Question

Problem 1 (10 marks) Answer the following questions regarding a Cobb-Douglas utility function U(X,Y)= X0.3 0.7 (a) Does this utility function exhibit diminishing marginal utility in X? Show why or why not. (b) Does this utility function exhibit diminishing marginal rate of substitution? Mathematically show and verbally explain why it has (or doesn't have) such property. Problem 2 (10 marks) Consider the following utility function U(X,Y)= X14734 Suppose that prices and income are given as following Px= 1 Py = 2 I=300 What is the optimal bundle (X*, Y*)?

Answer 1

Answer 2) • U(X,Y) = (X)^(1/4) .(Y)^(3/4)

•Budget line equation: I= Px.X+ Py.Y

• Budget line equation: 300= X+2Y

• At optimum MUx/Px= MUy/Py

• MUx=∆U/∆X = (1/4) (X)^ (-3/4).(Y)^(3/4)

• MUy= ∆U/∆Y=(3/4). (X)^(1/4).(Y)^(-1/4)

so, {(1/4)(X)^(-3/4).(Y)^(3/4)}/1= { (3/4)(X)^(1/4).(Y)^(-1/4)}/2

(Y/4) = (3/8).X.

Y= (3/2)X

Putting Y=(3/2)X in budget line equation

300= X+(2)(3/2)X

X*=75

Y*= (3/2)(75) = 112.5

Answer 1) U(X,Y) = X^0.3 .Y^0.7

a) MUx= ∆U/∆X

MUx=[ 0.3.(X)^ (-0.7) .(Y)^ 0.7 ]   

So, MUx>0 , which shows utility is increasing as ww are consuming more

*∆MUx/∆x = Finding slope of MUX

∆MUx/∆x= (0.3)(-0.7) (X)^(-1.7) (Y)^(0.7)

∆MUx/ ∆x= (-0.21)(X)^(-1.7)(Y)^(0.7)

(∆MUx/∆x ) < 0

so, slope of MUx is negative which shows that Utility increases at Diminishing rate .

b) MRSxy= MUx/MUy

• MUx= ∆U/∆X= (0.3) ( X^ -0.7)( Y^0.7)

•MUy=∆U/∆Y= (0.7)(X^0.3)( Y^ -0.3)

So, MRSxy= (3/7) (Y/X)

∆MRS/∆X= (3/7) [ {X.(∆MRS/∆X) - Y }/ X^2 ]

( We have differentiated using the division rule)

So , ∆MRS/ ∆X= [ -Y / (7/3)X^2- X]

∆ MRS/∆X < 0

So, MRS is diminishing.

This is because of law of diminishing marginal utility. When Consumer consumes more and more of ons commodity ,his MU for thay commodity keeps on declining and he is willing to give up less and less of other commodity in exchange of first commodity hence , MRS declines

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