Question
Imagine a simple economy with only two people, Leroy and Percy. If the Social Welfare Function is W = UL + UP, and the Utility Possibilities Frontier is UPF = UL + 2UP, what will be the societal optimum?
Explanation / Answer
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(?) 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(?) 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?u(ct)). Now, as ct is consumption per person, then we can define Ct = H?ct as the aggregate consumption of the generation at t and define U(?) as the "aggregate" utility of that generation, so U(Ct) = H?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(?) 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(?) 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