Question

1. Recall that, for a set of alternatives X, a choice structure is a pair (B,C), where B be a collection of subsets of X and C is a function defined on 'B. with C(B) CB for each B E B. C(B) is the set of alternatives chosen when the budget set is B. In the following examples, X = {a,b,c}. (a) State in each of the three cases whether the Weak Axiom of Re- vealed Preferences holds. Provide a counterexample if the answer is negative. i. B = {B1, B2} B = {a,b), B2 = {a,b,c} C(B) = {a}, C(B2) = {b} ii. B = {B} B = {a,b} C(B) = {a} iii. B = {B1, B2, B3} B = {a,b}, B = {b.c), B3 = {a.c} C(B) = {a}, C(B2) = {b}, C(B3) = {c} (b) For the same examples as above, state whether the Strong Axiom of Revealed Preferences holds. (c) For the same examples as above, state whether C can be generated by a complete preference relation. If your answer is positive, provide the preference relation. If your answer is negative, give a reason. (d) For the same examples as above, state whether C can be generated by a complete and transitive preference relation. If your answer is positive, provide the preference relation. If your answer is negative, give a reason.

Answer 1

(a)

1. No, Weak axiom of revealed preference does not hold in this case. Because in case of Budget set B1 he chose a in presence of b, which shows he prefers a over b but in Budget set 2 he chose b in presence of a, showing he prefers b over a. This is inconsistent with the weak axiom of revealed preference.

2. Yes, because he faced only one choice problem and anything chosen here will not affect either the weak axiom or strong axiom of revealed preference.

3. Yes, because when faced with budget set B1 he chose a in presence of b, showing he prefer a over b and with budget set B2 he chose b over c, showing he prefer b over c. This means he prefer a over b and b over c but when faced with budget set B3 he chose c over a, showing he prefers c over a. Which is consistent with weak axiom of revealed preference because it does not talk about the transitivity of the preferences.

(b) In addition to weak axiom of revealed preferences, strong axiom looks for the transitivity in preferences also. Transitivity says that, If a is preferred over b and b is preferred over c then it must be the case that a is preferred over c.

1. No...same reason as above

2. Yes..same reason as above

3. No, see that it does not follow the transitivity in preferences as discussed in solution of (a). He preferred a over b and b over c but also c over a , which violates the transitivity and hence will violate the strong axiom of revealed preferences.

(c) and (d) yes for both, consider the preference relation as,

B1 = {a,b} and C(B1) = a

B2 = {b,c} and C(B2) = b

B3 = {c,a} and C(B3) = a

It follows both completeness and transitivity.

# Note - Completeness says that for every pair {x,y} either x is preferred over y or y is preferred over x or both i.e, indifferent.