The second fundamental theorem points to a concern for distribution which lies outside economists’ usual obsession with efficiency. However, Welfare economics, or social choice theory, has emerged as a subfield in economics where the implications of various social preferences for resource allocation are studied. It is important to note that economists do not make value judgments (which is unscientific), but they analyze the consequences of various possible ethical stances.
Economists have chosen to analyze the social choice problem by asking whether it is possible to construct a social preference ordering with similar properties as those we like individual preferences to have, i.e., to be complete and transitive. What do we mean by “social preferences” anyway? Well, we can imagine that each person can express his/her preferences over various allocations, e.g., x, y or z. These allocations consist of lists of commodity bundles for each person in the economy. That is why I typed them with bold faced types; they are really vectors of commodity bundles. Now my ranking of these vectors may include the possibility that I prefer bundle x to bundle y, even if my own consumption bundle is the same in both, but you have less in x than in y. That is very nasty of me, but we’re not in a position to rule out any preferences à priori; in a democracy each person is supposed to have freedom of voice and opinion. In a democracy we must find some way to reconcile these different opinions (i.e., find a way to make social choices) in a process which is seemed to be fair, and which respects each individual’s freedoms. One possibility is to vote over different proposed vectors of allocations and let the majority decide. However, it was noted already by Condorcet in 1785 that majority voting is not transitive; it may give rise to cyclical majorities (social choices). In more technical language intransitivity means that there is no best choice. Consider the following preference profile for three agents and three alternatives:
u1(x) > u1(y) > u1(z)u2(y) > u2(z) > u2(x)u3(z) > u3(x) > u3(y)
In pair-wise voting between x and y; x wins, in a vote between y and z; y wins, and transitivity of social preferences requires that x is socially preferred to z; however, in a vote between x and z; z wins — The Condorcet Paradox. The problem here is that majority voting is vulnerable to manipulation, the so called “agenda setter”, has a lot of power in that he/she may structure the voting process so that alternatives which he/she does not like is sure to be eliminated (e.g. if the agenda setter is person 3, he/she may decide that the vote should first be between x and y, and the winner will be pitched against z → z wins).
So majority voting as a social choice rule is not perfect; but maybe there is other rules which obeys some basic requirements for fairness and of being “democratic” and which is able to give an ambiguously best choice without being subject to manipulation? The answer is no! This result was proven by Kenneth Arrow and is called Arrow′s impossibility theorem:
If a social choice rules satisfies the requirement of being complete, reflexive and transitive, and if it obeys the Pareto principle, and if it also exhibits independence of irrelevant alternatives, it must be dictatorial; i.e., the social preference rankings coincide with that of one individual.
This is an amazing (and maybe depressing) result, which shows that if we add some quite reasonable requirements on our social choice rule, it will turn out to be as un−democratic as can be. The proof of this theorem is beyond the scope of this course, but we may anyway discuss the requirements a little bit. We’ve already discussed the three first ones, but not the Pareto principle, and the “independence of irrelevant alternatives” requirements. The Pareto principle is quite simple; if everyone prefers x to y, the social preference ranking should express the same ranking. “Independence of irrelevant alternative” means that if we, for example, first ask about the rankings between x and y, and find that our social choice rule ranks x in front of y, but if we then add a third alternative (z), the social choice rule should not change the rankings of x and y. The problem here is that, if this requirement is not satisfied, an agenda setter who prefers y in front of x, could, by add a third alternative, change the outcome in his/her own favor.
The upshot of Arrow’s theorem is that we cannot hope for perfect choice rules, and politics will remain a “dirty business”. However, we will not dwell on this point here but will go on boldly anyway, remembering Arrow’s warning all along. (Arrow’s theorem does not say that things will go wrong in all voting schemes, just that there is no such scheme which is immune to the risk that things may go wrong.) Anyway, if we want to be on the safe side we have to skip some of Arrow’s requirements (the independence of irrelevant alternative may be the weakest of the requirements and could be jettisoned), if we want to go on to define a social welfare function corresponding to some (transitive) social preferences; such a function would be convenient since it would help us to pick out a “best” allocation among all the possible Pareto efficient allocations.
A social welfare function is a function of individual utility functions, which themselves are functions of vectors of allocations. I.e., the social utility of allocation x, then there are n persons in the economy, is defined as,
W(x) =W(u1(x), ...,un(x))
As is the case for individual utility functions, the social welfare function is just a weighting formula which produces a number which is higher for preferred bundles (vectors in this case). As a matter of fact, we can do any increasing transformation to the function we use, as long as it preserves the rankings. We will require that if x is preferred to y for all individuals, the social welfare function will give a higher number for x than for y, (the Pareto principle), but apart from that, it can look almost any way we like.
Moral philosophers have suggested various ethical principles to determine distribution of utility in society, which we can translate into different types of social welfare functions, e.g.,
• The utilitarian welfare function, is simply the sum of all individuals’ utility,
• The minimax, or Rawlsian, welfare function focuses on maximizing the utility of the worst off individual, and is expressed as,W(u1, ...,un) =min (u1, ...,un) .
• A generalization of the utilitarian welfare function is the weighted-sum-of-utilities welfare function,
The a′is are weights given to various individuals’ utility in social welfare. Of course, ex ante all persons have the same weight, but at some moment in time, different fortunes have produced different de facto positions in life, which may require the a′is to differ between people. We should also note that we’re free to make any ranking preserving transformations of the individuals’ utility functions we want, e.g., we may take the natural logarithm, and rewrite the last welfare function as,
The important point of this is that it gives us a Cobb-Douglas form of the function, with convex social indifference curves.
Equipped with a social welfare function we can proceed to maximize it, i.e., to pick out the best point on the contract curves running through the Edgeworth box. However, since we use individual utilities as variables in the welfare function, it would be nice if we could go from the quantity space in the box-diagram to a utility space, i.e., a diagram with the individual utilities on the axes. This turns out to be quite simple; let’s go back to Figure and look at the origin for person 1. At his point he has no consumption at all and it is natural to normalize his utility function so that it shows zero utility (i.e., u1(0, 0) = 0), now person 2 owns all available consumption goods at this point and reaches his maximum utility (u2(ω1, ω2) = umax 2, where (ω1, ω2) is the aggregate endowment of both goods). At the origin for person 2 we of course have the reverse situation, person 1 reaches his maximum utility and person 2 gets zero utility. Now if we move up the contract curve, into the interior of the box, from person 1′s origin, his utility increases and person 2′s utility decreases. If we plot all the utility combinations along the contract curve in the utility space we must end up with a downward sloping curve.
This is illustrated in figure shown below. The grey area shows the utility possibility set, and the north-east border of this set shows all Pareto-efficient set of points, or the utility possibility frontier, i.e., the points derived from the contract curve, as described. The utility possibility set is drawn as a convex set, but this is not implied by the contract curve, however, if it is we can apply the usual maximization techniques to find a social welfare maximum. To this effect I’ve also drawn three different indifference curves, corresponding to the three different social welfare functions discussed above, each one picking out a different social welfare maximum.
If society has Rawlsian preferences it will always end up at a position of complete equality (in utility terms); a Utilitarian society has linear social indifference curves (the slope of the indifference curve is −1), and will only end up at a position of complete equality if the utility possibility set is completely symmetric, i.e., if both persons have exactly the same preferences. In the figure I’ve assumed that this is not the case and that the Utilitarian social welfare maximum lies above the 45?-degree line, giving more utility to person 2. The Generalized Utilitarian function has convex indifference curves and shows therefore a diminishing marginal rate of substitution to an increase in the utility given to person 1, i.e., to sacrifice more of person 2′s utility, requires greater and greater additions to person 1′s utility. The social welfare maximum occurs here at point C, indicating a higher weight on person 1.
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