Market -Welfare The First Welfare theorem In the last lecture troduced: Pareto efficiency and general equilibrium. How do they relate? Theorem: The first welfare theorem states that every general equilibrium involves a Pareto efficient allocation. equilibrium. This point cannot lie on the contract curve(the dashed line)since it is not effice also a general The proof works by contradiction. Suppose there was a non-Pareto efficient allocation which Suppose it was a point like IA. If it is a general equilibrium as well, consumer A must be maximising given prices (illustrated by the budget line). But consumer B must be maximising also- say at point IB IA cannot be equal to Ig as it is not on the contract curve. Therefore Ig is a different point. But now the market for good 2 does not clear: x4+x2>44+42-a contradiction(this is not an equilibrium Market- welfare Market failure Assumptions need to be made for this theorem to work. There are three crucial ones. 2. Price Taking Behaviour: Each agent in the economy behaves as a price taker. any other agent No Externalities: Each agents consumption decision does not affect the utility of 3. Prices are Known: All the prices for each of the goods must be known to each of the agents. Importantly, the onsumers do not have different (asymmetric) information concerning the goods The first assumption is critical. The next lecture deals with the case of externalities in more depth The last is the minimal information requirement. Agents need only know price. They need not know the demand or output decisions of others. or how much of a good is available. They behar Market failure arises when any of these assumptions is not met. Externalities, market power and asymmetric information examples of market failure
Market — Welfare 1 The First Welfare Theorem • In the last lecture two concepts were introduced: Pareto efficiency and general equilibrium. How do they relate? • Theorem: The first welfare theorem states that every general equilibrium involves a Pareto efficient allocation. ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... . . . . . . . . . . . . . . . . . . . . . . . . ....................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................................................... . . . . . . . ......................................................................................................................................... ........................................................................................................................................... A B • • xA • x xB 2 A x 2 B • The proof works by contradiction. Suppose there was a non-Pareto efficient allocation which was also a general equilibrium. This point cannot lie on the contract curve (the dashed line) since it is not efficient. • Suppose it was a point like xA. If it is a general equilibrium as well, consumer A must be maximising given prices (illustrated by the budget line). But consumer B must be maximising also — say at point xB. • xA cannot be equal to xB as it is not on the contract curve. Therefore xB is a different point. • But now the market for good 2 does not clear: x 2 A + x 2 B > ω 2 A + ω 2 B — a contradiction (this is not an equilibrium). Market — Welfare 2 Market Failure • Assumptions need to be made for this theorem to work. There are three crucial ones. 1. No Externalities: Each agent’s consumption decision does not affect the utility of any other agent. 2. Price Taking Behaviour: Each agent in the economy behaves as a price taker. 3. Prices are Known: All the prices for each of the goods must be known to each of the agents. Importantly, the consumers do not have different (asymmetric) information concerning the goods. • The first assumption is critical. The next lecture deals with the case of externalities in more depth. • The last is the minimal information requirement. Agents need only know price. They need not know the demand or output decisions of others, or how much of a good is available. They behave “selfishly” given the known prices. • Market failure arises when any of these assumptions is not met. Externalities, market power and asymmetric information are all examples of market failure
Market -Welfare The Invisible hand What are the implications of this theorem? In a general equilibrium everyone maximises utility selfishly" given prices. Firms" selfishly" maximise profits. However, as a result of this selfish behaviour(given the assumptions discussed earlier) a socially desirable outcome arises. An allocation is achieved where no-one can be made better off without making someone else worse off. This is a formalisation of the invisible hand argument of Adam Smith. What role is there for government intervention in such an efficient world? There are two possibilities 1. Market failure: If one of the assumptions fail, the allocation may no longer be efficient. 2. Distributive Goals: Pareto efficiency says nothing about distributional fairness Market- welfare The Second Welfare Theorem An equilibrium is efficient, are efficient allocations always part of an equilibrium? Theorem: The second welfare theorem states that every Pareto efficient allocation can be supported by a general equilibrium set of prices given a suitable reallocation of the endowment Equilibrium Prices The demonstration proceeds constructively. Which set of prices port the above Pareto efficient allocation (r)as a general equilibrium? The answer is the given by the budget line that separates the two indifference curves. In order to support r the endowment would need to reallocated from w to w
Market — Welfare 3 The Invisible Hand • What are the implications of this theorem? • In a general equilibrium everyone maximises utility “selfishly” given prices. Firms “selfishly” maximise profits. • However, as a result of this selfish behaviour (given the assumptions discussed earlier) a socially desirable outcome arises. An allocation is achieved where no-one can be made better off without making someone else worse off. • This is a formalisation of the invisible hand argument of Adam Smith. • What role is there for government intervention in such an efficient world? There are two possibilities: 1. Market failure: If one of the assumptions fail, the allocation may no longer be efficient. 2. Distributive Goals: Pareto efficiency says nothing about distributional fairness. Market — Welfare 4 The Second Welfare Theorem • An equilibrium is efficient, are efficient allocations always part of an equilibrium? • Theorem: The second welfare theorem states that every Pareto efficient allocation can be supported by a general equilibrium set of prices given a suitable reallocation of the endowment. ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................................................... . . .......................................................... ...... ...... ... ...... ...... ...... ... ........................ . . . . . . . . . . . . . . .. .. .. ...... ...... . ........................................................................................................ ........................................................................................................ A B • x Equilibrium Prices • ω 0 ω • • The demonstration proceeds constructively. Which set of prices can support the above Pareto efficient allocation (x) as a general equilibrium? The answer is the given by the budget line that separates the two indifference curves. • In order to support x the endowment would need to reallocated from ω to ω 0
Market -Welfare Convexity and the Theorem The crucial assumption here is that of conuerify. Preferences need elk-behaved for the theorem to work What would happen if they were not? Consider the following case. A has well-behaved preferences. B does not Can a point like r be supported neral equilibrium? The budget line is a set of prices, separating th maximisIng rB maximises utility given these prices for consumer B. r maximises utility for consumer A. This is not a general equilibrium since the market for good 2 is not clearing Market- welfare Implications of the Theorem Notice that convexity is only required for the second theorem. The first theorem holds for any preferences What are the implications of the second theorem Distributional issues can be separated from efficiency issues. A government, operating in such a world can simply transfer endowments to achieve any distributional goals they might have and leave the market to attain efficiency Prices play two roles in the market, (i) allocative- relative scarcity of the two goods and(ii)distributive-how much agents can afford. These can be separated. Do not use prices to attain distributive goals, use endowments. Note: Although only pure exchange economies have been considered so far, everything goes through analogous way for a production economy. The assumptions need to apply both to consumers and firms
Market — Welfare 5 Convexity and the Theorem • The crucial assumption here is that of convexity. Preferences need to be well-behaved for the theorem to work. • What would happen if they were not? Consider the following case. A has well-behaved preferences. B does not. ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................................................... . . . ........................................................................................................ .............................................................................................. ...... ...... ...... ...... ............................................................................................................ ................. ............................. A B • x xB • • Can a point like x be supported as a general equilibrium? The budget line is a set of prices, separating the two indifference curves. A is maximising. B is not — they could do better, by choosing xB. • xB maximises utility given these prices for consumer B, x maximises utility for consumer A. This is not a general equilibrium since the market for good 2 is not clearing. Market — Welfare 6 Implications of the Theorem • Notice that convexity is only required for the second theorem. The first theorem holds for any preferences. • What are the implications of the second theorem? • Distributional issues can be separated from efficiency issues. A government, operating in such a world can “simply” transfer endowments to achieve any distributional goals they might have and leave the market to attain efficiency. • Prices play two roles in the market, (i) allocative — relative scarcity of the two goods and (ii) distributive — how much agents can afford. These can be separated. Do not use prices to attain distributive goals, use endowments. • Of course, such a transfer of endowments is not so simple — it may be impossible. • Note: Although only pure exchange economies have been considered so far, everything goes through in an analogous way for a production economy. The assumptions need to apply both to consumers and firms
Market -Welfare Efficiency and Welfare The contract curve is the set of allocations that are pareto efficient By applying the first and second welfare theorems a general equilibrium will lie on this curve and government could transfer endowments to achieve any of these points. The question is Which one Pareto efficiency says nothing about"fairness"or "justice.Indeed the allocation where He everything and consumer B gets nothing is Pareto efficient. It is probably not fair however. Suppose the government could rank the various allocations available simply attaching a number to each outcome. The one with the highest number would then be the best allocation from societies point of view. How should the government construct their ranking ower different allocations? In other words, how should the gowernment turn individual preferences into social welfare? How can preferences be aggregated? ket- welfare Suppose there are three agents(1, 2 and 3)in the economy and three possible allocations: a, b and c. Suppose the following table illustrates the preferences the three individuals have over the various allocations. c). So this social ordering is not transitive even though the utility function of each agent is. Is there a better way to rank the three alternatives? What properties should such a social welfare function have?
Market — Welfare 7 Efficiency and Welfare • The contract curve is the set of allocations that are Pareto efficient. • By applying the first and second welfare theorems a general equilibrium will lie on this curve and moreover a government could transfer endowments to achieve any of these points. The question is: Which one? • Pareto efficiency says nothing about “fairness” or “justice”. Indeed, the allocation where consumer A gets everything and consumer B gets nothing is Pareto efficient. It is probably not fair however. • Suppose the government could rank the various allocations available, simply attaching a number to each outcome. The one with the highest number would then be the best allocation from societies point of view. • How should the government construct their ranking over different allocations? In other words, how should the government turn individual preferences into social welfare? How can preferences be aggregated? Market — Welfare 8 Aggregating Preferences • Suppose there are three agents (1, 2 and 3) in the economy and three possible allocations: a, b and c. • Suppose the following table illustrates the preferences the three individuals have over the various allocations. Agent 1 Agent 2 Agent 3 a b c b c a c a b • How should their preferences be aggregated? Suppose a majority voting mechanism is proposed. Given a choice between a and b, a would win (agents 1 and 3 would vote for a). Given a choice between b and c, b would win (agents 1 and 2 would vote for b). Given a choice between a and c, c would win (agents 2 and 3 would vote for c). • So this social ordering is not transitive even though the utility function of each agent is. • Is there a better way to rank the three alternatives? What properties should such a social welfare function have?
Market -Welfare Arrows Impossibility Theorem Suppose the aggregation method(or social welfare function) had three properties: are complete, transitive and reflexive(unlike majority voting 2. If everyone prefers some allocation a to another. b, then a should be socially preferred to b 3. Social preferences between a and b should only depend on the way that agents rank a and b and not on the elative rank of any other irrelevant"allocation Theorem: Arrou's impossibility theorem states that any mechanism for the aggregation of preferences(social elfare function) which satisfies the above three properties is a dictatorship In other words, the social rankings correspond exactly with one individuals rankings -the social welfare function There is no"perfect"way to rank allocations. How can the government choose between allocations? Market- welfare Social Welfare Obviously, they use a less than"perfect "social welfare function. Either one of the properties is not satisfied or they use a dictatorship welfare function. Surely this is bad? Perhaps not Formally, a social welfare function(SWF)maps individual utility functions to a umber social welfare. Consider the Rawlsian and Utilitarian social welfare functions which are given respectively by: (u…,un)=∑ Consider the Rawlsian SWF. It satisfies property 1- since it is simply a utility function and hence complete, transitive and reflexive. It satisfies property 2. since if everyone prefers a to b then the minimum utility individual prefers a to b and hence the SwF ranks a above b It is independent of irrelevant alternatives, c, since all individuals(including the one with the smallest utility)rank and b independently of c. Hence, there must be a dictator. There is. The least well off agent. Is this so bad? The Utilitarian SWF also either breaks one of the three conditions or is a dictatorship. Exercise: Which?
Market — Welfare 9 Arrow’s Impossibility Theorem • Suppose the aggregation method (or social welfare function) had three properties: 1. Given a set of complete, transitive and reflexive preferences, aggregation should result in social preferences that are complete, transitive and reflexive (unlike majority voting). 2. If everyone prefers some allocation a to another, b, then a should be socially preferred to b. 3. Social preferences between a and b should only depend on the way that agents rank a and b and not on the relative rank of any other “irrelevant” allocation c. • Theorem: Arrow’s impossibility theorem states that any mechanism for the aggregation of preferences (social welfare function) which satisfies the above three properties is a dictatorship. • In other words, the social rankings correspond exactly with one individual’s rankings — the social welfare function is simply the utility function of a particular agent. • There is no “perfect” way to rank allocations. How can the government choose between allocations? Market — Welfare 10 Social Welfare • Obviously, they use a less than “perfect” social welfare function. Either one of the properties is not satisfied or they use a dictatorship welfare function. Surely this is bad? Perhaps not. • Formally, a social welfare function (SWF) maps individual utility functions to a number — social welfare. • Consider the Rawlsian and Utilitarian social welfare functions which are given respectively by: WR(u1, . . . , un) = min i ui and WU (u1, . . . , un) = Xn i=1 ui • Consider the Rawlsian SWF. It satisfies property 1 — since it is simply a utility function and hence complete, transitive and reflexive. It satisfies property 2, since if everyone prefers a to b then the minimum utility individual prefers a to b and hence the SWF ranks a above b. • It is independent of irrelevant alternatives, c, since all individuals (including the one with the smallest utility) rank a and b independently of c. Hence, there must be a dictator. There is. The least well off agent. Is this so bad? • The Utilitarian SWF also either breaks one of the three conditions or is a dictatorship. Exercise: Which?
Market -Welfare Welfare maximisation The government wishes to choose the best allocation according to the Swf- it wishes to maximise welfare Not all allocations are possible. The utility possibilities set is the set of feasible allocations. U U The feasible allocations lie within the sets U in the graph above. The boundary of this set is called the Pareto frontier. The isowelfare lines- lines of equal social welfare are illustrated for two exam The first example is for a Rawlsian SWF-it results in equality, ui ug. The second example is for a Utilitarian SWF-"the greatest good to the greatest number-it does not result in equality, ui>us Notice that every Pareto efficient point(a point on the Pareto frontier)is the maximum of some SWF. Market- welfare Fairness, Envy and Equity Another approach might be to propose a particular type of allocation-a fair one for example. What is a fair allocation? An envy free or equitable allocation is one where consumer A does not prefer the bundle consumer B gets and vice-versa. In the Edgeworth box the other consumers bundle is the"mirror image"bundle. would prefer to be at y- that is, neither nsumer wants to swap bundles with the other. Notice that w lies on this budget line also- equal endowments. In fact if bot ers start with an equal amount of the two goods each they will trade to a fair allocation. A appetitive equilibrium from equal division must be a fair allocation
Market — Welfare 11 Welfare Maximisation • The government wishes to choose the best allocation according to the SWF — it wishes to maximise welfare. • Not all allocations are possible. The utility possibilities set is the set of feasible allocations. ............. ............. ............. ............. ............. ........... . . . . . . ............. ............. ............. ............. ............. ............. ............. . . . . . ............................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................................... . . ............................................................................................................. . ........................................................................................................................................ . . ......... ........ .... . ......... ........ .... . .. .. ... ... ... ... ... ... ... ..... ....... ........ ......... .......... ............. ....................... .............................. . .. .. ... ... ... ... ... ... ... ..... ....... ........ ......... .......... ............. ....................... .............................. U U • • u1 u1 u2 u2 u ∗ 1 u ∗ 2 u ∗ 1 u ∗ 2 0 0 • The feasible allocations lie within the sets U in the graph above. The boundary of this set is called the Pareto frontier. The isowelfare lines — lines of equal social welfare are illustrated for two examples. • The first example is for a Rawlsian SWF — it results in equality, u ∗ 1 = u ∗ 2 . The second example is for a Utilitarian SWF — “the greatest good to the greatest number” — it does not result in equality, u ∗ 1 > u ∗ 2 . • Notice that every Pareto efficient point (a point on the Pareto frontier) is the maximum of some SWF. Market — Welfare 12 Fairness, Envy and Equity • Another approach might be to propose a particular type of allocation — a fair one for example. • What is a fair allocation? An envy-free or equitable allocation is one where consumer A does not prefer the bundle consumer B gets and vice-versa. In the Edgeworth box the other consumer’s bundle is the “mirror image” bundle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................................................... . . ....................................... . . . . . . . . . . . . . . .. .. .. ...... ...... . .......................................................................................................................................... .......................................................................................................................................... A B • x • • y ω • x is a fair allocation, (if) it is equitable and efficient. Neither consumer would prefer to be at y — that is, neither consumer wants to swap bundles with the other. Notice that ω lies on this budget line also — equal endowments. • In fact if both consumers start with an equal amount of the two goods each they will trade to a fair allocation. A competitive equilibrium from equal division must be a fair allocation
