Theory of Goodness, brought to you by Linear Transforms

Neil and Dennis are discussing how to determine moral worth, when you’re operating in our extremely utilitarian world. You know, to what degree we should care about foreign strangers and not just friends, how we can measure the good intention of anyone short of messianic-perfection (ie, who still occasionally acts in self-interest), and compensating for the different abilities/resources).

I’ve taken the most straw-man-like approach, and am arguing that all that matters is how much you give money to third world charities, and any other charitable action has at least some aspect of self-interest to it. Or maybe it’s just some irrationality, as Neil describes (under his luscious pie theory). But they’re clear on what is good to do, it’s just a question of how you judge others. Under my proposition, it would appear that your moral worth is how much you give to said charities – ie, making the richest of us automatically the most morally accomplished.

Instead, economics would tell me that a person’s moral worth is measured by how much they don’t give to charities (well the negative value of that, of course).

From the beginning, introduction to economics again.

Let’s say I have some money, and my entire enjoyment in the world is determined by: Cars, Computers, and Candy. I decide how to spend my money on those 3 things. Now the enjoyment I get out of each of those for each dollar I spend, are different functions. Specifically, Cars and Computers it’s pretty clear that there’s a certain amount of money I can spend on them, at which point I’ve gotten a machine that can do what I need it to, and from then on it’s only luxury. This is concavity. But let’s assume that for Candy, although I have a small value for it, each piece of Candy is as valuable to me as any other piece, no matter how much I have had already.

To intuit this, imagine I offered you a coin flip. On heads, I took your Computer, but on Tails I got you a second one of it. Would you take the coin flip? Of course not. Even if I offered you $50 to do so? Again, probably not. The second computer just isn’t worth much to you, compared to having 1 computer or none.

Now assume I did the same for your Easter candy. You may or may not take the coin flip, but you wouldn’t be passionate about it either way. If I threw in $5 for it, you’d say sure in a heartbeat.

So my “utility function” would be something like:
Square Root of (Money spent on Cars) plus 3 times Square Root of (Money spent on Computers) plus .1 times (Money spent on Candy).
Or
U = (C1)^.5 + 3*(C2)^.5 + .1*(C3)
where: C1+C2+C3=my total money

(Admittedly, I would in all fact be concave for Candy, but it’s possible to take 3 concave equations and transform them so that one of them is linear and the other two are still concave. Economists and physicists do this all the time, since it’s necessary in order to play with the equations in useful ways. What ways? Glad you asked…)

So this equation is useful for what it says about my spending habits. I will spend entirely on Cars and Computers, up until the point that $1 spent on a Cars or Computers gives me the exact same happiness back as $1 spent on Candy. Because I get less and less enjoyment from Cars and Computers, this has to happen eventually, and after that point any more $1 on Cars or Computer would be less than the happiness I’d get from spending $1 on Candy. So after that point, I will spend entirely on Candy. If anything happened, like my income doubled, all that extra money would just go directly to Candy. (Again, this is assuming I had enough money that I reached the diminishing returns for Cars and Computers).

Now we are in fact concave for all these things, and almost any consumer good we can think of. We all get full, or only have so much time to spend on gadgets, or whatnot. So finding the product we are linear for is very hard.

I would posit that Charity is most likely to be linear, at least if we are anything close to rational. Sure, giving gifts to one person produces diminishing returns, but saving a number of starving children doesn’t. We are probably indifferent between saving 1 starving child for sure, and a coin flip between saving none or 2. Would anyone disagree with this assumption? (At the very least, it is more linear than any other goal)

In particular, it’s very linear with respect to the actions of one person. Oxfam, or Doctors Without Borders, may approach diminishing returns for money after a point sure, but not in the thousands of dollars any of us would give. I would imagine that even the Bill Gates and the Sultan of Brunei could find a network of charities large enough to engulf their money without suffering much diminishing returns.

What does this mean? Well, to the degree that anyone controls all their money (one could argue that most uber-rich people do not given the nature of financial assets), and that people are rational (or at least honestly desire to do “good”), a person would spend a certain amount on things that benefit themselves – and every dollar beyond that goes to charity. This certain amount may be high, but it is capped. Ie, if you have given $1000 to Oxfam, and your income doubles from $30,000 to $60,000, My theory would imagine that you’d next give $31,000 to Oxfam. (Of course other things can change which affect the value you place on self-interested good, or Charity – just not income).

Which then comes down to, I measure your balance of self-interestedness and altruism as when $1 towards consumed goods gives you the same marginal value as $1 to charity. This would not be reflected in how much you gave to Charity, but the amount of the cap that you consumed.

Which gives me things like: poor people (especially those who give some to Charity), have the most benefit of the doubt, even if we don’t know necessarily that their cap is really low; it’s easy to believe that no one in the country really has altruistic drives (eg, that their altruism isn’t linear, and thus it’s about fulfilling some sort of compassion quota, and not how much good the money really does); I should think hard about where my personal cap is; and this is harder to analyze for public servants and people in general who see their work itself (or for politicians even their social context) as altruism (the easiest thing would be to find a comparison of what they could do privately, and how much more money they'd earn).