For each of the following utility functions, calculate
a. MRS (if applicable) - is it homothetic? Convex?
b. Draw an indifference curve (does not have to be exact - just get the general shape)
c. Marshallian Demand
d. Hicksian Demand
e. Indirect Utility Function
f. Expenditure Function
g. Total, Income and Substitution Effects for a small price change (i.e. using the derivative) for
good x (show Slutsky equation holds)
h. Own price elasticity (with respect to good x)
(You do not need to write out the Lagrangian explicitly for Marshallian and Hicksian demands, but
you can if you want extra practice. It might also be good practice to solve for the expenditure function
using the indirect utility function and vice versa and going from Marshallian to Hicksian and back using
the indirect utility function and the expenditure function as in the week 4 Monday lecture.)
1. $U(x,y) = x^{1/5}y^{4/5}$
2. $U(x,y) = \frac{1}{3}ln(x) + \frac{1}{4}ln(y)$ (hint: you do not have to calculate this one fully. Explain why it gives
the same or effectively the same - answers as the previous example)
3. $U(x,y) = 15x + 10y$
4. $U(x, y) = min(2x, 10y)$
5. $U(x,y) = x^{1/2} + y^{1/2}$ (skip g)
6. $U(x,y) = 2xy + \frac{1}{2}x^{2}$ (skip g, Consider corner solutions)
7. $U(x,y) = x^{2}+ y^{2}$ (Only do a-c. Be careful with this one. Why is it different from other examples
we've done?)
8. $U(x,y) = 6 ln(x) + 3y$ (Consider corner solutions)
