Economics Application - Cost Functions

# Cost Functions

In economics, a function $C$, called a cost function is often a polynomial, that is, $C(x) = a_0 + a_1x^1 + a_2x^2 + ... + a_nx^n$, where $\Delta C = \frac{C(x_2) - C(x_1)}{x_2 - x_1}$ describes the average rate of cost change of some commodity based on some quantity $x$. The function $C'$ is known as the marginal cost function and describes the instantaneous rate of cost change, that is:

(1)
\begin{align} \lim_{\Delta x \to 0} \frac{\Delta C}{\Delta x} = \frac{d}{dx} C \end{align}

For example, consider the cost function $C(x) = 175000 + 5x + 0.5x^2$. To find the marginal cost function, we would simply differentiate $C$ to get:

(2)
$$C'(x) = 5 + x$$

At a production of $x = 1000$ items, our marginal cost per item would be $1005$.

## Example 1

Find the marginal cost function $C'$ given that the cost function for expensive sports cars at a particular company is described by $C(x) = 19000 + 10x + 0.01x^3$, and determine the marginal cost at production level for 1500 units.

We first find the marginal cost function by differentiating $C$ as follows:

(3)
$$C'(x) = 10 + 0.03x^2$$

We now evaluate this function at $x = 1500$ to get $C'(1500) = 67510$. So the marginal cost at production level for $1500$ units is $67510$ per unit.