Paper: Samuelson, "Exact Consumption-Loan Model of Interest"

The model is a one of multiple, identical generations (OLG), with each generation living 3 periods, working in the first 2 and being retired in the last. Total lifetime savings has to equal zero (i.e. grandparents don’t go to the grave with their money and they don’t leave estates).

In the stationary case, the case where interest rates don’t change over time and population doesn’t grow, the interest rate must zero. This is the case because everyone is assumed to be the same. There are no “savers”, those that don’t mind consuming in the future rather than today, who would get positive interest rates from “spenders”, those that want to consume today. “Saver” and “spender” are relative, you’re only a “saver” in relation to all the “spenders”, terms and because everyone is the same there can’t be relative differences.

If you let the population grow (exponentially) but keep interest rates constant over time, you find the interest rate must be equal to the population growth rate. Of course, this means that if the population is declining, you’d see a negative interest rate… Hmmmm.

Further, with equal interest rates over time, interest rate equal to the population growth rate is socially optimal.

But will each generation trade with the others. Its easy to see in the 2 period case, generations will not trade. The old won’t buy bonds they’ll be dead before they mature. Thus, the young consume all of their product and the old, well, consume their product (poor them). Recall, that we had interest rates on bonds equal to the population growth. So, in this case with no trading, why would there be any interest rate (not to mention a sensible one) on bonds?

Samuelson goes on to show that the optimal amount of savings won’t be achieved in most settings and comments that this is where there is a role for government. He ends the essay by declaring that money is an asset that can be traded across generations, “as long as the new current generations of workers do not repudiate the old money, this gives workers of one epoch a claim on workers of a later epoch.” Money, beyond being a medium of exchange, is the “social compact” that brings the economy into optimality.


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