Intertemporal Justice

a dark satanic mill



(A) Rawlsian Social Welfare
(B) Rawlsian Altruism

(A) Rawlsian Social Welfare

So far, we have adhered to a (weighted) Benthamite structure for our social welfare function. But there are alternative structures. John Rawls's (1971) "maximin" social welfare function is one that comes to mind quite readily. We can adapt this to the intertemporal scenario quite simply. Instead of maximizing the sum of utilities of different generations, we maximize the minimum utility of any generation. Specifically, our social welfare function takes the form:

S = mint Ut(Ct}

So choosing consumption allocations that maximize social welfare function S translates into finding the allocation that maximizes the minimum utility, i.e.

max S = max{C} {mint Ut(Ct)}.

In other words, the social planner wants to improve the lot of the generation that is worst off.

The Rawlsian social welfare function may be commendable for its highly egalitarian structure within a generation, but it is trickier in its intertemporal form. Recall that, in the Ramsey exercise, savings reduce the utility of the current generation, but, via capital accumulation and growth, that implies higher utility for future generations. Thus, a Benthamite social welfare function will lose on one end but gain in the other. The Rawlsian social welfare function, however, just loses. If the worst-off generation saves anything, then social welfare as a whole is lower because the consequent gains in utility by other better-off generations will not be counted. Thus, the main peculiarity of the intertemporal Rawlsian social welfare function is that it cannot balance the utilities of current and future generations. As a result, in the absence of population growth or technical progress, it predicts that the optimal -- or "just" -- rate of savings will be zero.

[Note: this result was already alluded to by Tinbergen (1960) and Solow (1974). Phelps and Riley (1978) create intergenerational externalities by considering a Rawlsian overlapping generations model. In this case, the savings of one generation when young can be partly compensated in the future (when they are old), so net savings will not be zero necessarily.]

Now, Rawls (1971: p.284-93) was quite aware of this dilemma. Although he refrains from applying his "maximin" principle in the simple manner given above, he makes suggestive remarks to the effect that it might be applicable if "dynastic" utilities were incorporated. If "[t]he parties are regarded as representing family lines, say, with ties of sentiment across generations", then "the characterization of justice remains the same. The criteria for justice between generations are those that would be chosen in in the original position." (Rawls, 1971: p.292).

So, incorporating dynastic considerations once again, we now have a social utility function:

S = mint Ut(Ct, Ut+1)

so the utility function of generation t includes utility function of generation t+1.

Now, if we assume that generation t's utility function is additively separable and exhibits time preference -- and recalling that utility of the t+1 generation includes the utility of the t+2 generation -- then we can write this as:

Ut(Ct, Ut+1) = Ut(Ct) + b Ut+1(Ct, Ut+1)

so generation t explicitly recognizes that generation t+1 is itself altruistic towards generation t+2, etc. The maximin social welfare function would thus be:

S = mint [Ut(Ct) + b Ut+1(Ct+1, Ut+2)]

or, recursing further for Ut+2, etc., we obtain in the end:

S = mint åt =t¥ bt Ut(Ct)

So, maximizing this social welfare function means maximizing the smallest discounted infinite stream of utility. Note that it is the starting point of the stream, t, that matters.

Kenneth J. Arrow (1973) and Partha Dasgupta (1974), who started on this track, claimed that the dynastic Rawlsian form would yield dynamic inconsistency and "it is at least questionable that the sawtooth pattern [of dynastic inconsistency] corresponds to any intuitive idea of justice" (Arrow, 1973). We should note here that Arrow-Dasgupta result relies on the faulty manner in which they incorporate "dynastic" considerations. Their analysis was criticized by Guillermo Calvo (1978), who provided the form and analysis we use here.

To see dynamic inconsistency à la Calvo, examine Figure 1, where we have two consumption plans, C (in black) and C¢ (in red). Obviously, the worst off generation in both cases is the first.

Figure 1 - Dynamic Inconsistency with Rawlsian SWF

Now, if we take the dynastic perspective, from the first generation's point of view, it may very well be that C¢ is better than C. This is because the higher utility gained by generations 2 and 3 via C¢ will, in generation 1's altruistic calculation, be weighted heavier than the relatively lower utility generations 4, 5, 6, etc. will consequently get (if you think this is not obvious in Figure 1, you know we can easily adjust the path so that it is so). So, by the maximin criteria, C¢ has greater social welfare than C.

But now examine the same paths from the perspective of generation 3 onwards. In generation 3's view, the utility of generations 4 and 5 are given much more weight than they had in generation 1's perspective. So, from generation 3's perspective, even though they themselves get higher utility with C¢ than with C, the immediacy of the drop right after them means that C will be better than C¢ . This is dynamic inconsistency. Generation 1 will plan for path C¢ , but by generation 3, that plan will be dropped by generation 3, and path C will be adopted.

Now, Calvo (1978) shows that while dynamic inconsistency is possible it is not necessarily the result. He proves that if we use the dynastic Rawlsian social welfare function with a standard Neoclassical optimal growth model, we can obtain a dynamically-consistent solution.

(B) Rawlsian Altruism

We should note that John Rawls (1971) rejected time preference immediately. In his famous "original position...there is no reason for the parties to give any weight to mere position in time." (Rawls, 1971: p.294). He recognizes the mathematical need for time-preference to provide complete comparability, but remains unimpressed. "Unhappily I can only express the opinion that these devices simply mitigate the consequences of mistaken principles" (1971: p.297).

However, we find all this quite curious. Rawls abandoned the Benthamite structure, yet it is precisely the Benthamite structure that makes time preference "unjust". Time preference does not, in and of itself, imply "selfishness". That arises in combination with other elements of the utility function -- specifically, the assumption of additive separability.

To see this, suppose there is a cake that can be eaten by a father or a son, both of whom would receive 100 utils per slice of cake consumed. If we are maximizing the pure sum of utilities without discounting, then it would not be inconsistent for the father to split the cake equally with his son. However, with discounting, the father underestimates his child's utility, e.g. he believes that the son will only get 80 utils per slice consumed.

Several consequent courses of action are conceivable. For instance, the father might divide the cake so that his son gets a greater proportion of the cake to compensate for his "lower" estimated utility per slice (e.g. father receives 30% and the son 70% of the cake). In this case, time preference did not cause "selfishness" at all, but quite the opposite! This is "Rawlsian altruism". In contrast, if the father is a pure Benthamite where only the "dynasty's aggregate utility" matters, then the father will consume a greater portion of the cake himself because his contribution to "dynasty utility" is greater than the son's.

We can state Rawlsian altruism as follows. Let the father be generation t and the son be generation t+1, then instead of writing the father's utility as an additively separable utility function, we can write it as a Leontief discounted utility function:

Ut(Ct, Ut+1) = min {Ut(Ct), b Ut+1(Ct+1, Ut+2)}

and then recurse future generations through this, yielding:

Ut(Ct, Ut+1) = min {Ut(Ct), b Ut+1(Ct+1), b2 Ut+2(Ct+2), ...}

In this case, we actually get the result of infinite patience! Specifically, as 0 < b < 1, then t ® ¥ implies bt U(Ct+t ) ® 0, so the utility of the father at time t (i.e. the minimum of his dynasty) is the near-zero utility of the last descendent, far into the infinite future. He will thus consume nothing himself and allocate all of his income into the far future!

We see, then, that the time discount factor, in and of itself, does not mean the father is "selfish". It is really only when we combine it with a Benthamite altruism, i.e. an additively separable dynastic utility function, that we achieve that "unjust" result. Time discounting can mean that the older generation is selfish (Benthamite altruism) or that it acts like a "mother hen" towards its progeny (Rawlsian altruism). Which case arises turns out to depend on the functional form of the dynastic utility function.

One can counter, of course, that one should not incorporate time discounting into the Rawlsian maximin function as it doesn't yield impatience. In other words, when we place the discount factor b in the Rawlsian utility function we are not really incorporating "time-preference" but something else. Perhaps. But the original definition of time-preference, as posited by Böhm-Bawerk (1889), was constant underestimation of future utility. That is enough justification to include b , regardless of the functional form of the utility function. It yields impatience in the additively separable utility function, but altruistic "patience" in the Rawlsian maximin form.

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