Question

Emily's preferences can be represented by u(x,y)=x^1/4 y^3/4 . Emily faces prices (px,py) = (2,1) and her income is $120.

Her optimal consumption bundle is: __________ (write in the form of (x,y) with no space)

Now the price of x increases to $3 while price of y remains the same Her new optimal consumption bundle is: ____________ (write in the form of (x,y) with no space)

Her Equivalent Variation is: $ ____________

Answer 1

u(x,y)=x^1/4 y^3/4

Marginal utility of x(MUx) = (1/4)x^-3/4 y^3/4

Marginal utility of y(MUy) = (3/4)x^1/4 y^-1/4

MRS= MUx/MUy= (1/3)(y/x)= y/3x

Budget line: 2x+y= 120

px/py= 2

Optimal condition:

MRS= px/py

y/3x = 2

y= 6x Equation 1

Use this in Budget line:

2x+y= 120

2x+6x= 120

8x=120

x = 120/8= 15

y= 6x= 6(15)= 90

(x,y)= (15,90) Optimal bundle

Price of x increases to $3:

px'= 3

New budget line: 3x+y= 120

Optimal comdition:

MRS= px'/py

y/3x = 3

y= 9x Equation 2

Use this in Budget line:

3x+9x= 120

12x = 120

x = 10

y= 9(10)= 90

(x,y)= (10,90) Optimal bundle

Equivalent variation is the change in income to keep utility as initial:

Change in income= Change in price of x * initial optimal Quantity of x

Change in income= (3-2)(15)= $15

Equivalent variation is $15