Answer

Solution

1. 1

   $U$ is convex (sum of two convex functions), so on a line segment (the budget line) the max is attained at the endpoints. Candidates:

2. 2
   $$A = (w/p_1, 0), \quad B = (0, w/p_2).$$
3. 3
   $$U(A) = 4(w/p_1)^2,\quad U(B) = (w/p_2)^2.$$
4. 4

   The solution is $\{A\}$ if $U(A) > U(B)$, $\{B\}$ if reversed, and $\{A, B\}$ if $U(A) = U(B)$:

5. 5
   $$4(w/p_1)^2 = (w/p_2)^2 \iff 2/p_1 = 1/p_2 \iff p_1 = 2 p_2.$$
6. 6

   A single-point set is trivially convex; a two-point set $\{A, B\}$ is not convex.

7. 7

   Check options:

8. 8

   \begin{itemize}[leftmargin=*]

9. 9
   • (A) $p_1 = 2, p_2 = 1 \Rightarrow p_1 = 2 p_2$: tie. $X = \{(5,0), (0,10)\}$, not convex. TRUE.
10. 10
    • (B) $p_1 = 6, p_2 = 3 \Rightarrow p_1 = 2 p_2$: tie. $X = \{(5,0), (0,10)\}$, not convex. (B) claims convex --- FALSE.
11. 11
    • (C) $p_1 = 1, p_2 = 1$: $U(A) = 400, U(B) = 100$, so $X = \{(10,0)\}$, singleton, convex. (C) claims not convex --- FALSE.
12. 12
    • (D) $p_1 = 4, p_2 = 2 \Rightarrow p_1 = 2 p_2$: tie. $X = \{(5,0),(0,10)\}$, not convex. (D) claims convex --- FALSE.
13. 13

    \end{itemize}

14. 14

    Only (A) is true.

Answer structure / marking notes

Content note