Sean C. answered • 06/08/21

Sean C. -- Economics PhD student at CSU

Here we have a constrained optimization problem.

max_{q1,q2} {4q_{1}^{1/3}q_{2}^{2/3} : p_{1}q_{1}+p_{2}q_{2 }≤ x}

The Lagrangian is

L = 4q_{1}^{1/3}q_{2}^{2/3} + λ(x - p_{1}q_{1}+p_{2}q_{2})

Then the FOCs are

∂U/∂q_{1} = 4/3q_{1}^{-2/3}q_{2}^{2/3 }- λp_{1}=0

∂U/∂q2 = 8/3q_{1}^{1/3}q_{2}^{-1/3 }- λp_{2}=0

∂L/∂λ = x - p_{1}q_{1}-p_{2}q_{2}=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.

(4/3q_{1}^{-2/3}q_{2}^{2/3})/p_{1}= (8/3q_{1}^{1/3}q_{2}^{-1/3})/p_{2}

q_{1} = (2(p_{1}/p_{2})q_{2}^{-1})^{-1}

Substitute into budget constraint.

x - p_{1}(2(p_{1}/p_{2})q_{2}^{-1})^{-1 }- p_{2}q_{2}=0

Therefore, the Marshallian Demand is

q_{2} = 2/3(x/p_{2})

q_{1} = 1/3(x/p_{1})