I do not understand how Weitzman 1998 (https://scholar.harvard.edu/files/weitzman/files/recombinant_growth.pdf) arrives at the following expression (equation 5 in the above article).

$$ \alpha[N(1)-1/2] > 1 $$

Weitzman's work conceptualizes economic growth as combinatorial process that essentially can grow faster than any exponential growth. Therefore he introduces the following for the number of different binary pairings that can be made from N ideas or objects:

$$ \tag{1} C_2(N) = (N*(N-1))/2 $$

and this difference equation to represent the binary recombinant expansion process

$$ \tag{2} N(t+1) - N(t) = \alpha[C_2(N(t))-C_2(N(t-1))] $$

Then he also introduces something called the "reproducibility assumption" (which I do not fully grasp the meaning of) which somehow define the initial conditions of the system.

$$ \tag{3} \alpha[C_2(N(1))-C_2(N(0))] \geq N(1) - N(0) > 0 $$

Now he wants to prove a Lemma which states that this recombinant process grows faster than exponential and in doing so the first step is to substitute (1) and (2) into (3) and arrive at the initially stated expression. I do not understand how he does this.



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