By now, you have learned about using derivatives to analyze things such as population change and motion along a line. While that is all well and good, have you ever wondered if math could make you money?
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Jetzt kostenlos anmeldenBy now, you have learned about using derivatives to analyze things such as population change and motion along a line. While that is all well and good, have you ever wondered if math could make you money?
Well, it turns out, it can! You can use derivatives and rates of change to describe the simple business and economic concepts of change in cost, revenue, and profit.
The rate of change of a company's profit depends on
the rate at which the company sells its products,
the cost of making each product,
the cost of labor, and
the potential competition for these resources.
You can use these parameters to create an equation that models a company's revenue. Then, the derivative, or rate of change of that equation, is the rate at which the company earns revenue.
While cost and revenue rates of change help the business world make predictions on the future prospects of investments, they can also be useful to businesses themselves to describe how well they're doing or if changes need to be made.
So, what is the meaning of cost and revenue?
Cost represents the amount of money a company must spend to produce a certain commodity. The derivative of a company's cost function is called its marginal cost.
If \(C(x)\) is the cost of producing \(x\) number of items, then the marginal cost, \(MC(x)\), of producing \(x\) items is \(C'(x)\).
Mathematically,
\[ \begin{align}\text{Cost} &= C(x) \\\text{Marginal Cost} &= C'(x) = MC(x).\end{align} \]
Similarly, revenue represents the amount of money a company obtains from selling a certain commodity. The derivative of a company's revenue function is called its marginal revenue.
If \(R(x)\) is the revenue from selling \(x\) number of items, then the marginal revenue, \(MR(x)\), of selling \(x\) items is \(R'(x)\).
Mathematically,
\[ \begin{align}\text{Revenue} &= R(x) \\\text{Marginal Revenue} &= R'(x) = MR(x).\end{align} \]
So, what is the relationship between cost and revenue?
Well, profit represents the amount of money pocketed by a company – once its cost and revenue are tallied up. The formula for profit is subtracting the total cost to the company spent on producing the commodity from the total revenue obtained from selling the commodity.
If \(P(x) = R(x) - C(x)\) is the profit from making and selling \(x\) number of items, then the marginal profit, \(MP(x)\), from making and selling \(x\) items is \(P'(x)\).
Mathematically,
\[ \begin{align}\text{Profit} &= P(x) \\ &= R(x) - C(x), \\\text{Marginal Profit} &= P'(x) \\ &= MP(x) \\ &= R'(x) - C'(x).\end{align} \]
Unfortunately, there are no general formulas for cost, revenue, or profit. Depending on the question, you may either be tasked with determining these formulas based on context clues within a word problem, or the formula(s) may be given to you.
That being said, using the definition of the derivative, you can approximate
\[ \begin{align}\text{Marginal Cost} &= MC(x) \\&= C'(x) \\&= \lim_{h \to 0} \frac{C(x+h) - C(x)}{h} \\\end{align} \]
and
\[ \begin{align}\text{Marginal Revenue} &= MR(x) \\&= R'(x) \\&= \lim_{h \to 0} \frac{R(x+h) - R(x)}{h} \\\end{align} \]
and
\[ \begin{align}\text{Marginal Profit} &= MP(x) \\&= P'(x) \\&= \lim_{h \to 0} \frac{P(x+h) - P(x)}{h} \\\end{align} \]
by choosing an appropriate value for \(h\). But, what is an appropriate value for \(h\)?
Considering that the independent variable \(x\) represents physical items, \(h\) must be an integer greater than zero. So, since you want \(h\) to be as small as possible for the best approximation, substituting \(h = 1\) gives you the formulas:
\[ \begin{align}\text{Marginal Cost} &= MC(x) = C'(x) \approx C(x+1) - C(x), \\\text{Marginal Revenue} &= MR(x) = R'(x) \approx R(x+1) - R(x), \text{and} \\\text{Marginal Profit} &= MP(x) = P'(x) \approx P(x+1) - P(x).\end{align} \]
Looking at these formulas, it should become clear that:
\(C'(x)\) for any value of \(x\) can be thought of as the change in cost associated with the company making one more item.
\(P'(x)\) for any value of \(x\) can be thought of as the change in revenue associated with the company selling one more item.
\(P'(x)\) for any value of \(x\) can be thought of as the change in profit associated with the company making and selling one more item.
As you might have noticed in the definitions above, the derivatives, also known as the marginals, of the cost, revenue, and profit functions measure the change in the functions over time.
Just like the other applications of derivatives, the rate of change of the cost and revenue functions are the same.
For a review of average and instantaneous rate of change, see our article Rates of Change and Amount of Change Formula.
The average rate of change of a cost or revenue function measures how much the cost or revenue changes.
Let \(d = C(x)\) represent the total dollar amount spent on producing \(x\) items. The average rate of change in cost from the first item produced \(x_1\) and the last item produced \(x_2\) is
\[ \begin{align} \mbox{Average Rate of Change of Cost } &= \frac{\Delta d}{\Delta x} \\ &= \frac{C(x_2) - C(x_1)}{x_2-x_1}. \end{align}\]
What about the average rate of change of revenue?
Similarly, let \(p = R(x)\) represent the total dollar amount made from selling \(x\) items. The average rate of change in revenue from the first item sold \(x_1\) and the last item sold \(x_2\) is
\[ \begin{align} \mbox{Average Rate of Change in Revenue } &= \frac{\Delta p}{\Delta x} \\ &= \frac{R(x_2) - R(x_1)}{x_2-x_1}.\end{align} \]
For a walk through on how the average rate of change formula is derived, please refer to the Population Change article.
To find the exact rate of change of a cost or revenue at the point of producing or making one certain item, you find the instantaneous rate of cost or revenue change. By now, you should recognize that the instantaneous rate of cost or revenue change is synonymous with the derivative (or marginal) of the cost or revenue equation.
Again, let \(d = C(x)\) represent the total dollar amount spent on producing \(x\) items. The instantaneous rate of change of cost when producing item \(x\) is
\[ \begin{align} \mbox{Instantaneous Rate of Change of Cost } &= \lim\limits_{\Delta x \to 0} \frac{\Delta d}{\Delta x} \\ &= \frac{dd}{dx}. \end{align}\]
What about the instantaneous rate of change?
Similarly, let \(p = R(x)\) represent the total dollar amount made from selling \(x\) items. The instantaneous rate of change in revenue when selling item \(x\) is
\[ \begin{align} \mbox{Rate of Change in Revenue } &= \lim\limits_{\Delta x \to 0} \frac{\Delta p}{\Delta x} \\ &= \frac{dp}{dx}. \end{align}\]
These concepts are easier to understand with some examples.
Let's take a look at some examples involving cost, revenue, and things like the break even point.
A newspaper company has a fixed cost of production of \(\$80\) per edition, as well as a marginal distribution and printing materials cost of \(40¢\) per copy. Newspapers are sold at \(50¢\) per copy.
Solutions:
Let's take a look at another example.
Suppose a toy company produces \(x\) toys with a cost of \(C(x) = 10000 + 3x + 0.01x^2\) that encompasses the total overhead costs (like rent for the factory), materials, labor, and so on.
Solutions:
Cost is the amount of money a company spends on producing a certain commodity, while revenue is the amount of money a company earns from selling a certain commodity. We can use marginals to better study the changes in cost and revenue.
Cost is the amount of money a company spends on producing a certain commodity, while revenue is the amount of money a company earns from selling a certain commodity.
Cost is the amount of money a company spends on producing a certain commodity, while revenue is the amount of money a company earns from selling a certain commodity. Together, cost and revenue tell us how much profit a company makes.
As cost increases, more revenue is needed in order to produce a profit.
Think of a business that builds and sells toys. The more toys they sell, the more revenue they make. However, they'll likely also spend more money on materials to make the toys as well.
What is cost?
In the business world, cost typically represents the amount of money a company must spend to produce a certain commodity.
What is revenue?
Revenue typically represents the amount of money a company obtains from selling a certain commodity.
What is the relationship between revenue and cost?
Together, cost and revenue tell us how much profit a company makes.
What is profit?
Profit represents the amount of money pocketed by a company derived by subtracting the total cost to the company spent on producing the commodity from the total money obtained from selling the commodity.
What is marginal revenue?
If is a function for the cost of producing number of items, then the marginal revenue of producing items is .
What is marginal revenue?
If is a function for the cost of producing number of items, then the marginal revenue of producing items is .
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