Answer to Question #220014 in Microeconomics for isaac mensah

Question

1)     A consumer has a utility function given by

ln U = 5 ln x₁ + 3 ln x₂

if the budget constraint is given by

10x₁ + 14x₂ = 124, find

i)                  the optimal quantities of the two goods that the consumer should purchase in order to maximise utility, subject to the budget constraint.

ii)                the value of the consumer’s marginal utility of money at the optimum

iii)              the marginal rate of substitution (MRS) of x₁ for x₂ and determine its direction at the optimum

 

Answer 1

a. Solving ln U = 5 ln x₁ + 3 ln x₂

U=X₁⁵ + X₂³

Maximizing U=X₁⁵ + X₂³

Subject to 10X₁+14X₂=124

Introduce Lagrangian function

L= X₁⁵ + X₂³ + ƛ(124-10X₁-14X₂)

Differentiating L with respect to X_1, X₂ and ƛ

Differentiating L w.r.t. X₁

_1. = 5X₁⁴-10ƛ =0

Differentiating L w.r.t. X₂

2. =3X₂² - 14 ƛ = 0

Differentiating L w.r.t. ƛ

3. =124-10X₁-14X₂ =0

Rearranging 1 and 2

U_x/U_y = P_x/P_y

5X₁⁴ / 3X₂² = 10/14

X₁⁴/X₂² = 3/7

X₁ = 3

X₂=7

b. Consumer's marginal utility

MU_x= differentiating utility function w.r.t. to X₁

MU_x= 5X₁⁴

MU_y= differentiating utility function w.r.t. to X₂

MU_y= 3X₂²

c. MRS _x,y = MU_x / MU_y

MRS _x,y = 5X₁⁴ / 3X₂²

This is decreasing as x1 increases and x2 decreases, implying Decreasing MRS. Thus this is well-behaved.