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Has anyone developed a quantitative theory of personal finance in the following sense?
Most money advice falls back on rules of thumb; I'm looking for an approach that's made-up numbers all the way down.
The main idea would be to express utility as a function of financial quantities; an obvious candidate would be utility per unit time equals the log of money spent per unit time, making sure to count things like imputed rent on owned property as spending. Once you have that, there's an exact answer to the optimal risk/reward tradeoff in investments, how much to save/borrow, etc.
You may want to rephrase that :-)
No, I don't think so. For example, let's say your utility = log(wealth). That's a monotonous transformation, so if you want to maximize utility you just maximize your wealth. That doesn't answer the question of what is the appropriate risk/reward trade-off because you haven't even started talking about risk yet. And if you just want to maximize expected wealth you are open to being Pascal-mugged.