Question

5) Derive the demand function for x₁ and x₂ for each of the following utility functions.

a. U(x₁,x₂) = 5x₁x₂

b. U(x₁,x₂) = x^1/3₁ x^2/3₂

c. U(x₁,x₂) = x₁ + 3x₂

d. U(x₁,x₂) = {x₁2x₂}

Answer 1

Budget equation is
I=p1x1+p2x2

Utility function: U(x1,x2)=5x1*x2

Utility will be maximised when Marginal Rate of SUbstitution equals to price ratio

MRS=dU1/dU2=p1/p2

dU/dx1=5x2; dU/dx2=5x1

MRS=x2/x1=p1/p2

x1=(p2*x2/p1)

Using this expression into budget equation

I=p1(p2*x2/p1)+p2*x2
I=2p2*x2

x2=I/2p2 and x1=I/2p1

(x1,x2)=(I/2p1,I/2p2)..Demand function for x1 and x2 respectively

Ans B)

U(x1,x2)=x1^(1/3)*x2^(2/3)

MRS=(1/3(x2/x1)^(2/3))/(2/3(x1/x2)^(1/3)=(1/2)(x2/x1)=p1/p2

x2=2x1*p1/p2

I=x1*p1+x2*p2=x1*p1+2x1*p1=3x1*p1

(1/3)(I/p1)=x1..Demand for x1

(2/3)(I/p2)=x2..Demand for x2

Ans C)

U(x1,x2)=x1+3x2

MRS=1/3=p1/p2

Hence utility maximisation depends upon the price of good . we will not have any interior solution in perfect substitutes and Consumer would consume higher good with lower price and zero amount of good with higher prices(That is corner solution)

Ans D)

U(x1,x2)=(x1*x2)
MRS=x2/x1=p1/p2

x2=p1*x1/p2

I=p1x1+p2*(p1x1/p2)=2p1*x1

x1=0.5(I/p1) and x2=0.4(I/p2)