Convexity and the Value of Mixtures

The mixture test

Why convexity is plausible

What convexity buys in optimization

Testing convexity with numbers

Which statement defines convexity of preferences precisely?

Bundles $(2, 8)$ and $(8, 2)$ both lie on the level-16 curve of $u = x_1 x_2$. What is the utility of their midpoint $(5, 5)$?

What does strict convexity guarantee about the tangency solution of the consumer's problem?

Which preference pattern would violate convexity and threaten the uniqueness of tangency solutions?

For $u = x_1 x_2$, bundles $(3, 6)$ and $(6, 3)$ both lie on one indifference curve. Compute the utility of their midpoint $(4.5, 4.5)$, and write the number.

For the same utility, bundles $(1, 18)$ and $(18, 1)$ lie on the level-18 curve. Compute the utility of their midpoint $(9.5, 9.5)$, and write the number.

Bundles (1, 18) and (18, 1) both sit on the level-18 curve. Their midpoint scores 90.25, which exceeds 18. What does that prove?

How does strict convexity rule out two distinct optimal bundles on one budget line?

Which pattern violates convexity and threatens uniqueness of the tangency?

What is the risk-side reading of convexity of preferences over bundles?

Bundles (2, 8) and (8, 2) lie on the level-16 curve. Their average is (5, 5). Compute its utility and write the number.

Bundles (1, 4) and (4, 1) lie on the level-4 curve. Their average is (2.5, 2.5). Compute its utility with two decimals and write the number.