Question

STUCK WITH 3,4,5

U(q1,q2) = (q1)1/3+(q2)1/3

I = $108

1. Calculate optimal bundle (Bundle A) if p1=$4 p2=$1. Show on diagram (q1 in horizontal axis)

2. P1*= 9 p2=$1 calculate optimal bundle (Bundle B)

3. Calculate the income required so that we can obtain the same utility as Bundle A with the price in number 2. Calculate the new bundle (Bundle C)

4. What is the total effect, income effect and substitution effect on q1 because of the increase of p1 from $4 to $9?

5. What is the amount of compensating variation. What bundle will be consumed?

Answer 1

A)optimal bundle condition:

MU1/p1=MU2/p2

MU1=1/(3*q1^2/3)

P1=4

MU2=1/(3*q2^2/3)

P2=1

1/4(3*q1^2/3)=1/(3*q2^2/3)

Q2^2/3=4*q1^2/3

Q2=(4*q1^2/3)^3/2

Q2=4^3/2*Q1

Q2=8*q1

Budget constraint

I=p1*q1+p2*q2

108=4*q1+1*8q1

108=12q1

Q1=108/12=9

Q2=8*q1=8*9=72

Bundle A=(9,72)

B)we are going to to use again same condition but with different price,

1/9(3*q1^2/3)=1/(3*q2^2/3)

Q2=(9*q1^2/3)^3/2

Q2=9^3/2*Q1

Q2=27*q1

108=9q1+27q1

108=36q1

Q1=108/36=3

Q2=27*q1=27*3=81

Bundle B={3,81}

C)to obtain same utility,

U=9^1/3+72^1/3

The optimal condition with new price,

Q2=27Q1

9^1/3+72^1/3=q1^1/3+27Q1^1/3=q1^1/3+3*q1^1/3

4*q1^1/3=9^1/3+72^1/3

Q1^1/3=(9^1/3+72^1/3)/4

Q1=((9^1/3+72^1/3)/4)^3

Q1=(9+72+3*9*2+3*9*4)/64

Q1=(9+72+54+108)/64=243/64

Q2=27*243/64=6561/64

I=9*243/64+27*243/64=243/64(9+27)=243/64*36

I=2187/16

Additional income required=2187/16-108=459/16

Bundle C={243/64, 6561/64}

D)from bundle A to C is substitution effect showing effect of only price ratio change

From bundle C to B is income effect , showing due to change in price real income falls and its effect on optimal consumption bundle

From A to B is total effect of price change (substitution and income effect)

E) we already calculated that in 3) part .

Compensating variation is the additional money required to reach same utility level with new price.

Compensating variation=459/16

And bundle=Bundle C={243/64, 6561/64}

Note; there is another thing, equivalent variation that is change in income at initial price so that we can get same utility level that we will get after price change.

Hicksian approach uses equivalent variation and slutsky approach uses compensation variation to demonstrate income and substitution effect out of total price effect