Sam has preferences for weekly Video Games (V) and Sodas (S) described by the utility function U(V,S) = V2S2. Suppose the prices are denoted by pV and pS and Sam has income given by I

Assume that in Sam's optimal bundle, he consumes strictly positive quantities of both goods. a. Write out Sam's optimization problem and the associated Lagrangian expression. b. Compute the three critical value (first-order) conditions from the Lagrangian. c. Using your answer to b, find the expression for the optimal bundles as functions of the prices and income.

a. The optimization problem is
max V2S2.
s.t. pVV + pSS = I
The corresponding Lagrangian expression is :
L = V2S2 + λ[I-pVV - pSS]
b. The first order conditions are:
∂L/∂V = 2VS2 - λpV = 0
∂L/∂S = 2SV2 - λpS = 0
∂L/∂λ = I - pVV - pSS = 0
c. The first two expressions in b provide the MRS = MRT condition:
S/V = pV/pS
Rearrange this to S = V(pV/pS) and substitute into the third of the conditions
in (b) (AKA the budget constraint):
I - pVV - pVV = 0
Solving, we get:
V = I/(2pV).
Using the MRS = MRT condition above we get:
S = I/(2pS)