1. Frank Cowell: Welfare - Social Welfare function WELFARE: THE SOCIAL- WELFARE FUNCTION MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Welfare: Basics Welfare: Efficiency Almost essential Welfare: Basics Welfare: Efficiency Prerequisites March 2012 1

2. Frank Cowell: Welfare - Social Welfare function Social Welfare Function  Limitations of the welfare analysis so far:  Constitution approach Arrow theorem – is the approach overambitious?  General welfare criteria efficiency – nice but indecisive extensions – contradictory?  SWF is our third attempt  Something like a simple utility function…? Requirements March 2012 2

3. Frank Cowell: Welfare - Social Welfare function Overview... The Approach SWF: basics SWF: national income SWF: income distribution Welfare: SWF What is special about a social-welfare function? March 2012 3

4. Frank Cowell: Welfare - Social Welfare function The SWF approach  Restriction of “relevant” aspects of social state to each person (household)  Knowledge of preferences of each person (household)  Comparability of individual utilities utility levels utility scales  An aggregation function W for utilities contrast with constitution approach there we were trying to aggregate orderings A sketch of the approach March 2012 4

5. Frank Cowell: Welfare - Social Welfare function Using a SWF aa bb U  Take the utility-possibility set  A social-welfare optimum?  Social welfare contours  W defined on utility levels  Not on orderings  Imposes several restrictions… ..and raises several questions W (  a,  b,... ) March 2012 5

6. Frank Cowell: Welfare - Social Welfare function Issues in SWF analysis  What is the ethical basis of the SWF?  What should be its characteristics?  What is its relation to utility?  What is its relation to income? March 2012 6

7. Frank Cowell: Welfare - Social Welfare function Overview... The Approach SWF: basics SWF: national income SWF: income distribution Welfare: SWF Where does the social-welfare function come from? March 2012 7

8. Frank Cowell: Welfare - Social Welfare function An individualistic SWF  The standard form expressed thus W(  1,  2,  3,...) an ordinal function defined on space of individual utility levels not on profiles of orderings  But where does W come from...?  We'll check out two approaches: The equal-ignorance assumption The PLUM principle March 2012 8

9. Frank Cowell: Welfare - Social Welfare function 1: The equal ignorance approach  Suppose the SWF is based on individual preferences.  Preferences are expressed behind a “veil of ignorance”  It works like a choice amongst lotteries don't confuse  and  !  Each individual has partial knowledge: knows the distribution of allocations in the population knows the utility implications of the allocations knows the alternatives in the Great Lottery of Life does not know which lottery ticket he/she will receive March 2012 9

10. Frank Cowell: Welfare - Social Welfare function “Equal ignorance”: formalisation  Individualistic welfare: W (  1,  2,  3,...) use theory of choice under uncertainty to find shape of W  vN-M form of utility function:     u(x  ) Equivalently:          probability assigned to  u : cardinal utility function, independent of     utility payoff in state   A suitable assumption about “probabilities”? n h 1 W = —   h n h h=1 welfare is expected utility from a "lottery on identity“ payoffs if assigned identity 1,2,3,... in the Lottery of Life  Replace  by set of identities {1,2,...n h }:  h  h  h An additive form of the welfare function March 2012 10

11. Frank Cowell: Welfare - Social Welfare function Questions about “equal ignorance” hh identity | n h h |1 |1 |2 |2 |3 |3 |  Construct a lottery on identity  The “equal ignorance” assumption...  Where people know their identity with certainty  Intermediate case  The “equal ignorance” assumption:  h = 1/n h But is this appropriate?  Or should we assume that people know their identities with certainty?  Or is the "truth" somewhere between...? March 2012 11

12. Frank Cowell: Welfare - Social Welfare function 2: The PLUM principle  Now for the second  rather cynical  approach  Acronym stands for People Like Us Matter  Whoever is in power may impute:...either their own views,... or what they think “society’s” views are,... or what they think “society’s” views ought to be,...probably based on the views of those in power  There’s a whole branch of modern microeconomics that is a reinvention of classical “Political Economy” Concerned with the interaction of political decision-making and economic outcomes. But beyond the scope of this course March 2012 12

13. Frank Cowell: Welfare - Social Welfare function Overview... The Approach SWF: basics SWF: national income SWF: income distribution Welfare: SWF Conditions for a welfare maximum March 2012 13

14. Frank Cowell: Welfare - Social Welfare function The SWF maximum problem  Take the individualistic welfare model W(  1,  2,  3,...) Standard assumption  Assume everyone is selfish:  h = U h (x h ), h=1,2,...n h my utility depends only on my bundle  Substitute in the above: W(U 1 (x 1 ), U 2 (x 2 ), U 3 (x 3 ),...) Gives SWF in terms of the allocation a quick sketch March 2012 14

15. Frank Cowell: Welfare - Social Welfare function From an allocation to social welfare  From the attainable set... A A (x 1 a, x 2 a ) (x 1 b, x 2 b ) (x 1 a, x 2 a ) (x 1 b, x 2 b ) ...take an allocation  Evaluate utility for each agent  Plug into W to get social welfare  a =U a (x 1 a, x 2 a )  b =U b (x 1 b, x 2 b )  a =U a (x 1 a, x 2 a )  b =U b (x 1 b, x 2 b ) W(  a,  b )  But what happens to welfare if we vary the allocation in A ? March 2012 15

16. Frank Cowell: Welfare - Social Welfare function Varying the allocation  Differentiate w.r.t. x i h : d  h = U i h (x h ) dx i h marginal utility derived by h from good i The effect on h if commodity i is changed  Sum over i: n d  h =  U i h (x h ) dx i h i=1 The effect on h if all commodities are changed  Differentiate W with respect to u h : n h dW =  W h d  h h=1 Changes in utility change social welfare.  Substitute for d  h in the above: n h n dW =  W h  U i h (x h ) dx i h h=1 i=1 So changes in allocation change welfare. Weights from the SWF Weights from utility function marginal impact on social welfare of h’s utility March 2012 16

17. Frank Cowell: Welfare - Social Welfare function Use this to characterise a welfare optimum  Write down SWF, defined on individual utilities.  Introduce feasibility constraints on overall consumptions.  Set up the Lagrangean.  Solve in the usual way Now for the maths March 2012 17

18. Frank Cowell: Welfare - Social Welfare function The SWF maximum problem  First component of the problem: W(U 1 (x 1 ), U 2 (x 2 ), U 3 (x 3 ),...) Individualistic welfare Utility depends on own consumption The objective function  Second component of the problem: n h  (x)  0, x i =  x i h h=1 Feasibility constraint  The Social-welfare Lagrangean: n h W(U 1 (x 1 ), U 2 (x 2 ),...) -  (  x h ) h=1 Constraint subsumes technological feasibility and materials balance  FOCs for an interior maximum: W h (...) U i h (x h ) −  i (x) = 0 From differentiating Lagrangean with respect to x i h  And if x i h = 0 at the optimum: W h (...) U i h (x h ) −  i (x)  0 Usual modification for a corner solution All goods are private March 2012 18

19. Frank Cowell: Welfare - Social Welfare function Solution to SWF maximum problem  From FOCs: U i h (x h ) U i ℓ (x ℓ ) ——— = ——— U j h (x h ) U j ℓ (x ℓ ) Any pair of goods, i,j Any pair of households h, ℓ Any pair of goods, i,j Any pair of households h, ℓ MRS equated across all h. We’ve met this condition before - Pareto efficiency  Also from the FOCs: W h U i h (x h ) = W ℓ U i ℓ (x ℓ ) social marginal utility of toothpaste equated across all h.  Relate marginal utility to prices: U i h (x h ) = V y h p i This is valid if all consumers optimise  Substituting into the above: W h V y h = W ℓ V y ℓ At optimum the welfare value of $1 is equated across all h. Call this common value M Marginal utility of money Social marginal utility of income March 2012 19

20. Frank Cowell: Welfare - Social Welfare function To focus on main result...  Look what happens in neighbourhood of optimum  Assume that everyone is acting as a maximiser firms households…  Check what happens to the optimum if we alter incomes or prices a little  Similar to looking at comparative statics for a single agent March 2012 20

21. Frank Cowell: Welfare - Social Welfare function  Differentiate the SWF w.r.t. {y h }: n h dW =  W h du h h=1 Changes in income, social welfare n h dW = M  dy h h=1 n h =  W h V y h dy h h=1  Social welfare can be expressed as: W(U 1 (x 1 ), U 2 (x 2 ),...) = W(V 1 (p,y 1 ), V 2 (p,y 2 ),...) SWF in terms of direct utility. Using indirect utility function Changes in utility and change social welfare …...related to income change in “national income”  Differentiate the SWF w.r.t. p i : n h dW =  W h V i h dp i h=1. Changes in utility and change social welfare … n h = –  W h V y h x i h dp i h=1 from Roy’s identity n h dW = – M  x i h dp i h=1...related to prices Change in total expenditure. March 2012 21

22. Frank Cowell: Welfare - Social Welfare function An attractive result?  Summarising the results of the previous slide we have:  THEOREM: in the neighbourhood of a welfare optimum welfare changes are measured by changes in national income / national expenditure  But what if we are not in an ideal world? March 2012 22

23. Frank Cowell: Welfare - Social Welfare function Overview... The Approach SWF: basics SWF: national income SWF: income distribution Welfare: SWF A lesson from risk and uncertainty March 2012 23

24. Frank Cowell: Welfare - Social Welfare function Derive a SWF in terms of incomes  What happens if the distribution of income is not ideal? M is no longer equal for all h  Useful to express social welfare in terms of incomes  Do this by using indirect utility function V Express utility in terms of prices p and income y  Assume prices p are given  “Equivalise” (i.e. rescale) each income y allow for differences in people’s needs allow for differences in household size  Then you can write welfare as W(y a, y b, y c, … ) March 2012 24

25. Frank Cowell: Welfare - Social Welfare function Income-distribution space: n h =2 Bill's income Alf's income O  The income space: 2 persons  An income distribution  y 45° line of perfect equality  Note the similarity with a diagram used in the analysis of uncertainty March 2012 25

26. Frank Cowell: Welfare - Social Welfare function Extension to n h =3  Here we have 3 persons Charlie's income Alf's income Bill's income O line of perfect equality y  An income distribution. March 2012 26

27. Frank Cowell: Welfare - Social Welfare function Welfare contours  EyEy yaya ybyb  EyEy  y  An arbitrary income distribution  Contours of W  Swap identities  Distributions with the same mean  Anonymity implies symmetry of W  Equally-distributed-equivalent income  E y is mean income  Richer-to-poorer income transfers increase welfare.  equivalent in welfare terms equivalent in welfare terms   is income that, if received uniformly by all, would yield same level of social welfare as y. higher welfare higher welfare  E y  is income that society would give up to eliminate inequality March 2012 27

28. Frank Cowell: Welfare - Social Welfare function A result on inequality aversion  Principle of Transfers : “a mean-preserving redistribution from richer to poorer should increase social welfare”  THEOREM: Quasi-concavity of W implies that social welfare respects the “Transfer Principle” March 2012 28

29. Frank Cowell: Welfare - Social Welfare function Special form of the SWF  It can make sense to write W in the additive form n h 1 W = —    y h  n h h=1 where the function  is the social evaluation function (the 1/n h term is unnecessary – arbitrary normalisation) Counterpart of u-function in choice under uncertainty  Can be expressed equivalently as an expectation: W = E  y h  where the expectation is over all identities probability of identity h is the same, 1/n h, for all h  Constant relative-inequality aversion: 1  y  = —— y 1 –  1 –  where  is the index of inequality aversion works just like ,the index of relative risk aversion March 2012 29

30. Frank Cowell: Welfare - Social Welfare function Concavity and inequality aversion W (y)(y) income y (y)(y)  The social evaluation function  Let values change: φ is a concave transformation.  More concave  () implies higher inequality aversion  ...and lower equally-distributed- equivalent income  and more sharply curved contours lower inequality aversion higher inequality aversion  = φ() = φ() March 2012 30

31. Frank Cowell: Welfare - Social Welfare function Social views: inequality aversion  ½ ybyb yaya O  ybyb yaya O  ybyb yaya O   Indifference to inequality  Mild inequality aversion ybyb yaya O  Strong inequality aversion  Priority to poorest  “Benthamite” case (  = 0): n h W=  y h h=1  General case (  ): n h W =  [ y h ] 1-  / [1-i] h=1  “Rawlsian” case (  ): W = min y h h March 2012 31

32. Frank Cowell: Welfare - Social Welfare function Inequality, welfare, risk and uncertainty  There is a similarity of form between… personal judgments under uncertainty social judgments about income distributions.  Likewise a logical link between risk and inequality  This could be seen as just a curiosity  Or as an essential component of welfare economics Uses the “equal ignorance argument”  In the latter case the functions u and  should be taken as identical  “Optimal” social state depends crucially on shape of W In other words the shape of  Or the value of  Three examples March 2012 32

33. Frank Cowell: Welfare - Social Welfare function Social values and welfare optimum yaya ybyb  The income-possibility set Y  Welfare contours (  = ½)  Welfare contours (  = 0)  Welfare contours (  =  )  Y derived from set A  Nonconvexity, asymmetry come from heterogeneity of households  y * maximises total income irrespective of distribution    y*** gives priority to equality; then maximises income subject to that  Y y*y* y ***  y **  y** trades off some income for greater equality March 2012 33

34. Frank Cowell: Welfare - Social Welfare function Summary  The standard SWF is an ordering on utility levels Analogous to an individual's ordering over lotteries Inequality- and risk-aversion are similar concepts  In ideal conditions SWF is proxied by national income  But for realistic cases two things are crucial: 1. Information on social values 2. Determining the income frontier  Item 1 might be considered as beyond the scope of simple microeconomics  Item 2 requires modelling of what is possible in the underlying structure of the economy... ...which is what microeconomics is all about March 2012 34