Maximization Utility Problems

Question 1

for all (p,w) element of PW, the budget set B(p,w) is

Answer 1

compact and convex

Question 2

Step 1: Set up the utility maximisation problem and write down the
Lagrangian.

Answer 2

max
x1,x2

subject to p1x1 + p2x2 = w,

L(x1, x2, λ, p1, p2, w) = U(x) + λ(w − p1x1 − p2x2)

Question 3

Step 2:

Answer 3

Write down the first order necessary conditions for an interior maximum.

Question 4

Step 4)

Answer 4

Substitute the Marshallian demands back into the utility func-
tion to obtain the indirect utility function

Question 5

Step (3)

Answer 5

Solve the first order conditions to obtain the Marshallian (or
uncompensated) demand functions.
Answer: Dividing equation (1) by equation (2) (after taking negative to other side.

Question 6

Step 5

Answer 6

State Roy’s Theorem and verify that Roy’s Theorem does in-
deed give the same Marshallian demands that you found above

Question 7

Roy’s theorem

Answer 7

Roy’s Theorem states that if u is very well behaved
(or even as well behaved as u is in this exercise) and v is the
indirect utility function defined for u then the Marshallian de-
mand functions may be found as

Question 8

Treat prices and incomes as

Answer 8

fixed

Question 9

V(p, w ) is the

Answer 9

maximized value of the objective function in the
utility maximization problem