Answer to Question #273382 in Microeconomics for Swarswattie

Question

Stuart's utility function for goods X and Y is represented as U(X,Y)=X0.8Y0.2. Assume that his income is $100 and the prices of goods X and Y are $20 and $10, respectively.

 

(a) Express his marginal rate of substitution (MRS) between goods X and Y. As the amount of X increases relative to the amount of Y along the same indifference curve, does the MRS increase or decrease? Explain.

 

(b) What is his optimal consumption bundle (X*, Y*), given income and prices of the two goods?

 

(c) How will this bundle change when all prices double and income is held constant?  When all prices double AND income doubles?

 

(d) Derive the demand curve for good X and demand curve for good Y. 

Answer 1

Solution:

a.). MRS = $\frac{MUx}{MUy}$

U(X,Y)=X^0.8 Y^0.2

MUx = $\frac{\partial U} {\partial X} =$ 0.8X^-0.2Y^0.2

MUy = $\frac{\partial U} {\partial X}$ = 0.2X^0.8Y^-0.8

MRS = $\frac{0.8X^{-0.2} Y^{0.2} }{0.2X^{0.8}Y^{-0.8} }$

MRS(_X,Y) = $\frac{4Y}{X}$

The MRS decreases as the amount of X increases relative to the amount of Y along the same indifference curve. This is because of the law of diminishing marginal rates of substitution.

 

b.). Budget constraint: I = PxX + PyY

100 = 20X + 10Y

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

$\frac{4Y}{X} = \frac{20}{10}$

X = 2Y

Substitute in the budget constraint:

100 = 20(2Y) + 10Y

100 = 40Y + 10Y

100 = 50Y

Y = 2

X = 2Y = 2 x 2 = 4

The optimal consumption bundle (X, Y) = (4, 2)

c.). When all prices double and income doubles:

100 = 40X + 20Y

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

$\frac{4Y}{X} = \frac{40}{20}$

X = 2Y

Substitute in the budget constraint:

200 = 40(2Y) + 20Y

200 = 80Y + 20Y

200 = 100Y

Y = 2

X = 2Y = 2 x 2 = 4

The optimal consumption bundle (X, Y) = (4, 2)

 

d.). The demand curve for Good X:

Y = $0.25X\frac{Px}{Py}$

I = PxX + PyY

I = PxX + Py($0.25X\frac{Px}{Py}$)

I = PxX + 0.25XPx

I = 1.25XPx

X = $\frac{I}{1.25Px}$

 

The demand curve for Good Y:

X = $4Y\frac{Py}{Px}$

I = PxX + PyY

I = Px($4Y\frac{Py}{Px}$) + PyY

I = 4YPy + PyY

I = 5YPy

Y = $\frac{I}{5Py}$