🏛️ Market Structures, Monopoly & Price Discrimination

Market structure governs how pricing power, output allocation, and social welfare are determined. This chapter rigorously analyzes the spectrum from atomistic perfect competition to pure monopoly and multi-market price discrimination.

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1. ⚖️ Perfect Competition

In a perfectly competitive market:

1. Atomistic buyers and sellers (price-taking behavior: $P=\overline{P}$).
2. Homogeneous goods (${ϵ}_{cross}=\mathrm{\infty }$).
3. Perfect information.
4. Free entry and exit ($F{C}_{sunk}=0$ in long run).

1.1 Short-Run vs. Long-Run Equilibrium

• Short-Run Profit Maximization: $\underset{q}{max}\pi =P\cdot q-C(q)⟹P=MC(q)$ on the upward-sloping portion ($M{C}^{\prime }>0$).
  • Shutdown Condition (Short Run): Produce $q>0$ if $P\ge minAVC(q)$. If $P<minAVC$, optimal $q=0$.
  • Breakeven Condition: $P=minATC(q)$.
• Long-Run Zero Economic Profit: Due to free entry/exit, the market price equals minimum long-run average cost:$$P^{\ast }=minLRAC=LMC(q^{\ast })$$Allocative efficiency ($P=MC$) and productive efficiency ($P=minATC$) are simultaneously achieved.

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2. 🏰 Pure Monopoly & Pricing Power

A monopolist faces the entire downward-sloping market demand curve $P(Q)$. Total revenue is $TR(Q)=P(Q)\cdot Q$.

2.1 Marginal Revenue & Lerner Index

$$MR(Q)=\frac{dTR}{dQ}=P(Q)+Q\frac{dP}{dQ}=P(Q)[1+\frac{Q}{P}\frac{dP}{dQ}]=P(Q)[1-\frac{1}{|{ϵ}_d|}]$$

Setting $MR=MC$:

$$P[1-\frac{1}{|{ϵ}_d|}]=MC⟹\frac{P-MC}{P}=\frac{1}{|{ϵ}_d|}\text{(Lerner Index }L\text{)}$$

Key Monopolist Rule

A profit-maximizing single-price monopolist never produces where demand is inelastic ($|{ϵ}_d|<1$), because that would imply $MR<0$, which cannot equal positive marginal cost $MC>0$.

  Price ($)
    ▲
  P_m│         \
     │          \
  P_c│───────────┼─────── MC
     │  CS       │ \
     │  ┌────────┼──\
     │  │ Profit │DWL\
     │  └────────┴────\
     │           │     \  Demand
     │           │ MR   \
     └───────────┴───────┴────────► Quantity (Q)
                Q_m     Q_c

2.2 Deadweight Loss ($DWL$)

Because $P_m>MC$, output is restricted ($Q_m<Q_c$). The Harberger Deadweight Loss triangle is:

$$DWL=\frac{1}{2}(P_m-MC)(Q_c-Q_m)$$

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3. 🎯 Price Discrimination Taxonomy

Price discrimination allows a firm with market power to capture consumer surplus.

Type	Mechanism	Welfare Implication	Real-World Example
First-Degree (Perfect)	Charge each consumer their exact maximum willingness-to-pay ($WTP$).	$CS=0$, Profit is maximized, $DWL=0$ (Pareto efficient output!).	Personalized pricing, bespoke medical care
Second-Degree (Menu/Quantity)	Self-selection via nonlinear pricing schedules, volume discounts, versioning.	High-value consumers earn information rents.	Airline seating (Economy vs Business), electricity block tariffs
Third-Degree (Multi-Market)	Segment market into distinct demographic/geographic groups with differing $	\epsilon_i	$.

3.1 Third-Degree Optimization Condition

For markets $1$ and $2$:

$$M{R}_1(q_1)=M{R}_2(q_2)=MC(q_1+q_2)⟹P_1(1-\frac{1}{|{ϵ}_1|})=P_2(1-\frac{1}{|{ϵ}_2|})$$$$\frac{P_1}{P_2}=\frac{1-1/|{ϵ}_2|}{1-1/|{ϵ}_1|}$$

The market with more inelastic demand ($|{ϵ}_1|<|{ϵ}_2|$) is charged a higher price ($P_1>P_2$).

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4. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Two-Part Tariff Pricing

Problem: A monopoly tennis club serves $N=100$ identical consumers, each with individual demand $q_i(P)=20-2P$. The club's cost function is $C(Q)=2Q+500$, where $Q=\sum q_i$.

1. If the club charges a two-part tariff consisting of an annual membership fee $T$ and per-game fee $P$, determine the profit-maximizing $(P^{\ast },T^{\ast })$.
2. Calculate total club profit $\mathrm{\Pi }$ and consumer surplus per member.

Step-by-Step Rigorous Solution:

1. Set Per-Unit Price $P=MC$ to maximize total surplus:

   $$MC=\frac{dC}{dQ}=2⟹P^{\ast }=\mathrm{\$}2/\text{game}$$

2. Calculate individual consumption at $P^{\ast }=\mathrm{\$}2$:

   $$q_i^{\ast }=20-2(2)=16\text{ games/year}$$

3. Set Membership Fee $T^{\ast }$ equal to individual Consumer Surplus ($C{S}_i$): The demand curve intersects price axis at $P_{max}=10$.

   $$C{S}_i=\frac{1}{2}(P_{max}-P^{\ast })\cdot q_i^{\ast }=\frac{1}{2}(10-2)(16)=\frac{1}{2}(8)(16)=\mathrm{\$}64$$$$T^{\ast }=\mathrm{\$}64/\text{year}$$

4. Calculate Total Club Profit: Total output: $Q=100\times 16=1600$ games.

   $$\mathrm{\Pi }=N\cdot T^{\ast }+P^{\ast }\cdot Q-C(Q)=100(64)+2(1600)-[2(1600)+500]=6400-500=\mathrm{\$}5900$$

   Each member receives $C{S}_{net}=C{S}_i-T^{\ast }=64-64=\mathrm{\$}0$. Economic Result: The two-part tariff extracts 100% of consumer surplus while producing the allocatively efficient competitive output ($Q=1600$), yielding $DWL=0$.