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Seminar 1 General Equilibrium and Welfare Simona Montagnana Week 21 19 February 2017 Introduction Exchange economy model Who I am Simona Montagnana [email protected] http://people.bath.ac.uk/sm2446/ Office: 4.27 3East Office Hours Mondays from 13:00 to 15:00 Tuesdays from 13:00 to 15:00 Room: 2.16 3East Welfare Economics References Introduction Exchange economy model Welfare Economics References Questions 2. Write down a pure exchange economy model and answer the following: a. What is the core of the economy? Explain the difference between the concepts of contract curve and core of the economy. b. The Walrasian equilibrium must be in the core of the economy. True or false. Justify your answer. 3. Using an exchange economy general equilibrium model, explain carefully the implications of the two fundamental theorems of welfare economics for the role of the market and the social planner in a welfarist society. Introduction Exchange economy model Welfare Economics References Assumptions: I In the pure exchange economy model the only kind of economic agent is the consumer→ economic activity: consumption and trading (no production). I Each consumer i is described completely by her or his preference (or utility function, ui ), and her or his initial endowment of the k commodities wi . I Each consumer is assumed to behave competitively: the prices as given, independent of her or his actions. I Each consumer attempts to choose the most preferred bundle that she or he can afford. Introduction Exchange economy model Welfare Economics For simplicity: I Two goods 1 and 2, I Economic agent in the system: two consumers A and B, I A set of allocation (consumption bundle) for A and B: (x1A , x2A ), (x1B , x2B ) I Two utility functions (preferences) for A and B: u A (x1A , x2A ), u B (x1B , x2B ) I Consumer’s preferences are strongly monotone. References Introduction Exchange economy model Welfare Economics References I Two endowments for consumer A and B: w A (w1A , w2A ), w B (w1B , w2B ) I Total endowment: W = W A + W B = (w1A + w1B , w2A + w2B ) = (w1 , w2 ) I A feasible allocation is any distribution of good 1 and 2 among the consumers A and B (any combination is possible): (x1A , x2A ), (x1B , x2B ), s.t. x1A + x1B = w1A + w1B = w1 x2A + x2B = w2A + w2B = w2 I The initial endowment is a particular feasible allocation that consumers start with, and consists of the amount of each good that the consumers bring to the market. I The consumers trade the goods among themselves according to certain rules (they are price-takers). They will trade and end up with a final allocation. Introduction Exchange economy model Welfare Economics References Core & Contract curve a. What is the core of the economy? Explain the difference between the concepts of contract curve and core of the economy. Def. ”Core of an economy”: A feasible allocation x is in the core of the economy if it cannot be improved upon by any coalition (group of individuals acting on their own using only the total endowment of the group itself). In the case of two-consumers and two-goods the core is the subset of the Pareto set at which each agent does better than by refusing to trade. In other words, the core is simply that segment of the Pareto set that lies between the indifference curves that pass through the initial endowment W . Def. ”Contract curve”: the set of all Pareto efficient allocations. Introduction Exchange economy model Welfare Economics References Every allocation in the core is also in the contract curve, but not viceversa. Graphically: XB1 0B Pareto Set: C,D,E, all Pareto efficient points XB2 E XA2 Contract curve D C Core of Contract Curve W 0A Endowmentpoint XA1 W1 W2 Introduction Exchange economy model Welfare Economics References Differences: Remember that a Pareto efficient allocation is one for which there is no way to make all agents better off. Said another way, a Pareto efficient allocation is one for which each agent is as well off as possible, given the utilities of the other agents (Varian). Pareto optimality concept : I is only concerned with efficiency and has nothing to say about distribution of welfare. I depends only of the total endowment and not on the endowment point. I consumers’ preferences (IC). Core of an economy: I depends only of the endowment point. I consumers’ preferences (IC). Introduction Exchange economy model Welfare Economics References Walrasian equilibrium b. The Walrasian equilibrium must be in the core of the economy. True or false. Justify your answer. True. An allocation (x i )i∈N is in the core ifPthere isP no group of individuals S and (x̂ i )i∈N such that x̂ji = wji for each good i∈S i∈S j, and U i (x̂ i ) > U i (x i ) for each individual i ∈ S. Introduction Exchange economy model Welfare Economics References Proof: Assume that the Walrasian equilibrium x ∗i is not in the core. Then there is: S ⊂ I ⇒ (x̂ i )i∈S , s.t: P i P i i. x̂j = wj ∀j i∈S i∈S ii. U i (x̂ i ) > U i (x ∗i ) for each individual i ∈ S. But the definition of the Walrasian equilibrium implies: PP PP pj x̂ji > pj wji i∈S j ⇒ P j pj i∈S j X i∈S P x̂ji > P j pj P i∈S wji | {z } i∈S wji which contradicts the first equality (i). Economic intuition: the value of the alternative allocation strictly preferred by this coalition must be larger than their endowment, which means this allocation is not affordable for them. Introduction Exchange economy model Welfare Economics References 3. Using an exchange economy general equilibrium model, explain carefully the implications of the two fundamental theorems of welfare economics for the role of the market and the social planner in a welfarist society. Def. ”Welfare Economics or Normative Economics”: branch of economics that is concerned with the appropriate allocation of resources within an economy and with the establishment of criteria or norms that can evaluate or judge the policies that are likely to maximise social welfare. In the words of Baumol ”Welfare Economics has concerned itself mostly with policy issues which arise out of the allocation of resources, with the distribution of inputs among the various commodities and the distribution of commodities among various consumers” . Introduction Exchange economy model Welfare Economics References The First Fundamental Theorem of Welfare Economics says if preferences are stricktly monotonic, and if (x1∗ , x2∗ , p) is an equilibrium outcome, then the allocation (x1∗ , x2∗ ) is Pareto optimal. ⇒ every Walrasian equilibrium is a Pareto optimum. Conditions: I competitive market, that means the consumers are price taker ⇒ but the idea of competitive equilibrium make sense when there are enough agents. In the Edgeworth box there are only two agents, that can recognize their market power and use it to improve their position; I no externalities, each person’s utility depends, only on his or her own bundle of goods, I no asymmetric information. Introduction Exchange economy model Welfare Economics References Implications FTWE: I a private market that is competitive will result in Pareto efficiency all gains from trade will be exhausted; I a competitive market is a benchmark by which policy-makers can judge actual market outcomes; I the theorem assumes that there are no market imperfections such as monopoly, externalities and public goods; I agents need to know the prices of goods to make their consumption decisions. Introduction Exchange economy model Welfare Economics References The Second Fundamental Theorem of Welfare Economics says if the preferences are convex, continuous, and monotonic, a Pareto efficient allocation is a Walrasian equilibrium for some set of prices. ⇒ every Pareto optimum allocation can be decetralised as a Walrasian equilibrium. Conditions: I The preferences have to be convex, continuous, and monotonic. Implications: I The issue of efficiency and the issue of equality distribution are two distinct issues. I The market mechanism is distributionally neutral. I The price plays two different roles in the market system: an allocative and a distributive role. I The distributional issues should be resolved by changing the initial endowments (lump-sum transfer), and not by changing the competitive pricing mechanism. Introduction Exchange economy model Welfare Economics The idea of a Welfare Function is to consider all together the different consumers’ utilities, and construct some kind of social preferences: Individual welfare refers to the utility associated to a single consumer or economic agent. I Social welfare is the utility associated to society as a whole. I U"lity PossibilitySet U"lity Possibility Fron"er Social Op"mality References Introduction Exchange economy model Welfare Economics U"lity PossibilitySet Asocialplanner willselectan alloca"ononthe contractcurve, Eachpointonthe contractcurveis associatedwitha pair(U1;U2). U"lity Possibility Fron"er Thesetofpoints generatesthe U0lityPossibility Fron0er, Pointsaandbare Pareto-efficient, butpointcis inefficien. Social Op"mality Thealloca"onis chosenthat achievesthe highestlevelof socialwelfare, Pointoonthe u"litypossibility locusachievesthe highestsocial indifferentcurve. References Introduction Exchange economy model Welfare Economics References Social Welfare Function The form of the social welfare function (SWF) will determine the shape of the social indifference curves. The SWF captures the ethical objectives of the society. Along the social indifferent curve a social planner is indifferent among different points on the curve. I How do we represent, from a mathematical point of view, this trade off across people? the SWF is an aggregating function, that has the same features of the utility functions: W (ui , ..., un ) The social welfare function can be construct differently, depending on: I I social norms, policy maker or social planner’s aim. Introduction Exchange economy model Welfare Economics References Examples of Social Welfare Functions Pure utilitarian or Benthamin WF: a Social Welfare Function that care of the total amount of individual utility functions. With two consumers A and B: UB W (uA , uB ) = uA + uB In general: W (ui , ..., un ) = n X i=1 ui UA Utilities of each member are given equal weight. The isowelfare lines have slope of −1, indicating that the utilities of both are treated equally at the margin. Loosely speaking we can say that one extra unit of utility for a starving person is not seen to be of any greater value than an extra unit of utility for a millionaire. I Fair? Not quite. I I Introduction Exchange economy model Welfare Economics References A more generalized function as this form: With two consumers A and B: UB W (uA , uB ) = αuA + βuB In general: W (ui , ..., un ) = n X i=1 I αi ui UA Where different members of society ought to be given different weight ⇒ αn > 0 reflect the importance of the members of society (example: give greater weight to adults, hardworking people, etc.). Introduction Exchange economy model Welfare Economics References Minimax or Rawlsian WF: a Social Welfare Function that cares of the allocation of welfare of the worst off agent. With two consumers A and B: UB W (uA , uB ) = Min(uA , uB ) In general: W (ui , ..., un ) = Min(ui , ...un ) UA I This SW describes an equity seeking behaviour in the distribution of utility. I In this case the social planner will try to maximise the welfare of the least well-off person in the society: choosing allocations that maximizes the minimum utility. I Improvements in the utilities of the richest do not improve the SW. Introduction Exchange economy model Welfare Economics Social optimum To achieve a social optimum, we have to face with a maximization of society’s welfare, that is defined by two concepts: I efficiency: Concerned with the optimal production and allocation of resources given existing factors of production. I equity: Concerned with how resources are distributed throughout society. References Introduction Exchange economy model Welfare Economics References An allocation of resources is (socially) optimal, if it satisfies the following two criteria: 1. It must be an efficient allocation. I Efficiency - synonymous with Pareto optimality. I A resource allocation is Pareto optimal if it is impossible to make one member of society better off without making some other member or members worse off. 2. It must be an equitable allocation. I Equity is a normative concept, with no universal definition (what is equitable to you, not necessarily is equitable to me). Introduction Exchange economy model Welfare Economics References Readings Varian (2014), Intermediate Microeconomics: A Modern Approach (Ninth Edition), Ch. 32 and 34.