## Optimizing a growth model

For the discussion of math. Duh.

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### Optimizing a growth model

This problem was probably inspired by a variety of games, especially those of the cookie-clicker variety. In fact, I remember one such game which revolved around a mathematician proving theorems at an super-exponential rate.

There exists an initial quantity of goods.
These goods can be exchanged for machines which produce more goods. Perhaps, we can also exchange goods for factories which build these machines, and for goods for factory-building robots ... and so on.
Let x_0(t) be the quantity of goods at time t, and x_1(t) is the quantity of machines, x_2(t) the quantity of factories, and so on...
For notation, let goods be 0-items, machines be 1-items, factories be 2-items, and so on. In this system, (n+1)-items produce n-items.

We have (with constants a_i and b_i for all i > 0)

x_0' = a_1*x_1 - sum(y_i over i)

The first term comes from the good-producing machines.
The second term comes from the goods used to exchange for i-items

x_i' = (a_i+1)*(x_i+1) + b_i*y_i

The first term comes from the i-items producing (i+1)-items
The second term comes from exchanging goods for the i-items

x_n' = b_n*y_n

The first and only term comes from exchanging goods for the n-items

A natural question is then, given x_i(0) (a initial condition might be x_0(0) = 1 and x_i(0) = 0 for i > 0), how to maximize x_i(t) for some t.
With discrete time, it seems this could be written as a linear program and solved easily. We can also use this method to approximate continuous time. However, if t is large, this may become infeasible.
Also, this will not work when n is unbounded.
Another question would be how to achieve the fastest growth asymptotically. This question feels easier, but I'm not sure how to formalize it.

I'm wondering if there is any analytical solution to any of these questions, or any ideas about how the solution would might look like, or how one might approach the problem, or any interesting ideas at all.

Notice that the production chain in this problem is "linear". An interesting extension to this problem would be to solve it on a directed acyclic graph, or even a general directed graph, where for any vertices u and v, u-items produce v-items at a certain rate, and u-items can also be traded for v-items at a certain exchange rate (which does not allow arbitrage). The goal would then be to maximize a weighted sum of the amount of items of each vertex at either a fixed endpoint or asymptotically.