Assume that demand for three goods has been evaluated at the follow- ing three price vectors: p= (1,2,2); p…

Assume that demand for three goods has been evaluated at the follow- ing three price vectors: p= (1,2,2); p…

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Assume that demand for three goods has been evaluated at the follow-
ing three price vectors:
p¹= (1,2,2); p² = (2, 1,2); p = (2,2,1);
with income m= 12 in all cases. Consider two different data sets
specifying r¹ r(p¹, 12), x² = x(p², 12), and z³ = x(p, 12).
E
(a) Suppose that x¹ = (4,2, 2), r² = (2, 4,2), and ³ (2,2,4). Show
that there are no cycles in the affordability matrix. What does
this imply for existence of a utility function rationalizing these
choices?
(b) Suppose instead that r¹= (2,3,2),
(2,2,3), and 2³
(3,2,2). Show that there is a cycle in the affordability matrix.
What does this imply for existence of a utility function rational-
izing these choices?

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