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# The Present Value of Learning, Adjusted for Forgetting

 Optimal Open-Mindedness... Tim Taylor on Minimum Wage Inc...
Suppose learning marginal fact F increases your productivity by V.  What is the present value of learning F?  Economists will be tempted to mechanically apply the standard present value formula.  Using discrete time to keep things simple:

PDV(F)=V + V/(1+r) + V/(1+r)^2 + V/(1+r)^3 + ... = V/r.

If V=\$100, and r=5%, the present value of learning F is \$2000.

As Treebeard would say, however, "Don't be hasty."  When you learn F, this hardly implies that you will know F forever.  Every observant teacher knows that the opposite is closer to the truth: students usually quickly forget what they learn.

How quickly?  A large literature on summer learning loss finds that students lose roughly one month of learning for every three months they spend out of school.  As a major meta-analysis explains:
The meta-analysis indicated that the summer loss equaled about one month on a grade-level equivalent scale, or one tenth of a standard deviation relative to spring test scores.
This result probably doesn't hold along the entire range of learning.  As a previous post documented, students who master a subject by overlearning have excellent retention.  But the summer learning loss estimate seems fully applicable to marginal facts.  So let's apply it:

PDV(F)=V + .66V/(1+r) + .33V/(1+r)^2

[Note the absence of ellipses at the end of the equation!]

If V=\$100, and r=5%, the present value of learning F is just \$192.79.  The present value of learning adjusted for forgetting is over 90% less than the present value without forgetting.

And that is why economists must never forget to adjust for forgetting!  Arrested Development allusion intentional.

writes:

Mark Twain: The trouble with the world is not that people know too little, but that they know so many things that ain't so.

Maybe it's a good thing we forget what we "know."

I might buy the summer-learning-loss formula for random facts (which team was it that won the World Series?), but conditioning the fact being immediately and continuously relevant to productivity, the NPV formula makes a lot more sense. If F increase my productivity by \$100 this year, that implies some use/rehearsal that will aid retention as well as some incentive to remember.

Greg B writes:

I think an important thing to keep in mind is that repetition is key to maintaining memory. If we accept some level of human capital growth through education (even the very low values for such that you would accept), then the education which is relevant to productivity is composed of material which would be recalled on a regular basis.

I would imagine this is also a partial explanation for why overlearning increases retention so dramatically for basic skills. Algebra is required and used regularly in a standard calculus curriculum, so it is remembered not because the marginal learning at the edge of calculus provide a buffer that is forgotten first, but instead it is remembered because it is an oft-utilized tool-set; the necessity of repetition during calculus studies facilitates the retention of algebra learning.

Our current understanding of human memory is that it is functionally limitless, but our access mechanism is (by "design") limited by constraints based on how recent, relevant, and repeatedly used the memory at hand is.

writes:

Forgetting learned facts during summer (or completion of schooling) can be significantly mitigated by using spaced repetition software. Unfortunately, I only learned of this after completing university, so this analysis applies to the large majority of my formal education.

Vipul Naik writes:

The first formula should be V(1 + r)/r on the right side, though if r is small, your approximation is fine.

Slocum writes:

If F increase my productivity by \$100 this year, that implies some use/rehearsal that will aid retention as well as some incentive to remember.

Yes, and what's more -- there's a pretty good chance that the value of a given fact or skill will decline over time (perhaps because of changes in your industry or maybe because of changes in your role).

Brains have limited capacity, and forgetting is adaptive -- if the piece of knowledge continues to be useful and you continue to use it, then you won't forget it. But if stops being used, that's a good sign that it's no longer valuable (to you anyway), and forgetting is appropriate.

Of course, you grossly over-estimate how much learning increases productivity.

So specialize, specialize, specialize!

AS writes:

Perhaps this is still wrong--it still needs to be adjusted for the fact that we remember things that are more important to us. We're not forgetting just anything.

Jeff H writes:

I think you need to adjust the formula two address a couple things.

You don't always need to "know" the details of something. You only need to "know" enough to get you to the next level. For example, I don't remember every detail of my econometrics courses, but I run regressions almost every day. I needed to learn (a) enough to convince my professors that I knew enough of it to move on to actually using Stata to implement the methods and (b) to know the highlights of the non-common problems, methods, etc.

Basically, (a) means there is some hurdle I need to get over until I reach the next level. (Also, it doesn't have to just be something as mundane as convincing a professor.)

(b) means I don't have to have perfect recall, that is, I don't actually have to know all of the details, but I need to be familiar with them. This could be something more important like knowing the general distrubution of some function and knowing that a variable is unlikely to follow it, but not knowing everything about it.

In other words, the loss from not remembering (1) diminishes at a different rate from the benefits of remembering, so you should probably have different discount rates and (2) is only relevant for a shorter span, so maybe discount only half as long as the original material.

Also, there are some studies (can't link to any right now) that have shown that youth are now remembering less simply because they don't have to. With Google, etc., what's becoming more common is knowing how to get the information, not the information itself. It that case, you wnat the present value of knowing a relevant information, not all the information that is taught. So you probably want to overlearn stuff just to make the high-level topic stick.

writes:

Learning loss is one of the reasons that I think schools should focus on drilling simple principles into the students rather than all the complicated stuff they focus on. Even the principles of physics are pretty simple, it is the math that describes them that keeps real learning out of reach to average and below students and most everyone forgets that stuff anyway. It is useful to most everyone to know that in a vacuum an item dropped accelerates as it falls. It is rarely useful to know that the acceleration is 32 feet per second per second. We test the latter, an teach the former insufficiently.

MingoV writes:
... As Treebeard would say, however, "Don't be hasty." When you learn F, this hardly implies that you will know F forever....
The situation is complicated by more than just forgetting. Even if you remember F until the day you die, F may become outdated and lose most or all of its value.
daubery writes:

In so far as learning creates human capital I think this is much less of an issue. Eg. if I spend time learning how to be a nurse, then actually work as a nurse, I would bet that I will not forget my nursing skills much as I am constantly practising them.

And if learning isn't creating human capital - I'm memorising Virgil to show I have high compliance and boredom tolerance - then forgetting it doesn't matter. The [social] value is \$0 to start with, and the private value is showing I can tolerate boredom, which is a pre-existing personality trait, not knowledge I am going to gradually forget, so the private value should not fade out.

So a very interesting argument but I'm not sure it changes matters if we accept signalling to start with.

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