Rachel spends her income, Y, on Rock Shows (R) and Sunglasses (S) with

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.

ANSWER

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