Difficulty: Intermediate-Advanced | Prerequisites: Consumer optimisation, Edgeworth Box basics, Cobb-Douglas demand functions
This topic builds on the Edgeworth Box framework. Instead of checking whether a given price clears the market, you are now solving for the price that does. This is the core of general equilibrium analysis in a pure exchange economy: two consumers, two goods, no production, and the question is what prices make everyone's optimal choices compatible with the total supply. You should already know how to derive demand functions for Cobb-Douglas and Leontief utility.
A competitive equilibrium is the set of prices at which every consumer optimises and both markets clear simultaneously. You find it by normalising one price (the numeraire), expressing each consumer's demand as a function of the remaining price, and solving the market-clearing equation. Once you have the equilibrium price, plug it 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) total demand equals total supply for every good. In a two-consumer, two-good exchange economy, this means both markets clear at the same time.
In simple terms, this means: prices at which everyone gets what they want and nothing is left over or missing.
Numeraire
The good whose price is normalised to 1. In general equilibrium, only relative prices matter, so you can set P_Y = 1 and solve for P_X. This simplifies the algebra without losing any information.
Think of it as: measuring all prices in terms of one good, like using good Y as the "currency."
Walras' Law
The value of total excess demand across all markets is always zero. In a two-good economy, if the X market has excess demand of value V, the Y market must have excess supply of value V. Consequence: you only need to clear one market. The other clears automatically.
Market clearing condition
The equation that says total demand for a good equals total supply: D_A^X(P_X) + D_B^X(P_X) = ω_A^X + ω_B^X. Solving this equation for P_X (with P_Y = 1) gives the equilibrium price.
Price ratio
P_X / P_Y, the relative price of good X in terms of good Y. This is the slope of the budget constraint (in absolute value). In equilibrium, the price ratio equals the MRS for all consumers with interior solutions.
Choose one good as the numeraire. Convention in most problems (and in this course): set P_Y = 1.
You are now solving for the single unknown P_X.
Consumer A's income: M_A = P_X * ω_A^X + 1 * ω_A^Y
Consumer B's income: M_B = P_X * ω_B^X + 1 * ω_B^Y
These are functions of P_X, not fixed numbers.
For Cobb-Douglas utility U = X^a * Y^b:
Demand for X: X* = (a/(a+b)) * M / P_X
Demand for Y: Y* = (b/(a+b)) * M / P_Y
For perfect complements U = min(aX, bY):
Set aX = bY, substitute into budget constraint, and solve.
Express all demands in terms of P_X (since P_Y = 1 and M depends on P_X).
Pick one market (say X): D_A^X(P_X) + D_B^X(P_X) = ω_A^X + ω_B^X
This gives one equation in one unknown (P_X). Solve it.
Example from the homework: U_A = X^(4/5) * Y^(1/5), endowment (9, 4). U_B = X^(2/3) * Y^(1/3), endowment (3, 12). P_Y = 1.
M_A = 9P_X + 4. A's demand for X: (4/5)(9P_X + 4)/(5P_X) = (4/5) * M_A / P_X. Wait, let us be precise: a = 4/5, b = 1/5, so share on X = (4/5)/1 = 4/5. So D_A^X = (4/5) * (9P_X + 4) / P_X.
M_B = 3P_X + 12. B's demand for X: (2/3) * (3P_X + 12) / P_X.
Market clearing: (4/5)(9P_X + 4)/P_X + (2/3)(3P_X + 12)/P_X = 12 (total X endowment = 9 + 3).
Multiply through by P_X: (4/5)(9P_X + 4) + (2/3)(3P_X + 12) = 12P_X.
Expand: (36P_X/5 + 16/5) + (2P_X + 8) = 12P_X.
Multiply everything by 5: 36P_X + 16 + 10P_X + 40 = 60P_X.
46P_X + 56 = 60P_X, so 14P_X = 56, giving P_X = 4.
Plug P_X = 4 back into each consumer's demand functions.
A: M_A = 9(4) + 4 = 40. X_A = (4/5)(40)/4 = 8. Y_A = (1/5)(40)/1 = 8. Bundle: (8, 8).
B: M_B = 3(4) + 12 = 24. X_B = (2/3)(24)/4 = 4. Y_B = (1/3)(24)/1 = 8. Bundle: (4, 8).
Check: X_A + X_B = 12 = total X. Y_A + Y_B = 16 = total Y. Both markets clear.
Box dimensions: 12 (total X) by 16 (total Y).
Plot the initial endowment W at (9, 4) from A's origin.
Plot the competitive equilibrium D at (8, 8) from A's origin.
Draw the budget constraint through W with slope = -P_X/P_Y = -4.
Both consumers' indifference curves should be tangent to the budget line at point D.
Income from endowments (with numeraire P_Y = 1):
M_i = P_X \cdot \omega_i^X + \omega_i^YCobb-Douglas demand (general form for U = X^a * Y^b):
X_i^* = \frac{a}{a+b} \cdot \frac{M_i}{P_X}Y_i^* = \frac{b}{a+b} \cdot \frac{M_i}{P_Y}Market clearing condition (good X):
\sum_i X_i^*(P_X) = \sum_i \omega_i^XThis is one equation in one unknown (P_X) once P_Y = 1 is set.
Verification by Walras' Law:
If the X market clears, the Y market must also clear. You can verify:
\sum_i Y_i^*(P_X^*) = \sum_i \omega_i^YStudents often treat income as a fixed number when solving for equilibrium. Income is a function of P_X (because the consumer is selling their endowment at market prices). If you plug in a number for M before solving, you will get the wrong answer unless that number already accounts for the unknown P_X.
Students sometimes try to clear both markets simultaneously by writing two equations. You only need one market-clearing equation, because Walras' Law guarantees the other. Two equations in one unknown will either be redundant (correct) or contradictory (you made an error).
A common algebra mistake: forgetting to multiply through by P_X when the demand functions have P_X in the denominator. This leaves fractions that compound errors downstream.
Students occasionally confuse the equilibrium allocation with the endowment. The endowment is where consumers start. The equilibrium allocation is where they end up after trading at equilibrium prices. These are different points in the Edgeworth Box.
⚠️ Finding the competitive equilibrium price is one of the most heavily tested skills in this course. Expect a problem that gives you two utility functions and two endowments and asks you to solve for P_X.
⚠️ Always verify your answer by checking that both markets clear. If X clears but Y does not (or vice versa), you have an arithmetic error.
⚠️ Be prepared to draw the Edgeworth Box showing both the endowment and the equilibrium. The budget line must pass through both W and D, and both consumers' indifference curves must be tangent to the budget line at D.
⚠️ The exam may give you a non-equilibrium price first and ask you to show it does not clear, then ask you to find the price that does. This tests both skills in sequence.
True or False: In a two-good exchange economy, you need to solve two market-clearing equations to find the equilibrium price. (False. By Walras' Law, clearing one market is sufficient.)
Fill in the blank: Setting P_Y = 1 is called choosing a ______. (numeraire)
True or False: At competitive equilibrium, the budget constraint line in the Edgeworth Box must pass through the initial endowment point. (True. The endowment is always affordable at any prices.)
Fill in the blank: For U = X^(2/3) * Y^(1/3), the share of income spent on X is ______. (2/3)
True or False: If you double both P_X and P_Y, the equilibrium allocation changes. (False. Only relative prices matter.)
Q: Consumer A has U_A = X^(3/4) * Y^(1/4), endowment (6, 2). Consumer B has U_B = X^(1/2) * Y^(1/2), endowment (2, 6). P_Y = 1. Find the competitive equilibrium price P_X.
A: M_A = 6P_X + 2. D_A^X = (3/4)(6P_X + 2)/P_X. M_B = 2P_X + 6. D_B^X = (1/2)(2P_X + 6)/P_X. Market clearing: (3/4)(6P_X + 2)/P_X + (1/2)(2P_X + 6)/P_X = 8. Multiply by P_X: (3/4)(6P_X + 2) + (1/2)(2P_X + 6) = 8P_X. Expand: (9P_X/2 + 3/2) + (P_X + 3) = 8P_X. That gives 9P_X/2 + P_X + 3/2 + 3 = 8P_X, so 11P_X/2 + 9/2 = 8P_X, so 9/2 = 8P_X - 11P_X/2 = 5P_X/2, giving P_X = 9/5 = 1.8.
Q: Using the equilibrium price from above, what is each consumer's equilibrium bundle?
A: M_A = 6(1.8) + 2 = 12.8. X_A = (3/4)(12.8)/1.8 = 9.6/1.8 = 16/3. Y_A = (1/4)(12.8)/1 = 3.2. M_B = 2(1.8) + 6 = 9.6. X_B = (1/2)(9.6)/1.8 = 4.8/1.8 = 8/3. Y_B = (1/2)(9.6)/1 = 4.8. Check: 16/3 + 8/3 = 24/3 = 8 = total X. 3.2 + 4.8 = 8 = total Y. Both clear.
Q: Suppose you are given a price P_X = 1 for the problem above. Show that this is not an equilibrium.
A: At P_X = 1: M_A = 6 + 2 = 8. X_A = (3/4)(8)/1 = 6. M_B = 2 + 6 = 8. X_B = (1/2)(8)/1 = 4. Total demand for X = 10, but total supply = 8. Excess demand of 2 units. The market does not clear, so P_X = 1 is not an equilibrium price.
This connects to the First Welfare Theorem: every competitive equilibrium allocation is Pareto efficient, meaning it lies on the contract curve inside the Edgeworth Box. It also connects to the Second Welfare Theorem, which says any Pareto-efficient allocation can be achieved as a competitive equilibrium with appropriate redistribution of endowments. Understanding how prices adjust to clear markets here lays the groundwork for general equilibrium with production later in the course.
Competitive equilibrium, Walrasian equilibrium, general equilibrium, exchange economy, numeraire, market clearing, Walras' Law, price ratio, Edgeworth Box, contract curve, Pareto efficiency, First Welfare Theorem, Second Welfare Theorem, Cobb-Douglas demand, endowment economy, ECON 302, intermediate microeconomics, Penn State, McLeod