. While we will have gained market share, we will no longer be making a profit.

Technical note.

In business situations we often have cases where a change of quantity in the thousands only changes prices by pennies. Then our coefficients are close to zero and Excel may give formulas rounded to zero. In those cases we need to format the trendline to get more digits of accuracy.

Example 2.2.9: Problems with Using Big Numbers.

Figure $2.2.10.$ Video presentation of this example

We want to explore an issue that arises our coefficients are very small. We will have to be concerned with the number of significant digits in our coefficients.

We repeat the previous example, but with quantity and cost measured directly, rather than in millions. We should get the same answers, since we are using the same data.

Quantity	7,810,000	10,070,000	11,990,000	13,840,000	15,800,000
Demand Price	$12.07	$9.05	$7.60	$6.64	$5.64
Cost	$60,050,000	$70,090,000	$79,980,000	$89,900,000	$99,830,000

We face the same tasks. Assuming that price and cost are well modeled by linear equations, find the break-even points and explain what they mean with units included in the explanation.

Solution

We put the data into Excel and ask for best fitting lines.

As expected, a coefficient of each equation has been shifted by a factor of 1,000,000.

\begin{align*} \text{demand price} \amp =-8*10^{-7} q+17.478\\ \text{cost} \amp =5.0251 q+2*10^{7}\text{.} \end{align*}

These equations have only one digit of accuracy. In general that will not be accurate enough.

We enter these functions in new columns in the spreadsheet and then compute projective revenues and profit. We then use Goal Seek to find places where the projected profit is 0. The first break-even point goes from 1.83 million at price of $16.05 to 1.82 million at a price of $1602. The second break even point goes from 14.15 million units at a price to $6.45 to 13.75 million at a price of $6.48.

The solution is to right click (Command click on a mac) on the label and select “Format Trendline Label”. Then change category from general to number, and choose 10 decimal places. This gives us the equations:

\begin{align*} \text{demand price} \amp =-0.0000007796 q + 17.4782059302\\ \text{cost} \amp = 5.02506 q+ 20161700\text{.} \end{align*}

We then go through the same process at get our original answers back.

Exercises: Modeling Revenue, Costs, and Profit

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For Exercise $2.2.1–2.2.8$, given the equations of the cost and demand price function:

Identify the fixed and variable costs.
Find the revenue and profit functions.
Evaluate cost, demand price, revenue, and profit at $q_0\text{.}$
Find all break-even points.
Graph the profit function over a domain that includes both break-even points. Add a textbox and label to identify the first break-even point.

Exercise 1:

Given $demand\ price=-2 quantity+20$ and $cost=3 quantity+10\text{,}$ with $q_0=6\text{.}$

Answer

Identify the fixed and variable costs.
The fixed cost is $10 (the constant/fixed part of the cost function), and the variable cost is $3 per item.
Find the revenue and profit functions
\begin{align*} \text{Revenue}\amp=\text{demand price}*\text{quantity}\\ \amp=(-2 q+20)*q=-2 q^2 +20 q \end{align*}

\begin{align*} \text{Profit}= \text{revenue}-\text{cost} \amp=-2 q^2 +20q-(3q+10)\\ \amp=-2 q^2+17q-10\text{.} \end{align*}

2.3: Modeling Revenue, Costs, and Profit (2024)

Revenue and a review of demand price.

Cost.

Profit.

Break-Even Point.

Technical note.

Exercises: Modeling Revenue, Costs, and Profit

References