Question

Suppose an individual’s utility function for two goods X and Y is givenby U(X,Y) = X^(3/4)Y^(1/4)

Denote the price of good X by Px, price of good Y by Py and the income of the consumer by I.

1. a) (2 points) Write down the budget constraint for the individual.

2. b) (4 points) Derive the marginal utilities of X and Y.

3. c) (3 points) Derive the expression for the marginal rate of substitution of X for Y. Write

   down the tangency condition for the utility maximization problem of the individual.

{Very Important: In the questions below you must show the steps. In particular, you must clearly write the equations from which you get your answers. Otherwise, you are not going to get any credit.}

d) (6 points) Suppose Px = 3, Py = 1, and I = 100. Find out the utility maximizing amounts of X and Y consumed by the consumer.
e) (5 points) Now suppose Px falls to 2 while Py = 1, and I = 100. What are the new levels of demand for X and Y?

f) (5 points) Diagrammatically show the income and substitution effects of a change in Pxfrom 3 to 2 on the consumption of X. You must explicitly specify the optimal consumption bundles for the different sets of prices (appropriately graph and label the actual quantities) but you may qualitatively specify anything else that is deemed relevant (e.g. budget constraints or intermediary changes in X and Y).

g) (5 points) Which effect (substitution or income) will move you along a given indifference curve? And for this question, do you move to the left or right along the given indifference curve and describe what this means for the MRS (i.e. does the MRS increase or decrease?). Also, explicitly specify what the MRS is equal to before and after you move along the indifference curve.

Extra Credit: (5 points) Numerically derive the income and substitution effect of a change in the price of X from 3 to 2 on the consumption of X.

Answer 1

ч (ч. 3) = x 1/4 (9) , x+ 141 - 1 3 D) 194, - з * 4* Pча - 13/ч 3/4 1 tangency condition for utility maximitation х'' я ross = , MRS - х = 3 - 3 fit Piz = І Рx x + 1 = 1 | Ч | z = 10 С По ч 14) [