One of the critical and most debatable assumptions we have maintained thus far in our arguments is the assumption of *successive* generations. In other words, we have assumed that, every period, a new generation arises and the old one dies off. Generations precede and follow each other, but they do not overlap at any point. This is a very restrictive and unrealistic assumption but one that, unfortunately, is difficult to dispose of.

Models which allow successive generations to overlap with each other were first proposed by Maurice Allais (1947) and, independently, Paul Samuelson (1958). They noticed immediately that such a structure has some intriguing implications for intertemporal social welfare.

There are many ways of modeling overlapping generations. The simplest is the "two-period-life" version. In this case, each generation lives for *two* periods -- call it "youth" and "old age". At any time period, one generation of youths coexists with one generation of the elderly. At the beginning of the next period, the elderly die off, the youths themselves become elderly and a new generation of youths is born. Thus, there are two "overlapping" generations of people living at any one time.

Although we cover this in more detail elsewhere, our interest is in the social welfare implications of overlapping generations. To see this, let us attempt to construct a social welfare function when generations overlap. We assume a generation born at time period t (call it "generation t") lives for two periods: t and t+1. Let c_{t}^{t} and c_{t+1}^{t} denote the consumption in periods t and t+1 respectively by generation t. Let us denote by u^{t}(c_{t}^{t}, c_{t+1}^{t}) the intertemporal (two-period) utility function of generation t. Allowing for additive separability utility and personal myopia, we can write:

u

^{t}(c_{t}^{t}, c_{t+1}^{t}) = u^{t}(c_{t}^{t}) + b u^{t}(c_{t+1}^{t})

where b
is the personal discount factor. Now, this is for a single generation that is born at time t. As a new generation is born *every* time period t, then the
intertemporal social welfare function is:

S = u

^{0}(0, c_{1}^{0}) + å_{t=1}^{¥}_{ }u^{t}(c_{t}^{t}, c_{t+1}^{t})

where u^{0}(0, c_{1}^{0}) is the utility of the first generation of elderly people (born at t = 0), who have had no "youth". Notice that this is
intertemporal, so every generation, present and future, is given equal weight in this social welfare function (there was a small controversy between Abba
Lerner (1959) and Paul Samuelson (1959) over this). Thus, assuming the same personal discount rate across generations, we can plug in our explicit form:

S = b u

^{0}(c_{1}^{0}) + å_{t=1}^{¥}_{ }[u^{t}(c_{t}^{t}) + b u^{t}(c_{t+1}^{t})]

or, rearranging:

S = å

_{t=1}^{¥}_{ }u^{t}(c_{t}^{t}) + b å_{t=0}^{¥}_{ }u^{t}(c_{t+1}^{t})

By the Benthamite "equal capacity for pleasure" argument, let u^{t}(ｷ, ｷ) be the same across generations. This permits us to drop the t superscripts and rewrite
the social welfare function simply as:

S = å

_{t=1}^{¥}_{ }u(c_{t}) + b å_{t=0}^{¥}_{ }u(c_{t+1})

This is revealing. For any positive consumption path, this social welfare function S is *not* a finite sum, i.e. S = ¥
for any {c_{t}} > 0. Thus, not only are paths "non-comparable", but we cannot find a "social optimum". The
old problem re-emerges.

The overlapping generations construction yields interesting implications. Firstly, even when we incorporate personal myopia, we do *not* end up with finite social welfare sums. We
* cannot* appeal to the reality of individual discounting to solve the incomparability problem. To make the sums finite, to make consumption paths comparable, we require that the social planner start making evaluations of the relative social worth of different generations. Personal discounting will not do as a substitute. Thus, letting g
be the social planner's discount rate per generation, then we end up with:

S = å

_{t=1}^{¥}_{ }g^{ t-1}u(c_{t}) + b å_{t=0}^{¥}_{ }g^{ t-1}u(c_{t+1})

where, assuming 0 < g < 1, then S becomes finite and paths are now comparable. But g is an explicitly unethical discount. There is nothing obvious we can pluck out of society that can justify it. We must simply accept that our social planner is "morally challenged".

Secondly, the decentralization thesis
does not hold in overlapping generations. Specifically, it can be easily shown that in an overlapping generations model, the competitive equilibrium is *not* Pareto-optimal. This means that a social planner (or a government) can achieve a superior allocation than that yielded by the market. The social planner's solution (if we can find one) will be different from the market solution. The decentralization thesis breaks down.

However, there is a trick that is possible: namely, if we follow the "dynastic" logic employed earlier. Including intergenerational altruism and "bequests" in an overlapping generations model, as Robert Barro (1974) did, we can effectively replicate the traditional Ramsey-style infinite-horizon problem with successive generations and restore the decentralization thesis.

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