7.2.1 Monopolies

The corporate goal is always profit maximization. Starting from the simple notation of profit

is the optimality condition (FOC, first order condition)

which converts into the condition: marginal revenue = marginal costs:

Since the revenue is the product of price and quantity ( $p\left(q\right)\cdot q$ ), the marginal revenue $\mathit{GE}=\frac{d\left(p\left(q\right)\cdot q\right)}{\mathit{dq}}=p\left(q\right)+p^{\prime }\left(q\right)\cdot q$.
For a linear demand function applies $p\left(q\right)=\mathit{aq}+b.$ Please note: As an exception, we present the price as a function of quantity.
Therefore, $\mathit{GE}=2\mathit{aq}+b$, i.e. the marginal revenue falls twice as fast as the demand function.
The graph shows that the quantity resulting from the condition $\mathit{GE}=\mathit{GK}$ is smaller than the market equilibrium quantity and therefore a higher price can be charged. The difference between the marginal costs and the realized price benefits the seller and is therefore called monopoly surplus.

The monopoly quantity is smaller than the equilibrium quantity in the competitive market. Therefore, a welfare loss occurs in the amount of the blue triangle.