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

Frank Cowell: Microeconomics Welfare Principles Try to find general principles for running the economy. 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” We have already slipped in one notion of “desirability”  “Technical Efficiency”  …applied to the firm What about a similar criterion for the whole economy? What about a similar criterion for the whole economy?  A generalised version of efficiency.  Subsumes technical efficiency? And what about other desirable principles? And what about other desirable principles?

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

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

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

Frank Cowell: Microeconomics A definition of efficiency The basis for evaluating social states: v h (  ) the utility level enjoyed by person h under social state  A social state  is Pareto superior to state  ' if: 1. 1.For all h: v h (  )  v h (  ') 2. 2.For some h: v h (  ) > v h (  ') Note the similarity with the concept of blocking by a coalition “feasibility” could be determined in terms of the usual economic criteria A social state  is Pareto efficient if: 1. 1.It is feasible 2. 2.No other feasible state is Pareto superior to  Take the case of the typical GE model. Link to Blocking

Frank Cowell: Microeconomics Derive the utility possibility set   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 ) U  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 )  ...take an allocation   Evaluate utility for each agent   Repeat to get the utility possibility set

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

Frank Cowell: Microeconomics Finding an efficient allocation (1) Use essentially the same method. Use essentially the same method. Do this for the case where Do this for the case where  all goods are purely private  households 2,…,n h are on fixed utility levels   households 2,…,n h are on fixed utility levels  h Then problem is to maximise U 1 () subject to: Then problem is to maximise U 1 (x 1 ) subject to:  U() ≥ , h = 2, …, n h  U h (x h ) ≥  h, h = 2, …, n h  f = 1, …, n f   f (q f ) ≤ 0, f = 1, …, n f  i= 1, …, n  x i ≤ q i + R i, i= 1, …, n Use all this to form a Lagrangean in the usual way… Use all this to form a Lagrangean in the usual way…

Frank Cowell: Microeconomics Finding an efficient allocation (2) max L ( [x ], [q],  ) := U 1 (x 1 ) +  h h [U h (x h )   h ]   f  f  f (q f ) +  i  i [q i + R i  x i ] where x i =  h x i h q i =  f q i f Lagrange multiplier for each utility constraint Lagrange multiplier for each firm’s technology Lagrange multiplier for materials balance, good i

Frank Cowell: Microeconomics FOCs Differentiate Lagrangean w.r.t x i h. If x i h is positive at the optimum then: h U i h (x h ) =  i Likewise for good j: h U j h (x h ) =  j Differentiate Lagrangean w.r.t q i f. If q i f is nonzero at the optimum then:  f  i f (q f ) =  i Likewise for good j:  f  j f (q f ) =  j

Frank Cowell: Microeconomics Interpreting the FOC U i h (x h )  i ———— = — U j h (x h )  j for every firm: MRT = shadow price ratio for every household: MRS = shadow price ratio From the FOCs for any household h and goods i and j:  i f (q f )  i ———— = —  j f (q f )  j From the FOCs for any firm f and goods i and j: familiar example

Frank Cowell: Microeconomics ObOb OaOa x1x1 b x1x1 a x2x2 a x2x2 b Efficiency in an Exchange Economy   The contract curve   Alf’s indifference curves   Bill’s indifference curves   The set of efficient allocations is the contract curve.   Includes cases where Alf or Bill is very poor. Allocations where MRS 12 a = MRS 12 b Apply to uncertainty

Frank Cowell: Microeconomics Efficiency in an Exchange Economy: Uncertainty OaOa ObOb x RED a b x BLUE b a   Alf's indifference curves over prospects   Bill's indifference curves   The contract curve Allocations where MRS REDBLUE = MRS REDBLUE Allocations where MRS REDBLUE = MRS REDBLUE a b   The set of ex-ante efficient allocations is the contract curve.   Ex-post efficient allocations cover whole diagram

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

Frank Cowell: Microeconomics Efficiency and equilibrium The Edgeworth Box diagrams are suggestive. The Edgeworth Box diagrams are suggestive. Points on the contract curve… Points on the contract curve…  …are efficient  …could be CE allocations So, examine connection of market with efficiency: 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 Focus on two important theorems

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

Frank Cowell: Microeconomics Welfare theorem 2 Pick any Pareto-efficient allocation Pick any Pareto-efficient allocation Can we find a property distribution d so that this allocation is a CE for d? 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. 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: 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...

Frank Cowell: Microeconomics ObOb OaOa x1x1 b x1x1 a x2x2 a x2x2 b Supporting a PE allocation   The contract curve   Support the allocation by a CE.   This needs an adjustment of the initial endowment.   Beware: lump- sum transfers may be tricky to implement Allocations where MRS 12 a = MRS 12 b   An efficient allocation   Supporting price ratio = MRS [x] ^ [R]   The property distribution   A lump-sum transfer p1p2p1p2 p1p2p1p2

Frank Cowell: Microeconomics Supporting a PE allocation (production)   Firm f’s technology set 0   Supporting price ratio = MRT   f’s net output in the efficient allocation p1p2p1p2 p1p2p1p2   f’s net output in the allocation is profit- maximising for f. q2fq2f q1fq1f qfqf ^ what if preferences and production possibilities were different?

Frank Cowell: Microeconomics Household h makes “wrong” choice x2hx2h 0 x1hx1h xhxh ^ xhxh ~   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 p1p2p1p2 p1p2p1p2   Nonconvexity leads to “market failure”

Frank Cowell: Microeconomics Firm f makes “wrong” choice q2fq2f 0 q1fq1f qfqf ^ qfqf ~   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 p1p2p1p2 p1p2p1p2   Another example of “market failure”

Frank Cowell: Microeconomics Competitive Failure and Efficiency The market “failures” raise two classes of problem that lead to two distinct types of analysis: The market “failures” raise two classes of problem that lead to two distinct types of analysis: 1. The characterisation problem  Description of modified efficiency conditions…  …when conditions for the welfare theorems are not met. 2. The implementation problem  Design of a mechanism to achieve the allocation. These two types of problem pervade nearly all of modern micro economics. These two types of problem pervade nearly all of modern micro economics. It’s important to keep them distinct in one’s mind. It’s important to keep them distinct in one’s mind.

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

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

Frank Cowell: Microeconomics Indecisiveness of PE aa bb U   Construct utility-possibility set as previously   Two efficient points   Boundary points cannot be compared on efficiency grounds   Points superior to  A way forward? v   v     Points superior to  '.    and  ' cannot be compared on efficiency grounds

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

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

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

Frank Cowell: Microeconomics Applying potential superiority aa U    is not superior to  ' and  ' is not superior to .    * is superior to .   There could be lots of points accessible by lump-sum transfers.    ' is potentially superior to    Points accessible from  '. Difficulties v   bb v   v  *  

Frank Cowell: Microeconomics Problems with accessibility What prices should be used in the evaluation. What prices should be used in the evaluation.  those in ?  those in  ?  those in ?  those in  ' ? We speak only of potential income gains. We speak only of potential income gains.  compensation is not actually paid.  does this matter? If no income transfer takes place… 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... But this isn’t the biggest problem...

Frank Cowell: Microeconomics Re-examine potential superiority aa bb v   v   points accessible from  points accessible from    The process from  to  ', as before.   The process in reverse from  ' to .   Combine the two.    is potentially superior to  and …    is potentially superior to  !

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

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