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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
by cperciva 11y ago
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.
- yummyfajitas 11y agoGood 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.
- infinite8s 11y agoWhy is utility modeled concavely?
- nordsieck 11y agohttps://tgmstat.wordpress.com/2014/01/22/declining-marginal-utility-and-the-logarithmic-utility-function/ https://tgmstat.wordpress.com/2014/01/22/declining-marginal-...
- TJSomething 11y agoDue to the Law of diminishing marginal utility.
- yummyfajitas 11y agoMy 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.
- infinite8s 11y agoThanks, 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.
- cperciva 11y agoNot 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.
- tunesmith 11y agoI 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.
- evanpw 11y agoLogarithmic 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.