Is there a model for how much a philosopher produce in his lifespan? Is there a model for how much a philosopher produce in his lifespan?

If not, state the economical reason for why we pay the philosopher.
Here is a somewhat creative (and very imprecise) answer:

Let's assume that Philosophy creates no direct value for society (which seems implicit in your question). However, Mathematics and various other scientific disciplines to some extent originated from Philosophy. Most recently, Linguistics originated as a field of Philosophy. Based on this we can estimate the marginal product of a philosopher. You will need two numbers:

A: Take the marginal value of each invention generated by these fields. For example, speech recognition on your cell phone would not have been possible without Linguistics and thus Philosophy. You can estimate the marginal value of speech recognition by your willingness to pay for a phone with or without speech recognition. Do this for all products and services of an economy. (You can also take market prices and make some strong assumptions about the economy, such as competitive markets, etc.). Using time discounting, you can estimate the value of having a product one year earlier.

B: You must now assign the marginal contribution of a philosopher to the invention of linguistics to the speech recognition software. The marginal contribution seems to be infinite, since without linguistics there would be no speech recognition. However, you may simply assume that each additional Philosopher speeds up the process of the invention of linguistics by an equal amount. Thus, if you would have hired one more philosopher, the time needed to invent Linguistics would have been N/(N+1) the time N philosophers needed to actually invent linguistics.

Combining the information from A and B for all products gives you the marginal product of each philosopher. Note that this is only a rough answer and one would still need to carefully formulate the assumptions. However, it gives you a rough idea of how you could go about finding the marginal product of an otherwise unproductive philosopher.