A natural monopolist facing demand curve p = 18 2q with average costs AC = 12 0.5q sells output...

Given:

The demand and average cost function is:

{eq}\begin{align*} p &= 18 - 2q\\ AC &= 12 - 0.5q \end{align*} {/eq}

The total cost and the marginal cost functions are:

{eq}\begin{align*} TC &= AC \times q\\ &= \left( {12 - 0.5q} \right) \times q\\ &= 12q - 0.5{q^2}\\ MC &= \dfrac{{dTC}}{{dq}}\\ &= 12 - q \end{align*} {/eq}

In a two-part tariff system, one part is the fixed fee and the other part is the quantity times the per unit price of the good charged.

The fixed fee is set as equal to the consumer surplus gained if the market would have been under perfect competition.

To calculate the equilibrium price and quantity under perfect competition, put p=MC:

{eq}\begin{align*} p &= MC\\ 18 - 2q &= 12 - q\\ q &= 6\\ p &= 6 \end{align*} {/eq}

So, the consumer surplus when p is equal to $6 and q is equal to 6 units is:

{eq}\begin{align*} CS &= 0.5\left( {18 - p} \right)\left( {q - 0} \right)\\ &= 0.5\left( {18 - 6} \right)\left( {6 - 0} \right)\\ &= 36 \end{align*} {/eq}

Therefore, the fixed fee charged is $36.

The total revenue (TR) function for the monopolist under two part tariff is:

{eq}TR = 36 + pq {/eq}

Let the profit earned be represented by x.

{eq}\begin{align*} x &= TR - TC\\ &= 36 + pq - \left( {12q - 0.5{q^2}} \right)\\ &= 36 + 36 - 72 + 18\\ &= 18 \end{align*} {/eq}

So, the monopolist will earn a profit of $18.