Difficulty: Intermediate
Prerequisites: Edgeworth box fundamentals (see companion study notes), consumer optimisation with Cobb-Douglas and other utility functions, basic algebra for solving systems of equations.
A competitive equilibrium is a set of prices at which every consumer maximises utility and every market clears (total demand equals total supply for each good). In an exchange economy with no production, total supply is just the total endowment.
Finding a competitive equilibrium pulls together everything from consumer theory: you derive each consumer's demand function from their utility maximisation problem, then find the price(s) that make aggregate demand equal aggregate supply. This is one of the central analytical exercises in intermediate microeconomics, and it appears in various forms on nearly every exam covering general equilibrium.
You should already be comfortable computing optimal bundles for standard utility functions before tackling this material. The new skill is solving for the equilibrium price by imposing the market-clearing condition.
To find a competitive equilibrium, derive each consumer's demand as a function of prices, sum the demands, set total demand equal to total endowment for one good, and solve for the price ratio. Then plug the equilibrium prices back in to get each consumer's equilibrium allocation.
Competitive equilibrium
A price vector and an allocation such that (1) each consumer maximises utility given their budget constraint at those prices, and (2) markets clear: for every good, total demand equals total endowment.
In simple terms, it is the set of prices where everyone gets what they want and nothing is left over or missing.
Market clearing
The condition that total demand for a good equals total supply (total endowment in an exchange economy). Written as: Dₐˣ + D_bˣ = Wₐˣ + W_bˣ for good X.
Numeraire
A good whose price is fixed at 1, used as the unit of account. In a two-good economy, you typically set Pʸ = 1 and solve for Pₓ. Only relative prices matter for consumer choices, so normalising one price loses no information.
Think of it as choosing your currency: instead of solving for two prices, you express everything in terms of good Y.
Walras' Law
If there are n markets and n - 1 of them clear, the last one clears automatically. In a two-good economy, you only need to impose market clearing for one good; the other market will clear by construction.
Demand function (Marshallian demand)
The quantity a consumer demands of each good, expressed as a function of prices and income. For Cobb-Douglas U = Xᵃ·Yᵇ: Dˣ = [a/(a+b)]·(M/Pₓ). In an exchange economy, income M itself depends on prices, so demand is ultimately a function of prices and endowments alone.
First Welfare Theorem
Every competitive equilibrium allocation is Pareto efficient (under standard assumptions: complete markets, no externalities, price-taking behaviour). This is the formal link between market outcomes and efficiency.
Set the numeraire. Fix one price (typically Pʸ = 1) and solve for the other (Pₓ).
Compute each consumer's income as a function of Pₓ: Mᵢ = Pₓ·Wᵢˣ + 1·Wᵢʸ.
Derive each consumer's demand for both goods as functions of Pₓ, using the appropriate demand formula for their utility type.
Impose market clearing for one good (say good X): Dₐˣ(Pₓ) + D_bˣ(Pₓ) = total endowment of X.
Solve for Pₓ.
Plug Pₓ back in to compute each consumer's equilibrium demands for both goods.
Verify by checking that the other market (Y) also clears. By Walras' Law, it will.
Consumer A: U = X·Y, endowment (6, 1). Consumer B: U = X·Y³, endowment (2, 7). Total X = 8, Total Y = 8. Set Pʸ = 1.
Deriving A's demand:
U = X·Y is Cobb-Douglas with a = 1, b = 1, so a/(a+b) = 1/2.
Mₐ = 6Pₓ + 1.
Dₐˣ = Mₐ/(2Pₓ) = (6Pₓ + 1)/(2Pₓ).
Dₐʸ = Mₐ/2 = (6Pₓ + 1)/2.
Deriving B's demand:
U = X·Y³ is Cobb-Douglas with a = 1, b = 3, so a/(a+b) = 1/4.
M_b = 2Pₓ + 7.
D_bˣ = M_b/(4Pₓ) = (2Pₓ + 7)/(4Pₓ).
D_bʸ = 3M_b/4 = 3(2Pₓ + 7)/4.
Market clearing for X:
Dₐˣ + D_bˣ = 8
(6Pₓ + 1)/(2Pₓ) + (2Pₓ + 7)/(4Pₓ) = 8
Multiply through by 4Pₓ: 2(6Pₓ + 1) + (2Pₓ + 7) = 32Pₓ
12Pₓ + 2 + 2Pₓ + 7 = 32Pₓ
14Pₓ + 9 = 32Pₓ
9 = 18Pₓ
Pₓ = 0.50
Equilibrium allocations (Pₓ = 0.50, Pʸ = 1):
Mₐ = 6(0.50) + 1 = 4. Dₐˣ = 4/(2 × 0.50) = 4. Dₐʸ = 4/2 = 4. So A gets (4, 4) which is a change from their (6,1) endowment.
M_b = 2(0.50) + 7 = 8. D_bˣ = 8/(4 × 0.50) = 4. D_bʸ = 3(8)/4 = 24/4 = 6. Wait, let us re-check: D_bʸ = 3·M_b/(4·Pʸ) = 3·8/(4·1) = 6. Hmm, but total Y should be 8. Let us verify: Dₐʸ + D_bʸ = 4 + 6 = 10 ≠ 8.
This discrepancy arises because the correct demand formula for B requires care. For U = X·Y³ (a=1, b=3), B's demand for Y is D_bʸ = [b/(a+b)]·(M_b/Pʸ) = (3/4)·(8/1) = 6. And Dₐʸ = (1/2)·(4/1) = 2. Total Y demand = 2 + 6 = 8. That checks.
The error above was in Dₐʸ. Let us redo: Dₐʸ = [b/(a+b)]·(Mₐ/Pʸ) = (1/2)·(4/1) = 2, not 4. For A's utility U = XY (a=1, b=1), the demand for Y is Mₐ/(2Pʸ) = 4/2 = 2.
So the correct equilibrium: A gets (4, 2), B gets (4, 6). Total X = 8, total Y = 8. Markets clear.
When one consumer has linear utility (perfect substitutes):
MRS is constant. The equilibrium price ratio must equal that constant MRS (otherwise that consumer would demand only one good, which may or may not clear the market).
Use the other consumer's demand to pin down the allocation.
When one consumer has perfect complements:
Demand is X = Y = M/(Pₓ + Pʸ). Substitute into market clearing alongside the other consumer's demand and solve.
\text{Income in exchange economy: } M_i = P_X \cdot W_i^X + P_Y \cdot W_i^Y\text{Cobb-Douglas } U = X^a Y^b: \quad D^X = \frac{a}{a+b} \cdot \frac{M}{P_X}, \quad D^Y = \frac{b}{a+b} \cdot \frac{M}{P_Y}\text{Perfect complements } U = \min(X, Y): \quad D^X = D^Y = \frac{M}{P_X + P_Y}\text{Market clearing: } \sum_i D_i^X(P_X) = \sum_i W_i^X\text{Walras' Law: if } ED_X = 0 \text{, then } ED_Y = 0 \text{ automatically}Key shortcut for Cobb-Douglas U = Xᵃ·Yᵇ: the consumer spends exactly the fraction a/(a+b) of their income on good X and b/(a+b) on good Y, regardless of prices. This is the constant expenditure share property.
Competitive equilibrium is the theoretical backbone of how economists think about market prices. Every time you see a supply-and-demand diagram in the news, the "equilibrium price" is exactly this concept applied to a single market. In international trade, two countries trading goods reach (in theory) a competitive equilibrium where the terms of trade (the price ratio) clear both markets. The First Welfare Theorem is the formal basis for the economic argument that free markets tend to produce efficient outcomes, and also the starting point for understanding when and why markets fail (externalities, public goods, market power).
Students often try to solve for both Pₓ and Pʸ separately. In a two-good exchange economy, only the price ratio Pₓ/Pʸ matters. Set one price to 1 (the numeraire) and solve for the other.
A frequent error is forgetting that income depends on prices in an exchange economy. Unlike a standard consumer problem where M is given, here M = Pₓ·Wˣ + Pʸ·Wʸ changes when prices change. Leaving M as a fixed number when solving for equilibrium prices will give wrong answers.
Students sometimes check only one market and declare equilibrium. While Walras' Law guarantees the second market clears if the first does, you still need to verify your arithmetic by checking both markets. A mistake in your algebra might be hiding behind the assumption.
Confusing the demand formula exponents: for U = X·Y³ (a=1, b=3), the share spent on X is 1/4, not 1/3 or 3/4. The share is a/(a+b), where a and b are the exponents on X and Y respectively.
⚠️ "Find the competitive equilibrium price" is a standard 10-15 point exam question. The full method (set numeraire, derive demands, impose market clearing, solve) should be practised until automatic.
⚠️ You will often be asked to find equilibrium quantities for each consumer after finding the price. Do not stop at the price.
⚠️ Problems mixing different utility types (one consumer Cobb-Douglas, the other linear or perfect complements) are common. Know the demand formulas for each type cold.
⚠️ Walras' Law is frequently tested as a concept question: "If the market for X clears, what can you say about the market for Y?"
True or false: In a two-good exchange economy, you need to solve for both Pₓ and Pʸ independently. (False. Only the ratio matters; set one as numeraire.)
Fill in the blank: For Cobb-Douglas utility U = X²·Y, the fraction of income spent on X is ____. (Answer: 2/3)
True or false: Walras' Law says that if the X market has excess demand, the Y market must have excess supply. (True, in a two-good economy.)
Fill in the blank: A consumer with endowment (3, 5) facing prices Pₓ = 2, Pʸ = 1 has income M = ____. (Answer: 11)
True or false: At a competitive equilibrium, the allocation is always Pareto efficient. (True, by the First Welfare Theorem.)
Q: Consumer A has U = X·Y, endowment (6, 1). Consumer B has U = X·Y³, endowment (2, 7). Set Pʸ = 1. Find the competitive equilibrium price Pₓ.
A: Mₐ = 6Pₓ + 1, M_b = 2Pₓ + 7. A's demand for X: (6Pₓ + 1)/(2Pₓ). B's demand for X: (2Pₓ + 7)/(4Pₓ). Market clearing: (6Pₓ + 1)/(2Pₓ) + (2Pₓ + 7)/(4Pₓ) = 8. Multiply by 4Pₓ: 12Pₓ + 2 + 2Pₓ + 7 = 32Pₓ, so 14Pₓ + 9 = 32Pₓ, giving 18Pₓ = 9, Pₓ = 0.50.
Q: Using the equilibrium price from the previous question, find each consumer's equilibrium allocation.
A: Mₐ = 6(0.5) + 1 = 4. Dₐˣ = 4/(2 × 0.5) = 4. Dₐʸ = 4/(2 × 1) = 2. So A gets (4, 2). M_b = 2(0.5) + 7 = 8. D_bˣ = 8/(4 × 0.5) = 4. D_bʸ = 3 × 8/(4 × 1) = 6. So B gets (4, 6). Check: 4 + 4 = 8 (X clears), 2 + 6 = 8 (Y clears).
Q: In a two-good exchange economy, the market for X has excess supply of 5 units at the current price. What can you say about the market for Y?
A: By Walras' Law, the market for Y has excess demand. At the current prices, the value of excess demand for Y equals the value of excess supply for X.
Q: Consumer C has U = X^(1/3)·Y^(2/3), endowment (9, 3), and faces prices Pₓ = 2, Pʸ = 1. What bundle does C demand?
A: M = 2(9) + 1(3) = 21. Dˣ = (1/3)·(21/2) = 3.5. Dʸ = (2/3)·(21/1) = 14.
Competitive equilibrium builds directly on the Edgeworth box analysis from the companion study notes. The equilibrium allocation is one specific point on the contract curve: the one reached through price-mediated trade from the initial endowment.
The First Welfare Theorem connects this to welfare economics and policy analysis: if a competitive equilibrium is always Pareto efficient, then market failures (externalities, public goods, monopoly power) can be understood as violations of the theorem's assumptions. The Second Welfare Theorem, which you may encounter later, says any Pareto-efficient allocation can be supported as a competitive equilibrium with appropriate lump-sum transfers.
Competitive equilibrium, Walrasian equilibrium, general equilibrium, market clearing, excess demand, excess supply, numeraire, price ratio, Walras' Law, First Welfare Theorem, Second Welfare Theorem, Cobb-Douglas demand, expenditure share, constant expenditure share property, exchange economy equilibrium price, ECON 302, intermediate microeconomics, Penn State, McLeod