But, this is weird, what if you have an indifference curve like this:

http://www.peacefire.org/peanuts-and-root-beer.2.jpg

That’s still logically conceivable under the rule that utility has to be nondecreasing if your amount of either good increases… And even if the utility function has to be twice differentiable, I think you can still have two-humped indifference curves (can’t you?).

Then you can make the surrounding curves such that if Frieda’s income increases, she stays on one hump, but if her income decreases, she jumps to the other hump.

Then her utility function is twice differentiable, but the amount of root beer she purchases, as a function of income, is not even a continuous function. (And then you can structure the curves so that if the price of peanuts increases, she consumes less root beer, but if her income goes up, her root beer consumption could either increase or decrese.)

If not, then you can make indifference curves like these:
http://www.peacefire.org/peanuts-and-root-beer.jpg

In that curve, if the price of peanuts rises, then Frieda consumes less root beer. But if her income goes up, her root beer consumption goes down. On the other hand, in my graph, if her income goes *down*, her root beer consumption goes down as well.

Presumably when you say “Frieda’s root beer consumption moves in the same direction as her income,” that’s where you’re assuming it’s differentiable — that if her income moving one direction causes her root beer consumption to move one way, then her income moving in the opposite direction must cause her root beer consumption to move the opposite way.

Coming up with bizarre counterexamples is fun :)

You can draw indifference curves such that if the price of peanuts rises, Frieda reduces root beer consumption, and then if her income goes up by some amount, her root beer consumption goes up. On the other hand, you can also draw indifference curves such that if the price of peanuts rises, Frieda reduces root beer consumption, and then if her income goes up by some amount, her consumption of root beer goes *down*.

Example from real life: If the price of pork-flavored Ramen noodles goes up, a college student will increase their consumption of beef-flavored Ramen noodles. However, once the student graduates and their income goes (way) up, their consumption of beef-flavored Ramen noodles will often drop to zero as they start eating real food.

Would I be correct in assumping that the question should more precisely say, “Given a small positive delta change in Frieda’s income, will her root beer consumption go up or down by a small delta amount?” (Rather than asking how Frieda’s root beer consumption will change given an arbitrarily large change in Frieda’s income.)

Actually, it is definitionally most useful to have different names for different concepts. Utility maximization is one thing; efficiency is another. It’s good to keep them straight.

I’m not sure how Gates’ loud music applies here; that is a good argument for indulging the whims of those who control resources while this is a question of whether pure transfers are always to be considered neutral.