General Equilibrium - Illustration (Cobb-Douglas)

By Aleesha Mary Joseph

Let's begin with an easy one to understand the concept of General Equilibrium.

Suppose, U₁(X₁₁ , X₁₂ ) = X₁₁ X₁₂ and U₂(X₂₁ , X₂₂ ) = X₂₁ X₂₂ .

Let (e₁₁ , e₁₂ ) = (10,0) & (e₂₁ , e₂₂ ) = (0,10).

Normalize P₂ to be equal to 1.

Then note that individual 1’s income M₁ = 10*P₁ + 0*1 = 10P₁.

Similarly individual 2’s income M₂ = 0*P₁ +1 0*1 = 10.

Since the utility functions are cobb douglas, demand functions can be derived by equating MRS with P₁/P₂ and substituting the relevant expression in the budget constraint.

The demand functions so derived will be as follows:

X₁₁ = M₁ /2P₁ = 10P₁/2P₁ = 5

X₁₂ = M₁ /2P₂ = 10P₁/2P₂ = 5P₁ (Because P₂ is normalized to 1)

X₂₁ = M₂ /2P₁ = 10/2P₁ = 5/P₁

X₂₂ = M₂ /2P₂ = 10/2P₂ = 5 (Because P₂ is normalized to 1)

The Market clearing condition says

X₁₁ + X₂₁ = e₁₁ + e₂₁

5 + 5/P₁ = 10. 

On solving this we get P₁ = 1. Hence the competitive equilibrium price ratio P₁/P₂ = 1

Convince yourself that the same price ratio will clear the market clearing equation for good 2 as well ie X₁₂+ X₂₂ = e₁₂ + e₂₂ .

Substituting P₁ =1 in the demand functions, we get the competitive equilibrium allocation

As (X₁₁ , X₁₂ ) = (5,5)

     (X₂₁ , X₂₂ ) = (5,5)

With competitive equilibrium price P₁/P₂ as 1.

The articles to follow will cover min/max utility functions and lexicographic preferences. Stay tuned.

(Aleesha Mary Joseph graduated from St. Stephen's College in 2013. She is currently pursuing MA in Economics at Delhi School of Economics)