4.4 Classic example in economics, (revenue and costs)

Call Profit the difference between Revenue and Costs, that is

$$\begin{eqnarray*}\mbox{\rm Profit} & = & \mbox{\rm Revenue - Cost}. \\ \mbox{\rm P} &=& \mbox{\rm R - C}. \end{eqnarray*}$$

Assume that the firm wants to be a profit maximizer.

How many units should the units should the firm produce in order to maximize profits?

What should the relationship be between revenue and costs in order to maximize profits?

Maximize means find an optimum.

Use differentiation to find an optimum.

Differentiate the profit function, set it equal to zero and solve.

$$\begin{eqnarray*}\mbox{\rm Profit} & = & \mbox{Revenue - Cost}.\\ \mbox{\rm P(U)} & = & \mbox{R(U) - C(U)}. \end{eqnarray*}$$

$\frac{d}{dU}P(U) = \frac{d}{dU}R(U) - \frac{d}{dU} C(U)$.

Set the derivative equal to 0.

$$\fbox{$0 = \frac{d}{dU}R(U) - \frac{d}{dU} C(U)$ }$$

Solve:

$$\fbox{$\frac{d}{dU}R(U) = \frac{d}{dU} C(U)$ }$$

The derivative of the revenue function should equal the derivative of the cost function. Economist use the word marginal to express derivative ideas. So, produce a number of units at which marginal cost equals marginal revenue.

Do an example:

Costs equals: $5\,U$.

Revenue equals $40 + 100 \sqrt{U}$. (Recall that square root means to the power one half.)

Assume profit equals revenue minus costs.

How many units should be produced to maximize profits?

Find the function you want to optimize.

$$\fbox{$P = \overbrace{40 + 100\,\sqrt{U}}^{\mbox{\rm revenue}} - \overbrace{5\,U}^{\mbox{\rm cost}}$ }$$

Differentiate it.

$$\fbox{$\frac{dP}{dU} = 100\times 0.5\times U^{-0.5} - 5 $ }$$

Set the derivative equal to zero.

$$\fbox{$0 = 100\times 0.5\times U^{-0.5} - 5$ }$$

Solve for the quantity of interest.

$$\begin{eqnarray*}0 & = & 50 \frac{1}{\sqrt{U}} - 5 \\ 50 \frac{1}{\sqrt{U}} & =... ...ac{1}{\sqrt{U}} & = & 1\\ 10 & = & \sqrt{U} \\ 100 & = & U \\ \end{eqnarray*}$$

So produce 100 units to maximize profits.

How do we know that it is a a max. (1) Draw it. (2) Check second derivative is negative (more in math camp).