Answer to Question #153988 in Microeconomics for Rutendo

Question

Suppose a firm has the following Total Revenue and Total Cost functions: R(Q) = 100-Q²/100 ; C(Q) = 5000+2Q ; Find: the value of output Q that maximizes the profit of the firm. and also find the firm's profit

Answer 1

ANSWER

According to the marginal cost (denoted by MC) definition, the MC function can be calculated as the first order derivative of the Total cost function (denoted by TC) with respect to quantity (denoted by q). According to the marginal reevnue (denoted by MR) definition, the MR function can be calculated as the first order derivative of the Total revenue function (denoted by TR) with respect to q.

To find the output Q that maximizes the profit we should use the rule that $MR(Q)=MC(Q)$. The brief expmanation of this rule is based on the total profit function (denoted by PR).

It is easy to write the profit function as follows

$PR(q)=TR(q)-TC(q)$

Hence, the first order derivative of the PR with respect to q can be written as follows:

$dPR(q)/dq=dTR(q)/dq-dTC(q)/dq=MR(q)-MC(q)=0$

Hence,

$MR(Q)=MC(Q)$ where Q is the point where the profit is the highest.

According to the Question and explanation above we should compute the MR and MC:

$MR=dTR(q)/dq=d(100-q^2/100)/dq=-2*q/100=-q/50$

$MC=dTC(q)/dq=d(5000+2q)/dq=2$

Hence, $-Q/50=2$, and $Q=-100$ .

The profit when $Q=-100$ is equal to -4,800 and less than zero:

$PR(q=-100)=TR(q=-100)-TC(q=-100)=100-100^2/100-5,000+2*100=-4,800$

However, it is easy to see that quantity can not be less than zero.

Let's check other quantities:

When q=0 the total profit is equal -4,900.

When q=100 total profit is equal -5,200.

The quantity that maximizes the profit of the firm iz zero. The firm with aforementioned revenue and cost functions should be closed (or not produce).