Question

4. Suppose that Jana cares only about apples and lettuce. Her utility function is U = A0.5L0.5, where A is the number of apples and L is the number of heads of lettuce that she consumes. The price of apples is $1, and the price of lettuce is $4. Suppose that Jana must have 100 units of utility and wants to achieve this level of utility with the lowest possible expenditure.

a. How can Jana's expenditure minimization problem be expressed as a Lagrangian equation? b. Derive the first-order conditions for Jana's minimization problem.
c. What is the solution to Jana's minimization problem?
d. How much does this optimal solution cost?

Answer 1

(a)

U = 100 = A0.5L0.5

Budget line: M = A + 4L

Lagrangian function is:

V = A + 4L + (100 - A0.5L0.5)

(b)

Utility is maximized when V/A = 0, V/L = 0 and ​​​​​​​V/ = 0 (First Order Conditions: FOC).

V/A = 1 - [0.5 x (L/A)0.5] = 0

0.5 x (L/A)0.5 = 1

(L/A)0.5 = 2/ ........(1)

V/L = 4 - [0.5 x (A/L)0.5] = 0

0.5 x (A/L)0.5 = 4

(A/L)0.5 = 8/ ........(2)

​​​​​​​V/ = 100 - A0.5L0.5 = 0, so

A0.5L0.5 = 100​​​​​​​.........(3)

(c)

Dividing (1) by (2),

L/A = 1/4

A = 4L

Substituting in utility function,

L0.5(4L)0.5 = 100

40.5 x L0.5 x L0.5 = 100

2 x L = 100

L = 50

A = 4 x 50 = 200

(d)

Total cost = L x PL + A x PA = 50 x $4 + 200 x $1 = $200 + $200 = ​​​​​​​$400