General Equilibrium and Economic Welfare: Study Notes

General Equilibrium and Economic Welfare

Introduction

This chapter explores how markets interact in an economy, how resources are allocated efficiently, and how welfare is evaluated. It covers the concepts of general equilibrium, trading between individuals, competitive exchange, production and trading, and the balance between efficiency and equity.

Pareto Principle and Efficiency

Pareto Principle

• Pareto Principle: A criterion for evaluating welfare in general equilibrium without making interpersonal utility comparisons.

• Pareto Efficient Allocation: An allocation where no reallocation can make someone better off without making someone else worse off.

General Equilibrium

Partial vs. General Equilibrium

• Partial-Equilibrium Analysis: Examines equilibrium in a single market, ignoring interactions with other markets.

• General-Equilibrium Analysis: Studies how equilibrium is determined across all markets simultaneously, considering their interdependence.

Example: Corn and Soybean Markets

• Changes in one market (e.g., corn) can affect another (e.g., soybeans) due to resource allocation and consumer preferences.

Corn	Soybeans
Market Affected by Supply/Demand Shocks	Market Responds via Substitution or Resource Shift

Solved Problem: Coffee and Tea Markets

• Demand for coffee and tea depends on both their prices.

• Supply curves are independent (grown in different regions).

• Equilibrium is found by setting quantity demanded equal to quantity supplied for both goods and solving the system of equations.

• If a shock (e.g., freeze) shifts the coffee supply curve, equilibrium prices and quantities in both markets change.

Formula:

$Q^d_{coffee} = f(P_{coffee}, P_{tea})$ $Q^s_{coffee} = g(P_{coffee})$ $Q^d_{tea} = h(P_{tea}, P_{coffee})$ $Q^s_{tea} = k(P_{tea})$

Minimum Wages with Incomplete Coverage

Partial-Equilibrium Result

• Minimum wage causes quantity of labor demanded to be less than quantity supplied, leading to unemployment.

General Equilibrium with Incomplete Coverage

• If only some sectors are covered by minimum wage laws, workers may shift to uncovered sectors, affecting wages and employment in both sectors.

• Historical example: U.S. minimum wage law in 1938 led workers to move from covered industries to agriculture (uncovered).

Covered Sector	Uncovered Sector	Total Labor Market
Wage set above equilibrium	Wage determined by supply/demand	Aggregate employment adjusts

Solved Problem: Labor Tax in Covered Sector

• Government imposes a tax t per hour on labor in the covered sector.

• Firms pay wage + tax; workers receive wage.

• Equilibrium wages in both sectors are determined by labor mobility and market clearing.

• Total employment and sectoral employment are affected by the tax.

Formula:

$w_{covered} = w_{uncovered} + t$

Trading Between Two People

Endowments and the Edgeworth Box

• Endowment: The initial allocation of goods each person possesses.

• Example: Jane has 30 piles of firewood and 20 candy bars; Denise has 20 piles of firewood and 60 candy bars.

• The Edgeworth Box graphically represents all possible allocations between two people.

Mutually Beneficial Trades and the Contract Curve

• Assumptions: Utility maximization, standard-shaped indifference curves, no interdependence.

• At a Pareto-efficient allocation (on the contract curve):

  • Indifference curves are tangent.

  • Marginal rates of substitution (MRS) are equal.

  • No further mutually beneficial trades are possible.

• The contract curve is the set of all Pareto-efficient allocations.

Solved Problem: Identifying Pareto Efficiency

• To determine if an allocation is on the contract curve, check if no mutually beneficial trades are possible (i.e., indifference curves are tangent).

Competitive Exchange

First and Second Theorems of Welfare Economics

• First Theorem: Any competitive equilibrium is Pareto efficient.

• Second Theorem: Any Pareto-efficient allocation can be achieved as a competitive equilibrium, given an appropriate endowment.

Competitive Equilibrium

• With many buyers and sellers, individuals are price takers.

• Prices adjust so that quantity supplied equals quantity demanded in all markets.

Production and Trading

Production Possibility Frontier (PPF)

• PPF: Shows the maximum combinations of two goods that can be produced with given resources.

• The slope of the PPF is the marginal rate of transformation (MRT).

Formula:

$MRT = -\frac{dY}{dX}$

Comparative Advantage

• A person has a comparative advantage in producing a good if they have a lower opportunity cost than others.

• Example: Jane can produce 3 candy bars or 6 piles of firewood per day; Denise can produce 6 candy bars or 3 piles of firewood per day.

• Trade allows both to consume beyond their individual PPFs.

Solved Problem: Adding a Third Producer

• If Harvey can produce 5 piles of wood or 5 candy bars per day, the joint PPF expands, allowing for more efficient production and greater gains from trade.

Many Producers and the Shape of the PPF

• With many producers, the PPF is a smooth, concave curve.

• The marginal rate of transformation decreases (in absolute value) as production shifts toward one good.

• At the optimal product mix, the marginal rate of transformation equals the marginal rate of substitution for consumers.

Formula:

$MRT = \frac{MC_{candy}}{MC_{wood}}$

Efficiency and Equity

Efficiency

• The Pareto criterion ranks allocations where at least one person is better off and no one is worse off.

• Not all welfare-improving policies are Pareto superior; some create winners and losers.

Equity

• Social Welfare Function: Aggregates individual utilities to rank allocations for society as a whole.

• Utility Possibility Frontier (UPF): The set of utility combinations corresponding to Pareto-efficient allocations.

Voting and Arrow’s Impossibility Theorem

• Social preferences should be complete, transitive, and reflect only individual preferences between alternatives.

• Arrow’s Impossibility Theorem: No social decision rule can satisfy all fairness criteria simultaneously; democratic decision-making may fail to produce consistent social orderings.

Social Welfare Functions

• Utilitarian Welfare Function: Maximize the sum of individual utilities.

Formula:

$W = \sum_{i=1}^{n} U_i$

• Rawlsian Welfare Function: Maximize the utility of the least well-off individual.

Formula:

$W = \min \{ U_1, U_2, ..., U_n \}$

Efficiency Versus Equity

• Society may prefer an inefficient but more equitable allocation, depending on the social welfare function.

• Competitive equilibrium is always Pareto efficient but may not be equitable.

Applications

Minimum Wage and Price Gouging Laws

• Minimum wage and anti-price gouging laws can create covered and uncovered sectors, affecting overall market outcomes.

Covered Sector	Uncovered Sector	Total Market
Regulated by law	Unregulated	Aggregate effects depend on labor mobility

Summary Table: Key Concepts

Concept	Definition
Pareto Efficiency	No one can be made better off without making someone else worse off
General Equilibrium	Simultaneous equilibrium in all markets
Contract Curve	Set of all Pareto-efficient allocations
PPF	Maximum output combinations given resources
Comparative Advantage	Lower opportunity cost in producing a good
Social Welfare Function	Society's method for ranking allocations

Additional info: Some formulas, definitions, and context were expanded for clarity and completeness. Figures referenced (e.g., Edgeworth Box, PPF) are described in text due to lack of images.