Monopoly

Profits from Monopolist Market

Let $P=100-4q$ be demand function

$C=50+20q$ be cost function

$$\therefore R=p\cdot q=(100-4q)q=100q-4q^2$$ $$MR=100-8q$$ $$\begin{array}{rl}\therefore MR& =MC\\ 100-8q& =20\\ q& =10\end{array}\text{\hspace{1em}(For profit Maximization)}$$ $$\therefore P=100-4\times 10=60$$

Max $\Pi =60\times 10-\left[50+20\times 10\right]=\boxed{350}$

Thus, the monopolist have profit of $350$ by selling opt o/p of 10 units at a price of $50$ per unit.

Comparison with a competitive firm

Suppose, the firm is a competitive one, then, it follows a rule $\boxed{MC\; =\; P}$

$$\begin{array}{rl}\text{i.e., }20& =100-4q\\ q& =20\end{array}$$

Since the price is $P=20$ and the quantity $q=20$,

Max $\Pi =20\times 20-[50+20\times 20]=\boxed{-50}$

Thus, a consumer is getting lesser quantity for more price in monopoly than in perfect competition. Also firm is getting more profit in monopoly than in a perfect competition.

Price Discrimination

Suppose a monopolist discriminate the price he charges by selling a part of its o/p one price in one market and remaining part at another price in another makret. Suppose the monopolist sells its o/p $q_1$ in first market with price $p_1$ where elasticity is ${\eta }_1$ and sells its o/p $q_2$ in the second market with price $p_2$ where the elasticity is ${\eta }_2$.

$\therefore$ we have $M{R}_1=P_1\left(1-\frac{1}{{\eta }_1}\right)$ and $M{R}_2=P_2\left(1-\frac{1}{{\eta }_2}\right)$

Let $MC$ be the marginal cost $\therefore$ For profit max in first market $M{R}_1=MC$ and for profit max in second market $M{R}_2=MC$

Equate $MR$ in each market with $MC$

$$\therefore MC=M{R}_1=M{R}_2$$ $$P_1\left(1-\frac{1}{{\eta }_1}\right)=P_2\left(1-\frac{1}{{\eta }_2}\right)$$ $$\therefore \frac{P_1}{P_2}=\frac{\left(1-\frac{1}{{\eta }_2}\right)}{\left(1-\frac{1}{{\eta }_1}\right)}$$

$⟹$ For price discrimination price ratio must be maintained at $\frac{P_1}{P_2}=\frac{\left(1-\frac{1}{{\eta }_2}\right)}{\left(1-\frac{1}{{\eta }_1}\right)}$ $\therefore$ $P_1=P_2$ if ${\eta }_1={\eta }_2$. This price discrimination is not possible if the markets are of same elasticities so, a necessary condition for the price discrimination is that market must be of different elasticities

Suppose ${\eta }_1\ne {\eta }_2$ and

$$\begin{array}{rl}{\eta }_1& >{\eta }_2\\ \therefore \frac{1}{{\eta }_1}& <\frac{1}{{\eta }_2}\\ \therefore -\frac{1}{{\eta }_1}& >\frac{1}{{\eta }_2}\\ \therefore 1-\frac{1}{{\eta }_1}& >1-\frac{1}{{\eta }_2}\\ \therefore \frac{P_1}{P_2}& <1\end{array}$$

So, $P_1<P2$ In the first market, price is lesser than second market. Thus, he has to charge lesser price in more elastic market if price discrimination is to be followed.

Example

$P=100-4q$ and $C=50+20q⟹\frac{dc}{dq}=MC=20$ suppose monopolist tries price discrimination by selling his o/p in two markets with demand function

$$\begin{array}{rlr}{P}_1& =80-5q_1& P_2=180-20q_2;q_1+q_2=q\\ & & \\ P_1& =p_1{q}_1=(80-5q_1)q_1& \\ & =80q_1=5q_1^2& \\ & & \\ P_2& =p_2{q}_2=(180-20q_2)q_2& \\ & =180q_1=20q_2^2& \end{array}$$ $$M{R}_1=80-10q_{1}M{R}_2=180-40q_{2}MC=20$$

$\therefore$ For profit maximization $M{R}_1=MC$ & $M{R}_2=MC$

$$\begin{array}{rl}80-10q_1=20& ;180-40q_2=20\\ \boxed{q_1=6}& ;\boxed{q_2=4}\\ & \\ q=6+4& =10\end{array}$$ $$P_1=80-576=\boxed{10};P_2=180-20\times 4=\boxed{100}$$ $${\eta }_1=-\frac{p_1}{q_1}\cdot \frac{d{q}_1}{d{p}_1}=-\frac{50}{6}\cdot \left(-\frac{1}{5}\right)=\frac{5}{3}\approx 1.67$$ $${\eta }_2=-\frac{p_2}{q_2}\cdot \frac{d{q}_2}{d{p}_2}=-\frac{100}{4}\cdot \left(-\frac{1}{20}\right)=\frac{5}{4}\approx 1.25$$

Thus, the first market is more elastic than the second one so, he is charging lesser price in the first market than the second market by discrimination of price monopolist gets a profit of $450$ which is more than simple monopolist profit, $350$.