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I have not seen this question asked anywhere, so I'm posing it here in case anybody else (hopefully) can help me get to the answer. In a nutshell, my question is: how do we arrive at the saving function in a canonical OLG with Cobb-Douglas utility?

I'll explain my question in more detail: In Diamond's original OLG model, he describes saving as a function of the wage and the interest rate, so that:

$$ s_t=s_t(w_t,r_{t+1}). $$

More specifically, when utility is Cobb-Douglas:

$$ U = (1-\beta) \cdot ln (C_{1,t}) + \beta \cdot ln(C_{2,t+1}) $$

Diamond concludes that the saving rate is a fixed ratio of the wage rate, such that: $$ s_t= \beta \cdot w_t $$ The same conclusion can found in most online references that cover the canonical OLG (for example, here).

However, I have not been able to find an explanation anywhere of how one arrives at this equation for saving. Most references that are available online, like the one I've just linked to, simply present a formula with no explanation as to its derivation. Meanwhile, in his original paper, Diamond seems to jump to this conclusion (in page 1134) with no step-by-step explanation for how he arrived at this result for the Cobb-Douglas case.

Can anybody perhaps help me understand how this result comes about?

(My motivation for asking is that I want to understand what would happen to consumption in a canonical OLG model if we supplemented the wage with an exogenous source of income "A", so that consumption of the young at time "t" were given by: $C_{1,t} = w_t -s_t + A$).

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