Question

Q3: a. Consider a Cobb-Douglas Utility Function Find marginal utility of Y. Does marginal utility of depend on the level of consumption of X? how do you know? Explain b. Find slope of the indifference curve. c. Find MRS and draw the indifference curve for above function.

Answer 1

Ans. Let the Cobb Douglas utility function be,

U = X^a * Y^b

a) Marginal Utility of X, MUx = dU/dX = a*X^(a-1) * Y^b

and marginal utility of Y, MUy = dU/dY = b*X^a * Y^(b-1)

Thus, MUy dependes on consumption of X, this can be seen from the marginal utility function of Y. This is because there ia a constraint on income. So, more the good X is consumed, less will be the consumption of Y, thus, higher MUy while as the consumption of Y, the lower is the consumption of X, then the MUy is lower. This is in line with the law of diminishing marginal utility.

b) The slope of the indifference curve is the marginal rate of substitution,

MRS = -dY/dX = -(dU/dX)/(dU/dY) = -MUx/MUy = a/b * Y/X

c) The indifference curve will be downward sloping (as seen from the sign of MRS) and convex because more and more consumption of good X i.e. moving downwards on the indifference curve leads to a decrease in marginal utility of X and thus, MRS decreases.

Good Y ICO Good X