Why the Mathiness in Lucas (2009) Matters

Lucas (2009) makes a misleading claim about models of growth:

S1: {adding books} => {no change in growth}

David Andolfatto admits that S1 is false and the true statement is

{adding books} => {{no change in growth} or {sustained change in growth}}.

But he professes not to see why anyone should care that Lucas engages in this type of mathiness; that is, why anyone should care that Lucas makes a misleading verbal statement about the mathematics of growth theory.

Andolfatto misses what is at stake. Lucas does not make statement S1 because he cares about the math in other models. He makes it to hide a problem with the math of his own model. Hiding problems with the math of your own model is the kind of thing that usually we care about.

Recall how Lucas’s model works. Bits of knowledge are indexed by the positive real numbers. Someone who has the knowledge associated with the index value x > 0 has productivity equal to x units of output per year. Every piece of knowledge on the positive half line is already known by some person who has this index value.

Let average output per worker be y. Now pick a big number. Googolplex will do. There is a person we can call Mr. Googolplex, with productivity that is googolplex times y. When Mr. Googolplex bumps into another worker, he explains what he knows and the productivity of the other worker increases (on average) by a factor of googolplex.

In the model, this can’t happen often because Mr. Googolplex can only bump into only a few people each year. But suppose that Mr. Googolplex could write what he knows in a book and send a copy to everyone. If the book can record the words he would say when he bumps into someone in the hall, anyone who reads the book can enjoy the same increase in productivity that they’d get from a hallway encounter. So output in the economy will increase by a factor of googolplex. Googolplex is a big number so Mr. Googolplex and his potential readers have an incentive to make this happen.

Usually, in economic models, the theorist assumes that agents do things that they have an incentive to do. But to limit the growth rate in the model, Lucas has to hold fast to assumptions that severely limit how knowledge can pass from one person to the next.

S1 comes as Lucas starts to extend the basic model, on the way to what he describes as a model of schooling, which one might think would involve books in some way. At this juncture, S1 is pure misdirection. It points to one question: What do books do in other models? Its real job is to divert attention from a different question: What would books do in Lucas’s model own model?

S1 implies that what economists “know” is that bringing in books does not have any important effect on the rate of growth. What Lucas has to hide is that if he adds books to his model, growth increases by so much that the wheels fly off and the model shakes itself to pieces.