Consider a monopolist who is faced with the market demand curve P = 10 - Q. Its total cost is...

Given:

Market demand curve P = 10 - Q

Firm Total cost C = 2Q

Total Revenue (TR) equals to demand multiply by quantity:

{eq}\begin{align*} TR &= P \times Q\\ &= \left( {10 - Q} \right) \times Q\\ &= 10Q - {Q^2} \end{align*} {/eq}

So, the marginal revenue (MR) is:

{eq}\begin{align*} MR &= \dfrac{{d\left( {TR} \right)}}{{d\left( Q \right)}}\\ &= 10 - 2Q \end{align*} {/eq}

Marginal cost (MC) equals to:

{eq}\begin{align*} MC &= \dfrac{{d\left( C \right)}}{{d\left( Q \right)}}\\ &= 2 \end{align*} {/eq}

So, for profit-maximization, MC should be equal to MR. So, at the profit-maximizing condition:

{eq}\begin{align*} MC &= MR\\ 2 &= 10 - 2Q\\ Q &= 4 \end{align*} {/eq}

Putting quantity equals to four at the demand equation:

{eq}\begin{align*} P &= 10 - Q\\ P &= 10 - 4\\ P &= 6 \end{align*} {/eq}

So, price is equal to 6 at the profit-maximizing condition.

b.

If a monopolist use two-part tariff, then it produces where price equals marginal cost so price will be $2 per unit. So, a monopoly will produce:

{eq}\begin{align*} P &= MC\\ 10 - Q &= 2\\ Q &= 8 \end{align*} {/eq}

So, the producer now produces eight quantities.

Producer set the total entry fee equals to consumer surplus (CS).

{eq}\begin{align*} CS &= \dfrac{1}{2} \times \left( {10 - 2} \right) \times 8\\ &= 32 \end{align*} {/eq}

So, the producer will cost a total $32 as an entry fees and $2 as price per unit quantity.

c.

At B, the consumer surplus reduces to zero because they have to pay their entire consumer surplus as the entry fee. The producer gets better off because they now earn consumer surplus too. And social loss or deadweight loss decrease to zero in two tariff condition because monopoly produces, where price of commodity is equal to marginal cost.