We could cut down the number of points by looking at fairness. The centre of the box is obviously fair - both individuals get exactly the same amount of each good. But it is not efficient - it is not on the contract curve. Let us look at the part of that which we can reach starting from the centre.

Let us get rid of the indifference curves through the starting point.

We still have not got a unique point, but clearly all these points are fair.

Another way of thinking about it is in terms of no-envy. You could say that a situation is fair if neither envies the other - in the sense of wishing he/she had the other person's bundle. But consider the following graph. We have taken an extreme point on the section of the contract curve that is between the indifference curves through the centre of the box. Let us ask whether A envies B if exchange takes place at that point. Would A prefer to have B's bundle. The answer is 'no'. If he/she did she would be on a lower indifference curve than with his/her bundle.

We can push this argument further - right up to the point on the contract curve illustrated below. At that point A is indifferent between having his/her bundle and that of B. Hence the bit on the contract curve below the solid blue indifference curve of A and above the dashed blue indifference curve of A could be considered fair by individual A because he/she does not envy B's bundle.

We can carry out the same argument for B. Take the lowest point on the contract curve between the indifference curves passing through the centre of the box. Does B envy A if exchange takes place at that point? No - B would not prefer A's bundle - it would put him/her on a 'lower' indifference curve.

And we can push this argument right to the point illustrated in the graph below. So all the points above the solid red indifference curve of B and below the dashed red indifference curve of B could be considered fair by individual B because he/she does not envy A's bundle.

So fairness might lead to less uniqueness that we had hoped!

And what happens to this fairness argument if the starting position is at point S2 in the following graph?

Aggregation

Let us try a different approach. Let us ask if it is possible to have some way of deriving society's preferences from individual preferences? Is there some way of aggregating individual preferences to get society's preferences? In practice, voting is often used to decide what society implements. Is this a way? Consider a simple majority voting rule. But first define some notation. Let x, y and z denote possible allocations (complete descriptions of who gets what). If individuals have preferences over these different allocations, can these preferences be aggregated to give a Social Preference Functions (a Social Welfare Function). It would be nice if they could. But how? Might we use for example majority voting?

A prefers x to y to z

B prefers y to z to x

C prefers z to x to y

a majority (A and C) prefers x to y

a majority (A and B) prefers y to z

a majority (B and C) prefers z to x

We note that ranking things and then deciding on the basis of total rank may not help. Let us stay with the preferences above. First forget z.Then x and y tie. Now introduce z. Now y beats x if total rank used.

(2) If everyone prefers x to y then so should Society

(3) Society's preferences between x and y should depend only on individual preferences between x and y

Social Welfare Functions

Perhaps we should simply ASSUME a Social Welfare Function - after all, that is what politicians are for!

Let us suppose that there are N people in the society and let us denote the utility of Individual n by u_n. Let us simply say that society's welfare is some function W = f(u₁, u₂,,...,u_n) of the individual utilities. There are various possible forms that this function may take. The classic utilitarian form can be defined as the particular functional form W = u₁ + u₂ + ... + u_n. This effectively treats all people as equal - though it does asume that the utility values actually mean something (see lecture 5).

If we think that different people should have different weights (perhaps more weight because they are older) we could use the Weighted Utilitarian form can be defined as W = a₁u₁ + a₂u₂ + ... + a_nu_n.

The Rawlsian Utility function takes the form W = min(u₁, u₂,,...,u_n) - this is based on the argument that society's welfare should be based on the 'worst-off' member of society.

GIVEN a Welfare function - it follows rather trivially that an chosen outcome must be PARETO OPTIMAL (assuming, of course, that the welfare function is not decreasing in the individual utilities)

We use a pure exchange economy with two individuals, A and B, and two goods - just as in lecture 8. We draw the Edgeworth Box.

We then argue that society must choose a point along the contract curve and explore the possibilities along it. In figure 9.8 we write on the utility values for A and B at various points along the contract curve. We assume here that the utility functions take the form u = q₁^0.56q₂^0.24 for individual A and u = q₁^0.54q₂^0.36 for individual B

Figure 9.10.

Let us transfer these utility values to a utility possibility frontier.

Now let us add on a the contours of a Classical Utilitiarianism social welfare function.

We can identify the optimal point. Note that Individual B appears to do rather well.

Other SWFs

Note how sensitive the chosen point is to the social welfare function chosen. For example, consider the 'Nash' social welfare function W = u_Au_B

Again we can identify the optimal point.

You might like to think what would happen if we used the Rawlsian social welfare function.

Summary

Aggregation of preferences in general impossible.

Welfare functions are necessarily implicit in society decision making.

If sensible welfare function used then outcome is Pareto Efficient.