Lecture Notes, Econ G30D: General Equilibrium and the Welfare Theorems

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1 Lecture Notes, Econ G30D: General Equilibrium and the Welfare Theorems Martin Kaae Jensen November 30, 2008 Correspondence: Martin Kaae Jensen, University of Birmingham, Department of Economics, Edgbaston, Birmingham B15 2TT, UK. Voice: (+44/0) , Homepage: 0

2 1 Allocations and Pareto optimality We consider as before an economy with I consumers, J firms, and N goods. Recall from week 1 that X i denotes the consumption set of consumer i (usually taken to be N + ). Furthermore, let Y j N denote the production set of firm j. The production set describes all possible input-output combinations available to a given firm. In our favored description with production functions, we consequently have for a firm that produces good m {1,..., N } by means of the other goods: Y j = {y j N : y j f j (y j m 1,..., y j m 1, y j m+1,..., y j N )} (1) Notice the inequality in the previous expression. Even though a firm produces f j (y j 1,..., y j m 1, y j m+1,..., y j N ) units of the output, it is perfectly possible for it to supply a smaller amount y j m to the market since the firm can always just destroy some output or throw it away. 1 Definition 1 (Feasible Allocations) An allocation (x, y ) = (x 1,...,x I, y 1,..., y I ) is a collection of a consumption vector x i X i for each consumer i and a production vector y j Y j for each firm j. An allocation (x, y ) is feasible if aggregate demand equals aggregate supply: I I J x i = ω i + y j (2) i =1 Exercise: A Walrasian equilibrium allocation is always feasible. Why? i =1 Definition 2 (Pareto Optimality) A feasible allocation (x, y ) is Pareto optimal (or Pareto efficient) if there does not exist another feasible allocation ( x, ỹ ) that Pareto dominates it, i.e., if there does not exist a feasible allocation ( x, ỹ ) such that, u i ( x i ) u i (x i ) for all i = 1,..., I with at least one strict inequality. (3) The best way to think of Pareto optimality is as a minimum requirement of an allocation: If an allocation is not Pareto optimal, we would be able to allocate the economy s 1 Strictly speaking, this is an assumption (that firms can always dispense with output if they want to). This assumption is known as free disposal (actually free disposal says two things: (i) The one mentioned: that firms can always supply less than some given (feasible) amount for given inputs, and (ii) That if the firms can produce, say y j m then they can also produce this by using more of all factors (this, intuitively, means that they can always choose to be less efficient: workers take a few more breaks, machines are not oiled every day, and so on, hence even if more workers and more machines are used, no more output will be produced). j =1 1

3 resources in a different way such that everyone becomes at least as well off and at least one person becomes better off. Thus it may be said that if an allocation is not Pareto optimal, there is room for improvement. If, for example, it turned out that free market economies (=private ownership economies) led to WEAs that were not Pareto optimal, the conclusion would be that we could, at least in principle, do better. And not just better in some vague sense: Better in such a way that everyone would agree that it was at least as good and someone would find it strictly better. Clearly, that would be a strong argument against free markets. As we shall see next, WEAs are, however, Pareto optimal. 2 The Welfare Theorems Recall that a Walrasian equilibrium allocation (WEA) was defined before as a vector (x (p, W ), y (p )) where x (p, W ) = (x 1, (p, W 1 ),...,x I, (p, W I )) and W i = p ω i + J j =1 θ j i πj (p ) It is clear that W i is itself a function of prices only, i.e., W i = W i (p ) (precisely, W i (p ) = p ω i + J θ j j =1 i πj (p )). Hence we can also, and shall from now on write a Walrasian Equilibrium allocation s consumption component as a function of prices only: x (p ) = ((x 1, (p, W 1 (p )),...,x I, (p, W I (p ))) This notation is slightly more convenient because it is more compact. When p is a WE we can write the associated WEA as (x (p ), y (p )). The first theorem of welfare economics says that any WEA is Pareto optimal. The standard statement of this result concerns private ownership economies: Theorem 1 (First Welfare Theorem, Private Ownership) Assume that consumers utility functions are strongly monotone. Let (x (p ), y (p )) be a Walrasian equilibrium allocation in a private ownership economy. Then (x (p ), y (p )) is Pareto optimal. Yet, it is also true that any WEA in an economy with lump-sum transfers will be Pareto optimal: Theorem 2 (First Welfare Theorem, Economies with (Lump-sum) Transfers) Assume that consumers utility functions are strongly monotone. Let (x (p ), y (p )) be a Walrasian equilibrium allocation in an economy with transfers. Then (x (p ), y (p )) is Pareto optimal. 2

4 Even though, as discussed in the previous section, the first welfare theorem implies that private ownership economies (with or without transfers) pass the first acceptability check (Pareto optimality), it is important to realize the limitation of this result. It says nothing about fairness. To take an extreme example, Pareto optimality does not exclude a situation where one person ends up consuming everything. As long as this person s utility function is strongly monotone, we could not allocate things differently without taking something away from him, thus making him (strictly) worse off. In other words, we cannot find a Pareto dominating allocation. So what the first welfare theorem says is that WEAs are Pareto optimal. What it does not say is that a WEA will be fair in this or that sense of the term - it could be very unfair as in the example of the previous paragraph. Of course, opinions differ on what fair means. Some would say that whatever a free market economy ends up giving people is by definition fair because it accurately reflects their skills, talents, and luck. Others would say that anything that ends up distributing consumption in a grossly uneven way is unfair. Still, regardless of what one s pet allocation might be (in the sense that one thinks this is the most fair distribution) it ought to be Pareto optimal (why would we not want someone to become strictly better of when it does not hurt someone else?). This is where the second welfare theorem enters the debate. It says, roughly, that no matter which (Pareto optimal) allocation you prefer, an economy with transfers can give it to you. Theorem 3 (Second Welfare Theorem) Make all of the assumptions of Theorem 1 of the last set of lecture notes. Let (x, y ) be any Pareto optimal allocation. Then there will exist transfers (T 1,..., T I ) with i T i = 0 such that (x, y ) is the Walrasian equilibrium allocation of the economy with these transfers. In other words, (x, y ) = (x (p ), y (p )) where p is a Walrasian equilibrium of the economy with transfers (T 1,..., T I ). Needless to say, the second welfare theorem is a very forceful result: It says that no matter how you think an economy ought to distribute its resources; such a distribution can be accomplished by the free market once suitable lump-sum transfers have been carried out. For anyone who wants to understand modern economic policy it is crucial to understand the second welfare theorem. I ll talk more about this at the lectures, but here just two important points. Firstly note that the kind of transfers described in the second welfare theorem (as well as in the first welfare theorem for economies with transfers) are hardly realistic. They are of the lump-sum kind, and in reality taxes are not of this kind. Nonetheless, they remain an ideal precisely because they are non-distortionary (by which is meant precisely that they do not push the economy away from Pareto optimality). Other 3

5 kinds of taxes will as a general rule lead to WEAs that are not Pareto optimal - they are distortionary. Secondly, you may want to ask exactly how one would ever calculate the transfers T 1,..., T I associated with a specific Pareto optimal allocation. In reality, this cannot be done of course and would never even be attempted. But still, the main thrust of the result lives on: Say you prefer equal distributions. Then you can transfer incomes (presumably then in an egalitarian fashion) and leave the rest to the market economy. This will then produce a Pareto optimal allocation and chances are that you d be happy with the outcome. The key point here (as far as political opinions are concerned) is that you don t have to scrap the free market economy. Of course these last observations are modified, possibly even made irrelevant, by the fact that lump-sum transfers cannot be carried out in the real world. In the real world, you need to do these things via distortionary taxes, so you need to sacrifice efficiency in order to get fairness. 4