Difficulty: Intermediate
Prerequisites: Partial equilibrium analysis, utility maximisation, indifference curves, marginal rate of substitution (MRS). Comfortable with basic calculus (partial derivatives).
General equilibrium is where microeconomics stops looking at one market in isolation and starts asking what happens when every market clears at once. Up to this point in ECN 312, the focus has been on partial equilibrium, where you solve for price and quantity in a single market while holding everything else constant. General equilibrium drops that assumption and considers how consumers, firms, and markets interact simultaneously. If you have been away from the course for a few weeks, make sure you are solid on consumer optimisation, budget constraints, and the concept of marginal rate of substitution before diving in.
General equilibrium analysis examines how all markets in an economy reach equilibrium simultaneously, rather than one at a time. The Edgeworth box and contract curve show where two consumers can trade to a Pareto efficient outcome, and Walras' Law tells us that equilibrium prices are set where aggregate excess demand is zero. The First and Second Welfare Theorems together say that, under standard assumptions, competitive markets always produce efficient outcomes.
Partial equilibrium analysis
The study of equilibrium in a single market, holding conditions in all other markets constant. Consumer demand comes from utility maximisation subject to a budget constraint; producer supply comes from profit maximisation subject to costs.
In simple terms, you are zooming in on one market and ignoring spillovers to everything else.
General equilibrium
An analysis that considers all markets in the economy at the same time, accounting for the fact that what happens in one market affects others.
Think of it as the wide-angle lens version of equilibrium: every market clears simultaneously.
Price theory
Another name for partial equilibrium analysis, emphasising how interacting supply and demand determine the price in a particular market.
Edgeworth box
A graphical tool that represents all possible allocations of two goods between two consumers within a fixed total endowment. One consumer's origin is in the bottom-left corner, the other's is in the top-right.
Think of it as a rectangle where every point inside represents a way to split two goods between two people.
Contract curve
The set of allocations inside the Edgeworth box where the marginal rates of substitution of both consumers are equal (MRS₁ = MRS₂). Every point on the contract curve is Pareto optimal.
In simple terms, it is the line of "best possible deals" where neither person can be made better off without hurting the other.
Pareto optimal (Pareto efficient)
An allocation where no reallocation can make one person better off without making someone else worse off. An efficient outcome.
Think of it as: all the gains from trade have been used up.
The core
The segment of the contract curve that lies between the two consumers' initial-endowment indifference curves. Both consumers are at least as well off as they were before trading.
In simple terms, it is the range of fair deals, where neither person would prefer to walk away and keep their original bundle.
Budget line (in the Edgeworth box)
The line along which rational actors agree to trade, determined by the intersection of the indifference curves at the initial endowment. Its slope equals the price ratio of the two goods.
Price ratio
The slope of the budget line in the Edgeworth box. It tells you how many units of one good trade for one unit of the other.
Aggregate excess demand
The total amount by which quantity demanded exceeds quantity supplied across consumers at a given set of prices. In equilibrium, aggregate excess demand is zero.
Walras' Law
States that in a general equilibrium setting, prices adjust until aggregate excess demand is zero across all markets. If every market but one clears, the last one must clear too.
First Welfare Theorem
Market equilibria are Pareto efficient. A competitive market will exhaust all gains from trade.
In simple terms, if markets are competitive and all the standard assumptions hold, the outcome is efficient.
Second Welfare Theorem
When preferences are convex, a market equilibrium exists, and any Pareto efficient outcome can be achieved through competitive markets (with appropriate redistribution of endowments).
Think of it as: efficiency is always reachable, and society can choose which efficient outcome it wants by adjusting starting points.
Convex preferences
Preferences where averages are preferred to extremes. Graphically, the "better than" set above any indifference curve is a convex set. This is a standard assumption required for the Second Welfare Theorem.
Positive monotonic utility
The assumption that more of any good is always preferred to less. Utility increases when you get more of either good.
Partial equilibrium looks at one market at a time. You derive consumer demand from utility maximisation and producer supply from profit maximisation, then find the price that clears that single market.
This approach is also called price theory, because it shows how supply and demand interact to set the price for one particular good.
The limitation: markets do not exist in isolation. A price change in one market spills over into others, and firm profits flow back to consumers as income.
General equilibrium accounts for all of these interactions simultaneously. It asks: at what set of prices do all markets clear at once?
The Edgeworth box is the central graphical tool for two-consumer, two-good general equilibrium.
The box's width represents the total endowment of good X; its height represents the total endowment of good Y.
Consumer 1's origin is the bottom-left corner; Consumer 2's origin is the top-right corner (rotated 180 degrees).
Every point inside the box represents a feasible allocation, meaning the total of each good across both consumers equals the fixed endowment.
Each consumer's indifference curves can be drawn inside the box. Consumer 1's curves are oriented normally; Consumer 2's are flipped.
The contract curve is found by setting MRS₁ = MRS₂ and applying the feasibility constraints.
In the lecture's worked example, both consumers have the utility function uᵢ = xᵢ^(1/2) · yᵢ^(1/2).
Consumer 1 starts with endowment (2, 8). Consumer 2 starts with endowment (8, 2). Total endowments: X = 10, Y = 10.
For Consumer 1, MRS₁ = MUₓ / MUᵧ = y₁ / x₁.
For Consumer 2, MRS₂ = y₂ / x₂ (same form, because preferences are identical).
Setting MRS₁ = MRS₂: y₁/x₁ = y₂/x₂.
Substituting the feasibility constraints (x₂ = 10 − x₁, y₂ = 10 − y₁) and simplifying gives y₁ = x₁.
The contract curve in this example is the 45-degree line through the Edgeworth box.
The core is the subset of the contract curve where both consumers are at least as well off as they were at their initial endowments.
To find its endpoints, calculate each consumer's utility at their initial endowment, then find the point on the contract curve that gives the other consumer exactly that utility level.
In the example:
Consumer 1's initial utility: u₁ = 2^(1/2) · 8^(1/2) = 4. On the contract curve (y₁ = x₁), this gives x₁ = y₁ = 4. First endpoint: (4, 4).
Consumer 2's initial utility: u₂ = 8^(1/2) · 2^(1/2) = 4. In Consumer 1's coordinates: x₁ = 10 − 4 = 6, y₁ = 6. Second endpoint: (6, 6).
The core runs along y₁ = x₁ from (4, 4) to (6, 6).
Any trade that lands outside the core leaves at least one consumer worse off than they started, so rational agents will not agree to it.
Prices in the Edgeworth box are determined by the budget line that passes through the initial endowment point.
The budget line is found by setting the two consumers' utilities equal at the endowment constraint: x₁^(1/2) · y₁^(1/2) = (10 − x₁)^(1/2) · (10 − y₁)^(1/2).
Squaring and simplifying: x₁y₁ = (10 − x₁)(10 − y₁), which gives y₁ = 10 − x₁.
This is the budget line. Its slope is −1, meaning the price ratio pₓ/pᵧ = 1. The two goods trade one-for-one.
The equilibrium allocation is where the budget line intersects the contract curve: y₁ = 10 − x₁ and y₁ = x₁, so x₁ = y₁ = 5, and x₂ = y₂ = 5.
At the equilibrium price ratio, aggregate excess demand is zero: every consumer gets exactly what they want, and all goods are accounted for.
At any other price ratio, aggregate excess demand would be non-zero. Consumers would want to trade more, but those trades would not be mutually beneficial.
Walras' Law guarantees that the price ratio adjusts until excess demand disappears across all markets.
Both theorems rest on a set of assumptions: rational agents (the optimality principle), preferences that can be represented by utility functions, positive monotonic utility (more is better), convex preferences, and costless trade.
First Welfare Theorem: Every competitive market equilibrium is Pareto efficient. A competitive market exhausts all gains from trade. If the assumptions hold and markets are competitive, the outcome cannot be improved upon without making someone worse off.
Second Welfare Theorem: When preferences are convex, a competitive equilibrium exists. Further, any Pareto efficient outcome can be achieved as a competitive equilibrium, provided endowments are redistributed appropriately beforehand.
Combined statement: Given the standard assumptions, self-interested behaviour always drives the economy to a Pareto efficient market equilibrium.
The practical implication: the First Theorem says markets get you to an efficient point. The Second Theorem says if society prefers a different efficient point (for equity reasons, say), it can get there through lump-sum transfers followed by competitive trading, rather than by intervening in the market mechanism itself.
Utility function (Cobb-Douglas, symmetric)
u_i = x_i^{1/2} \, y_i^{1/2}Marginal utilities
\frac{\partial u}{\partial x_1} = \frac{1}{2} x_1^{-1/2} y_1^{1/2}\frac{\partial u}{\partial y_1} = \frac{1}{2} x_1^{1/2} y_1^{-1/2}Marginal rate of substitution
MRS = \frac{MU_x}{MU_y} = \frac{y_i}{x_i}Feasibility constraints
x_1 + x_2 = 10 \qquad y_1 + y_2 = 10Contract curve (for this example)
y_1 = x_1Budget line (for this example)
y_1 = 10 - x_1Equilibrium allocation
x_1 = y_1 = 5 \qquad x_2 = y_2 = 5Core endpoints
(4,\, 4) \text{ to } (6,\, 6) \text{ along } y_1 = x_1General equilibrium is the theoretical backbone behind computable general equilibrium (CGE) models used by governments and central banks to evaluate the economy-wide effects of tax changes, trade agreements, or environmental regulation. When policymakers want to know what happens to wages, prices, and output across all sectors after, say, a carbon tax, they build a general equilibrium model rather than a partial one.
The welfare theorems also underpin a core argument in economics: that competitive markets tend to produce efficient outcomes, and that if the outcome is inequitable, the preferred remedy is redistribution of endowments (lump-sum transfers) rather than price controls or quantity restrictions.
Students often think the contract curve depends on the initial endowments. It does not. The contract curve is determined entirely by preferences and the total endowment. The initial allocation determines the core (the segment of the contract curve both parties would agree to trade to), not the curve itself.
Students sometimes confuse Pareto efficiency with fairness. An allocation where one person has everything and the other has nothing can be Pareto efficient. Efficiency says nothing about equity.
A common error is to think the First Welfare Theorem says markets produce good outcomes. It says they produce efficient outcomes, which is a narrower claim. An efficient outcome can still be highly unequal.
Students mix up the two welfare theorems. A useful mnemonic: the First is about direction (markets lead to efficiency). The Second is about possibility (any efficient outcome is reachable through markets, if you start from the right endowment).
⚠️ Be able to derive the contract curve from scratch given a utility function and endowments. This is the most likely calculation question.
⚠️ Know how to find the core by computing each consumer's utility at their initial endowment, then finding those utility levels on the contract curve.
⚠️ Be prepared to state and distinguish the First and Second Welfare Theorems, including the assumptions each requires.
⚠️ Understand that the price ratio is the slope of the budget line and that equilibrium occurs where the budget line intersects the contract curve.
⚠️ Questions may ask you to identify whether a given allocation is Pareto efficient, in the core, or neither.
True or False: The contract curve depends on the initial endowments of the consumers.
False. It depends on preferences and total endowments only.
Fill in the blank: The set of Pareto optimal allocations that both consumers prefer to their initial endowments is called the ______.
The core.
True or False: The First Welfare Theorem guarantees that competitive equilibrium outcomes are fair.
False. It guarantees efficiency, not fairness.
Fill in the blank: Walras' Law states that equilibrium prices are set where ______ is zero.
Aggregate excess demand.
True or False: The Second Welfare Theorem requires that preferences be convex.
True.
Q: Two consumers have utility uᵢ = xᵢ^(1/2) yᵢ^(1/2). Consumer 1 has endowment (2, 8) and Consumer 2 has endowment (8, 2). Derive the contract curve.
A: Set MRS₁ = MRS₂. For both consumers, MRS = yᵢ/xᵢ. So y₁/x₁ = y₂/x₂. Substitute the feasibility constraints x₂ = 10 − x₁ and y₂ = 10 − y₁. Cross-multiply and simplify to get y₁ = x₁. The contract curve is the 45-degree line.
Q: Using the same setup, find the endpoints of the core.
A: Compute each consumer's utility at their initial endowment. u₁ = (2)^(1/2)(8)^(1/2) = 4. On the contract curve y₁ = x₁, so 4 = x₁^(1/2) · x₁^(1/2) = x₁, giving (4, 4). For Consumer 2, u₂ = 4 as well. In Consumer 1's coordinates: x₁ = 10 − 4 = 6, y₁ = 6, giving (6, 6). The core runs from (4, 4) to (6, 6).
Q: What is the equilibrium price ratio in this example, and what allocation does it produce?
A: The budget line is y₁ = 10 − x₁ (slope = −1), so the price ratio pₓ/pᵧ = 1. The equilibrium is where the budget line meets the contract curve: x₁ = y₁ = 5, x₂ = y₂ = 5.
Q: State the First Welfare Theorem and its key assumptions.
A: The First Welfare Theorem states that every competitive market equilibrium is Pareto efficient. It assumes rational agents, positive monotonic preferences, costless trade, and competitive markets.
Q: How does the Second Welfare Theorem differ from the First?
A: The First Theorem says competitive equilibria are efficient. The Second says any Pareto efficient outcome can be achieved as a competitive equilibrium, provided endowments are redistributed appropriately. The Second additionally requires convex preferences.
Q: A proposed allocation gives Consumer 1 the bundle (3, 3) and Consumer 2 the bundle (7, 7). Is this allocation Pareto efficient? Is it in the core?
A: It lies on the contract curve (y₁ = x₁ = 3), so it is Pareto efficient. However, Consumer 1's utility would be 3, which is less than their initial utility of 4. Consumer 1 would refuse this trade. It is not in the core.
This material connects directly to partial equilibrium and supply-and-demand analysis (earlier in ECN 312), since general equilibrium extends those ideas to multiple markets at once. The welfare theorems tie into welfare economics and the discussion of market failures: when the assumptions behind the theorems break down (externalities, public goods, market power, asymmetric information), competitive equilibria may no longer be efficient, which is the starting point for topics on government intervention and market design.
General equilibrium, partial equilibrium, price theory, Edgeworth box, contract curve, Pareto optimal, Pareto efficient, Pareto efficiency, the core, core of an economy, Walras' Law, Walrasian equilibrium, competitive equilibrium, aggregate excess demand, First Welfare Theorem, Second Welfare Theorem, welfare theorems, welfare economics, Cobb-Douglas utility, marginal rate of substitution, MRS, budget line, price ratio, endowment, feasibility constraint, convex preferences, monotonic preferences, gains from trade, ECN 312, intermediate microeconomics, intermediate macroeconomics, Arizona State University