Proportional giving was designed for people who didn't even necessarily want to be intrinsically motivated to give money (e.g. paying taxes or perhaps tithing to a church). If you want to raise money from such people, proportional donation aligns the incentives much better than threshold.
That said, there are a couple reasons why it's still useful for effective altruists:
The thing you mentioned about near mode.
As you get older, you gain more ability to buy utility at good prices: for instance, kids become increasingly expensive as they age.
It sets a norm that's easier for people to follow. For instance, fewer people would join Giving What We Can if the pledge were "give everything above $X" instead of "give 10% of your income".
It's more inclusive. Not everyone can give away everything above (e.g.) US$36k. A lot more people can give away 10% of income.
Nevertheless, many effective altruists (e.g. Toby Ord) do practice the fixed-income approach.
There's a difference between what the best course of action for you personally is, and the best recommendation to push towards society at large. The best recommendation to push for has different priorities: short message lengths are easier to communicate, putting different burdens on different people feels unfair and turns people off, and more onerous demands are less likely to be met.
"Give at least 10% of what you make" is low enough to get people on board, conveniently occupies a very nice Schelling point, short enough to communicate effectively, and high enough to get a lot out of the targets it hits. Furthermore, if you want to give more, you're still following the rule, so you can ask people to do the same without hypocrisy.
In short, it's a good social policy to push for and reward those who follow it. Personally, you should follow some kind of weighted utilitarianism, since if you get the utility function good enough then small errors in how you distribute your spending don't make much difference.
As an aside, an altruism-maximizer with a higher income may spend more money on themselves than one with a lower income - usually in the form of buying goods and services that make their income-generating ability better. Say, eating nourishing meals rather than the cheapest available one, so that their work performance goes up.
A compromise that I find appealing and might implement for myself is giving a fixed percentage over a fixed amount, with that fixed percentage being relatively high (well above ten percent). You could also have multiple "donation brackets" with an increased marginal donation rate as your income increases.
One factor you didn't mention is that significantly harder working people might need to spend more income to maintain their productivity. E.g. traders working 80-hour weeks might spend more on food because they don't have time to cook, more on clothes because their job requires signaling, more on airplane tickets because they have short-notice time-sensitive tasks requiring travel, etc. But there's still no reason this should work out to anything like proportional giving, so your point stands.
A pro of proportional giving is that it's dead simple to budget and keep in mind.
It can also get people to give a certain amount continually and keep them emotionally invested in charitable effort, while allowing them plenty of savings for contingencies and perhaps an eventual larger lump sum donation.
And even if it's not optimal, "indefensible" is a pretty strong claim.
I try to apply the hybrid philosophy of: "work hard, make money, give what you can afford to give today, give yourself a near-mode incentive to make more money, and create savings for the future".
The savings one is a big one you didn't mention. I feel like my future self has a high chance to smack me if I commit too much money now. It seems like the places I would give today can benefit more now from my donation than in the future, but I may find better places to spend the money or give it away in the future. Hence the hybrid.
An additional consideration is that a parliament of agents that cannot bind themselves to a contract would probably agree to split their resources in a way that reflects their political power. So if you model yourself as a parliament of agents, some egoist and some altruist, donating a fixed portion of your wealth and income can be "rational" in the sense that it is the result of your subagents behaving rationally in the context of politics. I think this is actually a reason people give a fixed portion of their income.
If you're an egoist, giving a fixed portion of your income has a fixed cost (since the marginally utility of money is approximately inversely proportional to what you have) and gives you a fixed amount of prestige, or signalling or whatever. That sort of thinking is useful for maximizing inclusive genetic fitness, so it's no surprise that we think that way.
This seems obviously correct to me. In my experience this is not obvious to everyone and many people find it a bit distasteful to talk about. I'm glad you bring it up.
I haven't really tried hard, but I think I would find it pretty difficult to get myself to behave this way.
The way I "resolve" this dissonance is by thinking in terms of a parliamentary model of me. Parts of me want to be altruistic and part of me is selfish and they sort of "vote" over the use of resources.
proportional giving is good because it's a kind of giving that you can get a lot of people to do. Not for weird math reasons. I agree with what you're saying given utilitarian calculations but I don't think you're doing the right calculation.
One effect of income is (supposedly) to incentivize work,
A way of modelling this is replacing u(x) with a u(X, L), where L is the number of non-work hours in a week, and replacing S with (168 - L)W, where 168 is the total number of hours in a week and W is your hourly wage. Then you maximize u((168 - L)W - g, L) + wtg with respect to L and g.
Given that there are competing incentives (altruism vs. egoism and others) it is clear that some trade-off is needed. An optimum solution might be too complex to be understood and accepted as fair by everyone. Thus we have to look for simple approximations.
The simplest are
1 give nothing: g(x)=0
2 give a fixed amount: g(x)=c
3 give a fixed fraction: g(x)=fx
4 give everything above a treshold: g(x)=x>c?x-c:0
5 give a fixed fraction above a treshold: g(x)=x>c?f(x-c):0
6 give a by second order polynomial: g(x)=ax^2+bx+c
7 give a by second order polynomial wi...
Executive summary: The practice of giving a fixed fraction of one's income to charity is near-universal but possibly indefensible. I describe one approach that certainly doesn't defend it, speculate vaguely about a possible way of fixing it up, and invite better ideas from others.
Many of us give a certain fraction of our income to charitable causes. This sort of practice has a long history:
Deuteronomy 14:22 Thou shalt truly tithe all the increase of thy seed, that the field bringeth forth year by year.
(note that "tithe" here means "give one-tenth of") and is widely practised today:
GWWC Pledge: I recognise that I can use part of my income to do a significant amount of good in the developing world. Since I can live well enough on a smaller income, I pledge that from today until the day I retire, I shall give at least ten percent of what I earn to whichever organizations can most effectively use it to help people in developing countries. I make this pledge freely, openly, and without regret.
And of course it's roughly how typical taxation systems (which are kinda-sorta like charitable donation, if you squint) operate. But does it make sense? Is there some underlying principle from which a policy of giving away a certain fraction of one's income (not necessarily the traditional 10%, of course) follows?
The most obvious candidate for such a principle would be what we might call
Weighted Utilitarianism: Act so as to maximize a weighted sum of utility, where (e.g.) one's own utility may be weighted much higher than that of random far-away people.
But this can't produce anything remotely like a policy of proportional giving. Assuming you aren't giving away many millions per year (which is a fair assumption if you're thinking in terms of a fraction of your salary) then the level of utility-per-unit-money achievable by your giving is basically independent of what you give, and so is the weight you attach to the utility of the beneficiaries.
So suppose that when your income, after taking out donations, is $X, your utility (all else equal) is u(X), so that your utility per marginal dollar is u'(X); and suppose you attach weight 1 to your own utility and weight w to that of the people who'd benefit from your donations; and suppose their gain in utility per marginal dollar given is t. Then when your income is S you will set your giving g so that u'(S-g) = wt.
What this says is that a weighted-utilitarian should keep a fixed absolute amount S-g of his or her income, and give all the rest away. The fixed absolute amount will depend on the weight w (hence, on exactly which people are benefited by the donations) and on the utility per dollar given t (hence, on exactly what charities are serving them and how severe their need is), but not on the person's pre-donation income S.
(Here's a quick oversimplified example. Suppose that utility is proportional to log(income), that the people your donations will help have an income equivalent to $1k/year, that you care 100x more about your utility than about theirs, and that your donations are the equivalent of direct cash transfers to those people. Then u' = 1/income, so you should keep everything up to $100k/year and give the rest away. The generalization to other weighting factors and beneficiary incomes should be obvious.)
This argument seems reasonably watertight given its premises, but proportional giving is so well-established a phenomenon that we might reasonably trust our predisposition in its favour more than our arguments against. Can we salvage it somehow?
Here's one possibility. One effect of income is (supposedly) to incentivize work, and maybe (mumble near mode mumble) this effect is governed entirely by anticipated personal utility and not by any benefit conferred on others. Then the policy derived above, which above the threshold makes personal utility independent of effort, would lead to minimum effort and hence maybe less net weighted utility than could be attained with a different policy. Does this lead to anything like proportional giving, at least for some semi-plausible assumptions about the relationship between effort and income?
At the moment, I don't know. I have a page full of scribbled attempts to derive something of the kind, but they didn't work out. And of course there might be some better way to get proportional giving out of plausible ethical principles. Anyone want to do better?