Answer to Question #232234 in Microeconomics for emma

Question

Mr Banda faces the following utility function from consuming nshima [X] and Rice [Y] 𝑈 = 3𝑋𝑌 1 3 Mr Banda’s budget allocation for nshima and rice is K240. The price of nshima is K60 per Kg while that of Rice is K20 per Kg. A. How much nshima and rice should Mr. Banda consume to maximize his utility? B. What is the total utility at the optimum? [4 Marks] C. What is the Lagrange multiplier? D. What will be the increase in utility when the budget allocation is increased to K241?

Answer 1

A.

Objective is maximize the utility function subject to budget constraint.

Set up the lagrange multiplier

$L = 3XY^{\frac{1}{3}} - λ(60X + 20Y - 240)$

First order condition:

dL / dX = 3Y^1/3 - 60λ = 0

$=> λ = (\frac{3Y^{\frac{1}{3}}} { 60}) -------------->eq(1)$

and

$\frac{dL }{ dY} = 3(\frac{1}{3}) X Y^{\frac{-2}{3}} - 20λ = 0$

$=> (\frac{X }{ Y^{\frac{2}{3}}}) - 20λ = 0$

$=> λ = (\frac{X }{ 20Y^{\frac{2}{3}}}) ------------> eq(2)$

and

$\frac{dL}{ dλ} = -(60X + 20Y- 240) =0$

=> 60X + 20Y = 240  -------------> Eq(3)

Set eq(1) = eq(2)

$(\frac{3Y^\frac{1}{3} }{ 60}) = (\frac{X}{ 20Y^{\frac{2}{3}}})$

$=> (\frac{Y\frac{1}{3} }{ 20}) = (\frac{X}{20Y^{\frac{2}{3}}})$

$=> Y^\frac{1}{3}\times Y^\frac{2}{3} =\frac{ 20X }{ 20}$

=> Y = X    -------------> Eq(4)

Substitute eq(4) in eq(3)

=> 60X + 20Y = 240

=> 60X + 20X =240

=> 80X = 240

$=> X = (\frac{240 }{ 80})$

=> X= 3

and

Y = X

=> Y = 3

Hence, the Banda should consume 3 kg of nshima and 3 kg of rice to maximize the utility.

B.

U = 3XY^1/3

Put X=3 and Y=3

=> U = 2(3) (3)^1/3

=> U = 8.65

The total utility at the optimum is 8.65

C.

λ is the lagrange multiplier

$λ = (\frac{3Y^\frac{1}{3} }{ 60}) ----> eq(1)$

Put Y = 3

$=> λ = \frac{(3) (3)^\frac{1}{3} }{ 60}$

=> λ = 0.072

The Lagrange multiplier value is 0.072

 

D.

Lagrange multiplier measure the change in optimal value of objective function due to change in constraint by an unit. 

The initial level of income was 240

The Lagrange multiplier value is 0.072 (i.e., λ = 0.072)

If the income level increase by 1 unit i.e., from 240 to 241, then the maximum utility value will increase by 0.072