1. A consumer has a quasilinear utility function with one modification:
u(x, y) = x + 40 ln y if x < 100
u(x, y) = 2x + 40 ln y if x >
Assume that the consumer is restricted to bundles (x, y) with x >
0, y >
(a) Suppose that p
= 1 and p
Solve for the consumer’s optimal bundle for all values
w of wealth in the following ranges: (1) 0 < w < 100; (2) w >
Here, it is necessary to check for optimal solutions for (1) x < 100, (2) x >
Within each region, the utility function is quasilinear, so an interior solution to the
consumer problem satisfies (du / dx) / p
= (du / dy) / p
, or du / dx = du / dy since
both prices are equal to 1.
For x < 100, du / dx = 1, du / dy = 40 / y, so an interior solution satisfies y
* = 40.
Walras’ Law, x
* = w – 40 at this interior solution with x < 100.
For x >
100, du / dx = 2, du / dy = 40 / y, so an interior solution satisfies y
* = 20.
Walras’ Law, x
* = w – 20 at this interior solution with x >
Now specialize analysis to different ranges of wealth.
RANGE 1: For w < 40, neither of these interior solutions is feasible.
Since du / dx is
constant while du / dy is decreasing, the consumer maximizes utility at a boundary
solution where she only consumes y.
The optimal bundle is (0, w).
RANGE 2: For 40 <
w < 100, the first interior solution is feasible, but it is not
possible to reach the point x = 100 where the utility function changes.
optimal bundle is (w – 40, 40).
RANGE 3: For w >
140, the first interior solution is not feasible (it would leave x >
100), but the second interior solution is feasible.
Since the change from region 1 (x <
100) to region 2 (x >
100) only increases utility, it will not be optimal to choose a
boundary solution in region 1.
The optimal bundle is (w – 20, 20).
(b) Is good x inferior or normal?
Explain your answer in words.
Good x is normal.
As wealth increases, the optimal choice of x increases.
point, there may be a discontinuous jump in x (e.g. from w – 40 to w – 20), but this
does not cause x to be inferior.
(c) Is good y inferior or normal?