Under many models of voting behavior, elections disproportionately (relative to present-day demographics) represent the interests of the elderly. This happens organically, without needing to consider anything like the accumulation of wealth, social capital, or other relatively “un-democratic” structural factors. We need only assume that people’s interests are correctly represented in policy (it’s sufficient for the median voter theorem to hold), and some degree of gerontocracy often follows.

Minimal model: Everybody votes for their future interests, with no discounting. So, a \(40\)-year-old places equal weight on the interests of somebody aged \(42\) and somebody aged \(79\). Since this is meant to be illustrative, let’s assume everybody lives exactly \(80\) years (and everybody knows this fact). Then the electoral weight assigned to the interests of someone of age \(Y < 80\) is \[ W(Y) = \int_0^{Y} \!dT\, P(T) \text. \] Here \(P(T)\) is the current population at age \(T\). Note that I’ve assumed that everyone can vote from birth!

Policy is set by the median voter, but does not reflect the interests of that voter at his current age. Rather it reflects his interests averaged between his current age and his death at \(80\). Perhaps more intuitively, policy is set by the age \(Y_*\) of the median bit of electoral weight: \[ \int_0^{Y_*} W(T)\, dT = \int_{Y_*}^{80} W(T) \,dT \] To give some intuition, consider two dramatically different population pyramids.

• A steady-state society with no growth (and, again, deterministic death at 80) has a flat pyramid \(P(T) = P_0\). The median age is \(40\); democracy will best reflect the interests of someone of age \(60\).
• A young society, with all ages equally distributed between \(0\) and \(40\). Democracy best reflects the interests of someone aged \(50\)—older than any current member of society.

Without discounting, younger societies (but holding life expectancy fixed) are more severely gerontocratic.

Future discounting: Now we let voters display time preference. In general we write a discount function \(f(T)\), normalized to \(f(0)=1\), indicating how much the voter cares about events time \(T\) in the future. Without needing to do any calculation, it’s clear that this is going to alleviate gerontocracy; how much depends largely on how steep the discount function is.

Naively, a rational discount function is just a decaying exponential \(f(T) = e^{-T/T_0}\). Empirical work shows that the typical discount function is in fact “hyperbolic”, which here just means that the decay rate is greater at early times and lesser at later times. That is, the ratio of utilities at two times \(\delta\) apart: \[ R(\delta; T) = \frac{f(T+\delta)}{f(T)} \] is dependent on how far in the future we’re making this measurement (and gets closer to \(1\) as we imagine things further in the future).

Note that in most democracies, there’s a good delay (several months) between the act of voting and the moment the newly-electeds take office. The delay between the act of voting and the arrival of consequences is even longer. That means that the effective discounting rate at work is the lesser, long-term one; in other words, the relevant discount function is fairly close to flat. I suppose this suggests an unconventional remedy to gerontocracy: increase the voters’ rate of time preference by shortening the time between the election and the arrival of consequences. The tradeoff, of course, is that government collectively shifts its time-horizon nearer.

Children: Now we add family relations into the mix, and attempt to account for the fact that people vote for the interests of their relatives. There are many possible versions of this model:

• Children consider their parents’ interest, and parents their children’s;
• Voters consider the interests of all ancestors (parents, grandparents) and descendents;
• Voters consider the interests of their household

In all of the above we also have to consider the weights that people assign to their relatives. It seems sensible to assume that these weights are reciprocal: if Sally places weight \(A\) on her own interests and weight \(B\) on those of John, then John weights his own interests \(A\) and Sally’s \(B\). As long as weights are reciprocal in this fashion, children and other family relations change voting patterns, but do not affect policy.