Question

Do all the problem. Also, please provide a formal justification/argument for each answer. 1. Assume a consumer has as preference relation ⪰ represented by a quasilinear utility function u(x1,x2 )=x1α +x2 for α∈(0 ,1), with x∈C=R+2 .

Answer the following:
A. Show the preference relation this consumer is convex and strictly monotonic. Are these preference partially strictly concave in good 1 ? How about partially strictly concave in good 2 ?
B. Write down the consumer's Marshallian optimization problem, and construct the first order conditions for this problem for an interior solution for consumption (i.e., optimal consumption of both goods are strictly positive).
C. (difficult question): for these preferences, are optimal solutions for both goods always strictly positive for all prices?
D. Are both goods normal for these preferences? If so, show this is the case. If not, explain why not.
E. Why is good 1 decreasing in it's own price p1 . Also, why is good 2 is decreasing in it's own price p2
.

Answer 1

The preference relation for the quasilinear utility function is convex and strictly monotonic. However, it is not partially strictly concave in good 1 or good 2.

Step-by-step explanation:

To show that the preference relation is convex and strictly monotonic, we need to understand the properties of the utility function u(x1, x2) = x1α + x2. Let's start with convexity. A preference relation is convex if whenever two bundles A and B are preferred to a third bundle C, any bundle on the line segment connecting A and B is also preferred to bundle C.

For this utility function, if A ≽ C and B ≽ C, then for any α∈(0 ,1), we can find a bundle on the line segment between A and B, say D = (x1, x2), which would satisfy D ≽ C. This means that the preference relation is convex.

Next, a preference relation is strictly monotonic if whenever A is preferred to B, any bundle with more of at least one good and no less of any other good is also preferred to B. For this utility function, if A ≽ B, then for any α∈(0 ,1), we can find a bundle with more of good 1 and no less of good 2, say D = (y1, y2), which would satisfy D ≽ B. This means that the preference relation is strictly monotonic.

As for partial strict concavity in good 1, we need to check if a bundle with more of good 1 and no less of good 2 is always preferred to a bundle with less of good 1 and no less of good 2. However, for this utility function, the preference relation is not partially strictly concave in good 1 because it does not satisfy this condition.

Similarly, we can also conclude that the preference relation is not partially strictly concave in good 2.