One of my current non-work projects is to learn German. My goal is to be able to read German; speaking it would be a bonus, but not as important to me. Having had a year of college German and listened responsively to the 5 levels of Pimsleur German (which I highly recommend), I don’t need more grammar for my purposes, just vocabulary. I was pleased to discover in a set of 4 “German Frequency Dictionaries” 10,000 German words listed in order of frequency with excellent example sentences, and part-of-speech lists. (I have to confess that the example sentences often emphasize interpersonal conflict!)

All I have to do is learn 10,000 words and I’ll be able to read German. (The books claim that only 5% of written German words are outside the top 10,000 words, and I figure those 5% are likely to be quite genre-specific and therefore quick to pick up for any given genre.)

But how hard should I work on learning German each day? Let me approach this as an abstract optimization problem that could be applied to other similar problems. (Brownie points are available for suggesting in a comment other problems the following math is relevant for.)

First, notation:

The idea is then to minimize the total cost, which equals the daily cost times the number of days to completion:

Minimizing the natural logarithm of the total cost has the same optimal speed:

The first-order condition for the logarithmic version of the problem is:

Rearranging the algebra, the first-order condition becomes:

This has a nice interpretation: the elasticity of the daily cost of speed should be set equal to the ratio by which the total cost (including both the cost of speed and the cost of delay) exceeds the cost of speed alone.

If the daily cost of speed is increasing in speed, the right-hand side of this first order condition is decreasing in x. If, in addition, the elasticity of the daily cost of speed is increasing in speed, then any solution to the first-order condition will provide a unique solution for the optimal speed. Not surprisingly, for a given functional form of the cost of speed, the optimal speed will be higher the higher the cost of delay.

To me, this suggests that I should study German relatively fast. I don’t like not being able to read German. I think I have identified the right technology for learning it, having done Pimsleur German, with the books I have and using the memory techniques I talk about in “The Most Effective Memory Methods are Difficult—and That's Why They Work.” And on any day that is only medium busy (a little easier to come by during this pandemic), I don’t think the elasticity of the daily cost rises much above one until I get above an hour a day on German study.

See if this kind of logic helps you with any practical decision that you have. Basically, it says we should get things done fast unless there is a relatively-quickly-increasing cost to speed.

The extension to multiple projects each with this kind of costs and benefits turns out to be very interesting. I plan to do another blog post or two on that in the new year.