For the following utility function, * Find the marginal utility of each good. *
ID: 2506102 • Letter: F
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