We claimed in the text that if preferences were monotonic,...

Name: Problem 3, Chapter 4: Intermediate Microeconomics: A Modern Approach
Uploaded: 2022-10-28T04:28:49-08:00
Duration: 4 min 7 s
Channel: Jorge Villanueva
Description: We claimed in the text that if preferences were monotonic, then a diagonal line through the origin would intersect each indifference curve exactly once. Can you prove this rigorously? (Hint: what would happen if it intersected some indifference curve twice?)

We claimed in the text that if preferences were monotonic, then a diagonal line through the origin would intersect each indifference curve exactly once. Can you prove this rigorously? (Hint: what would happen if it intersected some indifference curve twice?)

Key Concepts

Diagonal Line Through the Origin

A diagonal line through the origin in a two-good space represents bundles where the amounts of goods change equally. When preferences are monotonic, moving along this diagonal will either steadily increase or steadily decrease the level of utility, but will not create local extrema. This implies that if the diagonal were to intersect an indifference curve twice, one of the points would have to be strictly preferred over the other, as a higher quantity of both goods leads to a higher utility. This contradiction forces the conclusion that the diagonal intersects each indifference curve exactly once.

Indifference Curves

Indifference curves represent sets of bundles among which a consumer is indifferent; that is, all bundles on the same curve yield the same utility level. Under monotonic preferences, these curves must be downward sloping and cannot intersect each other in a way that violates the ordering of preferences. This ordering property ensures that any deviation along a line where both goods change identically (such as the diagonal line) cannot cross the same indifference curve more than once without contradicting the inherent order imposed by monotonicity.

Monotonic Preferences

Monotonic preferences imply that having more of any good is always better or at least not worse. Therefore, if one point dominates another—in the sense that it has higher amounts of all goods—it must yield a higher level of utility. This fundamental characteristic prevents the possibility of having two different bundles along a continuous line, like a diagonal, being indifferent to each other if one bundle clearly has more goods than the other.

Transcript

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