Monopoly

Only a single firm exists. Usually firms and not consumers determine the price, owing to the lack of coordination possible between the large number of consumers.

Causes of monopoly

Technical barriers. (hydroelectric power)
Legal barriers. (defense productions)

Markup

Proportional percentage of the price that is above the marginal cost.

At equilibrium $MC=MR$ .

And,

P\left(1+\frac{1}{e_{p}}\right) = MC\\ P+\frac{P}{e_{p}} = MC\\ \frac{P-MC}{P} = -\frac{1}{e_{P}}

The above quantity is called markup. Markup is inversely proportional to price elasticity.

Marginal cost is always positive, but profit is maximised at MC=MR. Thus, the monopolist sets price where marginal revenue is positive, and thus price is elastic.

Monopoly price and output

Example

Let demand curve be $q=a-b\cdot p$ and cost be $C=c+d\cdot q^2$

Total revenue is TR is,

\text{TR} = p\cdot q\\ =\frac{a-q}{b}\cdot q \\=\frac{aq-q^2}{b}

The marginal revenue MR is,

\text{MR} = \frac{d(\text{TR})}{dq}\\ = \frac{1}{b}(a-2q)

Total cost is

C=c+dq^2\\ \text{MC} = \frac{dC}{dq}=2dq

For maximising profit, $MC=MR$ . Solving, we get $q$ . Plug it back in the price-quantity equation to get $p$ .

Example

Let a monopolist have constant marginal cost, $MC: d$ . The cost is then $d\cdot q$ .

Let inverse demand function (price as a function of quantity) be $p=a-b\cdot q$

Then

TR=q(a-bq)\\=aq-bq^2 \\ MR=a-2bq\\ MC=d

When

MC=MC\\ a-2bq=d\\ q^*=\frac{a-d}{2b}\\ p^*=a-b\left(\frac{a-d}{2b}\right) = \frac{a+d}{2}

If cost is increased, what percentage of the cost will be passed on to the consumer? This is given by,

\frac{dp^*}{d(\text{MC})}=\frac{\delta p^*}{\delta d}=\frac{1}{2}

Thus, half the cost will be passed on to the consumer.

Sharon Kartika. Last modified: January 04, 2024.