For the following utility function, * Find the marginal utility of each good. *

Question

For the following utility function,

* Find the marginal utility of each good.

* Determine whether the marginal utility decreases as consumption of each good increases (i.e., does the utility function exhibit diminishing marginal utility in each good?).

* Find the marginal rate of substitution..

* Discuss how MRSxy changes as the consumer substitutes X for Y along an indifference curve.

* Derive the equation for the indifference curve where utility is equal to a value of 100.

* Graph the Indifference curve where Utility is equal to a  value of 100.

a. U(X,Y) = 5X + 2Y

b. U(X,Y) = X^0.33 Y^0.67

c. U{X,Y) = 10X^0.5 + 5Y

Explanation / Answer

a) U(X,Y)=5X+2Y

MUx=5, MUy=2

Marginal utilities are constant whether consumption of goods increases or decreases.

MRSxy=MUx/MUy=5/2=2.5

MRSxy is constant as the consumer substitutes X for Y

equation of indifference curve 5X+2Y=100

b) U(X,Y)=X^.33 Y^.67

MUx=.33X^(-0.67)Y^.67 MUy=0.67X^.33 Y^(-0.33)

So, as the consumption of X increases, MUx decreases.

Also, as the consumption of Y increases, MUy decreases.

So both the goods show diminishing marginal utility.

MRSxy=MUx/MUy=0.5Y^(0.33)/X^(0.67)

As the consumer substitutes X for Y, MRSxy increases.

equation of indifference curve X^.33 Y^.67=100

c) U(X,Y)=10x^0.5+5Y

MUx=5x^(-0.5), MUy=5

Marginal utility of X decreases as the consumption of X increases.

Whereas marginal utility of Y is constant.

So only X exhibits diminishing marginal utility.

MRSxy=MUx/MUy=x^(-0.5)

As the consumer substitutes X for Y, MRSxy increases.

equation of indifference curve 10x^0.5+5Y=100

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