Then I received a very inspiring comment (thanks, Reinhard!) to take into account fluctuations, and I did spent a happy morning extending the model towards an important concepts (in a hospital waiting room, mumbling over sketches of integrals): memory and knolwedge half-time.
Within this model then it is possible to understand then the importance of knowledge externalization.
The backside of this is that the mathematical apparatus gets quite heavy, I will describe in words the key ideas and only after that sketch the formalism.
Without saying before, the model was based on knowledge workers (no distinction between productivity and knowledge creation was made), this only simplifies the picture.
The simple model was static, instantanous. You need something, you demand knowledge, you get it - that's how it works with the help of social platforms (Luis was linking in twitter to a nice use case) - within seconds, well minutes.
But now the memory problem, the guy who had worked out the perfect answer for your question, ready-made and glossy, just left the company the other day (got hired by your competitor, won a million dollars, got driven over, your imagination ...). Most of you already what I am after, but for the sake of suspense, I am doing a little detour.
There are more sophisticated definitions of knowledge, but if we stick to a simple one: knowledge = information + context. Then we have to admit that knowledge is not eternal, it can be lost, and this can happen in two ways:
A The knowledge is not available anymore. The guy has just left the company the other day; this is why a key ingredient of Knowledge Management preaches the externalization of knowledge, put it into a database. But even in a catastrophical scenario the database can vaporize (which is hopefully more seldom than employee fluctuation).
B But there is another mechanism that results in the "loss" of knowledge: information loses its context, respectively the context losses its relevance. This is the concept of knowledge half-time (as an example might serve the dying of professions due to disruptive innovations)
So back to our friend, who left and enjoys his life at the beach, drinking coctails all day.
With respect to his knowledge for the company there are two scenarios:
A nothing externalized, everything only in his head: the half-time function of his knowledge for the company becomes just 0, 0(t), nothing left for the company.
B externalized to the database, there is the memory, the half-time function gives a good contribution for some period of time, until the context loses its relevance.
In mathematical terms:
With the concept of the knolwedge half-time h(t) we have prevented the model from exploding when introducing now the memory into the model (and summing up over the past):
The productivity in a time period is then build from working on own agenda, and using the memory of everything that has been shared and bears contextual relevance.
with p(t) = o(t) + r*s(t)
P = int[t1,t2] dt p(t)
= int[t1,t2] dt o(t) + r* int[-inf, t] d t' s(t') h(t')
ps: I am pretty aware that the formulae are hard to read (especially without the proper signs) and harder to understand
ps: for the sake of simplicity I have not introduced any memory in o(t), that is even more complicating the formulae without transporting the message.