Why self-study costs 30% more at A2 and 70% more at C1

Most estimates treat study method as a fixed multiplier: self-study costs some percentage more than a class, and that percentage is the same wherever you are.

Ours does not, and getting this wrong produced the single worst error this model has made.

The bug that forced the change

An early version applied a flat 3× penalty to app-only study at every level. It returned 1,782 hours for A2 Chinese.

A2 is simple everyday exchanges. Eighteen hundred hours is most of a bachelor's degree. Every unit test passed, because the tests checked relations — that harder languages take longer, that higher levels take longer — and every relation was intact. The number was simply, obviously, absurdly wrong, and nothing was checking absurdity.

The fix was not a smaller constant. It was recognising that a constant is the wrong shape.

Why the penalty grows

Each method now has two coefficients, one at the bottom of the range and one at the top, interpolated between:

Three things in that table are worth reading slowly.

Self-study starts cheap and gets expensive. 25% overhead at A1, 70% at C1. At the bottom you are memorising forms, and a good textbook is nearly as good as a teacher at presenting them. At the top you are producing novel sentences, and a textbook cannot tell you that the one you just produced was wrong.

App-only degrades faster and then stops. From 1.3× to 3.5×, and above B1 the model refuses to return a figure at all. Where an app actually stops covers that in full.

Immersion improves with level — the only method that does. 0.9× at A1, 0.75× at C1. Immersion is a poor way to learn your first hundred words and an unbeatable way to learn your last thousand, because the thing it supplies is precisely what the upper levels are made of: unpredictable input and constant correction.

What the shape means in practice

Spanish to B2, an hour a day, five days a week:

Self-study to B2 costs about 235 extra hours against a tutor. At an hour a day that is roughly eight months of your life.

Which suggests a genuinely useful strategy, and it is not "hire a tutor from day one":

Self-study is close to optimal at the bottom. 25% overhead for something free or nearly free is excellent value, and tutor time spent on your first two hundred words is expensive silence.

The overhead compounds where the levels get long. B2 and above is where the flat-rate assumption bankrupts you, and where a person becomes the cheapest thing you can buy.

The classic mistake is the reverse: paying for tutoring at A1 when a textbook would do, running out of money and patience, then attempting B2 alone — buying the expensive option exactly where it is worth least, and the cheap one exactly where it is worth least.

The honest caveat

These coefficients are estimates. FSI publishes hours for its own intensive classes and nothing else; it does not publish a self-study comparison, because it does not run self-study courses.

The relationships are well attested — that self-study is slower, that the gap widens at higher levels, that immersion improves with level — and the specific numbers are ours. They are marked as estimates in the code and in the caveats under every result.

The full methodology, including which figures are published and which are ours, and the calculator if you want to see the difference for your own case.