[throwawaytemp29]

Isn’t the integral of the blue curve higher though? Like if I want to maximize total utility over the displayed time the blue would be higher.

Also time has value, getting something earlier is generally better due to compound interest. Even some vague utility function like fun can display such a property of being better earlier, due to being able to remember the memory for longer.

[acjohnson55]

Yes, it's important to have some concept of time budget for leveling up and expected tenure in the role.

Sometimes, you really do need someone who will be a heavy hitter on day 1. Other times, you can afford to wait to let someone mature.

[jvanderbot]

It's wrong to integrate when the target value is the value of Y. The integral of everything youve learned and accomplished _is_ your y value. A double integral doesn't make obvious sense to me.

[jacoblambda]

Is it?

For example, if you take `y` to be quality of life, you obviously want the highest quality of life you can get but what really matters is the integral `Y` quality of life over the course of your lifespan.

A steeper slope that starts you with a much worse QOL isn't inherently better just because the end of your life is spent with a high QOL. Doubly so as depending on how age effects your ability to do the things you enjoy or the experiences you form/retain, the true function you care about (let's call it `z` and `Z`) may decrease the impact of `y` with time. Even more so when you don't know what lies in your future and/or how long you'll be around.

This applies to knowledge and utility as well. Your immediate utility `y` is an integral. It's the aggregation of your accumulated knowledge. However the integral of this, `Y` is the total utility throughout your life. You may be more immediately useful with the steeper red slope later on but you get more total work done with the shallower blue slope.

[xigoi]

The more sloped line will have a higher integral too, as long as you let it go to double the point where it crossed the other line.