Question

Suppose that Ken cares only about bathing suits (B) and flip-flops (F). His utility function is U = B0.8F0.2. The price of bathing suits is $30, and the price of flip-flops is $2. Ken has a budget of $150.
a. What Lagrangian equation can be used to solve Ken's utility maximization problem?
b. Derive the first-order conditions for the maximization problem.

c. What is the solution to Ken's maximization problem?

Answer 1

(a)

U = B0.8F0.2

Budget line: 150 = 30B + 2F

Lagrangian function is:

L = B0.8F0.2 + $1585316237399_blob.png$(150 - 30B - 2F)

(b)

Utility is maximized when $1585316237373_blob.png$L/$1585316237282_blob.png$B = 0, $1585316237364_blob.png$L/$1585316237353_blob.png$F = 0 and $1585316237353_blob.png$​​​​​​​L/$1585316237237_blob.png$$1585316237381_blob.png$ = 0 (FOC).

$1585316237236_blob.png$L/$1585316237234_blob.png$B = [0.8 x (F/B)0.8] - 30$1585316237377_blob.png$ = 0, so

0.8 x (F/B)0.8 = 30$1585316237377_blob.png$ ........(1)

$1585316237367_blob.png$L/$1585316237372_blob.png$F = [0.2 x (B/F)0.2] - 2$1585316237377_blob.png$ = 0, so

0.2 x (B/F)0.2 = 2$1585316237377_blob.png$............(2)

$1585316237376_blob.png$​​​​​​​L/$1585316237754_blob.png$$1585316237370_blob.png$ = 150 - 30B - 2F = 0, so

150 = 30B + 2F.........(3)

(c)

Dividing (1) by (2),

4 x (F/B) = 15

F = B x (15/4)

Substituting in budget line,

150 = 30B + 2 x B x (15/4)

150 = 30B + (15B/2)

300 = 60B + 15B

75B = 300

B = 4

F = 4 x (15/4) = 15