You allocate $24 per week for the purchase of cookies and apples at your school\

Question

You allocate $24 per week for the purchase of cookies and apples at your school's cafeteria.

Your utility from eating cookies and apples is given by: U(C,A) = 2C^(1/2) + A^(1/2)

Assume that cookies cost $1 each and apples cost $0.50.

(a) Set up the Lagrangian function associated with this problem.

Solve for the optimal proportion of cookies to apples.

Given your budget constraint, how many cookies and apples will you buy each week?

(b) Solve additionally for the Lagrangian multiplier.

What is your interpretation of this parameter?

What happens to this parameter if you allocate twice as much per week to the purchase of cookies and applies?

How do you interpret this change?

Explanation / Answer

L= 2c^1/2a1/2+ k(24-c-0.5a)
solving for the first order conditions, we get, c=12, a=24, k=2^1/2

lagrange multiplier is also called marginal utility of money. it is also called shadow price.
when budget is doubled, k=2, so lagrange multiplier also doubles. marginal utility of money doubles

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