Overlapping Generations

by elder-schmelder 20 Replies latest watchtower bible

  • elder-schmelder

    I think that this explains it best !!

    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 (?, ?) 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.

  • dgp

    Not to be mean, but I find it easier to understand it the old fashioned way. I did get, however, that they are making reference to "different" generations that overlap, not the "the same generation".

  • St George of England
    St George of England

    Do you think Jesus used these formulea when speaking in Matt 24?


  • undercover

    Yes, it's so simple. Jesus said it best himself...

    Matthew 24:34

    Truly I say to YOU that this generation - S = b u 0 (c 1 0 ) + å t=1 ¥ [u t (c t t ) + b u t (c t+1 t )] - will by no means pass away until all these things occur.

  • wasblind

    LOL, undercover

  • elder-schmelder

    I am glad that we finally got it right this time. I don't that they will have to change it again.


  • alanv

    All you have to do is show any person who knows nothing of the generation doctrine, and they will give the proper sensable meaning to the words Jesus said.

    Remember the only reason we have the current meaning of generation is because all the societies other efforts failed miserably. And it only took them 130 years.

  • Pistoff

    I know this was posted tongue in cheek, but found it interesting that this may be part of the basis for the WT's 'new light'.

    But.....it uses the term overlapping generationS, not one overlapping generation.

    Overlapping generation, not plural, is self contradictory; it tries to lengthen the time period, but does it by overlapping 2 generations and calling it one generation.


  • sir82
    I don't that they will have to change it again.

    ...at least, not until another 30 or 40 years pass, and even those who became anointed as youngsters during Fred Franz's lifetime are all pushing daisies.

    But then, that's a problem for Governing Body 3.0, isn't it?

    Meanwhile, Jackson, Morris, et al are busy packing for their first class flight to inspect the Fiji branch office, and trying to decide if they will bring back only one, or maybe better 2, cases of Shiraz...

  • cyberjesus

    I think the way they explained it yesterday at the DC is easier :-)

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