Answer

Solution

1. 1

   Since $u$ is a real-valued function on $\mathbb{R}_{++}^2$, the induced

2. 2

   preference is automatically complete and transitive. Since $u$ is

3. 3

   continuous, the preference is continuous.

4. 4

   Convexity. The upper contour set is

5. 5
   $$\{(x,y): x - \ln y \ge c\} = \{(x,y): x \ge c + \ln y\}.$$
6. 6

   The boundary $x = c + \ln y$ is concave in $y$ (second derivative

7. 7

   $-1/y^2 < 0$), so the region lying above it is not a convex set.

8. 8

   Equivalently, the Hessian

9. 9
   $$\nabla^2 u = \begin{pmatrix} 0 & 0 \\ 0 & 1/y^2 \end{pmatrix}$$
10. 10

    is positive semidefinite, not negative semidefinite, so $u$ is convex (not concave) in $y$ and the preference fails to be convex.

11. 11

    Hence the preference is complete, transitive, continuous, but not convex.

Answer structure / marking notes

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