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Comment by cperciva | Hacker News Reader

If I use credit, I can have utility log(5.5 / (1+r)), with r the interest rate. At 10% interest, that's 3.2. At 20% interest that's 3.0. At 50% interest that's 2.6.

Correct idea, incorrect math. The optimum is found with (11+r)/(2 + 2r) consumption in year 1 and (11+r)/2 consumption in year 2, for a total utility of 2 log(11+r) - 2 log(2) - log(1+r); at (10%, 20%, 50%) interest rates this yields (3.33, 3.26, 3.09) total utility.

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Good catch, I wildly oversimplified. That's what I get for doing arithmetic before coffee.

Key idea is that since utility is concave, smoothing it out is a win. Now just trade off the gains from smoothing against the losses from financing, and you are golden.

Why is utility modeled concavely?

https://tgmstat.wordpress.com/2014/01/22/declining-marginal-...

Due to the Law of diminishing marginal utility.

My utility from eating (in one time period) $800 worth of food is less than 2 x utility of eating $400 worth of food. Doubling my consumption doesn't double my happiness.

Concavity is a way of formalizing this intuition; it means, roughly speaking, that f(2x) <= 2f(x).

Econ uses log(x) as a simple model, but the same idea would apply to any convex function albeit with different arithmetic.

Thanks, that makes sense. I guess that's why flat income tax rates are considered "unfair" since they are modeled on a linear expectation of utility.

Not really. Completely flat percentage taxes (e.g., pay 10% of every dollar you earn) reduce everybody's utility by the same amount if you have a utility = log(dollars) model. Actual "flat taxes" (e.g., pay 10% of every dollar you earn after the first $10k) have a larger impact on the utility of high income earners (at high incomes you approach a change of log(0.9), while at low incomes the impact of the tax on utility reaches zero).

This is however an argument for poll taxes (e.g., "every adult must pay $1000") being unfair, as well as sales taxes on inferior goods (warning: technical term) if the regressive effects aren't compensated by a quasi-fixed-dollar tax rebate.

I don't understand this. If the utility of money is roughly logarithmic, then that would imply the most fair tax bracketing would be a logarithmic curve. An example would be the "logarithmic flat tax" I see mentioned sometimes - figure out how many times above the poverty rate you make, log-10 it, and multiply by some flat constant (they usually recommend 9 or 10). If you plug in a middle-class earner's revenue to that formula, their effective tax rate is far lower than it would be under any flat tax I've seen mentioned. Flat tax in comparison is very regressive.

Logarithmic utility of money implies that the last X% of your income is worth the same amount of utility, regardless of what your income is: log(x) - log[(1 - r)x] = log(x) - log(1 - r) - log(x) = -log(1 - r). But "does the same harm to every person" is not necessarily the right definition of "fair" in the context of taxes, though.

Can you expand on that? What is x, what is r?

And you are saying that the last 5% of a millionaire's income has the same utility as the last 5% of a different person's income that makes 25k/year?

Why are you sure that for you, $800 worth of food is less than 2 x utility of eating $400 worth of food? How would you convince somebody who thinks you are wrong (e.g. deluding yourself, or not telling the truth)?

You could ask them to buy twice as much food as they normally would every time they go to a restaurant or the grocery store. They can either eat until they feel uncomfortable and gain weight or they can start throwing a ton of food away, either way the utility per dollar spent goes way down past a certain point.

Note that this assumes that you are already able to spend an optimal amount on food.

What if you asked them to buy half as much food? Would their default mode be less than twice as good as starving?

In America, it is actually surprisingly difficult to buy 'half as much food' at many places. Most mid-scale (Approx 10-30 USD/plate) restaurants attempt to convince you the service is worth the price by giving you far too much food (and also not paying their employees enough either; hence tips, which BTW I'd be inclined to stop paying in 'living minimum wage' cities (15+/hr)).

You're assuming that 'food' is a single fungible commodity. Spending twice as much on food doesn't mean buying twice as much food, it could also mean buying better quality food and going to higher quality restaurants. I could easily both double and triple my current food spending without eating more or wasting food.

But taste/enjoyment isn't the only purpose of food. You're doubling enjoyment, but not doubling the utility.

If I gave you the choice of only eating $1 of Cup-Noodles every day for 30 days ($30 total), or one single $300 meal at a high class restaurant, and nothing for 29 days, which is the smarter choice?

It doesn't seem like it would apply for all instances and combinations of food. For me, I've had a $90 porterhouse, which was truly delicious, and was indeed worth more (not less) than if I had gotten one $45 alternative cut to eat then and a second one to take home. But extrapolated out for other food / time scales / diet preferences, buying two 5 pound packs of bacon for $40 gives me a lot more utility than $80 worth of pepperoni bits which are quite a bit more expensive per pound but very convenient as a ready-to-eat snack.

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