A consumer is observed making two choices. First, he faces prices p = (1,3) with wealth w = 8 but is limited to purchasing no more than 3 units of good 1. Given this budget set, he chooses the bundle z = (2,2). Second, he faces prices p' = (2,1) and wealth w' = 8 with no limit on his purchases of either good. Given this budget set, he chooses the bundle r' = (4,0). (a) (4) Identify all revealed preferences given these choices. Solution: Since I is affordable when r' is chosen, r' is revealed preferred to z. Since I exceeds the limit on good 1 when z is chosen, r is not revealed preferred to 1¹. (b) (2) Do this consumer's choices satisfy the Weak Axiom of Revealed Preference? Solution: Yes. WARP says that the two bundles cannot be revealed preferred to each other; here, only I' is revealed preferred to I. (e) (3) Could these choices result from the consumer choosing optimally given some mono- tone, convex, and continuous preferences? Explain your answer carefully; you do not have to provide a formal proof. Solution: For these to be optimal choices with such preferences, the indifference curve through z must lie entirely on or above the budget line associated with (p,w), and simi- larly for r' for the budget line associated with (p', w'). Because each of these bundles lies below the other budget line, this implies that the indifference curves must cross, which is impossible. Alternatively, note that (3,1) is a conver combination of r and r', so for conver preferences must be weakly preferred to r (the less preferred bundle between I and 2). But then the bundle (3,5/3) must be strictly preferred 1, contradicting that I is optimal given the initial budget set.
A consumer is observed making two choices. First, he faces prices p = (1,3) with wealth w = 8 but is limited to purchasing no more than 3 units of good 1. Given this budget set, he chooses the bundle z = (2,2). Second, he faces prices p' = (2,1) and wealth w' = 8 with no limit on his purchases of either good. Given this budget set, he chooses the bundle r' = (4,0). (a) (4) Identify all revealed preferences given these choices. Solution: Since I is affordable when r' is chosen, r' is revealed preferred to z. Since I exceeds the limit on good 1 when z is chosen, r is not revealed preferred to 1¹. (b) (2) Do this consumer's choices satisfy the Weak Axiom of Revealed Preference? Solution: Yes. WARP says that the two bundles cannot be revealed preferred to each other; here, only I' is revealed preferred to I. (e) (3) Could these choices result from the consumer choosing optimally given some mono- tone, convex, and continuous preferences? Explain your answer carefully; you do not have to provide a formal proof. Solution: For these to be optimal choices with such preferences, the indifference curve through z must lie entirely on or above the budget line associated with (p,w), and simi- larly for r' for the budget line associated with (p', w'). Because each of these bundles lies below the other budget line, this implies that the indifference curves must cross, which is impossible. Alternatively, note that (3,1) is a conver combination of r and r', so for conver preferences must be weakly preferred to r (the less preferred bundle between I and 2). But then the bundle (3,5/3) must be strictly preferred 1, contradicting that I is optimal given the initial budget set.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:A consumer is observed making two choices. First, he faces prices p = (1,3) with wealth
w = 8 but is limited to purchasing no more than 3 units of good 1. Given this budget set, he
chooses the bundle r = (2, 2). Second, he faces prices p' = (2,1) and wealth w' = 8 with no
limit on his purchases of either good. Given this budget set, he chooses the bundle z' = (4,0).
(a) (4) Identify all revealed preferences given these choices.
Solution: Since x is affordable when x' is chosen, a' is revealed preferred to x. Since
a' exceeds the limit on good 1 when x is chosen, x is not revealed preferred to a'.
(b) (2) Do this consumer's choices satisfy the Weak Axiom of Revealed Preference?
Solution: Yes. WARP says that the two bundles cannot be revealed preferred to each
other; here, only a' is revealed preferred to a.
(c)(3) Could these choices result from the consumer choosing optimally given some mono-
tone, convex, and continuous preferences? Explain your answer carefully; you do not
have to provide a formal proof.
Solution: For these to be optimal choices with such preferences, the indifference curve
through a must lie entirely on or above the budget line associated with (p, w), and simi-
larly for r' for the budget line associated with (p', w'). Because each of these bundles lies
below the other budget line, this implies that the indifference curves must cross, which
is impossible. Alternatively, note that (3,1) is a conver combination of x and x', so for
conver preferences must be weakly preferred to x (the less preferred bundle between a
and a'). But then the bundle (3,5/3) must be strictly preferred z, contradicting that
is optimal given the initial budget set.
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