Consumer Theory - Optimal Choice II

Question 1

How do you calculate and find optimal choice using the Lagrangian method?

Answer 1

STEP 1: Reorder the equation into a Lagrangian function (Utility function - λ (Budget constraint in brackets))

STEP 2: Find the 1st order conditions of each of lagrangian function

i) ∂L/ ∂X
ii) ∂L/ ∂Y
iii) ∂L/ ∂λ (Just the budget constraint with a minus)

STEP 3: Solve as a simultaneous equation to find optimal ratio
(i/ii)

STEP 4: Substitute the optimal ratio back into the budget constraint to figure out X* and Y*

(SEE NOTES FOR EXAMPLES)

Question 2

How does the Lagrangian Method link with the 1st method covered in the previous flashcards?

Answer 2

If we look at L(X,Y,λ) = U(X,Y) - λ(PxX + PyY - M)

i) i)∂L/ ∂X
ii) ∂L/ ∂Y
iii) ∂L/ ∂λ

Then i) and ii) implies
(∂U/ ∂X) / (∂U/ ∂Y) = Px/Py

This is the MRT=MRS condition

Question 3

How can we know that we are looking at utility maximisation instead of utility minimisation?

Answer 3

The budget constraint is linear
The indif curve is convex to the origin

If the indif curve is concave to the origin we are looking at utility minisation

Question 4

READING CHAPTERS

Answer 4

3.3 AND 3.4