Rachel spends her income, Y, on Rock Shows (R) and Sunglasses (S) with prices pR and pS. Rachel's preferences are given by the Cobb-Douglas utility function

U(X,Y) = R.8S.2
a. Write out the Lagrangian for Rachel's utility-maximization problem.
b. Use the Lagrangian to derive Rachel's optimal choice, (R*,S*).
c. For a given utility level, U0, derive Rachel's Expenditure function E(pR,pS,U0).
d. Use the Expenditure function to derive Rachel's compensated demand for Rock Shows.

a. The Lagrangian is
L = R.8S.2 + [Y – pRR – pSS]
b. The necessary conditions for utility maximization
LR = .8R-.2S.2 – pR = 0
LS = .2R.8S-.8 – pS = 0
L = Y – pRR – pSS = 0
The first two conditions above yield the MRS = MRT condition:
4S/R = pR/pS
Solve to get S = R(pR/4pS)
Plug into the 3rd condition above (the budget constraint) to get
Y = pRR + pRR/4 = 5pRR/4
Solve for the demand equations:
R* = 4Y/5pR
S* = Y/5pS
c. Using the Lagrangian for the corresponding minimization problem:
L = pRR + pSS + [U0 – R.8S.2]
The necessary conditions are
LR = pR – .8 R-.2S.2 = 0
LS = pS – .2R.8S-.8 = 0
L = U0 – R.8S.2 = 0
The first two conditions yield:
4SpS = RpR
which is the same as in part b. Substituting into the expenditure expression we get
E = pRR + pSS = pRR + pRR/4 = 5pRR/4
Rearranging terms we get
R = .8E/pR
And similarly
S = .2E/pS
The indifference curve expression yields
U0 = (.8E/pR).8(.2E/pS).2
Rearranging, we get
E = U0(pR/.8).8(pS/.2).2

d. Taking the derivative of the expenditure function with respect to pR:
R = ∂E/∂pR = (.8pS/.2pR).2U0