Question

Emily's preferences can be represented by u(x,y)=x^1/4 y^3/4 . Emily faces prices (p_x,p_y) = (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

Solution :-

Emily's preferences can be represented by

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

Emily faces prices (p_x,p_y) = (2,1) and

her income is = $120.

For Her optimal consumption bundle:-

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

= [ (1/4)x^-3/4 y^3/4] / [(3/4)x^1/4 y^-1/4]

= 1/4 x 4/3 [( y^4/4)/(x^4/4)]

= 1/3 ( y/x)

MRS = y/3x

budget line is :-

2x + y = 120 and

Px/Py = 2

Optimal condition :-

MRS = Px/Py

y/3x = 2

y = 2 * 3x

[ y = 6x ]...........( equation 1)

Budget line is :-

2x + y = 120

Put y = 6x

2x + 6x = 120

8x = 120

x = 120/8

[ x = 15 ]

Now, for y

y = 6x

Put x = 15

y = 6 * 15

[y = 90 ]

So, Her optimal consumtion bundle is :-

(x,y) = (15,90).

* Now the price of x increases to = $3  

while price of y remains the same :-

MRS = y/3x

Now,

Px' = 3

So, New budget line is :-

3x + y = 120

Optimal condition :-

MRS = Px'/Py

y/3x = 3

y = 3 * 3x

[ y = 9x ].......( equation 2)

New budget line :-

3x + y = 120

Put y = 9x

3x + 9x = 120

12x = 120

x = 120/12

[ x = 10 ]

Now, for y

y = 9x

Put x = 10

y = 9 * 10

[y = 90 ]

Her new optimal consumption bundle is:

(x,y) = (10,90).

* Her equivalent variation is:-

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

Initial optimal quantity of x = 15

Change in income = ( Change in price of x) * (initial optimal quantity of x )

Change in income = ( 3 - 2) * 15

= $15

So, Her equivalent variation is = $15.