How to find the marginal cost, marginal revenue, and marginal profit functions
What do marginal cost, revenue, and profit represent?
We’ve been looking at physical applications of derivatives, but there are also economics applications.
In this lesson, we’ll look at marginal cost, revenue, and profit. But before we jump into these marginal values, let’s look at cost, revenue, and profit in general.
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Cost, revenue, and profit
If a business wants to calculate the revenue generated, the cost incurred, and the profit gained by producing units of a product, it can use the specific formulas.
Revenue: ???R(x)=xp???
Cost: ???C(x)=F+V(x)???
Profit: ???P(x)=R(x)-C(x)???
In these formulas, ???p??? is the demand function for the product, so revenue is given by the product of demand and the number of units sold. ???F??? is fixed cost and ???V(x)??? is variable cost, so cost is the sum of the fixed and variable costs. The profit is then the difference between the revenue and the cost.
In other words, if a company is making ???100??? units of their product, the revenue function will tell them how much revenue will be generated by the ???100??? units, the cost function will tell them how much it’ll cost to produce the ???100??? units, and the profit function will find the total profit gained from producing and then selling the ???100??? units.
The marginal functions
Of course, every company wants to maximize its profits, but increasing the number of units they produce doesn’t always translate to higher profits.
For instance, if an airplane company is making as many planes each month as their current manufacturing space allows, they might need to build a second factory in order to make even one more plane per month. But building a second factory, to make only one more plane, won’t necessarily be profitable. On the other hand, if they build a second factory in order to produce ???100??? more planes each month, that might be a profitable decision.
The marginal revenue, cost, and profit functions are what the company can use to determine whether or not they should increase production. These marginal functions are the derivatives of their associated functions. So the marginal revenue function is the derivative of the revenue function; the marginal cost function is the derivative of the cost function; and the marginal profit function is the derivative of the profit function.
The marginal revenue function models the revenue generated by selling one more unit, the marginal cost function models the cost of making one more unit, and the marginal profit function models the profit made by selling one more unit.
This understanding of what the marginal functions model should make sense to us. Because these marginal functions are derivative functions, they model the slope of the original function, or the change per unit. So if we, for instance, find a marginal cost function as the derivative of the cost function, the marginal cost function should be modeling the change, or slope, of the cost function. And that slope is really just how much the original cost function is increasing or decreasing, per unit.
How to find the marginal cost, marginal revenue, and marginal profit functions
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Let’s do an example where we calculate the marginal cost, revenue, and profit of producing a specific number of goods
Example
A smart phone manufacturer knows that the cost of producing ???x??? phones is given by ???C(x)=6x^2+34x+2,500??? and that the demand function for their phones is ???p=60x???. Calculate the marginal cost, marginal revenue and the marginal profit of producing ???75??? phones.
To calculate marginal cost at ???75??? units, we take the derivative of the cost function and then evaluate the derivative at ???x=75???.
???C'(x)=12x+34???
???C'(x)=12(75)+34???
???C'(x)=934???
The marginal cost at ???x=75??? is ???\$934???, which means the additional cost associated with producing the ???76???th unit is ???\$934???.
Because these marginal functions are derivative functions, they model the slope of the original function, or the change per unit.
To calculate marginal revenue at ???75??? units, we need to find a revenue function, take its derivative, and then evaluate the derivative at ???x=75???.
The revenue equation is ???R(x)=xp??? where ???p??? is the demand function, ???p=60x???.
???R(x)=x(60x)???
???R(x)=60x^2???
Taking the derivative of revenue to get marginal revenue, we get
???R'(x)=120x???
Then we evaluate at ???x=75???.
???R'(x)=120(75)???
???R'(x)=9,000???
The marginal revenue at ???x=75??? is ???\$9,000???, which means the additional revenue associated with selling the ???76???th unit is ???\$9,000???.
Finally, to solve for marginal profit we need to find a profit function, take its derivative, and then evaluate the derivative at ???x=75???.
The profit equation is ???P(x)=R(x)-C(x)???, where ???R??? is the revenue function we found earlier and ???C??? is the cost function we were given.
???P(x)=\left(60x^2\right)-\left(6x^2+34x+2,500\right)???
???P(x)=54x^2-34x-2,500???
Taking the derivative of profit to get marginal profit, we get
???P'(x)=108x-34???
Then we evaluate at ???x=75???.
???P'(x)=108(75)-34???
???P'(x)=8,066???
The marginal profit at ???x=75??? is ???\$8,066???, which means the additional profit associated with producing and selling the ???76???th unit is ???\$8,066???.
Remember that marginal cost, revenue and profit are always only approximations. For that reason, ???P'(x)??? will not always be equal to ???R'(x)-C'(x)???.
