EA discussions often assume that the utility of money is logarithmic, but while this is a convenient simplification, itās not always the case. Logarithmic utility is a special case of isoelastic utility, a.k.a. power utility, where the elasticity of marginal utility is Ī·=1. But Ī· can be higher or lower. The most general form of isoelastic utility is the following:
u(c)={c1āĪ·ā11āĪ·Ī·ā„0,Ī·ā 1ln(c)Ī·=1
Some special cases:
When Ī·=0, we get linear utility, or u(c)=c.
When Ī·=0.5, we get the square root utility function, u(c)=2(ācā1).
When Ī·=1, we get the familiar logarithmic utility function, u(c)=ln(c).
For any Ī·>1, the utility function asymptotically approaches a constant as c approaches infinity. When Ī·=2, we get the utility function u(c)=1ā1/c.
Ī· tells us how sharply marginal utility drops off with increasing consumption: if a person already has k times as much money as the baseline, then giving them an extra dollar is worth (1/k)Ī· times as much. Empirical studies have found that Ī· for most people is between 1 and 2. So if the average GiveDirectly recipient is 100x poorer than the average American, then it could be 100 to 10,000 times as valuable to give money to GiveDirectly than to spend the same money on yourself.
The ratio of (jargon+equations):complexity in this shortform seems very high. Wouldnāt it be substantially easier to write and read to just use terms and examples like āa project might have a stair-step or high-threshold function: unless the project gets enough money, it provides no return on investmentā?
Or am I missing something in all the equations (which I must admit I donāt understand)?
Iām basically saying that the logarithmic utility function, which is where we get the idea that doubling oneās income from any starting point raises their happiness by the same amount, is a special case of a broader class of utility functions, in which marginal utility can decline faster or slower than in the logarithmic utility function.
a project might have a stair-step or high-threshold function: unless the project gets enough money, it provides no return on investment
All of the maths here assumes smooth utility returns to money; there are no step functions or threshold effects. Rather, it discusses different possible curvatures.
I wasnāt trying to imply that was the only possibility, I was just highlighting step/āthreshold functions as an example of how the utility of money is not always logarithmic. In short, I just think that if the goal of the post is to dispute that simplification, it doesnāt need to be so jargon/āequation heavy, and if one of the goals of the post is to also discuss different possible curvatures, it would probably help to draw rough a diagram that can be more-easily understood.
My fan fiction about what is going on in this thread:
A good guess is that ālog utilityā is being used by EAs for historical reasons (e.g. GiveWellās work) and is influenced by economics, where log is used a lot because it is extremely convenient. Economists donāt literally believe people have log utility in income, it just makes equations work to show certain ideas.
Itās possible that log utility actually is a really good approximation of welfare and income.
But sometimes ideas or notions get codified/ācanonized inappropriately and accidentally, and math can cause this.
With the context above, my read is that evelynciara is trying to show that income might be even more important to poor people than believed.
Sheās doing this in a sophisticated and agreeable way, by slightly extending the math.
So her equations arenāt a distraction or unnecessary mathematical, itās exactly the opposite, sheās protecting against mathās undue influence.
I was hoping for a more dramatic and artistic interpretation of this thread, but Iāll accept whatās been given. In the end, I think there are three main audiences to this short form:
People like me who read the first sentence, think āI agree,ā and then are baffled by the rest of the post.
People who read the first sentence, are confused (or think they disagree), then are baffled by the rest of the post.
People who read the first sentence, think āI agree,ā are not baffled by the rest of the post and say āYep, thatās a valid way of framing it.ā
In contrast, I donāt think there is a large group of people in category 4. Read the first sentence, think āI disagree,ā then understand the rest of the post.
But do correct me if Iām wrong!
Well, I donāt agree with this perspective and its premise. I guess my view is that it doesnāt seem compatible for what I perceive as the informal, personal character of shortform (like, ālive and let liveā) which is specifically designed to have different norms than posts.
I wonāt continue this thread because it feels like Iām supplanting or speaking for the OP.
Utility of money is not always logarithmic
EA discussions often assume that the utility of money is logarithmic, but while this is a convenient simplification, itās not always the case. Logarithmic utility is a special case of isoelastic utility, a.k.a. power utility, where the elasticity of marginal utility is Ī·=1. But Ī· can be higher or lower. The most general form of isoelastic utility is the following:
u(c)={c1āĪ·ā11āĪ·Ī·ā„0,Ī·ā 1ln(c)Ī·=1
Some special cases:
When Ī·=0, we get linear utility, or u(c)=c.
When Ī·=0.5, we get the square root utility function, u(c)=2(ācā1).
When Ī·=1, we get the familiar logarithmic utility function, u(c)=ln(c).
For any Ī·>1, the utility function asymptotically approaches a constant as c approaches infinity. When Ī·=2, we get the utility function u(c)=1ā1/c.
Ī· tells us how sharply marginal utility drops off with increasing consumption: if a person already has k times as much money as the baseline, then giving them an extra dollar is worth (1/k)Ī· times as much. Empirical studies have found that Ī· for most people is between 1 and 2. So if the average GiveDirectly recipient is 100x poorer than the average American, then it could be 100 to 10,000 times as valuable to give money to GiveDirectly than to spend the same money on yourself.
Source: āThe value of money going to different groupsā by Toby Ord
see: https://āāpapers.ssrn.com/āāsol3/āāpapers.cfm?abstract_id=1096202
The ratio of (jargon+equations):complexity in this shortform seems very high. Wouldnāt it be substantially easier to write and read to just use terms and examples like āa project might have a stair-step or high-threshold function: unless the project gets enough money, it provides no return on investmentā?
Or am I missing something in all the equations (which I must admit I donāt understand)?
Iām basically saying that the logarithmic utility function, which is where we get the idea that doubling oneās income from any starting point raises their happiness by the same amount, is a special case of a broader class of utility functions, in which marginal utility can decline faster or slower than in the logarithmic utility function.
All of the maths here assumes smooth utility returns to money; there are no step functions or threshold effects. Rather, it discusses different possible curvatures.
I wasnāt trying to imply that was the only possibility, I was just highlighting step/āthreshold functions as an example of how the utility of money is not always logarithmic. In short, I just think that if the goal of the post is to dispute that simplification, it doesnāt need to be so jargon/āequation heavy, and if one of the goals of the post is to also discuss different possible curvatures, it would probably help to draw rough a diagram that can be more-easily understood.
My fan fiction about what is going on in this thread:
A good guess is that ālog utilityā is being used by EAs for historical reasons (e.g. GiveWellās work) and is influenced by economics, where log is used a lot because it is extremely convenient. Economists donāt literally believe people have log utility in income, it just makes equations work to show certain ideas.
Itās possible that log utility actually is a really good approximation of welfare and income.
But sometimes ideas or notions get codified/ācanonized inappropriately and accidentally, and math can cause this.
With the context above, my read is that evelynciara is trying to show that income might be even more important to poor people than believed.
Sheās doing this in a sophisticated and agreeable way, by slightly extending the math.
So her equations arenāt a distraction or unnecessary mathematical, itās exactly the opposite, sheās protecting against mathās undue influence.
I was hoping for a more dramatic and artistic interpretation of this thread, but Iāll accept whatās been given. In the end, I think there are three main audiences to this short form:
People like me who read the first sentence, think āI agree,ā and then are baffled by the rest of the post.
People who read the first sentence, are confused (or think they disagree), then are baffled by the rest of the post.
People who read the first sentence, think āI agree,ā are not baffled by the rest of the post and say āYep, thatās a valid way of framing it.ā
In contrast, I donāt think there is a large group of people in category 4. Read the first sentence, think āI disagree,ā then understand the rest of the post. But do correct me if Iām wrong!
Well, I donāt agree with this perspective and its premise. I guess my view is that it doesnāt seem compatible for what I perceive as the informal, personal character of shortform (like, ālive and let liveā) which is specifically designed to have different norms than posts.
I wonāt continue this thread because it feels like Iām supplanting or speaking for the OP.