This is what I did, without the pedantry of the C.
The point is that because the constant is there, saying that utility grows logarithmically in money underspecifies the actual function. By ignoring C, you are implicitly using $1 as a point of comparison.
A generous interpretation of your claim would be to say that to someone who currently only has $1, having a billion dollars is twice as good as having $50000 -- in the sense, for example, that a 50% chance of the former is just as good as a 100% chance of the latter. This doesn’t seem outright implausible (having $50000 means you jump from “starving in the street” to “being more financially secure than I currently am”, which solves a lot of the problems that the $1 person has). However, it’s also irrelevant to someone who is guaranteed $50000 in all outcomes under consideration.
By comparing changes in utility as opposed to absolute values.
To the person with $50000, a change to $70000 would have a log utility of 0.336, and a change to $1 billion would have a log utility of 9.903. A change to $1 would have a log utility of −10.819.
The point is that because the constant is there, saying that utility grows logarithmically in money underspecifies the actual function. By ignoring C, you are implicitly using $1 as a point of comparison.
A generous interpretation of your claim would be to say that to someone who currently only has $1, having a billion dollars is twice as good as having $50000 -- in the sense, for example, that a 50% chance of the former is just as good as a 100% chance of the latter. This doesn’t seem outright implausible (having $50000 means you jump from “starving in the street” to “being more financially secure than I currently am”, which solves a lot of the problems that the $1 person has). However, it’s also irrelevant to someone who is guaranteed $50000 in all outcomes under consideration.
Then how do you suggest the person under discussion evaluate their working patterns if log utilities are only useful for expected values?
By comparing changes in utility as opposed to absolute values.
To the person with $50000, a change to $70000 would have a log utility of 0.336, and a change to $1 billion would have a log utility of 9.903. A change to $1 would have a log utility of −10.819.
I see, thanks.