Welfare: Efficiency

1. Prerequisites Almost essential Welfare: Basics Welfare: Efficiency MICROECONOMICS Principles and Analysis Frank Cowell December 2006

2. Welfare Principles • Try to find general principles for running the economy. • May be more fruitful than the constitution approach. • We have already slipped in one notion of “desirability” • “Technical Efficiency” • …applied to the firm • What about a similar criterion for the whole economy? • A generalised version of efficiency. • Subsumes technical efficiency? • And what about other desirable principles?

3. Overview... Welfare: efficiency Pareto efficiency Fundamental criterion for judging economic systems CE and PE Extending efficiency

4. A short agenda • Describe states of the economy in welfare terms... • Use this analysis to define efficiency. • Apply efficiency analysis to an economy • Use standard multi-agent model • Same as analysed in earlier presentations. • Apply efficiency concept to uncertainty • distinguish ex-ante and ex-post cases

5. The essential concepts • Social state • Describes economy completely. • For example, an allocation • Pareto superiority • At least as much utility for all; • Strictly greater utility for some • Pareto efficiency • Uses concept of Pareto superiority. • Also needs a definition of feasibility...

6. A definition of efficiency • The basis for evaluating social states: vh(q) the utility level enjoyed by personhunder social stateq • A social state q is Pareto superior to stateq' if: • For all h: vh(q)³ vh(q') • For some h: vh(q)> vh(q') Link to Blocking Note the similarity with the concept of blocking by a coalition • A social state q is Paretoefficient if: • It is feasible • No other feasible state is Pareto superior to q “feasibility” could be determined in terms of the usual economic criteria Take the case of the typical GE model.

7. A A U Derive the utility possibility set • From the attainable set... (x1a, x2a) (x1b, x2b) • ...take an allocation • Evaluate utility for each agent • Repeat to get the utility possibility set ua=Ua(x1a, x2a) ub=Ub(x1b, x2b)

8. constant ub Utility possibility set – detail ub • Utility levels of each person. rescale this for a different cardinalisation • U-possibility derived from A • Fix all but one at some utility level then max U of that person • Repeat for other persons and U-levels to get PE points • Efficient points on boundary of U • But not all boundary points are efficient an efficient point U ua Find the corresponding efficient allocation

9. Finding an efficient allocation (1) • Use essentially the same method. • Do this for the case where • all goods are purely private • households 2,…,nhare on fixed utility levels uh • Then problem is to maximise U1(x1) subject to: • Uh(xh) ≥ uh, h = 2, …, nh • Ff(qf) ≤ 0,f = 1, …, nf • xi ≤qi + Ri ,i= 1, …, n • Use all this to form a Lagrangean in the usual way…

10. Finding an efficient allocation (2) Lagrange multiplier for each utility constraint max L( [x], [q], l, m, k) := U1(x1) + åhlh[Uh(x h)  uh]  åf mfFf (q f) + åi ki[qi + Ri  xi] where xi = åh xih qi = åf qi f Lagrange multiplier for each firm’s technology Lagrange multiplier for materials balance, good i

11. FOCs • Differentiate Lagrangean w.r.t xih. If xih is positive at the optimum then: lhUih(xh) = ki • Likewise for good j: lhUjh(xh) = kj • Differentiate Lagrangean w.r.t qif. If qif is nonzero at the optimum then: mfFif(qf) = ki • Likewise for good j: mfFjf(qf) = kj

12. Interpreting the FOC • From the FOCs for any household h and goods i and j: for every household: MRS = shadow price ratio Uih(xh) ki ———— = — Ujh(xh) kj • From the FOCs for any firm f and goods i and j: for every firm: MRT = shadow price ratio Fif(qf) ki ———— = — Fjf(qf) kj familiar example

13. b a a b x2 x1 x1 x2 Efficiency in an Exchange Economy Ob • Alf’s indifference curves • Bill’s indifference curves • The contract curve • The set of efficient allocations is the contract curve. • Includes cases where Alf or Bill is very poor. Allocations where MRS12a = MRS12b Apply to uncertainty Oa

14. Allocations where MRSREDBLUE= MRSREDBLUE a b b a a b xBLUE xBLUE xRED xRED Efficiency in an Exchange Economy: Uncertainty • Alf's indifference curves over prospects Ob • Bill's indifference curves • The contract curve • The set of ex-ante efficient allocations is the contract curve. • Ex-post efficient allocations cover whole diagram Oa

15. Overview... Welfare: efficiency Pareto efficiency Relationship between competitive equilibrium and efficient allocations CE and PE Extending efficiency

16. Efficiency and equilibrium • The Edgeworth Box diagrams are suggestive. • Points on the contract curve… • …are efficient • …could be CE allocations • So, examine connection of market with efficiency: • Will the equilibrium be efficient? • Can we use the market to implement any efficient outcome? • Or are there cases where the market “fails”? • Focus on two important theorems

17. Welfare theorem 1 • Assume a competitive equilibrium. • What is its efficiency property? • THEOREM: if all consumers are greedy and there are no externalities then a competitive equilibrium is efficient. • Explanation: • If they are not greedy, there may be no incentive to trade. • If there are externalities the market takes no account of spillovers.

18. Welfare theorem 2 • Pick any Pareto-efficient allocation • Can we find a property distribution d so that this allocation is a CE for d? • THEOREM: if, in addition to conditions for theorem 1, there are no nonconvexities then an arbitrary PE allocation be supported by a competitive equilibrium. • Explanation: • If “lump sum” transfers are possible then we can arbitrarily change the initial property distribution. • If there are nonconvexities the equilibrium price signals could take us away from the efficient allocation...

19. b a a b x1 x1 x2 x2 ^ • [x] Supporting a PE allocation Ob • The contract curve • An efficient allocation [R] • Supporting price ratio = MRS • The property distribution • A lump-sum transfer • Support the allocation by a CE. • This needs an adjustment of the initial endowment. • Beware: lump-sum transfers may be tricky to implement p1  p2 Allocations where MRS12a = MRS12b Oa

20. ^ qf Supporting a PE allocation (production) q2f • Firm f’s technology set • f’s net output in the efficient allocation • Supporting price ratio = MRT • f’s net output in the allocation is profit-maximising for f. p1  p2 what if preferences and production possibilities were different? q1f 0

21. ~ xh ^ xh Household h makes “wrong” choice x2h • Household h ’s utility function violates second theorem • Suppose we want to allocate this consumption bundle to h. • Introduce prices • h's choice given this budget • Nonconvexity leads to “market failure” p1  p2 x1h 0

22. ^ qf ~ qf Firm f makes “wrong” choice q2f • Firm f ’s production function violates second theorem • Suppose we want to allocate this net output to f. • Introduce prices. • f's choice at those prices • Another example of “market failure” p1  p2 q1f 0

23. Competitive Failure and Efficiency • The market “failures” raise two classes of problem that lead to two distinct types of analysis: • The characterisation problem • Description of modified efficiency conditions… • …when conditions for the welfare theorems are not met. • The implementation problem • Design of a mechanism to achieve the allocation. • These two types of problem pervade nearly all of modern micro economics. • It’s important to keep them distinct in one’s mind.

24. Overview... Welfare: efficiency Pareto efficiency Attempt to generalise the concept of Pareto superiority CE and PE Extending efficiency

25. Why extend the efficiency concept? • Pareto efficiency is “indecisive.” • What about the infinity of PE allocations along the contract curve? • Pareto improvements may be elusive. • Beware the politician who insists that everybody can be made better off • Other concepts may command support. • "potential" efficiency • fairness • …

26. Indecisiveness of PE ub • Construct utility-possibility set as previously • Two efficient points • Points superior to q • Points superior to q'. • Boundary points cannot be compared on efficiency grounds v(q)  • q and q' cannot be compared on efficiency grounds  v(q) U ua A way forward?

27. “Potential” Pareto superiority • Define q to be potentially superior to q' if : • There is a q* which is actually Pareto superior to q' • q* is “accessible” from q. • To make use of this concept we need to define accessibility. • This can be done using a tool that we already have from the theory of the consumer.

28. The idea of accessibility • Usually “q* accessible from q” means that income total in q* is no greater than in q. • “if society can afford q then it can certainly afford q* ” • This can be interpreted as a “compensation rule” • The gainers in the move from q' to q... • ...get enough income to be able to compensate the losers of q' ®q. • This can be expressed in terms of standard individual welfare criteria • the CV for each person

29. A result on potential superiority • Use the terminology of individual welfare • CVh(q' ®q) is the monetary value the welfare change • for person h… • …of a change from state q' to state q • …valued in terms of the prices at q • CVh > 0 means a welfare gain; CVh < 0 a welfare loss. • THEOREM: a necessary and sufficient condition for q to be potentially superior to q' is Sh CVh(q' ®q) > 0. • But can we really base welfare economics on the compensating variation...?

30. Applying potential superiority ub • q is not superior to q' and q' is not superior to q. • q* is superior to q. • Points accessible from q'. v(q*) • There could be lots of points accessible by lump-sum transfers. • q' is potentially superior to q  v(q)   v(q) U ua Difficulties

31. Problems with accessibility • What prices should be used in the evaluation. • those in q? • those in q' ? • We speak only of potential income gains. • compensation is not actually paid. • does this matter? • If no income transfer takes place… • so that there are no consumer responses to it • can we accurately evaluate gains and losses? • But this isn’t the biggest problem...

32. Re-examine potential superiority • The process fromq to q', as before. ub • The process in reverse from q' to q. points accessible from q • Combine the two. • q is potentially superior to q and … • q is potentially superior to q! v(q)  points accessible from q  v(q) ua

33. Extending efficiency: assessment • The above is the basis of Cost-Benefit Analysis • It relies on notion of “accessibility” • a curious concept involving notional transfers? • more than one possible definition • “Compensation” is not actually paid • If equilibrium prices differ substantially, the rule may produce contradictions.

34. What next? • This has formed part of the second approach to the question “how should the economy be run?” • The first was “the constitution”. • The approach covers widely used general principles • Efficiency • Neat. Simple. But perhaps limited • Potential efficiency • Persuasive but perhaps dangerous economics/politics • A natural way forward: • Examine other general principles • Consider problems with applying the efficiency concept • Go to the third approach: a full welfare function.