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Prof. Dr. Olivier Bochet Room: A.314 Phone: 031 631 4176 E-mail: olivier.bochet@vwi.unibe.ch Webpage: http://staﬀ.unibe.ch/bochet Advanced Microeconomics Fall 2010 Lecture Note 2 General Equilibrium I General equilibrium is the study of the microfoundations of the aggregate. All agents and markets interact with one another. So it views the economy as an interrelated system in which equilibrium values of all variables are determined simultaneously. It is a theory of the determination of prices quantities in a system of perfectly competitive markets. The basic questions we aim to address are whether markets work well, do market equilibrium exist, what do we think about the outcome of a market process? We will ﬁrst introduce the notations and deﬁnitions that are useful for this part of the course. At the end of the note, we will discuss in detail several of the crucial assumptions under the theory of general equilibrium is built. One of the two crucial assumptions are the existence of a complete set of markets and price-taking behavior –i.e. the assumption that economic agents have no inﬂuence whatsoever on the quoted prices. 1 1 Basic notations and deﬁnitions 1.1 A review of properties of preference relations Let RL be the consumption set. A bundle x = (x1 , ..., xL ) ∈ RL is simply a + + list of quantities of each good. A preference relation denoted is a binary relation deﬁned over the consumption set RL . A preference relation describes + an (ordinal) ranking of the possible bundles in RL . For each x, y ∈ RL , x y + + reads as ”x is at least as good as y”; x y reads ”x is strictly preferred to y”; and x ∼ y reads as ”x is indiﬀerent to y”. For preference relation and each x ∈ RL , let LC( , x) ≡ {y ∈ RL : x y} be the lower contour set at + + x of ; U C( , x) = {y ∈ RL : y+ x} be the upper contour set at x of ; SU C( , x) = {y ∈ RL : y+ x} be the strict upper contour set at x of ; and IC( , x) = {y ∈ RL : x ∼ y} be the indiﬀerence set at x of . + We now introduce several properties of preference relations that will be useful for this note. 1. Preference relation is complete if for each x, y ∈ RL , either + x y, or y x, or both. 2. Preference relation is transitive if for each x, y, z ∈ RL , x + y and y z implies that x z. If property 1. and 2. are satisﬁed, we say that is rational. In the rest of this note, we take for granted that preference relations are rational. 3. Preference relation is locally non-satiated if for each x ∈ RL + and each > 0, there exists y ∈ RL such that y − x ≤ and y x. + The property of local non-satiation asserts that for any x, one can draw a closed ball of radius centered at x, B ,x ≡ {y ∈ RL : y − x ≤ }, and + ﬁnd a bundle y within that ball such that y x. Notice that the deﬁnition is silent regarding the direction in which one moves from x to ﬁnd y. For instance the deﬁnition does not rule out that y is located south-west of x –see Figure 1 in the appendix. However, notice that although the deﬁnition does not rule out that some goods are ”bad”, it is not possible that all goods are ”bad”. For suppose this is the case, then the best consumption point would be the 0 bundle, in contradiction with local non-satiation. One important 2 implication of local non-satiation is that it rules out thick indiﬀerence sets. This is shown in Figure 2. 4. Preference relation is monotonic if for each x, y ∈ RL such that + y x, we have y x. Monotonicity of preferences require that for y to be strictly preferred to x, y should contain more than x in every good. That is for each good , y > x . This is what means. Obviously, monotonicity of preferences implies local non-satiation. 5. Preference relation is strictly monotonic if for each x, y ∈ RL + such that y > x, we have y x. Strict monotonicity requires that for each good , y ≥ x with at least one strict inequality for one good. Obviously, strict monotonicity implies monotonicity. We now turn to convexity assumptions. 6. Preference relation is convex if for each x, y, z ∈ RL such that + y x and z x, then αy + (1 − αz) x for each α ∈ [0, 1] A direct implication of the assumption of convexity is that upper contour sets are convex. That is for each x ∈ RL , the set U C( , x) is a convex set. + Convexity is usually explained as diminishing marginal rates of substitutions → inclination of agents to diversify their consumptions. An important impli- cation of the assumption of convexity is that it rules out ”wavy” indiﬀerence curves. 7. Preference relation is strictly convex if for each x, y, z ∈ RL + such that y x and z x, then αy + (1 − αz) x for each α ∈ (0, 1) An important implication of the assumption of strict convexity is that it rules out indiﬀerence curves with ﬂat segments. Obviously, strict convexity implies convexity. See Figure 3. 3 8. Preference relation is continuous if for each sequence of pairs m m ∞ of bundles {(x , y }m=1 with lim xm = x, lim y m = y, and xm y m for m→∞ m→∞ all m, then x y. Thus, the preference relation is continuous if it is preserved under limits. An important implication of the assumption of continuity is that upper contour sets and lower contour sets are closed sets. A second important implication is that if the preference relation is continuous, there exists a utility function u : R+ → R such that for each x, y ∈ RL , x y ⇐⇒ u(x) ≥ L + u(y). That is, the function u represents the preference relation . A classical example of a preference relation that is not continuous is the lexicographic preference relation. Example 1 Lexicographic preferences are not continuous Consider the case L = 2 goods. Let be the lexicographic preference relation over good 1. Then for each x, y ∈ R2 + x y if either x1 > y1 , or if x1 = y1 and x2 ≥ y2 Notice in particular that by deﬁnition if x1 > y1 , we necessarily have x y. You can check that lexicographic preferences are rational, strictly monotonic and strictly convex. However they are not continuous as shown below. 1 Consider the two sequence xm = ( m , 0) and y m = (0, 2). Note that y m is a constant sequence since it will not vary with m. For every m ≥ 1, 1 we have xm y m since m > 0. But observe that lim xm = (0, 0) ≡ x and m→∞ lim y m = (0, 2) ≡ y. Hence in the limit we obtain that y x, a contradiction m→∞ with continuity. The lexicographic preference relation puts inﬁnite weight to inﬁnitesimal changes. An important feature of the lexicographic preference relation is that indiﬀerence sets are singletons, i.e. no two bundles x and y are indiﬀerent. Obviously, its lack of continuity implies that upper contour sets are not closed. 4 1.2 Exchange Economies There is a set of agents N = {1, .., n} and L inﬁnitely divisible goods. The consumption set of each agent i ∈ N is RL . For each i ∈ N , a bundle + xi = (x1 , ..., xL ) ∈ RL is simply a list of quantities of each good. A preference + relation for agent i, denoted i. Each i ∈ N has some initial endowment –a stock of resources– ωi = (ω1i , ..., ωLi ) ∈ RL \{0}. The aggregate endowment is + denoted ω = i∈N ωi . We assume that ω ∈ RL , i.e. each good is available ¯ ¯ ++ in some quantities. The endowment point ω = (ω1 , ..., ωn ) ∈ RLn is the list + of agents’ initial endowments. An exchange economy is E = ( i )i∈N , (ωi )i∈N . An allocation x = (x1 , ..., xn ) ∈ RLn is a list of bundles, one for each agent. An allocation + x ∈ RLn is feasible if i∈N xi ≤ ω . This deﬁnition of feasibility implicitly + ¯ implies that there is free disposal of goods, i.e. there is a technology available to destroy goods if necessary. The set of feasible allocations for economy E is FE ≡ {x ∈ RLn : i∈N xi ≤ ω }. The set of non-wasteful allocations is + ¯ ¯E ≡ {x ∈ RLn : ¯ F + i∈N xi = ω }. Notice that FE is nothing more than the ¯ Edgeworth box for E. We now want to investigate among all allocations that are feasible, the ones that are economically meaningful. For this, we introduce the central notion of economic eﬃciency, namely Pareto eﬃciency. Pareto eﬃciency: Allocation x ∈ FE is Pareto eﬃcient if there does not exist another allocation y ∈ FE such that yi i xi for each i ∈ N (1) yj j xj for at least one j ∈ N (2) By looking at Figure 4, we quickly notice that an allocation can be Pareto eﬃcient only when the intersection of the strict upper contour sets of agents is empty. That is, allocation x is eﬃcient if ∩ SU C( i , xi ) = ∅. i∈N At an interior allocation1 , indiﬀerence curves must be tangent at x if x is eﬃcient. This is illustrated in Figure 5. On the other hand, this claim is not true for allocations that lie on one of the boundary of the Edgeworth box as shown in Figure 6.2 There, the tangency condition is not necessary. In 1 An allocation is interior if for each i ∈ N , xi 0. 2 ¯ ¯ The boundary of the Edgeworth box is ∂ FE = {x ∈ FE : there exists i ∈ N for whom x i = ω for at least one }. ¯ 5 Figure 6, we clearly see that the slopes of the two indiﬀerence curves at x are not equal. However, it remains true that SU C( 1 , x1 ) ∩ SU C( 2 , x2 ) = ∅. Let us now look at a computational example of the set of Pareto eﬃcient allocations. Example 2 Computing the Pareto set ¯ Let n = L = 2, ω = (1, 1), and preferences of agents be represented by utility functions as follows u1 (x11 , x21 ) = x11 x21 1 2 u2 (x12 , x22 ) = (x12 ) 3 (x22 ) 3 Let’s start by asking ourselves the following question. Can an allocation at which one agents gets 0 of one of the good be Pareto eﬃcient? Notice that this agent would get a 0 utility level. Actually the indiﬀerence curves of these two utility functions never touch the axis –i.e. both axis are asymptotes. Therefore, each agent is indiﬀerent between getting the 0 bundle and any bundle that gives 0 unit of one of the good. We conclude that Pareto eﬃcient allocations are interior exception made of both origins –preferences being strictly monotonic, an agent getting all of the two goods while the other gets nothing is obviously eﬃcient. Hence at an interior eﬃcient allocations, we have −2 2 1 x21 (x12 ) 3 (x22 ) 3 M RS1 = M RS2 ⇐⇒ = 3 1 −1 x11 2 (x12 ) 3 (x22 ) 3 3 Rearranging, x21 x22 = (3) x11 2x12 We now want to compute an equation of x21 as a function of x11 , that describes the set of eﬃcient allocations (from the point of view of agent 1). Because preferences are monotonic, we know that eﬃcient allocations ¯ must be non-wasteful, i.e. any eﬃcient allocation x is in FE . Hence ¯ x11 + x12 = ω1 = 1 ¯ x21 + x22 = ω2 = 1 6 Substituting into equation (1) above, we have x21 1 − x21 = x11 2(1 − x11 ) We obtain that the set of eﬃcient allocations is described as x11 x21 (x11 ) = 2 − x11 What is the shape of this curve? We already know that it connects both origins. dx21 (x11 ) 2 = >0 dx11 (2 − x11 )2 And d2 x21 (x11 ) 4 = > 0 since x11 ≤ 1 dx211 (2 − x11 )3 The Pareto set is an increasing and convex curve –and it lies below the ◦ 45 line. This is shown in Figure 7. We now turn our attention to the outcomes delivered by market pro- cesses. Notice that in determining the Pareto set, the initial distribution of resources in the economy played no role whatsoever. Only the size of the Edgeworth box mattered. This will not be true of market processes since initial endowments will play a central role: along with prevailing prices, they determine the initial wealth of each agent. For each i ∈ N , let Bi (p) = {xi ∈ RL : p · xi ≤ p · ωi } be agent i’s budget + set at prices p. Walrasian equilibrium: Given an economy E, A price-allocation pair ∗ (p , x∗ ) ∈ RL \ {0} × RLn is a Walrasian equilibrium if the following two + conditions hold 1) x∗ = ω i ¯ i∈N Supply=Demand 2) For each i ∈ N , x∗ i xi for i all xi ∈ Bi (p) x∗ is maximal for i i over Bi (p) 7 There are two components in the deﬁnition of a Walrasian equilibrium. The ﬁrst item states that supply is equal to demand. The second one states that each agent i is utility maximizing at x∗ . Any bundle strictly preferred i to x∗ must be unaﬀordable. Notice that the deﬁnition does not rule out i that there may be more than one such maximizer –e.g. indiﬀerence curves that have ﬂat segment. Also nothing guarantees that an economy has a single Walrasian equilibrium. First, existence is not guaranteed (we will study later on the conditions that guarantee existence). Second, in general an economy may have several Walrasian equilibria. The following ﬁve ﬁgures illustrate the deﬁnition and the possibility for multiplicity of equilibria. Figure 8 shows an example in which the max- imization of preferences, given prices p and endowments ω, violates the supply = demand requirement contained in the deﬁnition of a Walrasian equilibrium. Figure 9 shows an example of a Walrasian equilibrium. Notice that the maximizers of each agent’s preferences, given p and ω, is a singleton. This is so because preferences are strictly convex. On the other hand, Figure 10 shows an example of a Walrasian equilibrium where preferences of agent 1 are convex but not strictly convex. We see that the set of best bundles that agent 1 can obtain, given p and ω1 , indeed contains x1 but this is not the only bundle that maximizes preferences with respect to agent 1’s budget constraint. In Figure 11, x is a Walrasian allocation that is on the boundary of the Edgeworth box. In Figure 12, x is a boundary allocation that is not Walrasian. If we restrict ourselves to looking only inside the Edgeworth box, then x seems Walrasian. But there are bundles that are aﬀordable and better than x1 for agent 1 and that lie outside of the Edgeworth box. Obviously these bundles violate feasibility but observe that the preference maximization part in the deﬁnition of a Walrasian equilibrium does not require that the preference maximizing bundle be feasible. This is a very important detail to understand. What happens outside of the box is crucial to understand what is going on inside! Finally, Figure 13 displays an example of an economy with multiple Wal- rasian equilibria. Notice that the number if equilibria is odd. This is a general observation –we will come back to it in the next lecture note. Example 3 The algebra of equilibrium 8 Let us go back to our previous example and add now the initial distribu- tion of resources in the economy with ω1 = (1, 0) and ω2 = (0, 1). In order to ﬁnd the (unique) Walrasian equilibrium of this economy, we need to ﬁrst compute the demand functions of both agents. Agent 1 : M ax u1 subject to p · x ≤ p · ω1 = p1 x11 ,x21 Since u1 is strictly increasing, the constraint must hold with equality. Also since u1 = 0 whenever the bundle contains 0 unit of one of the two goods, we know that the solution to the maximization problem is interior. Hence, at the optimum, x21 p1 M RS1 = = x11 p2 We know then that p2 x21 = p1 x11 . We substitute this information into the budget constraint and solve for the demand functions. We obtain that 1 x11 (p, p · ω1 ) = 2 p1 x21 (p, p · ω1 ) = 2p2 and p2 x12 (p, p · ω2 ) = 3p1 2 x22 (p, p · ω2 ) = 3 This takes care of the utility maximization part in the deﬁnition of a Walrasian equilibrium. Now at equilibrium, we must also have supply equal to demand. Let us normalize p1 = 1 –only relative prices matter. Let us look at the market for good 1 –we only need to solve for one of the two markets: whenever L − 1 markets are in equilibrium, then so is the nth market; can you say why? 1 p2 3 + = 1 ⇐⇒ p2 = 2 3 2 Hence the equilibrium price vector is p = (1, 3 ). We now compute the 2 demands at these prices and check that supply is indeed equal to demand. 9 We obtain that 1 x11 = 2 1 x21 = 3 1 x12 = 2 2 x22 = 3 The Walrasian equilibrium (p∗ , x∗ ) is 3 1, 2 , ( 1 , 1 ); ( 1 , 2 ) . 2 3 2 3 Given our investigation of markets processes, we may want to know more regarding this equilibrium allocation. In particular, we would like to know whether markets deliver an allocation that is eﬃcient. The equation of the Pareto set for this economy was x11 x21 = 2 − x11 1/2 At x∗ , x11 = 1 and hence x21 = 2−1/2 = 1 . We conclude that x∗ is 2 3 eﬃcient. This turns out to be a general observation under some very weak assumptions regarding the primitives of the economy. Theorem 4 (First welfare theorem) Let E be an economy. Suppose pref- erences are locally non-satiated. Then any Walrasian equilibrium is Pareto eﬃcient. Proof. Pick an economy E and pick (p∗ , x∗ ) a Walrasian equilibrium. As- ¯ sume by contradiction that x∗ is not eﬃcient. Then there exists y ∈ FE such ∗ ∗ that yi i xi for each i ∈ N , and yj j xj for at least one j ∈ N . The preference maximization part in the deﬁnition of a Walrasian equilibrium implies that yj j x∗ =⇒ p∗ · yj > p∗ · ωj . Local non-satiation implies also j an additional property: if yi i x∗ , then p · yi ≥ p · ωi . Suppose this is not i true. That is there exists yi i x∗ and p · yi < p · ωi . By local non-satiation, i there exists yi and > 0, arbitrarily small, such that yi − yi ≤ , yi i yi , and p · yi ≤ p · ωi . By transitivity yi i x∗ . But this is in contradiction with i 10 x∗ being a maximal element in agent i’s budget set. Hence the claim is true. i This gives us that p · yi > p · ωi = p · ωi = p · ω ¯ i∈N i∈N i∈N Thus, p· yi > p · ω ¯ i∈N This inequality can be true if and only if ¯ yi > ω i∈N / ¯ But then y ∈ FE , a contradiction with our initial assumption. We con- clude that x∗ is Pareto eﬃcient. Q.E.D. Figure 14 shows what is called the contract curve: the intersection be- tween the Pareto set and the set of allocations that are individually rational.3 Indeed, no one would accept to trade to an allocation that makes one worse than keeping one’s endowments. Contracts (trades) can only occur within the individually rational region. By the ﬁrst welfare theorem, trades must lead to an eﬃcient allocation of resources. The ﬁrst welfare theorem delivers a strong message. Under very weak as- sumptions, the outcome of the interactions of agents through markets gives an eﬃcient allocation of resources. This is nothing less than the ”invisible hand” described by Adam Smith. However, observe that we have no infor- mation regarding distribution of resources. All we know is that this is done in an eﬃcient manner. But could markets be biased toward some distributions, thereby favoring some agents (or group of agents) over others? The answer is negative and this will be the content of the second welfare theorem –the most powerful of the two. Before going to its deﬁnition, we ﬁrst conclude on the ﬁrst welfare theorem. The central assumption on the primitive of the economy is that preferences satisfy local non-satiation. Unfortunately, this assumption cannot be dispensed with, as show in Figure 15. There agent 3 An allocation x is individually rational if for each i ∈ N , xi Ri ωi . 11 1 has a thick indiﬀerence curve at x. Although x is a Walrasian, it is not eﬃcient because it is located inside the shaded area determined by agent 1’s thick indiﬀerence curve. We now look at the second welfare theorem and its basic message. We will leave it unproved at this stage. Its proof will be the object of the next lecture note in which we introduce economies with production –private ownership economies. Before stating formally the second welfare theorem, we deﬁne a more general notion of price equilibrium of which Walrasian equilibrium is a special case. Price equilibrium with transfers: Given an economy E, allocation x∗ is supportable as a price equilibrium with transfers if there exists p = 0 and a system of transfers T = (T1 , ..., Tn ) with i∈N Ti = 0 and such that 1) x∗ = ω i ¯ i∈N 2) For each i ∈ N , x∗ i i yi for all yi such that p · yi ≤ p · ωi + Ti This notion of equilibrium is indeed more general than the Walrasian equilibrium notion. The initial distribution of resources is not necessarily the one that prevails since the planner may perform wealth transfers across agents. However, if Ti = 0 for each i ∈ N , then the deﬁnition of a price equilibrium with transfers coincides with the one of a Walrasian equilibrium. Transfers Ti can be positive, negative or simply 0, but notice that transfers are balanced in the sense that they sum to 0. No wealth goes out of the system, and no additional wealth comes in. Theorem 5 (Second welfare theorem) Let E be an economy. Suppose pref- erences are continuous, convex and strongly monotonic. Then any Pareto eﬃcient allocation can be supported as a price equilibrium with transfers. Figure 16 illustrates the mechanic of the second welfare theorem. Its message is very powerful since it assesses that markets are unbiased. Any eﬃcient distribution of resources can be reached through the operation of markets. Simply redistribute wealth across agents in an appropriate manner, and let the invisible hand do its work. 12 Observe that the second welfare theorem relies on much stronger assump- tions that the ﬁrst welfare theorem. What happens if we drop convexity or strong monotonicity? Figure 17 and 18 exemplify the possible failures of the second welfare theorem if one of these two assumptions is not met. In Figure 17, agent 1 has preferences that are not convex –”wavy” indiﬀerence curves. Allocation x is eﬃcient since there is tangency between both agents’ indiﬀerence curves at x, but it cannot be supported as a Walrasian equilibrium. In Figure 18, we see another kind of failure. Because agent 2 likes only good 1 and holds all of good 1, the only possible Walrasian equilibrium is one in which there is no trade, i.e. ω is the Walrasian allocation. Obviously this allocation is Pareto eﬃcient. But it cannot be a Walrasian equilibrium. Suppose that p 0. Then agent 1 can always aﬀord a better bundle than x1 . On the other hand, having one of the price being equal to 0 generates an inﬁnite demand for at least one agent. This shows that not only ω cannot be supported but there exist no transfers that would make this possible. Notice also how the second welfare theorem is linked to the existence of Walrasian equilibria. 13 2 appendix 14 Figure 1: Local non-satiation I 15 Figure 2: Violation of local non-satiation 16 Figure 3: Convex and non-convex indiﬀerence curves 17 Figure 4: An allocation that is not Pareto eﬃcient 18 Figure 5: Tangency condition 19 Figure 6: No tangency on the boundary 20 Figure 7: Pareto set 21 Figure 8: Supply is not equal to demand 22 Figure 9: An interior Walrasian equilibrium I 23 Figure 10: An interior Walrasian equilibrium II 24 Figure 11: A boundary Walrasian equilibrium 25 Figure 12: A boundary allocation that is not Walrasian 26 Figure 13: Multiplicity of Walrasian equilibria 27 Figure 14: The contract curve 28 Figure 15: Failure of the 1st welfare theorem 29 Figure 16: The second welfare theorem 30 Figure 17: Non-convex preferences and the second welfare theorem 31 Figure 18: Monotonic preferences and the second welfare theorem 32