Monopoly

To sell one additional unit, the monopolist must move down its downward-sloping demand curve, which means lowering the price NOT just on the new unit, but on all units sold. So MR equals the price of the new unit MINUS the lost revenue from charging less on the previous units. That subtraction makes $MR < P$ for every unit after the first.

(a) For linear demand $P = a - bQ$, $MR = a - 2bQ = \boxed{100 - 4Q}$.

(b) Set $MR = MC$: $100 - 4Q = 20 \Rightarrow Q^* = 20$. Then P^* = 100 - 2(20) = \boxed{\60}. Output 20, price \60.

(a) Per-unit profit = P_M - ATC = 60 - 30 = \30$. Total profit =$$30 \times 20 = \boxed{$600}$.

(b) Competitive outcome: P = MC = \20 \Rightarrow Q_C = 100 - 2P)/2$? Use demand directly:$20 = 100 - 2Q \Rightarrow Q_C = 40$.

DWL is the triangle between demand and MC from $Q_M = 20$ to $Q_C = 40$:

• Base (along Q-axis) = $40 - 20 = 20$.
• Height (gap between $P_M$ on demand and MC) = $60 - 20 = 40$.

DWL = \tfrac{1}{2}(20)(40) = \boxed{\400}$.

(a) Because economies of scale are so large that one firm produces at lower ATC than two or more firms could; competition is unsustainable. The market "naturally" leads to a single producer.

(b) When ATC is continuously falling, MC lies BELOW ATC at every Q. Setting $P = MC$ means $P < ATC$, so per-unit revenue is less than per-unit cost — the firm loses money on every unit. Without a subsidy, it cannot survive.

(c) Fair-return regulation: set $P = ATC$ at the regulated quantity. The firm earns zero economic profit (covers all costs including opportunity cost). $Q$ is higher and $P$ lower than the unregulated monopoly outcome, but DWL is not eliminated entirely. This is the standard "second-best" solution for natural monopoly.

Let $TS^*$ denote the maximum (efficient) total surplus.

(i) Consumer surplus: Competitive (largest) > Single-price monopoly > Perfect PD (zero). Justification: PD captures every unit of CS for the firm; competitive markets give consumers the largest surplus.

(ii) Producer surplus: Perfect PD (largest, captures full $TS^*$) > Single-price monopoly > Competitive. Justification: PD turns all of CS into PS; the competitive firm earns zero economic profit in long run.

(iii) Total surplus: Competitive = Perfect PD ($= TS^*$, both efficient) > Single-price monopoly. Justification: PD is allocatively efficient because the monopolist serves every buyer whose willingness to pay exceeds MC, eliminating DWL — but transfers all surplus to the seller.

(iv) Deadweight loss: Single-price monopoly (positive DWL) > Competitive = Perfect PD (both zero). Justification: only the single-price monopoly restricts $Q$ below $Q^*$.