Question

each of the following situations,

3. U(X,Y) = X1/2Y1/2 M = $36; PY = $1; PX is initially = $1; the price of Good X increases to PX = $9.

4. U(X,Y) = X1/2Y1/2 M = $64; PY = $1; PX is initially = $16; the price of Good X decreases to PX = $1.

1）calculate the change in the quantity demanded of Good X that is due to the Substitution Effect

2）calculate the change in the quantity demanded of Good X that is due to the Income Effect

Answer 1

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

M = $36;

PY = $1; PX is initially = $1; the price of Good X increases to PX = $9.

MUx= (1/2)X^-1/2 Y^1/2

MUy= (1/2)X^1/2 Y^-1/2

MRS= MUx/MUy= Y/X

For optimal quantity:

MRS= Px/Py

Y/X = 1/1

Y= X Equation 1

Initial budget line: X+Y= 36

Use equation 1:

2X= 36

X= 18

New budget line after price increase: 9X+Y=36

Use equation 1:

9X+X=36

X'= 3.6 Optimal quantity after price increase

Compensated new income= M'= M+X(change in price)= 18 x 8=36+ 144

New budget line at new income and new price: 9X+Y= 180

9X+X= 180

X''= 18 Optimal quantity at new price and compensated income

Substitution effect= X''-X= 18-18= 0 (quantity demanded of Good X that is due to the Substitution Effect)

Income effect= X'-X''= 3.6-18= -14.4 (quantity demanded of Good X that is due to the Income Effect)

4.

U(X,Y) = X^1/2 Y^1/2

M = $64;

PY = $1; PX is initially = $16; the price of Good X increases to PX = $1.

MUx= (1/2)X^-1/2 Y^1/2

MUy= (1/2)X^1/2 Y^-1/2

MRS= MUx/MUy= Y/X

For optimal quantity:

MRS= Px/Py

Y/X = 16/1

Y= 16X Equation 1

Initial budget line: 16X+Y= 64

Use equation 1:

32X= 64

X= 2

New budget line after price decrease: X+Y=64

Use equation 1:

X+16X=64

X'= 3.76 Optimal quantity after price increase

Compensated new income= M'= M+X(change in price)=64 +2 x -15= 34

New budget line at new income and new price: X+Y= 34

X+16X= 34

X''= 2 Optimal quantity at new price and compensated income

Substitution effect= X''-X= 2-2= 0 (quantity demanded of Good X that is due to the Substitution Effect)

Income effect= X'-X''= 3.76-2= 1.76 (quantity demanded of Good X that is due to the Income Effect)