Microeconomics II: Edgeworth Box, Efficiency & Equilibrium

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APUNTES MICROECONOMICS II Bachelor's Degree in Business Administration and Management Pompeu Fabra University AprenAI 2026-2027 Clear notes on pure exchange economies, Edgeworth box, Pareto efficiency, the core, and step-by-step competitive equilibrium calculations with Walras' Law.

APREN.AI Contents 1 Market Fundamentals, Feasibility, and the Edgeworth Box Foundations: Market Supply, Demand, and Equilibrium Pure Exchange Economies and the Feasible Set Geometry of the Edgeworth Box 2 Pareto Efficiency, Contract Curve, and the Core Pareto Efficiency Concept and Mathematical Formulation Boundary Solutions vs Interior Solutions The Core of an Exchange Economy 3 Competitive Equilibrium in Exchange Economies Definition and Equilibrium Conditions Step-by-Step Procedure to Solve a Competitive Equilibrium 4 Formula Sheet Feasibility and Edgeworth Box Pareto Efficiency and the Core Competitive Equilibrium and Walras' Law Common Exam 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: Edgeworth Box, Efficiency & Equilibrium 1 1 Market Fundamentals, Feasibility, and the Edgeworth Box BY THE END YOU WILL BE ABLE TO Construct an Edgeworth box using initial individual endowments Formulate feasibility conditions for pure exchange economies THINK BEFORE YOU READ 1 How can two individual consumption spaces be combined into a single bounded diagram?

Foundations: Market Supply, Demand, and Equilibrium In standard microeconomic markets, consumer behavior is guided by preferences, yielding demand curves, while firm technology determines production costs and supply curves. With quasilinear utility of the form , where represents money or a numeraire good and reflects consumption utility, consumers order purchase units according to diminishing marginal utility to form the demand curve . Conversely, the supply schedule orders produced units according to increasing marginal costs.

A competitive equilibrium occurs at the intersection where market supply equals market demand: This market-clearing condition yields the competitive market price and quantity . u (q , m ) = i i i m + i v (q ) i i m i v (q ) i i D S S(Q ) = D(Q ) p Q Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 2 Gráfica del equilibrio competitivo a partir de costes y utilidades marginales. Pure Exchange Economies and the Feasible Set An exchange economy models direct trade between economic agents without production. Consider two agents, and , and two consumption goods, and . Each agent enters the market with an initial endowment vector denoted by .

Agents evaluate bundles using utility functions and representing convex preferences. Gráfica de curvas de indiferencia individuales para los agentes A y B. An allocation pair is classified as truly feasible when total consumption does not exceed the aggregate endowment of each good in the economy: A B X Y i {A, B} e = i (x , y ) i e i e u (x , y ) A A A u (x , y ) B B B ((x , y ), (x , y )) A A B B Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 3 When resources are fully utilized with strict equalities, allocations form the feasible set illustrated inside an Edgeworth box. Geometry of the Edgeworth Box The Edgeworth box synthesizes the two consumers into a single diagram.

The total horizontal width equals aggregate initial holdings of good ( ), while the total vertical height equals total holdings of good ( ). Agent is plotted from the lower-left origin , and Agent is oriented upside-down from the upper-right origin . Esquema de la caja de Edgeworth con la dotación inicial de bienes. Every internal coordinate represents a feasible split of aggregate resources. The initial endowment point serves as the benchmark against which voluntary trades are assessed. COMMON MISTAKE Agents measured from opposite origins can cause measurement confusion. Remember that consumption for agent increases as you move leftward and downward away from origin .

CHECK 2 How do the total dimensions of an Edgeworth box change if agent A's endowment of good X increases? {(x , y ), (x , y ) A A B B x + A x B x + A e x , y + B e A y B y + A e y } B e X x + A e x B e Y y + A e y B e A O A B O B e = (e , e ) A B B O B Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 4 OUTLINE Components of an Exchange Economy KEY IDEAS In quasilinear settings, competitive equilibrium equates supply based on marginal cost with demand based on marginal utility. Total endowments define the fixed width and height of the Edgeworth box. A feasible allocation does not consume more resources than the economy initially possesses. CHECK YOUR UNDERSTANDING 3.

What mathematical relationship characterizes competitive equilibrium between demand and supply lists? 4. How are the coordinates of agent B read inside the Edgeworth box? Quiz Circle the correct answer. The answers are at the end. 5. What determines the horizontal dimension of an Edgeworth box? 6. When is an allocation technically feasible in an exchange economy?

Exchange Economy Two agents and two commodities without production Endowments Initial resource vectors eA and eB Preferences Utility functions uA and uB Edgeworth Box Represents all feasible allocations Agent B's initial endowment of good Y a) Agent A's consumption of good X b) Total initial endowment of good X c) The relative price ratio px/py d) When consumption equals income at zero prices a) When total consumption does not exceed aggregate endowments b) When utility reaches its absolute mathematical bliss point c) When both agents receive identical shares of goods d) Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 5 7. What forms the supply schedule in the introductory model?

8. What does each internal coordinate within an Edgeworth box represent? 2 Pareto Efficiency, Contract Curve, and the Core BY THE END YOU WILL BE ABLE TO Formulate the constrained optimization problem for Pareto efficiency Distinguish between the general contract curve and the core THINK BEFORE YOU READ 9 Why wouldn't rational traders accept just any Pareto efficient allocation? Pareto Efficiency Concept and Mathematical Formulation An allocation is defined as Pareto efficient if there is no feasible reallocation that makes one agent strictly better off without simultaneously making another agent worse off.

Analytically, any internal Pareto efficient allocation is derived by maximizing the utility of agent subject to keeping agent at a constant baseline utility level within resource constraints: Units ordered by consumer marginal utility a) Indifference curves tangent to budget lines b) Total endowments divided by market price c) Units ordered according to production costs d) A disequilibrium production plan a) A unique set of market prices b) An unbounded consumption set c) A feasible distribution between both agents d) A B u ˉB x ,y A A max s.t.

u (x , y ) A A A u (x , y ) = B B B u ˉB x + x = x + x A B A e B e y + y = y + y A B A e B e Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 6 Esquema de la curva de contrato uniendo puntos de tangencia. Varying traces out the set of mutually tangential indifference curve points, forming the contract curve inside the Edgeworth box. Boundary Solutions vs Interior Solutions Tangency conditions identify interior allocations. However, efficiency can also occur on the boundary of the box. Tangency-based methods characterize Pareto efficient allocations inside the interior, but boundary allocations require checking non-negativity constraints where tangency does not necessarily hold.

The Core of an Exchange Economy While the contract curve includes all Pareto efficient points, voluntary trade requires that neither agent ends up worse off than at their initial endowment. The core is the subset of Pareto efficient allocations where both agents achieve at least the utility of their initial endowment: u ˉB u (x , y ) A A A u (x , y ) and u (x , y ) A A e A e B B B u (x , y ) B B e B e Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 7 Diagrama del núcleo resaltado sobre la curva de contrato desde la dotación.

The core corresponds geometrically to the segment of the contract curve located between the two indifference curves that pass directly through endowment point . COMMON MISTAKE Do not confuse Pareto efficiency with the core. All allocations in the core are Pareto efficient, but not all Pareto efficient allocations belong to the core, as some leave one agent worse off than their starting endowment. CHECK 10 Why does free voluntary trade restrict final outcomes to the core rather than the entire contract curve?

PROCESS Characterizing Pareto Efficiency to the Core 1 Initial Endowment Endowment point E sets the reference utility levels 2 Utility Maximization Maximize uA holding uB constant within feasible limits 3 〰 Contract Curve Locus of all tangency points across utility levels 4 Core Selection Efficient allocations strictly weakly pre ferred to E E Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 8 KEY IDEAS Pareto efficiency occurs where indifference curves are tangent inside the box, meaning no agent can be made better off without harming another. The contract curve links all Pareto efficient allocations.

The core is the segment of the contract curve that provides both agents at least their initial endowment utility. CHECK YOUR UNDERSTANDING 11. What is the optimization problem used to compute interior Pareto efficient allocations? 12. Where is the core located within an Edgeworth box? Quiz Circle the correct answer. The answers are at the end. 13. What defines a Pareto efficient allocation? 14. What characterizes the core of an exchange economy? 15. When does the standard tangency condition fail to identify Pareto efficient allocations? 16. What curve connects all interior Pareto efficient points?

Zero marginal rate of substitution for both goods a) Equiproportional distribution of total commodities b) Inability to improve one agent without worsening another c) Equal utility levels for all participants d) All allocations giving positive utility a) Allocations where marginal utilities equal zero b) Only allocations lying strictly on box boundaries c) Efficient allocations where no agent is worse off than endowment d) When efficient allocations occur on the boundary a) When both agents have identical incomes b) When preferences are quasilinear c) When total endowments are strictly positive d) The budget line a) The Engel expansion path b) The contract curve c) The market supply curve d) Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 9 3 Competitive Equilibrium in Exchange Economies BY THE END YOU WILL BE ABLE TO Calculate gross demand functions from utility functions and endowments Compute competitive equilibrium prices using a numeraire and market clearing THINK BEFORE YOU READ 17 Why do competitive equilibrium calculations determine price ratios rather than unique price levels?

Definition and Equilibrium Conditions A competitive equilibrium in an exchange economy consists of a price vector and a feasible allocation such that each agent chooses their optimal bundle subject to their individual budget constraint defined by their endowment value. Under monotonic preferences, agents exhaust their budget.

Walras' Law ensures that the aggregate value of excess demands is identically zero: Step-by-Step Procedure to Solve a Competitive Equilibrium Solving for equilibrium follows a structured three-step method: (p , p ) x y ((x , y ), (x , y )) A A B B p (x + x ) (x + x ) + x [ A B A e B e ] p (y + y ) (y + y ) = y [ A B A e B e ] 0 Derive Gross Demands: Solve the consumer problem for each agent by setting the Marginal Rate of Substitution equal to the price ratio and substituting into their budget constraint . 1 MRS = i p /p x y p x + x i p y = y i p x + x i e p y y i e Select a Numeraire: Normalize prices by fixing one price, typically .

2 p = x 1 Clear One Market: Equate aggregate demand to total endowment in good (or ). Solve for the relative price. By Walras' Law, the other market automatically clears. 3 X Y Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 10 EXAMPLE Consider endowments and with preferences and . Step 1: Gross Demands For agent : . Substituting into the budget constraint gives : Applying the identical Cobb-Douglas rule for agent with yields: Step 2: Numeraire Choice Set . Demands simplify to: Step 3: Market Clearing Equating total demand for good to total supply ( ): Thus, , which represents the same price ratio as .

COMMON MISTAKE Do not attempt to solve market clearing equations for both goods simultaneously as independent equations. Because of Walras' Law, market clearing equations are linearly dependent; clearing one market guarantees clearing in the second. CHECK 18 Why does setting px = 3 and py = 2 yield the exact same equilibrium demands as px = 1 and py = 2/3?

e = A (10, 20) e = B (10, 10) u (x , y ) = A A A x y A A u (x , y ) = B B B x y B B A MRS = A = x A y A p y p x p x = x A p y y A p x + x A p y = y A p (10) + x p (20) y 2p x = x A 10p + x 20p y x (p , p ) = A x y , y (p , p ) = 2p x 10p + 20p x y A x y 2py 10p + 20p x y B e = B (10, 10) x (p , p ) = B x y , y (p , p ) = 2p x 10p + 10p x y B x y 2p y 10p + 10p x y p = x 1 x (1, p ) = A y 5 + 10p , y (1, p ) = y A y + p y 5 10 x (1, p ) = B y 5 + 5p , y (1, p ) = y B y + p y 5 5 X 10 + 10 = 20 (5 + 10p ) + y (5 + 5p ) = y 20 10 + 15p = y 20 15p = y 10 p = y 3 2 p = x 1, p = y 3 2 p = x 3, p = y 2 Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 11 PROCESS Solving for Competitive Equilibrium 1 Gross Demands Solve MRS = px/py using individual bud get lines 2 Set Numeraire Fix px = 1 to express system in relative prices 3 Clear Market Set aggregate de mand equal to total endowment 4 Equilibrium Prices Determine py and ver ify equilibrium allocation KEY IDEAS A competitive equilibrium requires market feasibility and individual utility maximization on budget lines.

Walras' Law guarantees that clearing all but one market implies equilibrium in the remaining market. Equilibrium prices are determined up to a scale factor, allowing one good to be selected as numeraire. CHECK YOUR UNDERSTANDING 19. What is the role of a numeraire good in solving an exchange economy? 20. State the condition describing Walras' Law for an economy with two goods. Quiz Circle the correct answer. The answers are at the end. 21. What does Walras' Law state regarding aggregate excess demands? 22. How many market clearing conditions must be solved in a two-good economy? 23. In the example solved, what is the equilibrium price py when px = 1?

The market value of excess demands sums to zero a) Total excess demand is strictly positive b) Prices must equal marginal costs directly c) Both goods must have identical market prices d) Three conditions including production costs a) Both simultaneously as independent equations b) Only one, because the second clears automatically c) None, because prices are fixed exogenously d) 1 a) 1/2 b) 3/2 c) 2/3 d) Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 12 24. What happens if all prices are multiplied by a positive scalar k?

4 Formula Sheet Feasibility and Edgeworth Box Formula or Condition Meaning of Symbols When to Use : initial endowments of agent Constructing the total dimensions of the Edgeworth box : consumption allocations Checking feasibility of resource allocations in an exchange economy Pareto Efficiency and the Core Concept / Problem Mathematical Formulation When to Use Pareto Efficient Allocations and resource clearing ( for interior optima) Deriving the contract curve Core Allocations along the contract curve Identifying voluntary, unvetoed trade allocations relative to endowment Competitive Equilibrium and Walras' Law Rule / Procedure Mathematical Formulation When to Use Walras' Law Ensures aggregate value of excess demands equals zero; clearing one market clears the other Solving Competitive Equilibrium Compute gross demands via and budget constraints Set numeraire ( ) Clear market ( ) Solve for Finding equilibrium price ratios and allocations in 2x2 exchange economies Agents consume strictly zero of both goods a) Demands double for both goods b) The budget constraint rotates towards the origin c) Real consumption demands remain completely unchanged d) Width = x + A e x , Height = B e y + A e y B e x , y i e i e i {A, B} x + A x B x + A e x , y + B e A y B y + A e y B e x , y i i max u (x , y ) s.t.

u (x , y ) = x ,y A A A A A B B B u ˉB M RS = A M RS B u (x , y ) A A A u (x , y ) and u (x , y ) A A e A e B B B u (x , y ) B B e B e E p [(x + x A x ) B (x + A e x )] + B e p [(y + y A y ) B (y + A e y )] = B e 0 M RS = i p /p x y p = x 1 X x + A x = B x + A e x B e p y Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 13 Common Exam Pitfalls Independent market clearing: Do not treat both market clearing equations as independent; Walras' Law makes one equation redundant. Boundary vs. Interior solutions: The condition only identifies interior allocations; boundary efficiency requires non-negativity checks. Absolute vs.

Relative prices: Competitive equilibria determine only relative prices ( ); always normalize by choosing a numeraire. MRS = A MRS B p /p x y Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 14 Answers to the review questions 1. By rotating one agent's coordinate system 180 degrees and joining them such that the total dimensions equal aggregate goods. 2. The width of the Edgeworth box expands by that exact quantity because the width equals aggregate endowments . 3. It is characterized by the market-clearing condition . 4. They are measured relative to origin at the top-right corner, expanding leftward for good X and downward for good Y. 5. c) Total initial endowment of good X.

The width is determined strictly by the aggregate endowment . 6. b) When total consumption does not exceed aggregate endowments. Feasibility requires aggregate consumption of each good to be less than or equal to total available supply. 7. d) Units ordered according to production costs. The supply schedule arranges produced units according to marginal production costs. 8. d) A feasible distribution between both agents. Any point inside the box fully exhausts or distributes the fixed resource totals between agents A and B. 9. Because an allocation can be Pareto efficient while giving an agent less utility than their starting endowment. 10.

Because agents will veto any trade agreement that leaves them with less utility than simply consuming their initial endowment. 11. Maximize subject to and aggregate endowment balance constraints. 12. Along the contract curve inside the lens formed by the indifference curves passing through the endowment . 13. c) Inability to improve one agent without worsening another. By definition, Pareto efficiency means no further mutually beneficial improvements can be made. 14. d) Efficient allocations where no agent is worse off than endowment. The core contains Pareto optimal allocations respecting individual rationality relative to endowment . 15. a) When efficient allocations occur on the boundary.

The standard tangency method identifies interior points and does not fully describe efficiency along the boundary. 16. c) The contract curve. The contract curve represents the locus of points where indifference curves are mutually tangent. 17. Because only relative prices influence budget constraints; scaling all nominal prices leaves real budget sets unchanged. 18. Because demand functions in competitive exchange economies are homogeneous of degree zero in prices; only relative price ratios matter. 19. It anchors absolute price levels (e.g., ), allowing the determination of unique relative equilibrium prices. 20. . 21. a) The market value of excess demands sums to zero.

Walras' Law establishes that the total value of aggregate excess demands across all markets is identically zero. x + A e x B e S(Q ) = D(Q ) O B x + A e x B e S u A u = B u ˉB E E p = x 1 p [(x + x A x ) B (x + A e x )] + B e p [(y + y A y ) B (y + A e y )] = B e 0 Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 15 22. c) Only one, because the second clears automatically. By Walras' Law, clearing one market guarantees the second market clears automatically. 23. d) 2/3. Setting yields . 24. d) Real consumption demands remain completely unchanged. Demand functions are homogeneous of degree zero; scaling all prices does not change budget sets or choices.

10 + 15p = y 20 p = y 10/15 = 2/3 Microeconomics II: Edgeworth Box, Efficiency & Equilibrium 16