Marginal Functions

 Let $C(x)$, $R(x)$, and $P(x)$ denote, respectively, the total cost function, the total revenue function, and the total profit function.

Definition.

• $C(x+1)-C(x)$, the cost of producing the $x+1$st item,  is called the marginal cost at production level $x$.
• $R(x+1)-R(x)$, the revenue derived by the producer when $x+1$ items are sold, is called the marginal revenue at $x$.
• $P(x+1)-P(x)$, the producer's profit due to the $x+1$st item, is called marginal profit at $x$.
  

Let $E(x)$ be an economic function which represents any of the total cost, revenue, and profit functions and we assume that it is differentiable. In many real applications, the approximation
$$E'(x)\approx E(x+1)-E(x)$$
or
$$E'(x)\approx E(x)-E(x-1)$$
is valid. That is, $E'(x)$ is approximately the marginal economic function at level $x$ or level $x-1$.

Example. Given that the cost, in dollar, of producing $x$ short wave radios is $C(x)=x^2+80x+3500$,

1. Find the cost of producing; (a) the 100th radio; (b) the 101st radio.
2. Find $C'(100)$ and interpret this.
   

Solution.

1.  (a) It is the marginal cost at level 99, $C(100)-C(99)=279$ dollars. (b) It is the marginal cost at level 100, $C(101)-C(100)=281$ dollars. 
2. $C'(x)=2x+80$, so $C'(100)=280$ dollars. This can be used to approximate the marginal cost at level 99 or at 100. In either case, the error is 1 dollar and the percentage error in the approximation is 0.36.
   

Definition. The average cost function $\bar C(x)$ is defined by
$$\bar C(x)=\frac{C(x)}{x}$$

Example. Suppose that the relationship between price of and demand for a certain type of large color television set is given by the demand equation $10p+x=10000$, where $p$ is the per unit price in dollars, and $x$ is the number of sets demanded. If the producer's cost is $C(x)=60x+3000$,

1. Determine the revenue function.
2. Determine the marginal revenue function.
3. Determine the profit function.
4. Determine the marginal profit function.
5. What is the price for each color television when the marginal profit is zero?
6. Sketch the profit function, drawing the tangent line when the marginal profit is zero. What does this profit represent?
7. What can you conclude about the profit at the price obtained in 5?
   

Solution.

1. Solving the demand equation for price, we find $$p=-0.1x+1000$$ and so the revenue function is given by $$R(x)=px=-0.1x^2+1000x$$

2. The marginal revenue function at $x$ is $R(x+1)-R(x)$ but for practical purposes, it can be replaced by the approximation $R'(x)=-0.2x+1000$.

3. \begin{align*}P(x)&=R(x)-C(x)\\&=-0.1x^2+400x-3000\end{align*}

4. $P'(x)=-0.2x+400$.

5. When $P'(x)=-0.2x+400=0$, $x=2000$ i.e. the production level when the marginal profit is zero is 2000. The corresponding price per color television unit is $p=-0.1(2000)+1000=800$.

6.

 

 7. At $p=800$ ($x=2000$), the profit is maximized and the maximum value is $P(2000)=397000$ dollars.

In the preceding example, the profit is maximized when the marginal profit is zero. This is not a coincidence. The marginal profit is
$$P'(x)=R'(x)-C'(x)$$
If $P'(x)=0$, $R'(x)=C'(x)$ i.e. the marginal revenue equals the marginal cost. This means that the cost of producing one more item equals the revenue obtained by producing it, so there is no gain to be made by making the next item. As long as the marginal cost is less than the marginal revenue, it pays to keep producing more items. When the marginal cost becomes equal to the marginal revenue, it is time to stop producing more items.