Answer

Solution

1. 1

   Let bundles be $X^1=(100,100)$ at prices $p^1=(100,100)$ and $X^2=(120,x)$ at prices $p^2=(100,80)$.

2. 2

   Step 1: Is $X^1$ revealed preferred to $X^2$ at year-1 prices?\

3. 3

   $p^1\cdot X^1=20000$. $p^1\cdot X^2=100(120)+100(x)=12000+100x$.\

4. 4

   $X^2$ is affordable in year 1 iff $12000+100x\le 20000$, i.e.\ $x\le 80$. Then $X^1\succsim X^2$.

5. 5

   Step 2: Is $X^2$ revealed preferred to $X^1$ at year-2 prices?\

6. 6

   $p^2\cdot X^2=100(120)+80x=12000+80x$. $p^2\cdot X^1=100(100)+80(100)=18000$.\

7. 7

   $X^1$ is affordable in year 2 iff $18000\le 12000+80x$, i.e.\ $x\ge 75$. Then $X^2\succsim X^1$.

8. 8

   Inconsistency. Both reveal-preferences hold simultaneously iff

9. 9
   $$75\le x \le 80.$$
10. 10

    At the boundaries the alternative is just affordable (boundary WARP --- typically a tie), so the strict inconsistency region is $75<x<80$.

Answer structure / marking notes

Content note