How to Find Cost-minimizing Input

Theory

Optimum Production Level (OPL)

The optimum production level is the $x$ -value where the unit cost $U (x)$ is minimized. This happens when

U (x) = C^{'} (x)

Note! OPL is the production level that result in the lowest cost per unit, and therefore, the highest profit per unit. It may not be the highest total profit.

Theory

Interpretation of OPL

When $C^{'} (x) > U (x)$ , the cost of producing one more unit, $C^{'} (x)$ , is higher than the unit cost $U (x)$ .
When $C^{'} (x) < U (x)$ , the cost of producing one more unit, $C^{'} (x)$ , is lower than the unit cost $U (x)$ .
In the point where $C^{'} (x) = U (x)$ , the cost of producing one more unit, $C^{'} (x)$ , is equal to the unit cost $U (x)$ . That’s where you arrive at the optimum production level, OPL.

The unit cost and the marginal cost plotted together

Example 1

You have the cost function

C (x) = 4 x^{2} + 8 x + 16 .

Find the optimum production level (OPL). What is the cost at the OPL?

You know that you have reached the OPL when $C^{'} (x) = U (x)$ . Thus, you find $C^{'} (x)$ first:

C^{'} (x) = 8 x + 8

Then you find $U (x)$ :

\begin{array}{l} U (x) & = \frac{C (x)}{x} = \frac{4 x^{2} + 8 x + 16}{x} \\ = 4 x + 8 + \frac{16}{x} \end{array}

U (x) = \frac{C (x)}{x} = \frac{4 x^{2} + 8 x + 16}{x} = 4 x + 8 + \frac{16}{x}

Now you can set these equal to each other like in the formula, and solve for

x

\begin{array}{l} 8 x + 8 & = 4 x + 8 + \frac{16}{x} \\ 4 x & = \frac{16}{x} & | \cdot x \\ 4 x^{2} & = 16 & | \div 4 \\ x^{2} & = 4 \\ x = 2 & \land x = - 2 \end{array}

Because you are talking about production, it makes no sense to use negative numbers. It’s impossible to produce a negative number of units! Therefore, OPL equals 2.

To find the cost at the OPL, you insert the $x$ -value for OPL, $x = 2$ , into the cost function $C (x)$ :

\begin{array}{l} C (2) & = 4 {(2)}^{2} + 8 (2) + 16 \\ = 4 \cdot 4 + 16 + 16 \\ = 48 \end{array}

C (2) = 4 {(2)}^{2} + 8 (2) + 16 = 4 \cdot 4 + 16 + 16 = 48

For the

2

units, the cost at the OPL is

48

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