Intertemporal Social Welfare Functions

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Frank Ramsey (1928), following Pigou (1920), argued that "society" is composed of everybody in every generation, current and future, and that they all should be given equal weight in the social welfare function. So, suppose that every time period t = 0, 1, 2, 3, ..., a generation is born and another dies. Each generation has H people in it. Thus, Benthamite utilitarianism implies that the social utility of this society is:

S = å t=0¥ (å h=1H uth (cth))

where cth is the consumption of person h of generation t and uth(ｷ) is his utility function. Thus we are defining social utility as the (unweighted) sum of utilities of all people, current and future.

Let us make the traditional Benthamite "equal capacity for pleasure" assumption, so that all people, across generations and within generations, possess the same utility function, i.e. u(.) = uth(ｷ) for all h = 1, 2, ... H and t = 1, 2, .... This then reduces the social welfare function to S = å t=0¥ (å h=1H u(cth)). Furthermore, to get rid of the problem of allocation within a generation (or, given our assumption of "equality"), we can also assume that every household at time t gets the same consumption, i.e. cth = ct for all h = 1, 2, .., H, so that our social welfare function is further reduced to S = å t=0¥ (Hｷu(ct)). Now, as ct is consumption per person, then we can define Ct = Hｷct as the aggregate consumption of the generation at t and define U(ｷ) as the "aggregate" utility of that generation, so U(Ct) = Hｷu(ct). After all these maneuvers, we end up with a new social welfare function:

S = å t=0¥ U(Ct)

which can be conceived of as the sum of aggregate utilities of each generation. An allocation, then, can be defined as an infinite sequence of aggregate consumption bundles, i.e. C ={C0, C1, C2, ..., }. An example is shown in Figure 1. The "social optimum" is the allocation or sequence of consumption bundles that maximizes the social welfare function S.

Fig. 1 - A Consumption Allocation

However, the absence of time-preference implies incompleteness of the social preference orderings. Specifically, if we have two sequences, C and C¢ , it may be impossible to "compare" them and say which one is "socially better". Why this is so should be immediately evident: with an infinite time horizon, it is quite possible that there are feasible consumption paths such that S = ¥ , even if u(ｷ) is bounded above every step of the way and the economic constraints are doing their job. If two different consumption paths each yield an S which is infinite, then they become incomparable -- as two infinities cannot be arithmetically ordered.

Alternative methods of comparing paths with infinite sums have been proposed. A famous one is the "overtaking criterion" of Christian von Weizsäcker (1965) and Hiroshi Atsumi (1965), and later refined by David Gale (1967). Specifically, this proposes to "convert" the infinite-horizon to a finite-horizon problem, and then check if one program dominates another for subsequent extensions of the horizon. Specifically, consider two paths C and C¢ which are of infinite length. Now, impose a finite time, T, and compare the two paths up to that finite time. By imposing the final time period, the truncated social welfare measure becomes finite for any path, i.e. å t=0T U(Ct) < ¥ for all C if U(ｷ) is bounded. Thus, all consumption paths become comparable up to this final time period. So, suppose that comparing two paths C and C¢ , we find that for finite time T:

å t=0T U(Ct) > å t=0T U(Ct¢ )

In which case we will be tempted to argue that, at least up to T, the path C is socially better than the path C¢ . Now, if we can prove that this continues to hold true if we extend the end-period T to T+1, T+2, T+3,... etc., then we can actually come around to concluding that C is a socially better consumption allocation than C¢ , even though both are infinitely long. Even if we cannot compare both infinite sums directly, we can compare their finite equivalents, and then approximate the infinite case by gradually increasing the horizon.

Formally, by the "overtaking criterion", path C is said to be "better" than C¢ if there is a time period T* such that for all T ³ T*, åt=0T U(Ct) > åt=0T U(Ct¢ ). We say a path C* is "socially optimal" if there is a T* such that, for all T ³ T*, å t=0T U(Ct*) ³ å t=0T U(Ct) for all other feasible paths C.

However, the overtaking criterion does not solve all our problems. The issue of possible non-comparability of paths continues to lurk. For instance, it is quite possible that there is no T* such that the inequality holds for all T ³ T*. For instance, suppose consumption path C yields the utility stream {1, 0, 2, 0, 3, 0, ..., } and path C¢ yields utilities {0, 2, 0, 3, 0, 4, ...}. These are not comparable by the overtaking criterion: if we set the final horizon, T, at an odd time period, then C is better than C¢ ; but if we set T at an even time period, then C¢ is better than C. Thus, there is no T* for which one path will be consistently better than another for all T after T* (cf. Koopmans, 1965).

The overtaking criterion only permits a partial ordering over consumption paths. However, a partial ordering might be enough for most purposes. Put more precisely, as our example has shown, the overtaking criterion may find us an "optimal" path C* in the sense that it cannot be bettered by another path, but that does not imply that C* is itself better than every other feasible path.

[Note: A variation on this theme is the "agreeable criterion" proposed by Peter J. Hammond and James A. Mirrlees (1973). Loosely, given two infinite-horizon paths, C and C¢ , if we can agree that whatever happens in these paths after a particular time period T is "inconsequential" to us, then we can order these paths according to their truncated values. A survey of criteria can be found in McKenzie (1986).]

Partial ordering only does a partial job -- and when considering issues like "social optimality", that may not be good enough. We want a complete ordering. Tjalling Koopmans's (1965) suggestion was to introduce the notion of utility discounting -- what has become known as "time preference". By this he meant that the sum of social utility should be weighed so that earlier generations are "more socially valuable" than latter generations.

Specifically, the social welfare function could be rewritten in the form first suggested by Paul Samuelson (1937):

S = åt=0¥ bt U(Ct)

where 0 < b < 1 is the time discount factor. The further away a generation is, the less its utility matters for social utility. This maneuver yields the result that all consumption allocations over an infinite horizon will yield finite sums for S, i.e. S < ¥ for all possible paths C. With time discounting, then, all paths become arithmetically comparable and we can thus find a social optimum simply by comparing the sums.

However, following Pigou, Frank Ramsey (1928) considered time discounting as "a practice which is ethically indefensible and arises merely from the weakness of the imagination" (Ramsey, 1928). But Ramsey recognized that by omitting time discounting, the problem of non-comparability of infinite sums emerged. In its stead, as we have seen, he introduced the reverse social welfare function in the following manner:

R = åt=0¥ (B - U(Ct))

where B is the "bliss" level of utility. An allocation C is "better" than an allocation C¢ if:

å t=0¥ (B - U(Ct)) < å t=0¥ (B - U(Ct))

Consequently, for a social optimum, we wish to find the consumption allocation for which R is minimized.

How does the Ramsey device solve our comparability problem? Well, suppose that path C achieves "bliss" after the tth generation and that bliss is maintained for every generation forever after. In contrast, suppose that a different path C¢ achieves bliss at generation t+1, and then bliss is maintained forever after. As the utilities for our society are the same for both paths after t+1, so we can effectively ignore the utilities achieved after t+1. All that matters for comparison between C and C¢ is the utilities that are achieved up to generation t+1. These are finite sums which can be arithmetically ordered.

Of course, the question imposes itself: how do we know that "bliss" will be achieved for consumption paths at some point in time? We don't. This is the Achilles heel of the Ramsey device. On the one hand, it might not matter: as long as the paths converge asymptotically to bliss, that might be enough to compare two paths. On the other hand, they might not converge fast enough for comparisons to be made. Thus, the Ramsey criterion only permits a partial ordering of consumption allocations. As a result, it is common to impose "convergence to bliss" by either assuming "utility saturation" or, more indirectly, "capital saturation" in finite time. If these are imposed, then convergence to bliss within a finite time horizon will be possible and thus complete comparability.

Let us now turn to continuous time. If we assume people are continuously being born and dying (at equal rates, so that the population stays constant), we can convert all our previous social welfare functions into continuous time form as shown in Table 1:

 Type Discrete Time Continuous Time Benthamite sum S = å t=0¥ U(Ct) S = ò 0¥ U(C(t)) dt Samuelson "discount" S = åt=0¥ bt U(Ct) S = ò 0¥ U(C(t))e-r t dt Ramsey "bliss" device R = åt=0¥ [B - U(Ct)] R = ò 0¥ [B - U(C(t))] dt

Table 1 - Intertemporal Social Welfare Functions

In continuous time, an "allocation" is no longer merely an infinite sequence, but can be characterized as a function over time, C is a function C: [0, ¥ ] ® R. However, for comparability, we are still interested in ensuring that our social welfare functions S or R achieve finite values for every consumption allocation C. As it turns out, the Samuelson social welfare function is always finite. Formally, if U(ｷ) has an upper bound and r > 0, then the integral S = ò 0¥ U(C(t))e-r t dt will have a finite value.

However, as shown by Sukhamoy Chakravarty (1962), we are not ensured that for a social welfare function without this time-discount factor, this will be true. In particular, we are curious about the Ramsey function R. Now, a consumption path acceptable to the Ramsey criteria cannot exceed bliss, so we know for certain that R ® 0 as t ® ¥ . However, this is not sufficient to ensure comparability, for R may not approach 0 fast enough. In other words, even if we assume that U(ｷ) is bounded above, so that U(C(t)) £ B for all C(t), we can still have it that U(C(t)) approaches B asymptotically at too slow a rate so that in fact there is no consumption path that ensures that the integral converges. In other words, B may not be achieved for any finite C. Ramsey (1928), however, argued (without proof) that there would be at least one program that would ensure convergence.

But neither Chakravarty (1962) nor Koopmans (1965) proves that time preference will do the trick either. They assumed that time preference in the social planner's function would be sufficient to yield an optimal solution, but offered up no proper "proof" of this. Intertemporal social welfare functions imply that we have to deal with infinite-dimensional commodity spaces. As we discuss elsewhere, the mathematics of infinite-dimensional spaces can be quite complicated. As a result, this question was left somewhat vague until the 1980s, when Donald J. Brown and Lucinda M. Lewis (1981), Michael Magill (1981) and A. Araujo (1985), addressed the issue again -- this time with a fully-developed mathematical arsenal. They confirmed the intuition: positive time preference is a sufficient (albeit not always necessary) condition for the existence of optimal consumption paths.

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