Chapter 34 Welfare Key Concept: Arrow’s impossibility theorem, social welfare functions Limited support of how market preserves fairness.

Chapter 34 Welfare We want to say a bit about the distribution of welfare across people. Recall the Pareto efficiency is silent on this. Moreover, there are typically many Pareto optimal allocations. This leads us to some analysis regarding welfare and welfare function.

The idea is we would like to have a way to aggregate different consumer’s rankings to get a social preference. If we cannot do this, we may want to “add together” different consumers’ utilities.

We first turn to the issue of how to aggregate consumers’ preferences. Denote x a particular allocation (a description of what every individual gets of every good).

Note that each consumer is assumed to have preferences over the entire allocation of goods so we can model concepts such as equity. It is still possible that consumers might not care about what other people have though in this rather general model. Hence we are not imposing a restriction.

Social preference: If we know how all the individuals rank various allocations, we would like to use this information to develop a social ranking of the various allocations.

Consider two possible mechanisms: majority voting rank-order voting (1 for the best, 2 for the second best, and so on). Then we sum up across people to get the scores of each alternative.

majority voting rank-order voting Both are practical mechanisms. What could be the potential problems with them?

majority voting Social preferences may not be transitive, hence there will be no best alternative. Which outcome society chooses will depend on the order in which the vote is taken.

majority voting x vs. y first, winner vs. z → x, z z vs. x first, winner vs. y → z, y

Table 33.1

rank-order voting Only x and y are available → tie z is introduced → y

Table 33.2

Majority voting can be manipulated by changing the order on which things are voted to yield the desired outcome. Rank-order voting can be manipulated by introducing new alternatives that change the final ranks of the relevant alternatives.

What mechanisms are immune to this kind of manipulation? First list desirable things that we would want social decision mechanism to satify:

Social preference must: Be complete, reflexive and transitive (just as individual preference). If everyone prefers x to y, then social preference must also rank x ahead of y. The preferences between x and y depend only on how people rank x versus y, and not on how they rank other alternatives.

Arrow’s Impossibility Theorem: If a social decision mechanism satisfies these three properties, then it must be a dictatorship: all social rankings are the rankings of one individual.

Given the negative result of Arrow’s Impossibility Theorem, we have to drop some desirable feature. Property 3 is likely dropped first. In that situation, certain kinds of rank-order voting may become possible.

Alternatively, we could use the cardinalities of individual utility functions. And add up them one way or another. A social welfare function is some function of individual utility functions, W(u1, …, un). It gives a way to rank different allocations that depends only on individual utilities.

If each individual has a utility function ui(), we can add up each individual’s utility so that x is socially preferred to y if Σi=1nui(x)>Σi=1nui(y) where n is the number of individuals. This is the classical utilitarian welfare function where W(u1, …, un)=Σi=1nui The idea is…

Another often-mentioned is the Rawlsian social welfare function or W(u1, …, un)=min{u1, …, un}. The idea is...

These different social welfare functions are ways to compare individual utilities. Each represents different ethical judgments about the comparison between different agents’ welfare. Economists typically have no say on this except the social welfare function should be increasing in each consumer’s utility.

Once we have a welfare function we can write down the welfare maximization. Suppose we have k goods, n persons. Max W(u1(x) …, un(x)) st. Σi=1n xi1=X1 … Σi=1n xik=Xk

First observation: A maximal welfare allocation must be Pareto efficient. Suppose not, then there would be some other feasible allocation that Pareto improves. But since the social welfare function is increasing in each agent’s utility, this other allocation must have a larger welfare. This contradicts that we originally had a welfare maximum.

Graphically, we could plot the utilities of all allocations Graphically, we could plot the utilities of all allocations. This gives us the utility possibility set. The boundary of this set give us the Pareto efficient allocations or utility possibility frontier.

We could also draw the iso-welfare curves. To maximize social welfare, we are pushing the iso-welfare to the farthest on the utility possibility set. It end ups at the utility possibility frontier.

Fig. 33.1

Second observation: any Pareto efficient allocation must be a welfare maximum for some welfare function.

Fig. 33.2

Welfare maximum ↔ Pareto efficiency If individual’s preference only depends on the bundle of goods he gets (it does not depend on the entire allocation) or W(u1(x1) …, un(xn)) then welfare maximum ↔ Pareto efficiency ↔ market equilibrium

The welfare function approach is general enough to accommodate many kinds of moral judgment. Another approach is to start with some specific moral judgment and then examine the implications.

Let us think about fairness. We start defining what is fair. Then use our understanding of economic analysis to investigate its implications.

Suppose you want to divide some goods fairly among n equally deserving people. How would you do it? Most people probably will divide equally. Why is equal division appealing? One appealing feature is that it is symmetric.

But an equal division may not be Pareto efficient But an equal division may not be Pareto efficient. If agents have different tastes, they will generally desire to trade away from the equal division. But after trading, do we still preserve some goodness of symmetry?

What is so good about symmetry? Symmetry itself may not be so desirable. What is desirable is presumably, symmetry makes sure no one envies the other.

So we want no-envy and Pareto efficient. We can now formalize the ideas.

An allocation is equitable if no agent prefers any other agent’s bundle of goods to his or her own. If some agent i does prefer some other agent j’s bundle of goods, then we say that i envies j. Finally, if an allocation is both equitable and Pareto efficient, then we say it is a fair allocation.

To determine whether an allocation is equitable, just look at the allocation that results if two agents swap bundles. If this swapped allocation lies below each agent’s indifference curve through the original allocation, then the original allocation is an equitable allocation.

Fig. 33.3

The allocation is equitable and efficient, and hence fair. But will fair allocations typically exist?

Suppose we start from an equal division allocation and then allow people to trade according to the market mechanism. Is the end outcome fair? Everyone will choose the best bundle he can afford at the equilibrium prices (p1, p2).

First, is it equitable? Suppose not, let us say A envies B. Then (xB1, xB2) sA (xA1, xA2). Thus p1xB1+p2xB2 >p1wA1 +p2wA2. This is not possible since A and B started with the same bundle.

We also know that the equilibrium must be Pareto efficient.

A competitive equilibrium from equal division is fair. The market mechanism will preserve certain kinds of equity. That is, if the original allocation is equally divided, the final allocation must be fair.

Chapter 34 Welfare Key Concept: Arrow’s impossibility theorem, social welfare functions Limited support of how market preserves fairness.