Chapter summary

The two theorems of welfare economics provided important insights into economic policy. These insights are explored further in this chapter. It is argued that if it could be applied the Second Theorem would resolve the problems of economic policy. Why it cannot do so becomes apparent when the practical application of the lump-sum transfers required to decentralise is considered. It is also possible to question the merit of Pareto efficiency as a method of judging the acceptability of a given equilibrium since it avoids interpersonal comparisons of welfare. The chapter considers what can be achieved if interpersonal comparisons are permitted. The results of this chapter do not mean the two theorems are without value. Much of the subject matter of public economics takes as its starting point the practical shortcomings of these theorems and attempts to find a way forward to something that is applicable. Knowledge of what could be achieved if the optimal lump-sum transfers were available provides a means of assessing the success of what can be achieved and shows ways in which improvements in policy can be made.

2.1 Optimality with a single consumer

With a single consumer there is no doubt as to what is good and bad from a social perspective: it is sufficient to rely entirely on the consumer’s own judgement. If they prefer one outcome to another then so must society. This provides a significant simplification of the discussion of optimality.

In this single consumer context the equilibrium reached by the market simply cannot be bettered. Such a strong statement cannot be made when further consumers are introduced since issues of distribution between consumers then arise. However, what will remain is the finding that the competitive market ensures that firms produce at an efficient point on the frontier of the production set and that the chosen production plan is what is demanded at the equilibrium prices by the consumer.

• Efficiency and optimality are equivalent with a single consumer
• Highest isoprofit curve and budget constraint are coincident
• Competitive equilibrium is optimal

12.2 Social optimality

In designing policy, a policy maker would wish to achieve Pareto efficient equilibrium. The theorem shows this is achieved by making the economy competitive, selecting the equilibrium that is to be reached and providing each consumer with the correct level of income. The only policy employed is a lump-sum redistribution of endowments to ensure that consumers have the required incomes. If this
approach could be applied in practice, economic policy analysis reduces to lump-sum redistribution of endowments.

If an allocation that was not Pareto efficient was selected, then the welfare of at least one consumer could be raised without harming any other. Such gains should not be left unexploited. Applying this argument reduces the set of relevant alternatives to the Pareto efficient allocations and a choice must be made between these.

Consider the set of Pareto efficient allocations described by the contract curve. ach point on the contract curve is associated with a utility level for consumer 1 and a utility level for consumer 2. As the move is made from the south-west corner of the Edgeworth box to the north-east corner, the utility of consumer 1 rises and that of 2 falls. Plotting the utility levels on the contract curve generates as a locus in utility space - the utility possibility curve. Points such as a and b are Pareto efficient so it is not possible to raise both consumers’ utilities simultaneously. Point c is inefficient and lies inside the locus. This corresponds to a point off the contract curve.

The social planner is assumed to assess the welfare of society by an aggregate of the individual consumers’ welfares. Denote the function that undertakes this aggregation by W (U 1 , U 2 ) – this is a Bergson- Samuelson social welfare function. The social planner considers the attainable allocations of utility described by the utility possibility frontier and chooses the one that provides the highest level of social welfare.

The optimal point on the utility possibility locus is that which achieves the highest indifference curve. This point can then be traced back to an allocation in the Edgeworth box which is the socially optimal division of resources for the economy.

Having chosen the socially optimal allocation, the Second Theorem is applied. Lump-sum transfers between consumers are imposed to ensure that the incomes of the consumers are sufficient to allow them to purchase their allocation. Competitive economic trading then takes place. The equilibrium of the competitive economy then gives the chosen Pareto efficient allocation.

The interpretation of this construction shows that use of the Second Theorem allows the economy to achieve the outcome most preferred by its social planner. Given the economy’s limited initial stock of resources, this socially optimal allocation is both efficient and, relative to the social welfare function, equitable. In this way, the application of the Second Theorem can be said to solve the economic problem since the issues of both efficiency and equity are resolved and there is no better outcome attainable. Clearly, if this reasoning is applicable, all that a policy maker need do is choose the allocation, employ the transfers suggested by the theorem and ensure that the economy is competitive.

• Contract curve maps to utility possibility frontier
• Social indifference curve selects optimal allocation
• Optimum is achieved by decentralisation

12.3 Lump-sum transfers

In the application of the Second Theorem, lump-sum transfers are the only tool of policy that is required. In order for a transfer, or tax, to be lump-sum the consumer involved must not be able to affect the size of the transfer by changing their behaviour. The concern here is the use of optimal lump-sum transfers. Optimality requires that the equilibrium which occurs after the transfers have been carried out is the socially optimal one. To calculate the transfers the social planner must be able to predict the equilibrium that will emerge for all possible income levels. This requires
knowledge of the consumers’ preferences.

This shows the proposed taxes are not incentive compatible and indicates the potential for misrevelation of characteristics. Such problems will always be present in any attempt to base the transfers on unobservable characteristics. The Second Theorem relies on the use of optimal lump-sum transfers but such transfers are unlikely to be available in practice since the government cannot observe the relevant characteristics and relies on the consumers to reveal them. However, under the optimal tax policy it is not in the consumer’s interest to truthfully reveal their characteristics.

• A lump-sum tax is not affected by behaviour
• An optimal tax achieves the planner’s objectives
• Lump-sum taxes are not incentive compatible

12.4 Aspects of Pareto efficiency

The criticisms of the previous section undermine the practical relevance of the second theorem. To assess the policy value of the First Theorem it is necessary to study Pareto efficiency.

Consider a set of alternative economic states, S = {s 1 , s 2 ,...} and a set H, indexed h = 1,...,H, of consumers. The states should be thought of as different allocations of commodities between the consumers. A state s i is Pareto efficient if there is no other state that is Pareto preferred to it.

Now consider the division of a fixed quantity of a single good between two people where both people prefer more to less. In such circumstances every division of the good is Pareto efficient.

From this simple example it is possible to infer two deficiencies of Pareto efficiency. Firstly, extreme allocations, such as giving all the good to one person, can be Pareto efficient. Consequently, Pareto efficiency does not imply equity or fairness. Secondly, there can be many Pareto efficient allocations. In fact, for the division of a good between two people, Pareto efficiency gives no guidance except for showing that nothing should be thrown away.

The contract curve in the Edgeworth Box shows the set of Pareto efficient allocations. There is generally an infinite number of these. Once again the Pareto preference ordering does not select a unique optimal outcome.

A further failing of the Pareto preference ordering is that it does not always provide a complete ranking of the alternative states.

• Pareto efficiency avoids interpersonal comparability
• Extreme allocations can be Pareto efficient
• Pareto ranking can be incomplete

12.5 Welfare functions

A social welfare function was used to derive the socially optimal allocation. The welfare function was described as means to socially-rank different allocations of utility between consumers. What was not done was to provide a convincing description of where such a function could come from or of how the ranking could be constructed.

The first possibility is that the social welfare function captures the views of some central authority or dictator. Under this interpretation there can be two meanings of the individual utilities that enter the function. One is that they are the dictator’s perception of the utility achieved by each consumer for their level of consumption. This provides a consistent interpretation of the social welfare function but problems arise in its relation to the underlying model. To see why this is so, recall that the Edgeworth box, and the contract curve within it, were based upon the actual preferences of the consumers. This leads to a potential inconsistency between this construction and the evaluation using the dictator’s preferences. The second meaning of the utilities is that they are the actual utilities of the consumers. This leads directly into the central difficulty faced in the concept of social welfare.

An alternative interpretation of the social welfare function is that it captures some ethical objective that society should be pursuing. There are two famous examples to illustrate this. The utilitarian philosophy of aiming to achieve the greatest good for society as a whole translates into a social welfare function that is the sum of individual utilities. In this formulation, it does not matter how utility is distributed between consumers in the society. Alternatively, the Rawlsian philosophy of caring only for the worst-off member of society leads to a level of social welfare determined entirely by the minimum of that in society. With this objective, the distribution of utility is of paramount importance.

The final view that can be taken of the social welfare function is that it takes the preferences of the individual consumers (represented by their utilities) and aggregates these into a social preference. This aggregation process would be expected to obey certain rules; for instance if all consumers prefer one state to another, it should be the case that the social preference also prefers the same state. The structure of the social welfare function then emerges as a consequence of the rules the aggregation must obey.

• A social welfare function represents social preferences
• Utilitarian and Rawlsian philosophies
• Function can be the outcome of aggregation

12.6 Applying Arrow’s theorem

Although they appear very distinct in nature, both majority voting and the Pareto criterion are examples of procedures for aggregating individual preferences into a social preference. It has been shown that neither is perfect. The Pareto preference order can be incomplete and is unable to rank some of the alternatives. Majority voting always leads to a complete social preference order but this may not be transitive. What Arrow’s impossibility theorem shows is that such failings are not specific to these aggregation procedures. All methods of aggregation will fail to meet one or more of its conditions so it identifies a fundamental problem at the heart of generating social preferences from individual preferences.

What is at the heart of the impossibility is the limited information contained in individual utility functions. Effectively, all that is known are the individuals’ rankings of the alternatives - which is best, which is worst, and how they line up in between. Using information on how strongly individuals feel about the alternatives can be successful. It is interesting that strength
of preference comparisons can be used in informal situations, but this does not demonstrate that it can be incorporated within a scientific theory of social preferences.

• Pareto efficiency and social welfare functions as examples of aggregation
• Arrow’s theorem says they must fail
• The consequences of limited information

12.7 Interpersonal comparability

Earlier in this chapter it was noted that Pareto efficiency was originally proposed because it provided a means by which it was possible to compare alternative allocations without requiring interpersonal comparisons of welfare. It is also from this avoidance of comparability that the failures of Pareto efficiency emerge. This point is also at the core of the impossibility theorem. To proceed further, this section first reviews the development of utility theory in order to provide a context and then describes alternative degrees of utility comparability.

READING: Hindriks and Myles 2004: Chapter 13: Optimality and Comparability

Chapter 13 Optimality and Comparability

If an allocation that was not Pareto efficient was selected, then it would be possible to raise the welfare of at least one consumer without harming any other. p298 sic?

.. a Bergson-Samuelson social welfare function. Basically, given individual levels of happiness it imputes a social level of happiness. Embodied within it are the equity considerations of the planner.

Having chosen the socially optimal allocation, the reasoning of the Second Theorem is applied. Lump-sum taxes are imposed to ensure that the incomes
of the consumers are sufficient to allow them to purchase their allocation conforming to point o. Competitive economic trading then takes place. The chosen socially optimal allocation is then achieved through trade as the equilibrium of the competitive economy. This process of using lump-sum taxes and competitive trade to reach a chosen equilibrium is called decentralization. p299

In order for a tax to be lump-sum, the consumer on whom the tax is levied must not be able to affect the size of the tax by changing their behavior. Most tax instruments encountered in practice are not lump-sum. Income taxes cannot be lump-sum by this definition since a consumer can work more or less hard and vary income in response to the tax. Similarly, commodity taxes cannot be lump-sum since consumption patterns can be changed. Estate duties are lump-sum at the point at which they are levied (since by definition the person on which they are levied is dead and unable to choose any other action) but can be affected by changes in behavior prior to death (for instance by making gifts earlier in life). p300-301

The main points of the argument can now be summarized. To implement the Second Theorem as a practical policy tool it is necessary to employ optimal lump-sum taxes. Such taxes are unlikely to be available in practice or to satisfy all the criteria required of them. The taxes may be costly to collect and the characteristics upon which they need to be based may not be observable. When characteristics are not observable, the relationship between taxes and characteristics can give consumers the incentive to make false revelations. It is therefore best to treat the Second Theorem as being of considerable theoretical interest but of very limited practical relevance. p302

two deficiencies of Pareto efficiency can be inferred. Firstly, since no improvement can be made on an allocation where none is wasted, extreme allocations ... are Pareto efficient. Another failing of the Pareto preference ordering is that it is not always able to compare alternative states. In formal terms, it does not provide a complete ordering of states. The basic mechanism at work behind this example is that the Pareto preference order can only rank alternative states if there are only gainers or losers as the move is made between the states. p304-5

The basic mechanism at work behind this example is that the Pareto preference order can only rank alternative states if there are only gainers or losers as the move is made between the states. If some gain and some lose, as in the choice between s 1 and s 2 ... then the preference order is of no value. Such gains and losses are invariably a feature of policy choices and much of policy analysis consists of weighing-up the gains and losses. In this respect, the Pareto efficiency is inadequate as a basis for policy choice. p305

An alternative interpretation of the social welfare function is that it captures some ethical objective that society should be pursuing. Here the social welfare functions is determined by what is viewed as the just objective of society. There are two major examples of this. The utilitarian philosophy of aiming to achieve the greatest good for society as a whole translates into a social welfare function that is the sum of individual utilities. In this formulation, only the total sum of utilities counts, so it does not matter how utility is distributed between consumers in the society. Alternatively, the Rawlsian philosophy of caring only for the worst-off member of society leads to a level of social welfare determined entirely by the minimum of that in society. With this objective, the distribution of utility is of paramount importance. Gains in utility achieved by anyone other than the worst-off consumer do not improve social welfare p307

The Pareto preference order can be incomplete and is unable to rank some of the alternatives. Majority voting always leads to a complete social preference order but this may not be transitive. What Arrow’s theorem has shown is that such failings are not specific to these aggregation procedures. All methods of aggregation will fail to meet one or more of its conditions so it identifies a fundamental problem at the heart of generating social preferences from individual preferences. p308

The starting point is to define the two major forms of utility. The first is ordinal utility which is the familiar concept from consumer theory. Essentially, an ordinal utility function is no more than just a numbering of a consumer’s in-difference curves, with the numbering chosen so that higher indifference curves have higher utility numbers.... The second form of utility is cardinal utility. Cardinal utility imposes re- strictions beyond those of ordinal utility. With cardinal utility you can only transform utility numbers by multiplying by a constant and then adding a con-
stant, so an initial utility function U becomes the transformed utility Ũ = a+bU where a and b are the constants. Any other form of transformation will affect the meaning of a cardinal utility function. p310

.. for each form of comparability there is a specification of social welfare function that is consistent with the information content of the comparable utilities. To explain what is meant by consistent, recall that comparability is described by a set of permissible transformations of utility. A social welfare function is consistent if it ranks the set of alternative social states in the same way for all permissible transformations of the utility functions. p312

The first point to establish is that it is possible to find a social welfare function that is consistent with ordinal level comparability but that none that is consistent with ordinal non-comparability. What level comparability allows is the ranking of consumers by utility level (think of placing the consumers in a line with the lowest utility level first). p312