Microeconomics II: Notes & Formula Sheet (UPF)

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APUNTES MICROECONOMICS II Bachelor's Degree in Business Administration and Management Pompeu Fabra University AprenAI 2026-2027 Complete exam-oriented revision notes for Microeconomics II at UPF: general equilibrium, welfare theorems, externalities, monopoly, and oligopoly models.

APREN.AI Contents General Equilibrium 1 Pure Exchange Economies and the Edgeworth Box From Partial Equilibrium to a 2×2 Pure Exchange Economy Geometry of the Edgeworth Box Pareto Efficiency and the Contract Curve Voluntary Trade, Rational Outcomes, and the Core 2 Walrasian Equilibrium and the Welfare Theorems Competitive Price-Taking and Walrasian Equilibrium Excess Demand Functions and Walras's Law Properties of Walrasian Equilibrium First Fundamental Theorem of Welfare Second Fundamental Theorem of Welfare Market Failures 3 Externalities and Property Rights Nature and Taxonomy of Externalities Consumption Externalities in the Modified Edgeworth Box Property Rights and the Coase Theorem 4 Production Externalities, Pigouvian Taxes, and the Commons The Two-Firm Production Externality Model Social Optimum and Efficiency Conditions Remedies: Mergers, Tradable Permits, and Pigouvian Taxes The Tragedy of the Commons 5 Public Goods — Provision, Efficiency, and Collective Decision Classification of Goods Discrete Public Goods and Voluntary Contribution Breakdown Continuous Public Goods: The Samuelson Condition Quasi-Linear Preferences and Independence of Wealth Collective Choice, Voting Paradoxes, and the Median Voter 6 Monopoly — Price Setting, Welfare Loss, and Regulation Profit Maximization and the Price Elasticity Rule Welfare Inefficiency and Deadweight Loss Regulatory Instruments: Taxes, Subsidies, and Price Caps Natural Monopoly and the Pricing Dilemma Microeconomics II: Notes & Formula Sheet (UPF) 1 Price Discrimination 7 First-Degree and Third-Degree Strategies Foundations and Taxonomy of Price Discrimination Third-Degree Price Discrimination across Market Segments First-Degree Price Discrimination Implementation Mechanisms: Bundling and Two-Part Tariffs 8 Second-Degree, Bundling, and Spatial Competition The Screening Problem and Menu Design Binding Constraints and Information Rent Efficiency at the Top and Downward Distortion Commodity Bundling Spatial Competition and Hotelling's Linear City 9 Factor Markets — Factor Demand, Monopsony, and Vertical Monopolies Factor Demand and Output Market Power Monopsony in Input Markets Minimum Wage Under Competition vs.

Monopsony Vertical Monopolies and Double Marginalization 10 Oligopoly — Sequential and Simultaneous Competition in Quantities and Prices Stackelberg Duopoly: Quantity Leadership Cournot Duopoly: Simultaneous Quantity Competition Price Leadership: Sequential Price Competition Bertrand Duopoly and the Bertrand Paradox Repeated Interaction and Collusion Sustainability 11 Formula Sheet General Equilibrium & Exchange Externalities & Public Goods Monopoly, Regulation & Price Discrimination Factor Markets & Monopsony Oligopoly Models Common Pitfalls Answers to the review questions These notes may contain errors or omissions.

They are study support and do not replace the official syllabus or your teachers: check important information. Microeconomics II: Notes & Formula Sheet (UPF) 2 1 Pure Exchange Economies and the Edgeworth Box BY THE END YOU WILL BE ABLE TO Differentiate general equilibrium from partial equilibrium in a 2x2 pure exchange economy. Construct an Edgeworth box using individual endowments and identify feasible consumption bundles. Derive the contract curve analytically by equating marginal rates of substitution. Identify the core as the subset of Pareto optimal allocations satisfying individual rationality.

THINK BEFORE YOU READ 1 Can trade between two rational individuals leave both parties better off without increasing the total physical amount of goods? Standard microeconomic consumer analysis examines a single market in isolation, holding prices and actions in all other markets constant. In contrast, general equilibrium analysis investigates the interaction and simultaneous clearing across all interconnected markets within an economy.

General Equilibrium 1 Pure Exchange Economies and the Edgeworth Box 2 Walrasian Equilibrium and the Welfare Theorems Microeconomics II: Notes & Formula Sheet (UPF) 3 OUTLINE General Equilibrium Analysis Framework From Partial Equilibrium to a 2×2 Pure Exchange Economy In partial equilibrium models, demand and supply curves determine equilibrium price and quantity under the ceteris paribus assumption. However, when economic shifts alter income or relative prices across multiple goods, spillovers across markets occur. To formalise these multi-market interactions, we consider a benchmark pure exchange model with two consumers ( ) and two goods ( ), without production.

Each individual begins with an initial basket of physical goods known as an endowment. We write individual 's initial endowment vector as . The total available endowment of good across the whole economy is the aggregate sum: A consumption bundle for consumer is denoted . A complete state of the economy is described by an allocation, which is a pair of bundles specifying the consumption of both goods for both individuals.

General Equilibrium Simultaneous clearing across all interrelated markets Pure Exchange Model Fixed aggregate endowments without active production Edgeworth Box Geometric space of feasible allocations Contract Curve Pareto efficient allocations where MRSA equals MRSB Market Price Clearing Decentralized Walrasian equilibrium with competitive prices The Core Efficient allocations respecting individual rationality i {A, B} j {1, 2} i Ω = i (ω , ω ) i 1 i 2 j ω = j ω + A j ω for j B j {1, 2} i X = i (x , x ) i 1 i 2 (X , X ) A B Microeconomics II: Notes & Formula Sheet (UPF) 4 FEASIBLE ALLOCATION An allocation is defined as feasible if total consumption of each commodity does not exceed total available endowment in the economy: Under non-satiated preferences, consumers strictly prefer more to less.

Consequently, resources are not wasted, and feasibility holds with exact equality: . CHECK 2 Why does non-satiation imply that equilibrium consumption bundles use up the aggregate endowment entirely with equality? Geometry of the Edgeworth Box The Edgeworth box provides a compact geometric representation of all feasible allocations, individual preferences, and endowments in a two-consumer, two-good exchange economy. The total width of the box equals aggregate endowment of good 1 ( ), while the total height corresponds to aggregate endowment of good 2 ( ). The bottom-left corner serves as the origin for consumer A ( ), where consumption increases upwards and to the right.

The top- right corner serves as the origin for consumer B ( ), inverted so that consumption for consumer B increases downwards and to the left. Any point inside or on the boundaries of the box represents an exhaustive division of the aggregate endowment between A and B, satisfying and . (X , X ) A B x + A 1 x B 1 ω + A 1 ω and x + B 1 A 2 x B 2 ω + A 2 ω B 2 x + A j x = B j ω + A j ω B j ω = 1 ω + A 1 ω B 1 ω = 2 ω + A 2 ω B 2 O A O B x = B 1 ω 1 x A 1 x = B 2 ω 2 x A 2 Microeconomics II: Notes & Formula Sheet (UPF) 5 Diagram of an Edgeworth box displaying endowments, indifference curves, and mutual improvement lens.

The initial endowment vector is plotted as a specific coordinate within the box. Individual preferences are represented by standard indifference curves. For consumer A, indifference curves are convex towards , with utility increasing towards the upper-right corner. For consumer B, indifference curves are convex towards , with utility increasing towards the lower-left corner. CHECK 3 How do we locate consumer B's consumption bundle from an arbitrary point inside the Edgeworth box? Pareto Efficiency and the Contract Curve A change from an initial allocation to another is a Pareto improvement if it makes at least one agent strictly better off without making any other agent worse off.

An allocation is Pareto efficient (or Pareto optimal) if no further Pareto improvements are possible. Formally, a Pareto efficient allocation solves the following constrained optimisation problem, maximising consumer A's utility while fixing consumer B at a target utility level : W = (Ω , Ω ) A B O A O B u ˉB u (x , x ) x ,x ,x ,x A 1 A 2 B 1 B 2 max A A 1 A 2 subject to u (x , x ) = B B 1 B 2 u ˉB x + x = ω + ω A 1 B 1 A 1 B 1 x + x = ω + ω A 2 B 2 A 2 B 2 Microeconomics II: Notes & Formula Sheet (UPF) 6 For an interior solution where both agents consume positive amounts of both goods, efficiency requires that their indifference curves are mutually tangent.

At this tangency point, the consumers' marginal rates of substitution coincide: Graph showing indifference curve tangencies along the contract curve in an exchange economy. The set of all Pareto efficient allocations within the Edgeworth box forms a continuous path connecting origin to origin , known as the contract curve (or the Pareto set). ANALYTICAL DERIVATION OF THE CONTRACT CURVE Consider an economy with aggregate endowments and . Let utilities be and . The contract curve is the straight line stretching between origins inside the box.

MRS = A MRS B = μ A 2 μ A 1 μ B 2 μ B 1 O A O B ω = 1 20 ω = 2 30 u (x , x ) = A A 1 A 2 x x A 1 A 2 u (x , x ) = B B 1 B 2 x x B 1 B 2 Compute marginal rates of substitution: and . 1 MRS = A x /x A 2 A 1 MRS = B x /x B 2 B 1 Substitute market clearing equations into : and . 2 MRS B x = B 1 20 x A 1 x = B 2 30 x A 2 Equate marginal rates of substitution: . 3 = x A 1 x A 2 20 x A 1 30 x A 2 Cross-multiply and simplify: . 4 x (20 A 2 x ) = A 1 x (30 A 1 x ) A 2 20x = A 2 30x A 1 x = A 2 x 2 3 A 1 x = A 2 1.5x A 1 Microeconomics II: Notes & Formula Sheet (UPF) 7 COMMON EXAM PITFALL Students often forget to substitute the feasibility conditions ( ) into before equating it to .

Leaving or in the final equation fails to specify the contract curve in terms of consumer A's coordinates, resulting in lost marks. Voluntary Trade, Rational Outcomes, and the Core While the contract curve identifies all Pareto efficient allocations, voluntary trade from an initial endowment introduces individual rationality constraints. An allocation is a rational outcome if voluntary trade provides each agent with at least as much utility as their initial endowment basket: In the Edgeworth box, these constraints delimit a lens-shaped region bounded by the indifference curves passing through the endowment point .

Every point strictly inside this lens represents a mutual Pareto improvement over endowment. THE CORE The core is the segment of the contract curve that lies within the lens formed by the endowment indifference curves.

It satisfies both Pareto efficiency and individual rationality: Concept Mathematical Condition Geometric Meaning in the Box Feasible Allocation Any point within the boundaries of the box Pareto Improvement with strict inequality for at least one Movement into the interior of the mutually preferred lens Contract Curve Locus of mutual indifference curve tangencies connecting origins The Core and Segment of the contract curve intercepted by the endowment lens Pricing mechanisms and competitive decentralized exchange along this curve are examined when studying general competitive outcomes (see «General Equilibrium — Walrasian Equilibrium and the Welfare Theorems»).

CHECK 4 Can an allocation be Pareto efficient but fail to belong to the core? x = B j ω j x A j MRS B MRS A x B 1 x B 2 u (x , x ) A A 1 A 2 u (Ω ) and u (x , x ) A A B B 1 B 2 u (Ω ) B B W Core = {(X , X ) A B Contract Curve u (X ) A A u (Ω ) and u (X ) A A B B u (Ω )} B B x + A j x = B j ωj u (X ) i i u (W ) i i MRS = A MRS B MRS = A MRS B u (X ) i i u (Ω ) i i Microeconomics II: Notes & Formula Sheet (UPF) 8 KEY IDEAS A pure exchange economy analyses the reallocation of fixed initial endowments among individuals without active production. The dimensions of the Edgeworth box are determined strictly by aggregate endowments: and .

Pareto efficiency in an interior allocation requires equal marginal rates of substitution: . The contract curve represents the locus of all Pareto efficient allocations connecting both consumer origins. The core consists of the subset of the contract curve that provides both consumers with at least the utility of their initial endowment. CHECK YOUR UNDERSTANDING 5. What is the key geometric condition identifying an interior Pareto efficient allocation in the Edgeworth box? 6. If consumer A has endowment and consumer B has , what are the dimensions of the Edgeworth box? 7.

Why does voluntary bilateral trade restrict final equilibrium allocations to the core rather than the entire contract curve? GENERAL EQUILIBRIUM 2 Walrasian Equilibrium and the Welfare Theorems BY THE END YOU WILL BE ABLE TO Define Walrasian competitive equilibrium and formulate its market clearing conditions. Derive Walras's Law from individual budget constraints and apply price normalization via a numéraire. Formulate and prove the First Fundamental Theorem of Welfare using revealed preference logic. Contrast the First and Second Welfare Theorems and identify why non-convex preferences violate the Second.

THINK BEFORE YOU READ 8 Can an economic outcome be completely unfair and unequal yet perfectly Pareto efficient at the same time? ω = 1 ω + A 1 ω B 1 ω = 2 ω + A 2 ω B 2 MRS = A MRS B Ω = A (4, 1) Ω = B (2, 3) Microeconomics II: Notes & Formula Sheet (UPF) 9 Competitive Price-Taking and Walrasian Equilibrium In general equilibrium analysis, we model markets where an auctioneer or market mechanism announces prices. Economic agents act as price-takers. Each consumer treats the announced price vector as fixed and chooses an optimal consumption bundle subject to their budget constraint.

To visualize the space of feasible allocations and voluntary exchange, see «General Equilibrium — Pure Exchange Economies and the Edgeworth Box». DEFINITION A Walrasian equilibrium (or competitive equilibrium) in an exchange economy with consumers and goods consists of a price vector and an allocation bundle for each individual such that: When the economy is not in equilibrium, price adjustments occur. Excess demand puts upward pressure on prices, whereas excess supply causes prices to fall until demand equals supply across all markets. CHECK 9 Why does a competitive equilibrium require agents to take prices as given rather than negotiating them bilaterally?

Excess Demand Functions and Walras's Law Let denote consumer 's gross demand for good . The net demand or individual excess demand is defined as . Summing across individuals gives the aggregate excess demand for good , denoted by . Market clearing in a two-good economy means that and . Because every consumer exhausts their budget constraint, individual spending identically equals the value of their endowment: Summing this condition over all individuals establishes Walras's Law: Walras's Law is an accounting identity that holds for any price vector, not just equilibrium prices. It yields two immediate structural consequences: J N p = (p , p , … , p ) 1 2 N x = j (x , … , x ) 1 j N j j {1, …

, J} Utility Maximization: Each consumer maximizes utility subject to their endowment value: 1 x = j arg U (x ) s.t. p x = x R j + N max j j j p ω j j {1, … , J} Market Clearing: Aggregate demand matches aggregate endowment across all goods: 2 x = j=1 J i j ω i j=1 J i j {1, …

, N} x (p , p ) i j 1 2 j i e (p , p ) = i j 1 2 x (p , p ) i j 1 2 ω i j i z (p , p ) = i 1 2 e (p , p ) j i j 1 2 z (p , p ) = 1 1 2 0 z (p , p ) = 2 1 2 0 p x (p , p ) + 1 1 j 1 2 p x (p , p ) = 2 2 j 1 2 p ω + 1 1 j p ω 2 2 j p e (p , p ) + 1 1 j 1 2 p e (p , p ) = 2 2 j 1 2 0 p z (p , p ) + 1 1 1 2 p z (p , p ) = 2 2 1 2 0 Microeconomics II: Notes & Formula Sheet (UPF) 10 Clearing of the -th market: If markets are in equilibrium (so for all ) and prices are strictly positive, the remaining -th market must also clear ( ). Price normalization and numéraire: Equilibrium depends exclusively on relative prices.

Setting one good as the numéraire with price allows expressing the system as , where . If aggregate excess demand functions are continuous—guaranteed when individual consumers have continuous demand functions derived from strictly concave utility functions (such as Cobb-Douglas)— a competitive equilibrium exists and is unique.

PROCESS Walrasian Price Adjustment Mechanism 1 Price Announcement Auctioneer announces tentative relative price vector p 2 Consumer Optimization Each agent maxi mizes utility subject to their endowment value 3 Aggregate Excess Demand Excess demand is evaluated; prices ad just up or down 4 Market Clearing Relative prices adjust until aggregate ex cess demand equals zero CHECK 10 If an economy has 3 markets and we know that markets 1 and 2 clear at positive prices, what must be true of market 3? Properties of Walrasian Equilibrium A Walrasian equilibrium possesses three fundamental geometric and structural properties within the Edgeworth box: 1.

Belonging to the Core: Because trade is voluntary, no individual accepts a market trade that leaves them worse off than their initial endowment . The equilibrium allocation must lie within the core (see «General Equilibrium — Pure Exchange Economies and the Edgeworth Box»). 2. Tangency of Indifference Curves: Each price-taking consumer maximizes utility by equating their marginal rate of substitution to relative market prices: and . Consequently, in an interior equilibrium, consumers share an identical marginal rate of substitution: .

k k 1 z = i 0 i < k k z = k 0 p = 1 1 z (1, p) + 1 p z (1, p) = 2 0 p = p /p 2 1 ω MRS = A p /p 1 2 MRS = B p /p 1 2 MRS = A MRS = B p /p 1 2 Microeconomics II: Notes & Formula Sheet (UPF) 11 Graph illustrating consumer tangency and budget line at competitive equilibrium. 3. Feasibility and Market Clearing: The equilibrium point represents an intersection between both consumers' optimal choices along a shared budget line passing through the initial endowment point . COMMON MISTAKE Do not confuse Walras's Law with market clearing. Market clearing requires for each good individually at equilibrium prices .

In contrast, Walras's Law states that the aggregate value of excess demand holds identically at all price vectors, even in disequilibrium. First Fundamental Theorem of Welfare The welfare theorems evaluate the efficiency of competitive outcomes. DEFINITION The First Fundamental Theorem of Welfare states that any competitive market equilibrium is Pareto optimal, provided consumers are price-takers with locally non-satiated preferences and markets are complete. We prove this result by contradiction using revealed preference logic. Suppose the competitive equilibrium allocation supported by price vector is not Pareto optimal.

Then there exists an alternative feasible allocation such that: W z (p ) = i 0 p p z (p) = i i 0 X = ((x , x ), (x , x )) 1 A 2 A 1 B 2 B p = (p , p ) 1 2 Y = ((y , y ), (y , y )) 1 A 2 A 1 B 2 B Microeconomics II: Notes & Formula Sheet (UPF) 12 Without loss of generality, assume agent strictly prefers while agent finds at least as good as : Since consumer maximized utility at subject to their budget constraint, bundle must have been unaffordable: Similarly, for consumer , bundle must cost at least as much as their endowment: Summing both expenditure inequalities across individuals yields: However, feasibility of allocation requires total consumption to equal total endowment, which implies that the total value of must equal the total value of endowments.

This creates an outright mathematical contradiction. Therefore, no feasible allocation can strictly Pareto-dominate the competitive equilibrium. Limitations: The First Welfare Theorem breaks down when markets fail. Violations occur in the presence of externalities (see «Market Failures — Externalities and Property Rights»), public goods (see «Public Goods — Provision, Efficiency, and Collective Decision»), or market power (see «Monopoly — Price Setting, Welfare Loss, and Regulation»). Furthermore, the theorem guarantees efficiency, but it is silent regarding equity; an allocation where one agent holds nearly all resources can still be Pareto optimal.

CHECK 11 Why does the First Welfare Theorem fail to guarantee a fair or equitable society? Second Fundamental Theorem of Welfare While the First Theorem shows that market equilibria are efficient, the Second Fundamental Theorem of Welfare addresses the reverse question: can any desired Pareto efficient allocation on the contract curve be achieved via competitive markets? DEFINITION The Second Fundamental Theorem of Welfare states that if consumers have convex preferences (concave utility functions), any Pareto efficient allocation can be supported as a competitive market equilibrium following an appropriate initial redistribution of endowments via lump-sum transfers.

y + 1 A y = 1 B ω + 1 A ω and y + 1 B 2 A y = 2 B ω + 2 A ω 2 B A Y B Y X (y , y ) 1 A 2 A A (x , x ) and (y , y ) 1 A 2 A 1 B 2 B B (x , x ) 1 B 2 B A X yA p y + 1 1 A p y > 2 2 A p ω + 1 1 A p ω 2 2 A B yB p y + 1 1 B p y 2 2 B p ω + 1 1 B p ω 2 2 B p (y + 1 1 A y ) + 1 B p (y + 2 2 A y ) > 2 B p (ω + 1 1 A ω ) + 1 B p (ω + 2 2 A ω ) 2 B Y Y Microeconomics II: Notes & Formula Sheet (UPF) 13 The intuition relies on the existence of a common tangent budget line. At any Pareto efficient point , indifference curves are tangent with slope equal to . If preferences are convex, this tangent line acts as a separating hyperplane.

Redefining the endowment to any point along this line allows the decentralized market mechanism to achieve allocation as an equilibrium. When preferences are non-convex, this result fails. An indifference curve with non-convex segments can be tangent to the budget line at multiple points, causing consumers to strictly prefer an alternative bundle over the intended point . Graph showing the failure of the Second Welfare Theorem under non-convex preferences.

Dimension First Welfare Theorem Second Welfare Theorem Direction Equilibrium Pareto Optimal Pareto Optimal Equilibrium (with redistribution) Key Assumption Price-taking, locally non-satiated preferences Convex preferences (concave utility functions) Policy Implication Markets achieve efficiency without central planning Equity and efficiency can be separated via lump-sum transfers X MRS = p /p 1 2 X Y X Microeconomics II: Notes & Formula Sheet (UPF) 14 OUTLINE The Fundamental Welfare Theorems KEY IDEAS A Walrasian equilibrium consists of market-clearing prices and utility-maximizing individual allocations.

Walras's Law states that the aggregate value of excess demand across all markets is identically zero at any price vector: . If markets clear at positive prices, Walras's Law guarantees that the -th market clears as well. The First Welfare Theorem proves that every competitive equilibrium is Pareto optimal, though it makes no claim about equity. The Second Welfare Theorem establishes that any Pareto efficient allocation can be supported as a competitive equilibrium through lump-sum redistribution, provided preferences are convex. CHECK YOUR UNDERSTANDING 12. What is the difference between Walras's Law and the market clearing condition? 13.

Why are convex preferences necessary for the Second Fundamental Theorem of Welfare? Quiz Circle the correct answer. The answers are at the end. 14. What determines the horizontal width of an Edgeworth box in a two-good pure exchange model?

Welfare Theorems Link competitive markets and Pareto efficiency First Welfare Theorem Equilibrium implies Pareto optimality Price Taking Requires no market power or externalities Second Welfare Theorem Pareto optimality supported via redistribution Convex Preferences Ensures supporting separating hyperplane p z (p) + 1 1 p z (p) = 2 2 0 k 1 k Consumer A's income divided by good 1 price a) Relative price ratio of both goods b) Aggregate endowment of good 1 c) Total wealth of consumer A d) Microeconomics II: Notes & Formula Sheet (UPF) 15 15. Which condition must hold for an interior allocation to be Pareto efficient in pure exchange? 16. How is the core defined relative to the contract curve?

17. What happens to consumer B's bundle when moving towards the top-right corner of the Edgeworth box? 18. In an economy with two goods, Walras's Law implies that if market 1 has excess demand: 19. What does Walras's Law allow economists to do when solving an exchange economy? 20. The First Fundamental Theorem of Welfare shows that competitive equilibria are: 21. Which assumption is strictly necessary for the Second Welfare Theorem but not the First?

equals a) MRS A MRS B equals for both goods b) MU A MU B equals across all goods c) x A x B Utility of A equals utility of B d) The entire contract curve connecting both origins a) The tangency points located on the box boundary b) All feasible points giving equal utilities to both agents c) The contract curve subset satisfying individual rationality d) Consumer B's consumption decreases toward zero a) Consumer B's marginal rate of substitution increases b) Consumer B consumes the entire aggregate endowment c) Consumer B's consumption increases for both goods d) Both markets must be in equilibrium a) Market 1 prices must drop immediately b) Market 2 must have excess supply c) Market 2 must have excess demand d) Drop one market clearing condition a) Assume indifference curves are linear b) Set all absolute price levels arbitrarily c) Ignore consumer budget constraints entirely d) Pareto optimal under price taking a) Independent of consumer preferences b) Equitable in wealth distribution c) Guaranteed to clear without prices d) Price-taking behavior of consumers a) Complete information across agents b) Convexity of consumer preferences c) The existence of initial endowments d) Microeconomics II: Notes & Formula Sheet (UPF) 16 3 Externalities and Property Rights BY THE END YOU WILL BE ABLE TO Differentiate consumption and production externalities according to who is affected Analyze the smoker-non-smoker problem using a modified Edgeworth box State the Coase Theorem and derive the invariance condition using quasi-linear preferences Distinguish between Pareto efficiency and utility distribution when assigning property rights THINK BEFORE YOU READ 22 Can two individuals bargain to a socially optimal level of pollution on their own without government regulations?

Standard competitive general equilibrium models rely on price mechanisms to allocate every scarce commodity. However, when an agent's actions directly alter another agent's well-being outside the price system, market prices fail to convey proper social signals. Market Failures 3 Externalities and Property Rights 4 Production Externalities, Pigouvian Taxes, and the Commons Microeconomics II: Notes & Formula Sheet (UPF) 17 OUTLINE Topic 2 Market Failures Framework Nature and Taxonomy of Externalities An externality is a cost or benefit imposed unintentionally on a third party without passing through the price system.

Because individuals interact constantly as social beings, their individual choices frequently spill over to others. DEFINITION An externality occurs whenever an agent's choice directly affects another agent's production set or utility function without monetary compensation through market prices. Externalities can be categorized across two fundamental dimensions: Sign of the effect: A positive externality confers unpriced benefits (such as population vaccination, perspective innovations in art, or nearby beekeeping), leading to socially suboptimal underprovision.

A negative externality imposes uncompensated costs (such as noise pollution, environmental smoke, or traffic congestion), resulting in excessive generation. Identity of the affected party: A consumption externality directly affects consumers, whereas a production externality affects firms and producers (see «Market Failures — Production Externalities, Pigouvian Taxes, and the Commons»). Crucially, the classification depends strictly on who is affected, not on who generates the spillover. Market Failures Breakdown of the First Welfare Theorem Externalities Unpriced side effects on third parties Consumption Spillovers Direct utility impacts (e.g.

smoking) Production Spillovers Cost functions altered across producers Public Goods Non-rivalrous and non- excludable goods Coasean Bargaining Property rights assignment solutions Pigouvian Remedies Corrective taxation and standards Microeconomics II: Notes & Formula Sheet (UPF) 18 Under the presence of an externality, the First Fundamental Theorem of Welfare (see «General Equilibrium — Walrasian Equilibrium and the Welfare Theorems») breaks down. Because decision- makers focus entirely on private marginal costs and private marginal benefits rather than social values, the resulting competitive outcome fails to achieve Pareto efficiency.

CHECK 23 Why does a competitive market overproduce activities that generate negative externalities? Consumption Externalities in the Modified Edgeworth Box Consider two agents, Person and Person , sharing an enclosed space (such as a train compartment). There are two goods: private money and smoke . For Person (the smoker), both money and smoke provide positive utility. For Person (the non-smoker), money is a good, but smoke is a bad that reduces utility. Because Person and Person must breathe the identical atmosphere, smoke is non-excludable inside the space: both individuals consume the exact same amount of smoke .

Money, in contrast, is partitioned privately between them: This structure requires a modified Edgeworth box (see «General Equilibrium — Pure Exchange Economies and the Edgeworth Box»). The horizontal dimension measures the fixed total endowment of money, while the vertical axis represents smoke measured in the same upward direction for both consumers. Diagram of the modified Edgeworth box for smoke and money. If Person holds total freedom to smoke without regulation, smokes until their private marginal utility of smoke reaches zero, ignoring the utility loss inflicted on Person . Consequently, the unregulated outcome diverges sharply from Pareto efficiency.

A B M S A B A B S [0, 1] M + A M = B Ω M A A B Microeconomics II: Notes & Formula Sheet (UPF) 19 Chart of bargaining equilibria under alternative initial property rights endowments. As illustrated in the diagram, initial endowments define whether smoking starts from zero or maximum exposure: Endowment : initially owns the right to clean air ( ). Any smoke generated requires to compensate , yielding a negotiated equilibrium at point . Endowment : possesses the unrestricted right to smoke ( ). Moving towards an efficient outcome requires to pay to reduce smoke, arriving at point . CHECK 24 Why is the smoke axis in the modified Edgeworth box oriented in the same direction for both agents?

Property Rights and the Coase Theorem Market inefficiency arises because property rights over the environmental asset (clean air) are missing. When clean air has no owner, using it carries a zero private price despite carrying a positive social opportunity cost. DEFINITION Property rights are legally enforced entitlements defining the ownership, usage, and transferability of a scarce asset. E B S = 0 A B X E A S = 1 B A X Microeconomics II: Notes & Formula Sheet (UPF) 20 Once legal entitlements are assigned, trade in rights internalizes the externality. The smoker must buy the right to pollute from the non-smoker, or the non-smoker must pay the smoker to abstain.

Under trade, the externality is converted into a priced commodity, restoring the First Fundamental Theorem of Welfare. FOR REMEMBERING Assigning property rights transforms an unpriced external impact into an exchangeable good, ensuring that agents internalize full marginal costs through voluntary negotiation. To analyze the invariance of the equilibrium externality level, consider agents with quasi-linear preferences: where are constants, , and . Because the marginal utility of money is constant ( ), the marginal rate of substitution depends purely on smoke: Due to the absence of income effects on smoke demand, the indifference curves are horizontal translations of each other.

Consequently, points of tangency between the indifference curves align across a horizontal contract curve. Chart showing a horizontal contract curve under quasi-linear preferences. This result underpins the formal statement of the Coase Theorem.

u (M, S) = A aM + v (S), u (M, S) = A B bM + v (S) B a, b > 0 v (S) > A 0 v (S) < B 0 MU = A,M a MRS = A;M ,S = MU A,S MU A,M v (S) A a Microeconomics II: Notes & Formula Sheet (UPF) 21 COASE THEOREM Under well-defined property rights, zero bargaining transaction costs, and quasi-linear preferences in money, voluntary negotiation leads to the unique socially efficient externality level, regardless of which party is originally assigned the property rights. COMMON ERROR Do not confuse efficiency invariance with distributional neutrality.

While the physical level of smoke is identical regardless of who owns the rights, the distribution of wealth and individual utility levels depends critically on who holds the initial legal entitlement. Initial Rights Assignment Direction of Monetary Compensation Equilibrium Smoke Level Utility Distribution Right to Clean Air ( ) pays for the right to smoke (Efficient) achieves higher utility; is poorer Right to Pollute ( ) pays to reduce smoke (Efficient) achieves higher utility; is poorer CHECK 25 Why does the contract curve become a horizontal straight line under quasi-linear utility?

KEY IDEAS An externality is an unpriced side effect of an economic action that bypasses the price mechanism, invalidating the First Fundamental Theorem of Welfare. The classification of externalities depends on who is affected (consumers or producers) rather than who generates them. In an Edgeworth box with smoke and money, smoke is consumed identically by both parties, requiring a shared directional vertical axis. Assigning clear property rights establishes a market for the externality, enabling agents to reach Pareto efficient allocations via negotiation.

Under quasi-linear preferences, the Coase Theorem ensures that the equilibrium quantity of the externality is invariant to the initial allocation of rights, though final utility distribution is not. CHECK YOUR UNDERSTANDING 26. What is the key structural difference between the smoke-money Edgeworth box and the standard exchange Edgeworth box? 27. Under what preference condition is the equilibrium level of the externality strictly invariant to the initial assignment of property rights in the Coase Theorem?

S B A B S B A A B A S A B Microeconomics II: Notes & Formula Sheet (UPF) 22 MARKET FAILURES 4 Production Externalities, Pigouvian Taxes, and the Commons BY THE END YOU WILL BE ABLE TO Derive the private and socially optimal pollution levels in a two-firm production externality model. Calculate the optimal Pigouvian tax equating marginal abatement cost to marginal external damage. Differentiate the open-access equilibrium condition (AP = AC) from the socially optimal exploitation rule (MP = MC) in the Tragedy of the Commons. THINK BEFORE YOU READ 28 Why does a shared pasture end up overgrazed even when all farmers act entirely rationally?

The Two-Firm Production Externality Model A production externality occurs when an agent's production decisions directly affect another firm's production function or costs outside the price system. To analyze this failure, consider an upstream steel mill and a downstream fishery . Both operate in competitive output markets with exogenous prices and . Firm produces steel output and generates pollution emissions as a byproduct. Pollution can be adjusted (for instance, via filtering). Its cost function is . Emitting more waste allows cheaper production, so . Firm catches fish quantity with cost . Discharged pollution harms fishing conditions, meaning .

In the decentralized market, firm chooses both steel and emission level to maximize private profit: The first-order conditions for steel and pollution yield: Downstream firm takes the incoming level of pollution as given and solves: Because firm ignores the damage inflicted on firm , it expands emissions until its private marginal abatement savings reach zero, leading to an overproduction of pollution.

S F p s p f S s x c (s, x) s x c (s,x) s 0 F f c (f, x) f > x c (f,x) f 0 S s x Π (s, x) = s,x max S p s s c (s, x) s p = s , = s c (s , x ) s x c (s , x ) s 0 F x Π (f) = f max F p f f c (f, x ) f p = f f c (f , x ) f S F Microeconomics II: Notes & Formula Sheet (UPF) 23 CHECK 29 Why does the competitive market fail to deliver an efficient emission level? Social Optimum and Efficiency Conditions A social planner maximizes joint surplus : Evaluating this problem at the socially optimal allocation gives three first-order conditions: Here, represents the private marginal cost of abatement (marginal savings from emitting), whereas represents the external marginal damage.

Social efficiency equates the marginal benefit of pollution to the marginal damage. Because , the competitive pollution level satisfies . Graph showing socially optimal versus privately chosen pollution levels. W(s, f, x) = Π (s, x) + S Π (f, x) F p s + s,f,x max s p f f c (s, x) s c (f, x) f ( , , ) s ^ f ^ x ^ p = s , p = s c ( , ) s s ^ x ^ f f c ( , ) f f ^ x ^ = x c ( , ) s s ^ x ^ x c ( , ) f f ^ x ^ MC ( , ) + S s ^ x ^ MC ( , ) = F f ^ x ^ 0 MC (s, x) = S x c (s,x) s MC (f, x) = F x c (f,x) f MC (f, x) > F 0 x > x ^ Microeconomics II: Notes & Formula Sheet (UPF) 24 NUMERICAL COMPARISON Let , , for , and . Under private competition: Firm sets , and . Profit is . Firm sets . Profit is .

Joint profit is . Under social optimization: Outputs satisfy , , and . Solving simultaneously: , which implies . Joint profit equals . CHECK 30 Why does joint profit increase when pollution is reduced from 4 to 2?

Remedies: Mergers, Tradable Permits, and Pigouvian Taxes Three institutional arrangements can correct this market failure: p = s 12 p = f 10 c (s, x) = s s + 2 (4 x)2 0 x 4 c (f, x) = f f + 2 xf S 12 = 2s s = 6 2(4 x ) = 0 x = 4 Π = S 72 36 0 = 36 F 10 = 2f + x f = = 2 10 4 3 Π = F 30 9 12 = 9 36 + 9 = 45 = s ^ 6 10 = 2 + f ^ x ^ 2(4 ) = x ^ f ^ 10 = 4(4 ) + x ^ = x ^ 16 3 x ^ = x ^ 2 = f ^ 4 12(6) + 10(4) 4 2 6 2 (4 2) 2 2(4) = 48 > 45 Internalization via Merger: Unifying the two entities into a single enterprise forces management to maximize joint profits, naturally internalizing the damage. In practice, spatial dispersion or multilateral impacts across many firms can prevent mergers.

1 Establishment of Property Rights and Permit Markets: As established by the Coase Theorem (see «Market Failures — Externalities and Property Rights»), defining property rights over the river allows decentralized trade at unit price . If the fishery holds clean water rights, it sells pollution allowances; if the steel mill holds emission rights, the fishery pays it to abate. With quasi- linear profits, competitive equilibrium in the permit market establishes , restoring the unique socially optimal pollution level regardless of initial ownership. 2 q = x c s q = x c f x ^ Pigouvian Taxation: The policymaker levies a Pigouvian tax per unit of emissions directly on firm .

3 t S Microeconomics II: Notes & Formula Sheet (UPF) 25 OUTLINE Remedies for Negative Production Externalities When subject to tax rate , firm maximizes , producing first-order condition: To align private incentives with the social optimum, the regulator sets the tax equal to marginal external damage at the optimum: In the numerical example, setting yields , eliminating the deadweight loss. Graph illustrating the deadweight loss corrected by a Pigouvian tax.

Decentralized Solutions Market-based mechanisms without direct taxation Corporate Merger Internalizes damage within a single firm Tradable Permits Coasean bargaining with clear property rights Public Interventions Government regulatory instruments Pigouvian Tax Tax rate equals marginal external damage t S max p s s,x s c (s, x) s tx = x c (s, x) s t t = = x c ( , ) f f ^ x ^ MC ( , ) F f ^ x ^ t = = f ^ 4 2(4 x) = 4 x = 2 = x ^ Microeconomics II: Notes & Formula Sheet (UPF) 26 COMMON MISTAKE Setting the Pigouvian tax equal to marginal external damage evaluated at the competitive equilibrium overtaxes the firm.

The optimal rate must always equal marginal damage evaluated at the socially efficient output level . The Tragedy of the Commons Garrett Hardin (1968) formalized the Tragedy of the Commons as the degradation of a non-exclusive common-pool resource under open access. Consider a common pasture where villagers graze cows. Let denote total cows and the total milk produced. Milk price is normalized to , and each cow costs constant unit price . Because cows graze on fixed land, adding cows congests pasture fertility, meaning production per cow decreases in . Under unrestricted open access, individual villagers purchase cows as long as net private returns are non-negative.

This free entry equilibrium drives profit per cow to zero, equating average product to cost: By contrast, a single owner managing the resource maximizes aggregate surplus , yielding the optimality condition: x t x ^ v f(v) p = 1 a v f(v) v AP ( ) = v ˉ = v ˉ f( ) v ˉ a = AC max f(v) v av MP (v ) = f (v ) = a = MC Microeconomics II: Notes & Formula Sheet (UPF) 27 Graph of efficient output versus open-access equilibrium in common resources. Because average product strictly exceeds marginal product when average product is downward- sloping, open access leads to excessive entry and resource depletion: .

Dimension Open Access Equilibrium Social / Managed Optimum Decision Rule Resource Utilization Excessive stock ( ) Optimal stock ( ) Total Economic Rent Fully dissipated ( ) Maximized ( ) Externality Cause Uninternalized congestion on others' yields Marginal congestion accounted for CHECK 31 Why does open access fully dissipate economic rents? > v ˉ v AP (v) = AC = v f(v) a M P (v) = M C f (v) = a > v ˉ v v Π = 0 Π > 0 Microeconomics II: Notes & Formula Sheet (UPF) 28 KEY IDEAS A negative production externality drives a wedge between private marginal cost and social marginal cost, inducing competitive overproduction of pollution.

The socially efficient level of emissions equates the marginal abatement cost of the polluter to the marginal damage inflicted on the victim: . Internalization can occur privately through business mergers or Coasean property rights bargaining, and publicly through Pigouvian taxation. The optimal Pigouvian tax rate must equal the marginal external damage evaluated at the socially efficient output: . Under open access, the Tragedy of the Commons arises because free entry equates average product to unit cost ( ), whereas efficiency requires equating marginal product to marginal cost ( ). CHECK YOUR UNDERSTANDING 32. What mathematical relationship characterizes the optimal Pigouvian tax rate? 33.

Why does the open-access equilibrium fail to maximize aggregate resource rents? Quiz Circle the correct answer. The answers are at the end. 34. What defines a consumption externality? 35. Why does the First Fundamental Theorem of Welfare fail when externalities are present? 36.

According to the Coase Theorem under quasi-linear utility, changing initial property rights changes: MC = S MC F t = MC ( , ) F f ^ x ^ AP = AC MP = MC It is produced exclusively by consumer agents a) It affects consumers regardless of who generates it b) It involves monetary transfers between firms c) It is always a positive economic impact d) Markets become natural monopolies a) Prices fail to account for social costs b) Agents refuse to maximize private utility c) Consumers always face budget deficits d) Only the technology of the polluting agent a) Final utility levels but not externality quantity b) Neither utility levels nor externality quantity c) The optimal externality quantity and utility levels d) Microeconomics II: Notes & Formula Sheet (UPF) 29 37.

Why is the contract curve horizontal in the smoker example with quasi-linear utility? 38. In the competitive steel-fishery model, why does the steel firm overproduce pollution? 39. What is the socially optimal Pigouvian tax rate per unit of emission? 40. Which equilibrium condition determines total resource exploitation under open access? 41. Why does a merger between polluter and victim achieve Pareto efficiency? 5 Public Goods — Provision, Efficiency, and Collective Decision BY THE END YOU WILL BE ABLE TO Classify goods into the four categories defined by rivalry and excludability. Formulate discrete public good provision games and explain voluntary contribution failure.

Derive and apply the Samuelson condition for optimal continuous public good supply. Analyze voting outcomes using single-peaked preferences and the Median Voter Theorem.

Both agents have identical indifference curves a) Money has zero marginal utility for both b) Smoke cannot be produced beyond unit capacity c) Marginal rate of substitution depends only on smoke d) The fishery sets a prohibitive access price a) Its marginal cost of abatement is negative b) The price of steel exceeds its marginal cost c) It ignores the damage imposed on the fishery d) Marginal external damage at the social optimum a) Total revenue of the affected downstream fishery b) Average production cost of the steel firm c) Marginal external damage at competitive equilibrium d) Average product equals marginal product a) Average product equals unit cost b) Marginal product equals unit cost c) Marginal revenue equals marginal damage d) It eliminates diminishing returns in production a) It exempts the companies from Pigouvian taxation b) The merged firm internalizes all production externalities c) It grants monopoly power over both markets d) Microeconomics II: Notes & Formula Sheet (UPF) 30 THINK BEFORE YOU READ 42 Why does voluntary donation often fail to fund essential projects even when everyone agrees they are worth the cost?

Classification of Goods Economic goods are categorized according to two technical properties: excludability and rivalry. Excludability: A good is exclusive if producers or owners can prevent individuals from consuming it when they do not pay. Rivalry: A good is rivalrous if one agent's consumption diminishes the quantity or quality available to other consumers, or prevents concurrent use. Dimension Rival Non-Rival Exclusive Private Goods (food, clothing) Club Goods (cable TV, subscription services) Non- Exclusive Common Goods (fish stocks, public bins) Pure Public Goods (national defense, street lighting) A pure public good satisfies both non-excludability and non-rivalry.

Because no agent can be excluded and consumption does not deplete the resource, all individuals consume identical quantities of the public good, even though their subjective valuations may differ. Bad outcomes, such as environmental pollution, follow the same formal logic as public bads. OUTLINE Classification of Goods CHECK 43 Why is clean outdoor air classified as a pure public good rather than a common good?

Rival Goods Consumption diminishes availability Private Goods Excludable and rival Common Resources Non-excludable and rival Non-Rival Goods Joint consumption without depletion Club Goods Excludable and non-rival Pure Public Goods Non-excludable and non-rival Microeconomics II: Notes & Formula Sheet (UPF) 31 Discrete Public Goods and Voluntary Contribution Breakdown Consider an indivisible project requiring a fixed cost . An agent's willingness to pay (WTP) or reservation price represents the maximum amount individual would surrender to obtain the good, satisfying the indifference condition: where is initial wealth, 1 denotes provision, and 0 denotes non-provision.

To guarantee individual consistency (individual rationality), an agent's financial contribution cannot exceed this threshold: . Financial feasibility requires total contributions to cover project costs: . Provision achieves Pareto efficiency whenever aggregate willingness to pay meets or exceeds the cost: If this benchmark holds, cost shares can be structured such that for every participant, ensuring mutual welfare gains relative to non-provision. EXAMPLE Two flatmates consider purchasing a shared living room item costing , splitting expenditures equally at 10 each. If and , efficiency dictates purchase because .

Under equal cost sharing, agent 2 declines because contribution 10 exceeds valuation 5. Uncoordinated voluntary mechanisms collapse despite potential net social value. Non-excludability fosters the free-rider problem: agents face incentives to understate valuations or withhold contributions while benefiting from others' funding. Consider two symmetric agents with wealth , reservation values , and project cost .

Agent 1 \ Agent 2 Buy (Contribute 50) Do Not Buy (Contribute 0) Buy (Contribute 50) (110, 110) (60, 160) Do Not Buy (Contribute 0) (160, 60) (100, 100) Both agents possess a dominant strategy to refrain from funding, driving the decentralized equilibrium to and foregoing the Pareto optimal allocation . CHECK 44 Why does voluntary financing fail in the normal-form public good game when individual WTP is 60 and cost is 100? Continuous Public Goods: The Samuelson Condition When the level of public good can vary continuously, production requires resource cost with positive marginal costs.

For individuals, economy-wide wealth balance equates aggregate private expenditures and public production outlays to initial resources: c r i i u (w i i r , 1) = i u (w , 0) i i w i g i r i g i g i c ρ r ρc i=1 n i g i r > i g i c = 20 r = 1 15 r = 2 5 r + 1 r = 2 20ρ20 w = 1 w = 2 100 r = 1 r = 2 60 c = 100 (100, 100) (110, 110) G c(G) n Microeconomics II: Notes & Formula Sheet (UPF) 32 In standard private goods, Pareto efficiency equates each consumer's marginal rate of substitution to the marginal rate of transformation ( ). For non-rival public goods, an identical unit enters every utility function simultaneously. Efficiency mandates the Samuelson condition: where .

If private numéraire consumption has a marginal cost of unity, the condition simplifies to equating aggregate marginal willingness to pay to marginal cost: Vertical summation of individual MRS schedules determining efficient public good level G*. Geometrically, aggregate social demand for public goods requires vertical summation of individual marginal valuation schedules across identical output levels, contrasting with horizontal summation used for rival private goods. ERROR FREQUENT Do not sum public good quantities horizontally across individuals when finding market equilibrium.

Because public goods are non-rival, all consumers share the exact same quantity ; efficiency requires summing their marginal rates of substitution vertically at each level of . pρ x + i=1 n i c(G) = ρ w i=1 n i MRS = i MRT MRS = i=1 n i MRT MRS = i ρu /ρx i i ρu /ρG i ρ MU (G) = i=1 n i MC(G) G G Microeconomics II: Notes & Formula Sheet (UPF) 33 Quasi-Linear Preferences and Independence of Wealth Optimal public provision generally varies with wealth allocations.

Under quasi-linear preferences ( ), marginal utilities with respect to are independent of private consumption : Consequently, determines the efficient level of strictly from individual sub-utility functions , completely independent of income distributions or initial endowments. Collective Choice, Voting Paradoxes, and the Median Voter Because markets under-provide public goods, collective choice mechanisms determine allocation. Majority voting across multi-dimensional or unrestricted choices can induce circular preferences known as Condorcet cycles.

With three voters evaluating three alternatives , , and : Voter A: Voter B: Voter C: Pairwise contests generate intransitive social majorities ( defeats , defeats , and defeats ), eliminating any stable winner. Stability emerges when individual preferences are single-peaked along a unidimensional domain: utility strictly diminishes as the policy moves in either direction away from an agent's ideal point. Comparison between single-peaked preferences (A) and multi-peaked preferences (B).

U (x , G) = i i a x + i i v (G) i G x i MRS = i a i 1 dG dv (G) i ρMRS = i MC(G) G v (G) i x y z x A y A z y B z B x z C x C y x y y z z x Microeconomics II: Notes & Formula Sheet (UPF) 34 Under single-peaked preferences, pairwise cycling disappears, enabling the Median Voter Theorem. The policy peak favored by the median voter defeats every competing alternative in majority contests. In political competition along a single left-right ideological line, vote-maximizing candidates converge to this median policy. Direct voting models assume truthful reporting. When valuations are private knowledge, consumers face incentives to misreport preferences to lower their tax shares.

Resolving this distortion requires mechanism design schemes that align individual incentives with truthful revelation to implement Pareto efficient allocations. CHECK 45 Why does the Median Voter Theorem fail if voter preferences exhibit multiple utility peaks? KEY IDEAS Goods are categorized by excludability and rivalry; pure public goods exhibit both non- excludability and non-rivalry. Decentralized voluntary provision collapses due to free riding, as individual non-contribution acts as a dominant strategy. Pareto optimal provision of continuous public goods requires the Samuelson condition: .

Quasi-linear utility guarantees that efficient public good provision is completely independent of wealth distribution. Single-peaked preferences prevent Condorcet voting cycles and guarantee that majority voting selects the median voter's preferred policy. CHECK YOUR UNDERSTANDING 46. What is the key geometric difference between aggregating market demand for private versus public goods? 47. How does quasi-linear utility affect the relationship between income distribution and the Samuelson condition? Quiz Circle the correct answer. The answers are at the end. 48. Which set of characteristics defines a pure public good?

MRS = i MRT Rival and excludable a) Rival and non-excludable b) Non-rival and excludable c) Non-rival and non-excludable d) Microeconomics II: Notes & Formula Sheet (UPF) 35 49. What condition characterizes Pareto efficiency for a continuous public good? 50. What primary voting instability is avoided when preferences are single-peaked? 51. Why do decentralized voluntary markets underprovide public goods? 6 Monopoly — Price Setting, Welfare Loss, and Regulation BY THE END YOU WILL BE ABLE TO Derive optimal monopoly output and price using marginal revenue and price elasticity of demand. Compute consumer surplus, producer surplus, and deadweight loss relative to the competitive benchmark.

Evaluate regulatory mechanisms including unit subsidies, lump-sum transfers, and price caps. Determine output and viability under marginal cost and average cost pricing in a natural monopoly. THINK BEFORE YOU READ 52 Why does a single seller that can set any price it wants never choose a price on the inelastic portion of demand? A monopoly is a market structure with a single seller facing the entire market demand. Unlike a competitive firm that takes price as given, a monopolist exercises market power and chooses price and quantity simultaneously along the market demand curve.

Σ = 1 / MRT a) MRS i MRS = MRS = MRT b) = MC for every i c) MRS i Σ = MRT d) MRS i Median voter convergence a) First-degree price discrimination b) Unanimous consensus c) Condorcet cycling paradoxes d) Rivalry between non-paying consumers a) Diminishing returns to scale b) Free riding due to non-excludability c) Absence of government subsidies d) Microeconomics II: Notes & Formula Sheet (UPF) 36 OUTLINE Topic 3: Market Power and Market Structure Profit Maximization and the Price Elasticity Rule A competitive firm faces a flat demand curve where , making marginal revenue equal to price. In contrast, a monopolist faces a downward-sloping inverse demand curve with .

Total revenue is , and total profit is . Differentiating with respect to quantity yields the first-order condition: Monopoly & Regulation Price setting, welfare loss, price caps Price Discrimination 1st, 2nd, and 3rd degree strategies Factor Markets & Monopsony Input pricing and vertical structures Oligopoly Theory Cournot, Bertrand, and Stackelberg models p (y) = 0 p(y) p (y) < 0 R(y) = p(y)y π(y) = p(y)y C(y) MR(y) = p(y) + yp (y) = c (y) = MC(y) Microeconomics II: Notes & Formula Sheet (UPF) 37 Gráfica de los efectos de cantidad y precio sobre el ingreso marginal.

When quantity expands from to , total revenue experiences two opposite effects: Output effect: The firm gains additional revenue by selling extra units at price . Price effect: The firm loses revenue because the price falls from to on all infra-marginal units previously sold. Factoring out connects marginal revenue directly to the price elasticity of demand : DEFINITION The Lerner Index measures monopoly markup over marginal cost as a proportion of price: . Because marginal cost is positive ( ), the firm only produces where . This requires , meaning a profit-maximizing monopolist always operates where demand is elastic ( ).

In the inelastic range ( ), raising the price reduces quantity by a smaller percentage, which increases revenue while cutting total costs. CHECK 53 Why will a monopolist never choose an output level where demand is inelastic? QA QB P B P A P B p(y) ε = Δp/p Δy/y MR(y) = p(y) 1 + p (y) = ( p(y) y ) p(y) 1 ( ε 1 ) L = = p p M C ε 1 MC 0 MR 0 1 1/ ε 0 ε 1 ε < 1 Microeconomics II: Notes & Formula Sheet (UPF) 38 Welfare Inefficiency and Deadweight Loss In a competitive benchmark, firms produce until , maximizing total social surplus (see «General Equilibrium — Walrasian Equilibrium and the Welfare Theorems»).

Because a monopolist restricts quantity below this benchmark to sustain a higher price, an allocative inefficiency arises. Gráfica de la pérdida irrecuperable de eficiencia en el monopolio. EXAMPLE Consider inverse demand and cost , so . Regime Quantity Price Consumer Surplus Producer Profit Total Surplus Monopoly 30 60 450 1350 1800 Social Optimum (Competitive) 45 45 1012.5 1012.5 2025 p = MC p(x) = 90 x c(x) = 2 x2 MC(x) = x Monopoly outcome: Setting gives and . Profit is . Consumer surplus is . Total surplus is . 1 MR = MC 90 2x = x x = m 30 p = m 60 π = m 30(60) = 2 302 1350 CS = m = 2 30(90 60) 450 TS = m 1800 Competitive benchmark: Setting gives and . Profit is . Consumer surplus is .

Total surplus is . 2 p = MC 90 x = x x = 45 p = 45 π = 45(45) = 2 452 1012.5 CS = = 2 45(90 45) 1012.5 TS = 2025 Deadweight loss: The uncaptured social surplus is . 3 DWL = TS TS = m 2025 1800 = 225 Microeconomics II: Notes & Formula Sheet (UPF) 39 CHECK 54 How does deadweight loss manifest in the market? Regulatory Instruments: Taxes, Subsidies, and Price Caps Governments can eliminate deadweight loss using either fiscal incentives or direct price caps. Unit Subsidies Combined with Lump-Sum Taxes To induce the monopolist to produce the efficient quantity , the regulator introduces a per-unit tax . The firm chooses output where : The regulator provides a unit subsidy of 45 per unit.

This subsidy shifts the perceived marginal cost down and expands output. However, the government incurs an outlay of . The firm earns operating profit of . The government can fully fund the subsidy outlay without distorting output by levying a fixed lump-sum tax equal to . The net firm profit becomes , which matches the competitive level. Price Cap Regulation A price ceiling alters the effective demand and marginal revenue curves. At quantities where the ceiling binds, the…