Firms

Activity: production.

Objective: maximise profit.

\text{Total revenue} = P\cdot Q

\text{Cost} = C(Q)

\text{profit}= \Pi= \text{TR} - C=P\cdot Q-C

Additional revenue obtained by selling an extra unit

\text{MR} = \frac{d(\text{TR})}{dQ}=P

Demand curve:

Q=f(P)

Inverse demand curve:

P = g(Q)

Then,

\text{TR} = P(Q)\cdot Q

Since, $\Pi = PQ-C$ , first order maxima is given by,

\frac{d\Pi}{dQ} = 0\\ \implies \frac{d\Pi}{dQ}=\frac{\text{TR}}{dQ}-\frac{dC}{dQ}=0\\\implies \text{MR-MC}=0\\\implies \text{MR=MC}

Thus, a firm will maximise its profits when marginal revenue is equal to the marginal cost.

e_{p} = \frac{\left(\frac{dQ}{Q}\right)}{\left(\frac{dP}{P}\right)} = \frac{dQ}{dP}\cdot\frac{P}{Q}

and, $TR = P\cdot Q$ , and $MR = \frac{d(TR)}{dQ}$ . And, $Q=f(P)$ , so $P=f^.(Q)$ . Then,

\text{MR}=\frac{d(\text{TR})}{dQ} \\=\frac{d}{dQ}(P\cdot Q)\\= P+Q\cdot\frac{dP}{dQ}\\=P\left(1+\frac{dP}{DQ}\cdot \frac{Q}{P}\right)\\=P\left(1+\frac{1}{\frac{dQ}{dP}\cdot\frac{P}{Q}}\right)\\\boxed{\text{MR}=P\left(1+\frac{1}{e_{p}}\right)}

When $e_p=-1$ , $\text{MR}=0$ . Thus, when price is unitary elastic, marginal revenue is $0$ .

If change in demand is higher than price change it is defined as elastic. Else inelastic.

If $e_p>-1$ , then $\frac{1}{e_p}<-1$ and thus $\text{MR}<0$ . Thus, when price is inelastic, marginal revenue is negative.

If $e_p<-1$ , then $\frac{1}{e_p}>-1$ and thus $MR>0$ . Thus, when price is elastic, marginal revenue is positive.

Sharon Kartika. Last modified: January 04, 2024.