Question

1.    Given the Utility function U(X,Y) = X^.5 + Y^.5

a.    Write mathematical expressions for marginal utility of x and marginal utility of y

b.    Does the consumer the assumption of non-satiation (more is better) desire more of x and y?

c.    If the quantity of Y is held constant, does the marginal utility of x increase, remain constant or diminish as x increases? Prove your answer

d.    Derive an expression for the marginal rate of substitution of x for y.

e.    If Price of X equals 1 and price of Y equals 1 and income equals $200, what utility maximizing quantity of x and quantity of y will the consumer choose.

Answer 1

Ans. Utility function, U = X0.5 Y0.5

a) Marginal Utility of X,MUx = dU/dX = 0.5*(Y/X)0.5

Marginal Utility of Y, MUy = dU/dY = 0.5*(X/Y)0.5

b) Yes, as can be seen in marginal utility function of X and Y, utlity increases with increase in consumption of X and Y i.e. Utility function is increasing function of X and Y.

c) Differentiation of MUx with respect to X gives,

d2U/dX2 = -0.25 Y0.5X-1.5

It can be seen from the above function that increase in X would lead to decrease in marginal utility of X. Thus, marginal utility diminishes as X increases.

d) Matginal rate of substitution, MRS = dY/dX = MUx/MUy = Y/X

e) At equilibrium,

MRS = Price of X / Price of Y

=> Y/X = 1

=> X = Y

Substituting this in budget constraint,

X + Y = 200

=> 2X = 200

=> X = Y = 100 units

Thus, consumer will consume 100 units of X and Y each to maximize utility.

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