Question

QUESTION; Given Biwei’s utility function U=4XY, where X is consumption of beer and Y is consumption of pizza.
For this utility function, the marginal utility of X is MUx = 4Y; the marginal utility of Y is MUY = 4X.

1) Suppose Y = 3. Calculate Biwei’s utility for X = 2, 3, 10, and 11. For a given level of Y, does good
X display diminishing marginal utility?

2) Suppose X = 3. Calculate Biwei’s utility for Y = 2, 3, 10, and 11. For a given level of X, does good
Y display diminishing marginal utility?

3) Find three different bundles containing X and Y that give Biwei 48 units of satisfaction. Plot the
three bundles and connect them with an indifference curve. What happens to the marginal rate of
substitution between X and Y as consumption of X increases?

4) Does the principle of diminishing MRS depend on the diminishing marginal utility of X and Y?

Answer 1

U = 4XY

1) Y =3

For X = 2

U = 4 * 3 * 2 = 24

For X = 3

U = 4*3*3 = 36

For X = 10

U = 4*3*10 = 120

For X = 11

U = 4*3*11 = 132

Now, MUx = 4Y

Differentiating MUx wrt X gives us 0

So, for given level of Y, X does not display diminishing marginal utility

2) X = 3

For Y = 2

U = 4 * 3 * 2 = 24

For Y = 3

U = 4*3*3 = 36

For Y = 10

U = 4*3*10 = 120

For Y = 11

U = 4*3*11 = 132

Now, MUy = 4X

Differentiating MUx wrt Y gives us 0

So, for given level of X, Y does not display diminishing marginal utility

3) Following are the three different bundles that gives 48 units of satisfaction:

(3, 4) , (2, 6) , (4, 3)

:(2,6) n the IC D 2 + ts 1 2 3 4

MRSx,y = MUx / MUy

= 4Y / 4X

= Y / X

So, as consumption of X increases, MRS falls.

4)

Principle of diminishing MRS holds when the second derivative is less than 0

MRS = Y / X

Differentiating again wrt X gives = - Y / X²

since X and Y are positive so, double derivative of MRS is < 0

Hence, the principle holds true