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Price, profit, and the cost of market power

Find the monopolist’s global optimum and compare it with the welfare-maximizing quantity.

Demand & technology

Demand intercept · A

Demand slope · B

Variable cost scale · c

Returns in variable production

Cost exponent · ρ

Avoidable overhead · F

Show average total cost (ATC)

ρ > 1 makes MC rise; ρ < 1 makes MC fall. The firm can stay closed and avoid F. Closing gives zero output and zero profit.

Monopoly and the efficient benchmark

P(q) = 100 − 1q · C(q) = 20 + 10q^1

Axes stay fixed as parameters change.

DemandMRMCProfit · green areaCS · blue areaDWL · orange area

Monopoly output45

Monopoly price55

Profit2,005

Consumer surplus1,012.5

Deadweight loss1,012.5

Efficient output90

Work out the optimum

1 · Marginal revenue meets marginal cost

Positive-output candidate: MR = MC

\begin{aligned}\pi(q)&=(A-Bq)q-cq^\rho-F,\quad q>0\\ \mathrm{MR}(q)&=A-2Bq\\ \mathrm{MC}(q)&=c\rho q^{\rho-1}\\ A-2Bq&=c\rho q^{\rho-1}\end{aligned}

Choose the best positive candidate, then compare with π(0) = 0. For ρ ≠ 1 the displayed root is computed numerically; the equations remain exact.

Compare candidate profit with closure

\begin{aligned}q_{\mathrm{candidate}}&=45\\ q_m&=45\\ p_m&=100-1(45)\\&=55\\\mathrm{ATC}(q_m)&=10.4444\end{aligned}

2 · Calculate the areas

Profit and consumer surplus

\begin{aligned}\pi_m&=[p_m-\mathrm{ATC}(q_m)]q_m=2,005\\ \mathrm{CS}_m&=\frac12(A-p_m)q_m=1,012.5\\ W_m&=\mathrm{CS}_m+\pi_m=3,017.5\end{aligned}

These operating formulas apply when qₘ > 0. At closure all three values are zero.

3 · Efficient benchmark

Efficient surplus less monopoly surplus

\begin{aligned}W(q)&=Aq-\frac{Bq^2}{2}-cq^\rho-F\\W_e&=\max\!\left\{0,\max_{q>0}W(q)\right\}\\ P(q_e)&=\mathrm{MC}(q_e)\quad\text{at an operating optimum}\\ W_e&=4,030\\ \mathrm{DWL}&=W_e-W_m=1,012.5\end{aligned}

With increasing returns, marginal-cost pricing may not cover total cost. This is a welfare benchmark, not necessarily a sustainable competitive equilibrium.

What “returns to scale” means here

With one variable input z at unit price 1 and technology q = (z/c)^1/ρ, producing q requires z = cq^ρ. Thus ρ > 1 means decreasing returns in variable production, ρ = 1 constant returns, and ρ < 1 increasing returns. The separate fixed overhead F also lowers average cost as it is spread over more units.

Overhead is avoidable when the firm stays closed

\begin{aligned}C(q)&=\begin{cases}0,&q=0,\\F+cq^\rho,&q>0,\end{cases}\\ \mathrm{MC}(q)&=10q^{0}\\ \mathrm{ATC}(q)&=\frac{20}q+10q^{0},\quad q>0\end{aligned}

When both allocations operate, fixed cost cancels from the welfare comparison: DWL = A(qₑ−qₘ) − ½B(qₑ²−qₘ²) − c(qₑ^ρ−qₘ^ρ).