Question 7
Suppose we have three goods and we know that the consumer has strictly convex and strictly monotonic preferences. If we can run an unlimited number of experiments where we expose the consumer to prices \((p_{1},p_{2},p_{3})\) and give him income \(y\), can we fully learn the consumer’s preferences?
Correct: We can always entirely learn the consumer’s preferences.
False: We might not be able to learn the consumer’s preferences.
Explanation:
In three dimensions, the budget set is given by \(\{\left(x_{1},x_{2},x_{3}\right):p_{1}x_{1}+p_{2}x_{2}+p_{3}x_{3}\leq y\}\), where \(y\) is the consumer’s disposable money. Geometrically, this set is bounded by a plane (with equation \(p_{1}x_{1}+p_{2}x_{2}+p_{3}x_{3}=y\)). The diagram below displays one such plane (in green).
Since the consumer has strictly convex and strictly monotonic preferences, the indifference sets are given by surfaces like the blue surface in the diagram below: