Difficulty: Intermediate | Prerequisites: Cobb-Douglas optimisation, budget constraints, Edgeworth box basics (see companion notes)
The companion notes covered how to find each consumer's optimal bundle at given prices and how to check whether those prices clear the market. This set of notes tackles the next question: what price actually produces equilibrium? In a two-good exchange economy with Cobb-Douglas consumers, you can solve for the equilibrium price ratio algebraically by setting total demand equal to total supply in one market (Walras' Law guarantees the other clears automatically).
This is the core computational skill for the general equilibrium portion of Econ 302 and appears in almost every exam.
Normalise one price to 1 (the numeraire). Write each consumer's demand as a function of the unknown price. Set total demand equal to total supply in one market and solve. Once you have the equilibrium price, plug it back in to get each consumer's equilibrium demands.
Competitive equilibrium
A price vector and an allocation such that every consumer maximises utility subject to their budget constraint, and every market clears (total demand = total supply for each good). In a two-good, two-consumer exchange economy this means finding one price ratio.
In simple terms, it is the price 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 always fix one price and solve for the other. The standard convention is to set Py = 1 and solve for Px.
Think of it as: picking your unit of account. You are measuring "how many units of Y does one unit of X cost?"
Market clearing condition
The equation that says total demand for a good equals total supply. For good X: X_A*(Px) + X_B*(Px) = total endowment of X. You solve this equation for Px.
Demand function (in an exchange economy)
Each consumer's optimal quantity of a good, expressed as a function of prices and their endowment-based income. For Cobb-Douglas consumers, these are simple closed-form expressions.
From the homework: two consumers, two goods (X and Y).
Consumer A: endowment (12, 3), utility U_A = X^(2/3) · Y^(1/3).
Consumer B: endowment (12, 5), utility U_B = X^(1/4) · Y^(3/4).
Normalise Py = 1. Solve for Px.
Total endowment of X = 12 + 12 = 24. Total endowment of Y = 3 + 5 = 8.
I_A = Px(12) + 1(3) = 12Px + 3.
I_B = Px(12) + 1(5) = 12Px + 5.
Using the Cobb-Douglas spending-share rule:
Consumer A (exponents 2/3 and 1/3):
X_A = (2/3) · I_A / Px = (2/3)(12Px + 3) / Px = 8 + 2/Px.
Y_A = (1/3) · I_A / 1 = (1/3)(12Px + 3) = 4Px + 1.
Consumer B (exponents 1/4 and 3/4):
X_B = (1/4) · I_B / Px = (1/4)(12Px + 5) / Px = 3 + 5/(4Px).
Y_B = (3/4) · I_B / 1 = (3/4)(12Px + 5) = 9Px + 15/4.
Pick the X market (Walras' Law means the Y market will clear automatically if X does).
X_A + X_B = 24
(8 + 2/Px) + (3 + 5/(4Px)) = 24
11 + 2/Px + 5/(4Px) = 24
2/Px + 5/(4Px) = 13
Combine the fractions: 8/(4Px) + 5/(4Px) = 13/(4Px) = 13.
13/(4Px) = 13
4Px = 1
Px = 1/4.
Plug Px = 1/4 back into the demand functions.
Consumer A:
X_A = 8 + 2/(1/4) = 8 + 8 = 16.
Y_A = 4(1/4) + 1 = 1 + 1 = 2.
Consumer B:
X_B = 3 + 5/(4 · 1/4) = 3 + 5 = 8.
Y_B = 9(1/4) + 15/4 = 9/4 + 15/4 = 24/4 = 6.
X market: 16 + 8 = 24. Total endowment = 24. Clears.
Y market: 2 + 6 = 8. Total endowment = 8. Clears.
Both markets clear, confirming Px = 1/4 is the competitive equilibrium price.
Box dimensions: 24 (width, total X) by 8 (height, total Y).
Endowment W from A's origin: (12, 3). From B's origin: (12, 5).
Budget line passes through W with slope = -Px/Py = -1/4.
Competitive equilibrium D from A's origin: (16, 2). From B's origin: (24 - 16, 8 - 2) = (8, 6).
Both A and B's indifference curves are tangent to the budget line at D, confirming it lies on the contract curve.
This method of finding equilibrium prices by normalising a numeraire and clearing markets is the backbone of computable general equilibrium (CGE) models used in trade policy and tax analysis. The algebra here is the two-person version of what governments use to forecast the effects of tariff changes.
\text{Numeraire convention: set } P_Y = 1, \text{ solve for } P_X\text{Cobb-Douglas demand for } X: \quad X_i^* = \frac{\alpha_i}{\alpha_i + \beta_i} \cdot \frac{I_i}{P_X}\text{Income from endowment: } I_i = P_X \cdot X_i^w + P_Y \cdot Y_i^w\text{Market clearing: } \sum_i X_i^*(P_X) = \sum_i X_i^wStudents sometimes try to clear both markets simultaneously with two unknowns. You only need one equation, because Walras' Law guarantees the other market clears. The second unknown was removed when you set the numeraire.
A frequent error is writing demand as a function of income alone, forgetting that income itself depends on Px through the endowment. The income term 12Px + 3 changes as Px changes.
Some students try to set MRS_A = MRS_B to find the equilibrium. This gives you the contract curve, not the competitive equilibrium. The equilibrium also requires that both consumers can afford their bundles at the equilibrium price.
Students often forget to verify their answer. Always check that total demands sum to total endowments in both markets.
⚠️ Finding the competitive equilibrium price is the single most likely exam question from this unit. Practise the full procedure until it is automatic.
⚠️ The Edgeworth box for the equilibrium must show the budget line through W with the correct slope, the equilibrium point D, and proper labelling of both origins.
⚠️ You may be asked to verify that the equilibrium is Pareto efficient. At the competitive equilibrium, MRS_A = MRS_B = Px/Py, so it lies on the contract curve.
⚠️ Watch your fractions. The most common arithmetic mistakes come from mishandling terms like 5/(4Px) when combining with 2/Px.
True or False: In a two-good exchange economy, you need to solve two market clearing equations to find the equilibrium.
False. One equation suffices; Walras' Law handles the other.
Fill in the blank: If Py = 1 and the equilibrium price of X is 1/4, then one unit of X costs ____ units of Y.
1/4.
True or False: At a competitive equilibrium, the MRS of both consumers equals the price ratio.
True.
Fill in the blank: For U = X^(1/4) · Y^(3/4), the consumer spends ____% of income on good X.
25%.
True or False: Choosing a different numeraire changes the equilibrium allocation.
False. It changes the absolute prices but not the price ratio or the quantities traded.
Q: Consumer A has U = X^(2/3) · Y^(1/3), endowment (12, 3). Consumer B has U = X^(1/4) · Y^(3/4), endowment (12, 5). Py = 1. Find Px at the competitive equilibrium.
A: Demands for X: X_A = 8 + 2/Px, X_B = 3 + 5/(4Px). Market clearing: 11 + 13/(4Px) = 24. Solving: 13/(4Px) = 13, so Px = 1/4.
Q: Using the equilibrium price from above, what is Consumer A's equilibrium bundle?
A: I_A = 12(1/4) + 3 = 6. X_A = (2/3)(6)/(1/4) = 16. Y_A = (1/3)(6)/1 = 2. Bundle: (16, 2).
Q: What is Consumer B's equilibrium bundle?
A: I_B = 12(1/4) + 5 = 8. X_B = (1/4)(8)/(1/4) = 8. Y_B = (3/4)(8)/1 = 6. Bundle: (8, 6).
Q: Verify that the Y market clears at these demands.
A: Y_A + Y_B = 2 + 6 = 8. Total endowment of Y = 3 + 5 = 8. The market clears.
Q: What is the slope of the budget line in the Edgeworth box at equilibrium?
A: Slope = -Px/Py = -(1/4)/1 = -1/4.
Q: Suppose you had set Px = 1 as the numeraire instead. What would the equilibrium Py be?
A: Since the equilibrium price ratio is Px/Py = 1/4, fixing Px = 1 gives Py = 4. The allocation is identical.
This material builds directly on the Edgeworth box and optimal-bundle techniques in the companion notes. Once you can find the competitive equilibrium, the next topic is the First and Second Welfare Theorems, which tell you that competitive equilibria are Pareto efficient and that any Pareto efficient allocation can be supported as a competitive equilibrium with appropriate transfers.
The market clearing technique here also connects to supply-and-demand analysis in partial equilibrium. The difference is that in general equilibrium, income is endogenous (it depends on prices through the endowment), which creates the feedback loop that makes the algebra slightly more involved.
Competitive equilibrium, Walrasian equilibrium, market clearing, numeraire, exchange economy, Cobb-Douglas demand, spending share rule, Edgeworth box, contract curve, Pareto efficiency, First Welfare Theorem, general equilibrium, price ratio, budget constraint, endowment income, intermediate microeconomics, ECON 302, Penn State