Question

2. For each of the three utility functions below, answer these questions: • Does the marginal utility of good x diminish, remain constant, or increase as the consumer buys more x, holding good y constant? Justify. • Does the MRS of x for y diminish, remain constant, or increase as the consumer substitutes good y for more of good x to the right along an indifference curve? Justify. a. ?(?, ?) = ? ?? ? , where 0 < a < 1 and b > 0 b. ?(?, ?) = ? ? + ?? , where a >1 and b > 0 (10 points) c. ?(?, ?) = ?? + ? ? , where a > 0 and 0 < b < 1

Answer 1

Solution:

Marginal utility of good X, MUx = $\partial U(x, y)/\partial x$

Marginal rate of substitution of x for y, MRSxy = MUx/MUy ;

where MUy is the marginal utility of good y (MUy = $\partial U(x, y)/\partial y$ )

a) U(x, y) = x^ay^b

MUx = a*x^a-1y^b

Since, 0 < a <1, (a-1) < 0 or (1-a) > 0

So, MUx = a*y^b/(x^1-a). Clearly, MUx is diminishing in x (since xⁿ is in denominator where n is something positive), holding the y constant.

Also, MUy = b*x^ay^b-1

So, MRSxy = MUx/MUy = (a*x^a-1y^b)/(b*x^ay^b-1)

So, MRSxy = (a/b)*(y/x), b > 0 (as already given)

Clearly, as one moves to the right along an indifference curve, substituting good y by more of good x, MRSxy decreases (y in numerator which decreases, x in denominator which increases, so overall MRS decreases), implying that as a consumer moves more and more to the right (that is consume more and more of good x), he/she is willing to give up less and less of good y.

b) U(x, y) = x^a + by , a > 1, b > 0

MUx = a*x^a-1 , since a > 1, a-1 is positive. So, as x increases, MUx also increases.

MUy = b (which is a constant greater than 0)

MRSxy = MUx/MUy = (a/b)*x^a-1 , b > 0

So, with (a - 1) > 0, MRSxy increases as more and more good x is consumed (and less of good y is consumed as from the MRS equation we can see that consumption of good y does not directly affects MRS value). So, moving right along an indifference curve increases MRSxy.

c) U(x, y) = ax + y^b , a > 0, 0 < b < 1

MUx = a (which is a constant greater than 0)

So, in this case, marginal utility of x remains constant as the consumer buys more of good x, holding good y constant.

MUy = b*y^b-1

Since, b < 1, (b-1) < 0, or (1-b) > 0. So, we can write this as: MUy = b/(y^1-b)

MRSxy = MUx/MUy = a/(b/(y^1-b))

MRSxy = a*y^1-b/b , b > 0

Clearly, as one consumes more of good x, and less of good y, MRS falls (good x consumption doesn't affect MRS directly, but MRS is directly proportional to good y consumption as can be seen from the formula above. So, lower good y generates lower MRSxy). So, as consumer moves to the right along an indifference curve, MRSxy diminishes.