The Compensation Principle and Social Welfare Function

1. The Compensation Principle and Social Welfare Function Chapter 3

2. Incompleteness of Pareto Criterion • Pareto criterion is useless as a criterion for social choices in many real-world situations since most policy changes produce both gainers and losers • We employ two approaches to handle the inability of Pareto criterion to handle mixed outcomes • Compensation principle • Social welfare function

3. The Compensation Principle • Hicks (1939) and Kaldor (1939) • Consider a project that moves economy from state A to state B • This movement produces both gainers and losers • Incomes can be costlessly redistributed across individuals

4. Kaldor Compensation Criterion The project is desirable according to Kaldor compensation criterion if gainers can compensate losers in state B in such a way that everyone becomes better off compared to state A

5. Kaldor Criterion: An Example State B after redistribution State A State B $150 Since both John and Bill are better off in state B after the redistribution compared to state A, the project that replaces state A with state B is desirable according to the Kaldor criterion

6. Hicks Compensation Criterion The project is desirable according to Hicks compensation criterion if the would-be losers are unable to bribe the would-be winners not to make the move from state A to state B

7. Pareto and Compensation Criteria • The compensation principle is stated in terms of potential compensation rather than actual compensation • If compensation were required, the compensation principle would be equivalent to Pareto principle (consider example for Kaldor compensation criterion) • Considering the hypothetical compensation allows one to focus on the efficiency aspects of the policy change • In other words, a policy change is desirable according to the compensation criterion if total revenue resulting from the policy change exceeds total cost

8. Compensation Principle and General Equilibrium We can further illustrate the meaning and limitations of the compensation principle by considering redistributions of income between two households in the framework of general equilibrium

9. Utility Possibilities Frontier • Each point on the UPF corresponds to some general equilibrium • Imagine a redistribution of income among households • Quantity demanded by both households change for both goods • Shortages and surpluses created • Prices adjust to establish a new equilibrium • This new equilibrium will be another point on the PPF

10. Change in Consumer Choice • At A3 a consumer faces the budget constraint B3 • Budget constraint B2 corresponds to the case of more expensive housing and a different income • Consumer choice under B2 is A2 • Housing becomes more expensive because of the mismatch in demands of the two households • Excess demand/excess supply for housing and all other goods change the slope of the budget constraint

11. Utility Possibilities Frontier V U = utility of household 1 V = utility of household 2 AA = budget line in state A BB = budget line in state B Utility Possibilities Frontier VA Good 2 VB VB U A UB UB UA UA VA A Good 1

12. Utility Possibilities Frontier: Properties • All points on the utility possibilities frontier satisfy the Pareto condition, i.e. you cannot increase both households’ utilities by moving along this frontier away from any point on it • Any movement along the frontier involves redistribution of wealth (any improvement in one household’s welfare necessarily requires a reduction in the other household’s welfare) • No two points on the utility possibilities frontier can be compared by Pareto or the compensation criterion

13. Compensation Criterion in General Equilibrium Setting V 1. Initially economy is at point O, which is Pareto-inefficient since it is not on the utility possibilities frontier A 2. A move to point B is a Pareto improvement for both households B 3. A move to point A or C is a Pareto improvement for at least one of the households C O D Utility Possibilities Frontier What about movement to point D? U

14. Compensation Criterion in General Equilibrium Setting V 1. A movement from O to D is NOT a Pareto-improvement since utility of household 2 (or V) goes down 2. According to (Kaldor) compensation criterion, a movement from O to D is an improvement because we can move along the utility possibilities frontier (by redistributing wealth among the two households) to point B, which is a Pareto improvement compared to O A B 3. Remember: those compensations are hypothetical! The move from O to D is still NOT a Pareto improvement C O D Utility Possibilities Frontier U

15. Comparing Pareto-Efficient Allocations Compensation criteria always rank Pareto-optimal states above Pareto- inefficient states BUT: Different Pareto-optimal (efficient) allocations are not comparable according to compensation criteria

16. Compensation Principles: Limitations V 1. Frontier PP represents the old technology, while frontier RR represents the new one R C 2. B is preferred to A according to the new technology since a movement along the RR utility possibilities frontier to C will result in a Pareto improvement relative to A (Kaldor) P 3. However, A is also preferred to B since a movement along the PP frontier to D will result in a Pareto improvement as well (Hicks) A D 5. Thus, the compensation principle cannot completely order social states B R P U

17. R C P A D B P R Kaldor-Hicks Compensation and Scitovsky’s Paradox • Hicks: the losers of the project could not bribe the winners not to undertake the project (so a move to B is NOT desirable since such a bribing is possible) Change: PP  RR (AB) Kaldor: B > A Hicks: A > B Kaldor: Since winners at B can compensate the losers (by moving to C), a movement to B IS desirable

18. Scitovsky Paradox and Incompleteness of Kaldor-Hicks • One solution: require that AB is desirable according to Kaldor, but not to Hicks • While this solution works in some specific cases, these entail non-transitive preferences • In general, Kaldor-Hicks compensation criterion does not work in the general case, so we say its ranking of economy states is incomplete

19. Social Welfare Function • Whenever there is a utility conflict among households, we need more than a Pareto or compensation principle in order to be able to rank social states • Such a complete and consistent ranking of social states is called a social welfare ordering • If the social welfare ordering is continuous, it can be translated into a social welfare function • Social welfare function relates individual utility levels to one number called social welfare level so that the combinations of individual utility levels that translate into higher levels of social welfare are preferred to the combinations that result in lower levels of social welfare.

20. Social Welfare Functions: Properties • Welfarism: social welfare depends only on the utility levels of the households (distribution not involved) • Social welfare function is increasing in each household’s utility level (ceteris paribus), so that an isolated increase of any household’s utility level increases welfare of the whole societysocial welfare indifference curves are negatively sloped • Social welfare indifference curves are convex to the origin • Anonymity: It does not matter who gets a high or low level of utility

21. Social Welfare Indifference Curves V 1. Social welfare increases as we move North-East from the origin so that a move from A to B increases social welfare 2. Note that even if moving from A to B makes household 2 lose (V decreases) and no actual or hypothetical compensation is paid, the move is still socially desirable A B W3 W2 W1 U

22. Social Choice using Utility Possibilities Frontier and Social Welfare Function V 1. Maximum achievable welfare is attained at the tangency of the utility possibilities frontier and the highest attainable social welfare indifference curve A 2. Using the social welfare function, we have reduced the infinite number of possible general equilibria to a single equilibrium point Utility possibilities frontier U

23. A Simple Utility Function • Consider an unweighted average of the individual household utilities: W=U1+U2 • This welfare function is called utilitarian or Benthamite. It is double the unweighted average of the individual household utilities • Assume both households are equipped with ordinal utility functions (as opposed to cardinal utility functions).

24. Ordinal and Cardinal: a Re-Cap • The values of ordinal utility functions are only used for the comparison of two different utility levels attained by the same individual • In contrast, the values of cardinal utility functions can be interpreted so that comparisons between different individuals become possible • Because it is very difficult to measure utility in a cardinal way, in most of the cases the ordinal utility functions are being used

25. Individual Utility Functions • We consider two different individual utility functions: • Ranking of the social states will depend on the kind of individual utility function we use for each household

26. Ranking of States by Different Welfare Functions • According to the first welfare function, state 1 and state 2 are given the same rank • According to the second welfare function, state 2 is preferable to state 1

27. Rankings by Social Welfare Function Households rank different states consistently according to their individual utility functions Depending on the social welfare function, rankings of social states can be different Do welfare functions make sense?

28. Do Welfare Functions Make Sense? • The ranking of states produced by the social welfare function depends on the way in which we choose to represent individual preferences • According to one social welfare function society is indifferent between state 1 and state 2, while the other social welfare function says that state 2 is preferable to state 1 • It thus does not make much sense to define a social welfare function

29. Arrow’s Impossibility Theorem • If utility functions are ordinal and non-comparable, then the only possible consistent social welfare function is a dictatorship (i.e. a welfare function that coincides with the ordinal utility function of some individual regardless of the preferences of the others). • Arrow’s impossibility theorem for the case of two households: In order to rank social states we must use either the utility function of household 1 or the utility function of household 2

30. Consistent Preferences • Preferences must be consistent, not self-contradictory • Preferences are consistent if they are • Complete • Transitive • Completeness: we can rank any two states • Transitivity: if A>B and B>C, it must follow that A>C

31. An Example: Majority Voting • Three groups of individuals: the poor, the middle class, and the rich • The three groups vote among three tax schedules: A, B and C that raise the same revenue • The poor prefer schedule A, the middle class prefer schedule B, while the rich prefer schedule C • The poor and the rich prefer A to B: a majority prefers A to B • The poor and the middle class prefer B to C: a majority prefers B to C • By transitivity, the majoritymust prefer A to C • But: the middle class and the rich (another majority!) will vote for C

32. Whither Democracy? The use of a democratic voting process in an attempt to formulate social welfare function can produce contradictory welfare criteria

33. Bergson-Samuelson Social Welfare Functions • The following three properties have to be satisfied for welfare functions to produce a consistent ranking of social states: • Full measurability of household utilities (utilities that are cardinal) • Welfarism (social welfare only depends on the level of individual utilities) • Pareto principle

34. Partial Measurability and Comparability • Utility functions can be made cardinal by allowing for affine transformations • An example of an affine transformation: • F=32+1.8C, F stands for Fahrenheit, C for Celsius • Generally, U=a+bu, where a and b are constants, is an affine transformation • In this way we can measure social utility as a function of one individual’s utility

35. Choosing a Specific Form of Social Welfare Function • Under most measurability and comparability assumptions no unique social welfare function emerges • The choice of a specific form involves a further ethical judgment on aggregation of individual utilities

36. Utilitarianism • Society’s welfare is equal to the sum of the individual members’ utilities • This view is called utilitarianism • Jeremy Bentham (1748-1832) • Joseph Priestley (1768): ”The good and happiness of the members, that is the majority of the members of the state, is the great standard by which every thing relating to that state must finally be determined." • “The greatest happiness of the greatest number”

37. Utilitarian Utility Function • U=U1+U2 • Implicitly assumed: • Cardinal measurability • At least partial comparability • For example, U1=a+bU2 where b is common, but a is different for different households

38. Utilitarian Indifference Curves U2 1. Society is willing to give up one unit of household 1’s utility for a gain of one unit of household 2’s utility. 1 unit 2. This holds regardless of the level of utility of either one of the two households !! Inequality of incomes is NOT the same as inequality of utilities: marginal utility from an additional 1$ may differ across individuals W2 W1 1 unit U1

39. Convex Social Welfare Indifference Curves U2 1. Society is willing to accept a decrease in the utility of the poor only if there is a much larger increase in the utility of the rich 2. Poverty is understood in terms of utility units, NOT incomes 3. If U1 is utility of the poor, decreasing utility of the poor results in large increases in U2, utility of the rich W2 W1 U1

40. John Rawls Welfare of society only depends on the welfare of the poorest household Society gains nothing from improving welfare of rich household However: a policy change making a poor person a little richer, while making a rich guy richer by a lot, is desirable Naturally, egalitarians do not like Rawlsian ideas

41. Rawlsian Utility 1. Welfare of society only depends on the utility of the poorest households W2 W1 2. Society gains nothing from improving the welfare of the richer household 45

42. Maximization of Welfare by Reallocating Income • If the social welfare function produces (strictly) convex social welfare indifference curves, social welfare can be increased by reallocating income between households if their marginal utilities of income differ • Re-cap: marginal utility of income shows the amount by which one’s utility increases for each additional dollar received • Social welfare is maximized by distributing total income in such a way that all households get the same utility from an extra dollar

43. Redistributing Income and General Equilibrium V At point A, the extra utility the payee gets from a dollar is equal to the loss of utility to the payer A Utility possibilities frontier U

44. L-Shaped Preferences How do we determine the maximum welfare state when preferences are L-shaped? We do not talk about tangencies, but we look for a point on the UPF such that the welfare indifference curve corresponds to the maximum utility level

45. Social Welfare and the Distribution of Income • So far, we have implicitly assumed that the higher the initial income, the less is the utility derived from an extra dollar (diminishing marginal utility of income) • If all households have identical utility functions, an equal distribution of a given total amount of income maximizes social welfare

46. Lorenz Curve • Lorenz curve plots the percentage of total income earned by various income groups within the population • With complete equality, the poorest 25% have 25% of total (national) income, the poorest 50% have 50%, and so on • If inequality is great, the poorest part of the population gets a disproportionately small percentage of total income, while the rich get the most

48. Examples of Lorenz Curves % of income 1. If one Lorenz curve (A) lies North-West of another (B), the distribution of income in B is more equal compared to A 2. If two Lorenz curves cross each other, ranking is impossible (e.g. A and C) A B % of population sorted by wealth C

49. Comparing Income Distributions If one Lorenz curve lies North-West of another, the distribution of income corresponding to the first curve is more equal compared to the second Lorenz curve ranking is incomplete

50. Equality and Welfare • It can be shown that for a quite broad class of social welfare functions, moving towards a more equal distribution of incomes will increase social welfare • Marginal utility of one dollar for the poor is greater than that for the rich • However: • Achieving a certain distribution of income entails some costs • Disincentive effect (highly productive people may choose to work less if their income is reduced)