This article is a summary of "A Mathematical Theory of Saving", written in 1928 by British economist, philosopher, and mathematician Frank Ramsey (1903–1930). The paper poses the question of how a society should split its economic output between consumption for the present generation and investment for future generations, in order to maximize the aggregate benefits for all generations – in Ramsey's words, "How much of a nation’s output should it save?" The paper is a precursor of the modern theory of patient philanthropy. I've summarized the SEP article in this post because I think it's important for people interested in longtermism to understand this paper and its conclusions. I invite all of you to share your thoughts on it in the comments below.

The Ramsey rule

Ramsey's paper uses a growth model in which economic output is a function of only capital (and only one type of capital) – so it assumes that changes in the labor stock, human capital, technological progress, and natural resources don't affect economic output. The economy is described by the differential equation

$\frac{dK\left(t\right)}{dt}=F\left(K\right(t\left)\right)-C\left(t\right)$,

where $C\left(t\right)$ is consumption at time $t$ and $K\left(t\right)$ is the capital stock at time $t$. Utility of consumption is given by $U\left(C\right)$, which is monotonically increasing and strictly concave (so $U^{\prime }\left(C\right)>0$ and $U^{\prime \prime }\left(C\right)<0$ for all $C$).

The goal is to maximize aggregate utility over all $t$,

$V={\int }_0^{\infty }U\left(C\right(t\left)\right)e^{-\delta t}dt$,

where $\delta \ge 0$ is an (optional 😉) pure discount rate. (Ramsey was strongly against the idea of discounting the well-being of future generations simply because they are in the future, but we can instead let $\delta$ be the "hazard rate", or exogenous probability that the world ends at any given point in time.)

It follows that the pattern of economic growth over time is optimal when the social rate of return on investment $F_K\left(t\right)$ (the rate at which additional money invested in capital yields additional economic output) is always equal to the social discount rate

$F_K\left(t\right)=\rho \left(t\right)=\delta +\sigma \left(C\right(t\left)\right)g\left(C\right(t\left)\right)$,

where $\sigma \left(C\right)$ is the elasticity of marginal utility of consumption and $g\left(C\right(t\left)\right)=\frac{dC\left(t\right)/dt}{C\left(t\right)}$ is the percent rate of economic growth at time $t$. This formula is called the Ramsey rule.

Implications

The Ramsey rule implies that economies should reinvest very large portions of their gross economic output. To show this, we let $U\left(C\right)$ be the isoelastic utility function

$U\left(C\right)=\{\begin{array}{cc}\frac{C^{1-\sigma }-1}{1-\sigma }& \sigma \ge 0,\sigma \ne 1\\ \ln \left(C\right)& \sigma =1\end{array}$

where the elasticity of marginal utility of consumption $\sigma >0$ is constant. We also assume that the production function is a linear function of capital, $F\left(K\right)=\mu K$.

Per the Ramsey rule, we want the social rate of return to be equal to the social discount rate:

$F_K\left(t\right)=\delta +\sigma \left(C\right(t\left)\right)g\left(C\right(t\left)\right)$.

Since $F_K=\mu$ and $\sigma \left(C\right)$ is constant, we can simplify this to $\mu =\delta +\sigma g\left(C\right(t\left)\right)$. It follows that $g\left(C\right)$ is also constant, i.e. $C\left(t\right)$ grows exponentially over time.

To ensure optimal growth, we want consumption and capital to grow at the same rate. If consumption grows faster than capital, then the economy will eat into its capital stock, exhausting it in a finite amount of time. On the other hand, if the capital stock grows faster than consumption, then the economy is underutilizing it.

The optimal savings rate, the fraction of total output that should be reinvested in capital stock during each period of time, is given by $s^{\ast }=(1-\delta /\mu )/\sigma$. If $\delta =0$, this reduces to $s^{\ast }=1/\sigma$.

This equation implies that societies should save very large portions of their economic output for future generations in order to ensure optimal growth. For example, if we assume that the rate of return on investment $\mu =0.05$ (5% per year), $\sigma =1$, and the pure discount rate $\delta =0.001$ (0.1% per year), then society should save a whopping 98% of its yearly economic output. Even with a more modest pure discount rate, like 3% per year, society still ought to save 40% of its output.