Corners 1. A consumer has utility function $u(x_1, x_2) = x_1 + \log(x_2)$ and solves
$V(w, p) = \max_{x \ge 0} u(x)$ s.t. $px \le w$
a. Show that $Du(x_1, x_2) = (1, 1/x_2)$, that $\partial^2 u/\partial x_1 \partial x_1 = 0$, and that $\partial^2 u/\partial x_2 \partial x_2 < 0$.
b. Find the set of $(x_1, x_2)$ such that $\partial u/\partial x_1 < \partial u/\partial x_2$.
c. At prices $p = (p_1, p_2)$, find the set of $(x_1, x_2)$ such that the utils per dollar
spent on good 1 are smaller than the utils per dollar spent on good 2.
d. Give the isoquants of $u(\cdot, \cdot)$ and their slopes.
e. Put the previous two pieces of the problem together to find the consumer's
demad functions $x_1^*(p, w)$ and $x_2^*(p, w)$.
f. Find the possible patterns of the income expansion paths, that is, graph
$x^*(p, w) \in \mathbb{R}^2_+$ as $w$ increases.
