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See an example of how the answer we provide you can look like below.

Problem set 5.2

Question a)

We start by using the Lagrangian method in order to obtain the expression below:

L =c₁^θ + c₂^θ + λ(1-p₁c₁ – p₂c₂)

In order to find an expression for c₁/c₂ we first need the first-order conditions:

An general expression:

dL/dc_i = c_i^θ-1 – λp_i = 0

Which gives

c_i = (λp_i)^1/(θ-1)

In this case when we have two goods and we can write:

dL/dc₁ = c₁^θ-1 – λp₁ = 0
c₁ = (λp₁)^1/(θ-1)

dL/dc₂ = c₂^θ-1 – λp₂ = 0
c₂ = (λp₂)^1/(θ-1)

Now we can show that:

c₁/c₂ = (p₁/p₂)^1/(θ-1)

Since θ = (σ-1)/σ

We can rewrite the expression above. We get:

c₁/c₂ = (p₁/p₂)^–σ