Question

For each of the following utility functions, (Total 12 Marks) • ?nd the marginal rate of substitution, • discuss how MRS XY changes as the consumer substitutes X for Y along an indi?erence curve, • derive the equation for the indi?erence curve where utility is equal to a value of 100, and • graph the indi?erence curve where utility is equal to a value of 100.

(c) U(X,Y ) = X^2 + Y^2

Answer 1

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