Answer

Solution

1. 1

   Compute the cost matrix $C_{ij}=p^{i}\cdot x^{j}$:

2. 2
   $$\begin{array}{c|ccc} & x^{1} & x^{2} & x^{3}\\\hline p^{1} & 30 & 26 & 32\\ p^{2} & 32 & 30 & 26\\ p^{3} & 26 & 32 & 30 \end{array}$$
3. 3

   \textbf{At $p^{1}$:} $C_{11}=30$, $C_{12}=26\le 30$, so the consumer could afford $x^{2}$ but chose $x^{1}$ $\Rightarrow x^{1}\succ x^{2}$.

4. 4

   \textbf{At $p^{2}$:} $C_{22}=30$, $C_{23}=26\le 30$, so $x^{2}\succ x^{3}$.

5. 5

   \textbf{At $p^{3}$:} $C_{33}=30$, $C_{31}=26\le 30$, so $x^{3}\succ x^{1}$.

6. 6

   This yields the cycle $x^{1}\succ x^{2}\succ x^{3}\succ x^{1}$, which directly violates transitivity (and indeed the Strong Axiom of Revealed Preference). Consequently the observed choices cannot be rationalised by a complete-and-transitive (i.e.\ rational) preference relation. Among the four options, the one consistent with the data ruling out rationality is that preferences are not both complete and transitive --- in particular they fail transitivity, and any consistent assignment over $\{x^{1},x^{2},x^{3}\}$ has to drop completeness as well to avoid the cycle. Option (C) is therefore the intended answer.

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