Homework #2
1.24 Dene u to be multiplicatively separable if there are continuous u with u(x) = u (x ). Show that a
preference order on RL has a multiplicatively separable representation if and only if is completely
+
separable.
Answer: If
follows that
is
Homework #3
2.17 Show that when utility is homothetic, every cross-price elasticity k is independent of income.
Answer: Proposition 2.3 shows x(p, m) = mx(p, 1). Let (p) = x(p, 1), so x(p, m) = m(p). Then
xk /p = mk /p . The cross-price elastiticy is then
Homework #1
1.1 Suppose preferences are lexicographic on R2 . Find the indierence curve I(x) and strictly preferred set
+
P (x) for any x R2 .
+
Answer: The indierence curves have the form I(x) = cfw_x. The preferred set is P (x) = cfw_y : y1 > x1 ,
or y1
Homework #8
12.1 Suppose u1 (x1 ) = (x1 )1/3 (x1 )2/3 and u2 (x2 ) = (x2 )1/3 (x2 )2/3 , with endowments 1 = (7, 1) and 2 =
1
2
1
2
(3, 1). Find the core.
Answer: Since the consumers have identical Cobb-Douglas preferences, the Pareto set is the diagonal
Homework #4
3.17 Suppose u is homothetic. Show that the true cost-of-living index is independent of u0 (and so independent
of m).
Answer: The true cost-of-living index is PT (p0 , p1 , u0 ) = e(p1 , u0 )/e(p0 , u0 ). Let u(x0 ) = u0 . Given
u1 , nd > 0 su
Homework #7
8.8 Consumers 1 and 2 both have utility ui (xi , xi ) = xi xi . Their endowments are 1 = (1, 0) and 2 = (0, 1)
1
2
1 2
while the rm shares are 1 = 1/3, 2 = 2/3. There is one rm with technology set Y = cfw_(y1 , y2 ) : y1
e
0, y2 y1 . Using go
Homework #5
5.1 Let f (z) =
z be the production function. Find the cost function and the conditional factor
demands. Then maximize prot (when possible).
Answer: We form the Lagrangian L = wz (z q) +
z . The resulting rst-order conditions
are w = + . Thu
Homework #6
7.3 Suppose F is uniformly distributed over [1, a] for a > 1. Calculate the risk premium for the following
utility functions.
a) u(x) = x3 .
b) u(x) = x1/2 .
c) u(x) = ln x.
Answer: Note that the probability density for all three parts is 1/(a
Micro II Final, December 11, 2013
1. A consumer has utility u(x, y) = x + y. Prices are given by the vector p R2 and income is m > 0.
+
a) Find the indirect utility function.
b) Find the expenditure function.
Answer:
a) Note that the marginal utility is i