FOC - Microeconomics

Question

Consumer’s utility function is U=4q^1/3 q^2/3 and the prices for good 1 and good 2 are p1 and p2, respectively. If the consumer uses the Lagrange method to maximize her utility, then the only answer option that is part of the First Order Conditions (FOC) is?

Answer 1

Here we have a constrained optimization problem.

max_q1,q2 {4q₁^1/3q₂^2/3 : p₁q₁+p₂q₂ ≤ x}

The Lagrangian is

L = 4q₁^1/3q₂^2/3 + λ(x - p₁q₁+p₂q₂)

Then the FOCs are

∂U/∂q₁ = 4/3q₁^-2/3q₂^2/3 - λp₁=0

∂U/∂q2 = 8/3q₁^1/3q₂^-1/3 - λp₂=0

∂L/∂λ = x - p₁q₁-p₂q₂=0

Can also be done by calculating the marginal rate of substitution.

Then solve for q1 and q2 by substituting out λ to obtain the tangency condition.

(4/3q₁^-2/3q₂^2/3)/p₁= (8/3q₁^1/3q₂^-1/3)/p₂

q₁ = (2(p₁/p₂)q₂^-1)^-1

Substitute into budget constraint.

x - p₁(2(p₁/p₂)q₂^-1)^-1 - p₂q₂=0

Therefore, the Marshallian Demand is

q₂ = 2/3(x/p₂)

q₁ = 1/3(x/p₁)