Question

consumer has money income of USD 400 and consumes x and y with price of x and y are equals 2 and 5 respectively . U=X^1/3 Y^2/3

A) Calculate the demand function for x and y

B) calculate the utility maximize levels of x and y

C) compute the maximize utility that consumer drive from consumption of x and y

Answer 1

Solution:

A.). Demand function for X:

$\frac{MUx}{MUy} = \frac{Px}{Py}$

$\frac{Y}{Px} = \frac{X}{Py}$

Y = $\frac{Px}{Py}X$

M = PxX + PyY

M = PxX + Py($\frac{Px}{Py}X$)

M = PxX + PxX

M = 2PxX

X = $\frac{M}{2Px}$

 

Demand function for Y:

$\frac{MUx}{MUy} = \frac{Px}{Py}$

$\frac{Y}{Px} = \frac{X}{Py}$

X = $\frac{Py}{Px}Y$

M = PxX + PyY

M = Px($\frac{Py}{Px}Y$) +PxX

M = PyY + PyY

M = 2PyY

Y = $\frac{M}{2Py}$

 

B.). Derive MRS_XY = $\frac{MUx}{MUy} = \frac{Px}{Py}$

U= X^1/3Y^2/3

MUx = $\frac{\partial U} {\partial X} = \frac{1} {3}X^{-\frac{2}{3} } Y ^{\frac{2}{3}}$

MUy = $\frac{\partial U} {\partial X} = \frac{2} {3}X^{\frac{1}{3} } Y ^{-\frac{1}{3}}$

 

Set $\frac{MUx}{MUy} = \frac{Px}{Py}$

$\frac{1} {3}$X^-2/3Y^2/3 $\div$ $\frac{2} {3}$X^1/3Y^-1/3= $\frac{2} {5}$

X = 1.23Y

Budget constraint: M = PxX + PyY

400 = 2X + 5Y

Substitute the value of X in budget constraint to derive Y:

400 = 2(1.23Y) + 5Y = 2.46Y + 5Y = 7.46Y

Y = 53.6

X = 1.23Y = 1.23(53.6) = 65.9

The quantities of X and Y which maximize utility (Uxy) = (65.9, 53.6)

 

C.). Maximum Utility:

 

U= X^1/3Y^2/3

U = 65.9^1/353.6^2/3

U = 57.4

Maximum Utility that the consumer can derive from x and y = 57.4