Question

A firm's demand function for a certain good is given by. Its total cost function is . What output level maximizes the firm's profit?

Answer 1

Given the demand:

P= $100e ^{-0.10}$

Step 1: We can calculate the total revenue function:

$TR= P×Q = 100e^{-0.10} Q$

Step 2: Differentiate the total revenue to get the marginal revenue:

$$MR =\frac {\partial TR} {\partial Q} =100e^{-0.1Q}-10e^{-0.1}Q$$

We have the total cost function as:

$TC = 100^{e-0.10} + 50.$

Step 3: Differentiate the total cost to get the marginal cost:

$MC = \frac{\partial TC}{\partial Q} = — 10^{e -0.1} Q$

Step 4: Equate the marginal cost and the marginal revenue to get the optimal quantity

$=-10e^{-0.10}= -10Qe ^{-0.10} + 100e ^{-0.10}$

$=110e^{-0.1Q}= 10Qe^{-0.1Q} = 11$

$Q=11$

The optimal quantity is 11