[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.

[CuriousCosmic]

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.

[jvanderbot]

You're taking an example that doesn't make sense in the context of the article. TFA is most likely intended to mean "progress" of some sort. The double integral of velocity means what?

[CuriousCosmic]

    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. 

Does this (from my reply to you) not cover that exact circumstance?

TFA mentions this as well:

    For example I often hear conversations the first week of class where somebody will be bemoaning, "Oh so-and-so knows blah-blah-blah, how am I ever going to catch up to them?" Well, if you're one of the people who knows blah-blah-blah it's bad news for you because honestly everyone is going to catch up really quickly. Before you know it that advantage is going to be gone and if you aren't learning too you're going to be behind.

or

    Another example is hiring. Before I came back to academia a couple of years ago I was out doing startups. What I noticed is that when people hire they are almost always hire based on experience. They're looking for somebody's resume trying to find the person who has already done the job they want them to do three times over. That's basically hiring based on Y-intercept.

These examples of the `y` are knowledge or immediate utility/skill. Integrate these and you get `Y` which can be viewed as the application of that knowledge or skill over a period of time. Aka total contributions over the span of your lifetime or over the span of your employment/involvement.

Point being that while TFA is right that "a little bit of slope makes up for a lot of y-intercept", you can still have a smaller integral if the numbers don't happen to work out. TFA is an encouragement to try hard and push yourself to get ahead without being discouraged but it makes the assumption that someone with a steep learning/skill slope and low starting experience will eventually match the person with experience but a shallow skill slope. This works if you extrapolate out to infinity but if someone is only going to work for you for 2-5 years, you have to actually do the math to figure out which will likely perform the best.

[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.