Question

Jane's utility function is represented as: U equals F to the power of 0.5 end exponent C to the power of 0.5 end exponent, F is quantity of food and C is quantity of clothing. If her budget constraint is represented as: 120 = 2F + C, her optimal bundle of consumption should be

Answer 1

Answer:

Since F = 30 and C = 60, the optimal bundle of consumption is to consume 30 units of food and 60 units of clothing.

Explanation:

Maximize: $U = F^{0.5} C^{0.5}$

Subject to: 120 = 2F + C

Setting up the Langrangian function, we have:

$U = F^{0.5} C^{0.5}$ + ∧(120 - 2F - C) .................... (1)

Where ∧ is used to represent lamda.

Obtain partial differentials of equation (1) with respect to F, C and ∧ as follows:

$dU/dF = 0.5F^{-0.5} C^{0.5}$ - 2∧ = 0 .................. (2)

$dU/dC = 0.5F^{0.5} C^{-0.5}$ - ∧ = 0.......................(3)

du/d∧= 120 - 2F - C = 0 ................................(4)

Divide equation (2) by (3) to eliminate ∧ and rearrange, we have:

C/F = 2

C = 2F ............................................................... (5)

Substitute C = 2F into equation (4), we have:

120 - 2F - 2F = 0

120 = 4F

F = 120/4

F = 30

Substitute F = 30 into equation (5), we have:

C = 2 * 30

C = 60

Since F = 30 and C = 60, the optimal bundle of consumption is to consume 30 units of food and 60 units of clothing. At this bundle, the consumer will maximize his utility.