Theo Todman's Web Page - Notes Pages
(Text as at 12/08/2007 10:17:46)
Therefore, scaling factors are required to ensure that no individual is left out.
- ¥ < SWXjMij < + ¥ , whenever - ¥ < X < + ¥. M will vary with the good X and the population W under consideration. Some goods are optional luxuries while others are essentials. Also, the quantity of good G already possessed by the individuals j e W will affect the marginal utility of a further Xj of G to j.Because of the decreasing marginal utility of any good, in all cases, U(Xj) / Xj ® 0 as Xj ® ¥.In the case of any essential good G of utility X, of which the Xj represent the j's total holdings, U(Xj) ® - ¥ as Xj ® 0.A correction will be needed to prevent U(Xj=0) = -¥ for even an essential G. This is because on occasions, such as in time of war or where X is insufficient to satisfy all of W, we will have no option but to sacrifice some individuals.A further correction will usually need to be added to protect the interests of the person i making the ethical decision. That is, Mij increases as j ® i. This protective zone may also apply to the interests of others of his choice (such as his family), though not to the exclusion of all others. Both moderate selfishness & directed altruism are to be considered ethical.If j e W is a member of a future generation, Mij ® 0 as generations progress.There is no reason why any mathematical function should satisfy these criteria. However, it is to be noted that a logarithmic function satisfies c & d above, and that a negative exponential satisfies f & g, these being functions of quantity of good (X) and distance of relationship respectively. The only problem we have addressed above is one of partition. We have not yet discussed the issue of the choice of production of goods or of the utilitarian difference between goods.A critical issue is that, with resource R at our disposal, we have the ability to produce goods G, H or I . . . (or a combination of goods). How do we decide what to produce? Our calculation so far shows how the effective quantity (utility) of a good may vary with its distribution, but not how to arbitrate between goods: how to weigh G versus H, for instance.This is a complex question because, as we have noted, the marginal utility of Xj to j depends on j as well as (quantitatively & qualitatively) on X & the underlying good G.
- Let us call the scaling factor the utilitarian metric, Mij and let us assume that the population (including future generations) capable of being affected by the potential actions of a particular individual i is W, that the quantity or value of a quantifiable good G available to i is X, and that the utilitarian metric applicable to i is Mi. Then, the total utility, U(X, W), resulting from i's apportionment of X is:-
U(X, W) = SWXjMij, where SXj = X and SW sums over all j e W.
- What form does the utilitarian metric take? We can set some bounds as follows:-
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