Frank Cowell: EC426 Public Economics MSc Public Economics 2011/12 Policy Design: Social Insurance Frank A. Cowell 17 October 2011

Frank Cowell: EC426 Public Economics Role of social insurance Insurance and social insurance Insurance and social insurance  Why should a social approach be necessary?  Is this just classic market failure?  Will society succeed where the market fails? Redistribution Redistribution  A role for society  Why do it through an insurance mechanism? Paternalist management Paternalist management  Merit good argument: health insurance?  Failure of perception: life insurance? Overview of subject: Overview of subject:   Barr (1992) Barr (1992)   Diamond (2003)

Frank Cowell: EC426 Public Economics Design issues Game theoretic approach: Game theoretic approach:  Incomplete information  Asymmetric information 1+n players 1+n players  1 Government  n Citizens Government makes first move Government makes first move  Designs a menu of opportunities Citizen then makes choices Citizen then makes choices  in the light of the “menu” offered  Could be work, saving, etc More on design theory next week More on design theory next week

Frank Cowell: EC426 Public Economics Overview... Insurance model Adverse selection Distributional objectives Policy Design: Social Insurance Choice in the face of risk Moral hazard

Frank Cowell: EC426 Public Economics Insurance problem: outline You are endowed with a risky prospect You are endowed with a risky prospect  Value of wealth ex-ante is y 0  There is a risk of loss L  If loss occurs then wealth is y 0 – L You can purchase insurance against this risk of loss You can purchase insurance against this risk of loss  Cost of insurance is   In both states of the world ex-post wealth is y 0 –  Use a standard state-space diagram Use a standard state-space diagram

Frank Cowell: EC426 Public Economics Attainable set: insurance x BLUE x RED   P y y _ _ _ A   Endowment   Full insurance at premium    All these points belong to A   Can you overinsure?   Can you bet on your loss? unlikely to be points here   P 0 y 0 – L y 0  L –  partial insurance

Frank Cowell: EC426 Public Economics Insurance problem: preferences Also from standard model of risk-taking Also from standard model of risk-taking Satisfy von-Neumann-Morgenstern axioms Satisfy von-Neumann-Morgenstern axioms  Can use concept of expected utility  If  is probability of loss…  …slope of indifference curve where it crosses the 45º line is – [1–  Competitive market means actuarially fair insurance Competitive market means actuarially fair insurance  Slope of budget line given by –[1– ...  Strong implication for equilibrium, but… Outcome will depend on individual perceptions Outcome will depend on individual perceptions

Frank Cowell: EC426 Public Economics A Equilibrium and perceptions x BLUE x RED y y _ _ P* P* ll ll P 0     Attainable set, insurance problem _   P   Equilibrium – correct perception   Equilibrium – wild over-optimism   Equilibrium - mild over-optimism

Frank Cowell: EC426 Public Economics Overview... Insurance model Adverse selection Distributional objectives Policy Design: Social Insurance Problems with different risk classes Moral hazard

Frank Cowell: EC426 Public Economics Adverse selection: market model? Adverse selection models use one of 2 paradigms: Adverse selection models use one of 2 paradigms: Monopoly provision Monopoly provision  Can draw up menu of contracts  Limited only by possibility of non-participation or misrepresentation Competitive model with free entry into the market Competitive model with free entry into the market  Numbers determined by zero-profit condition  What type of equilibrium will emerge?  Will there be an equilibrium? Take multiple-agent (competitive) version of model Take multiple-agent (competitive) version of model Graphical representation

Frank Cowell: EC426 Public Economics A single risk type x BLUE x RED 0   a-type indifference map 1  a   a. 1  a   a. y – L y a _ A  L −    income and possible loss y a _ y   actuarially expected income   actuarially fair insurance with premium    a-type attainable set   Slope is same on 45  line   Gets “flatter” as  increases   Full insurance guarantees expected income   Endowment point P 0 has coordinates (y, y – L) P0 P0

Frank Cowell: EC426 Public Economics The insurance problem: types An information problem can arise if there is heterogeneity of the insured persons An information problem can arise if there is heterogeneity of the insured persons Assume that heterogeneity concerns probability of loss Assume that heterogeneity concerns probability of loss  a-types: low risk, low demand for insurance  b-types: high risk, high demand for insurance  Types associated with risk rather than pure preference Each individual is endowed with the prospec Each individual is endowed with the prospect P 0 Begin with full-information case Begin with full-information case

Frank Cowell: EC426 Public Economics P0 P0 Efficient risk allocation x BLUE x RED 0 y – L y   Endowment point   Attainable set and equilibrium, a-types   a-type indifference curves   b-type indifference curves 1  a   a. 1  a   a. 1  b   b. 1  b   b.    b >  a   Attainable set and equilibrium, b-types   A b-type would prefer to get an a-type contract if it were possible P *b P *a

Frank Cowell: EC426 Public Economics Possibility of adverse selection x BLUE x RED   Indifference curves (y, y  L) 0   Endowment   a-type (low-risk) insurance contract   b-type (high-risk) insurance contract   If b-type insures fully with an a-type contract   If over-insurance were possible...

Frank Cowell: EC426 Public Economics Pooling Suppose the firm “pools” all customers Suppose the firm “pools” all customers Same price offered for insurance to all Same price offered for insurance to all Proportions of a-types and b-types are (  ) Proportions of a-types and b-types are (  ) Pooled probability of loss is therefore Pooled probability of loss is therefore   a  b   a   b Can this be an equilibrium? Can this be an equilibrium?

Frank Cowell: EC426 Public Economics Pooling equilibrium? x BLUE x RED   Endowment & indiff curves 0   Pure a-type, b-type contracts   Pooling contract, low    a-type’s choice with pooling   Pooling contract, high    Pooling contract, intermediate  P0 P0   b-type’s unrestricted choice   b-type mimics an a-type   A profitable contract preferred by a-types but not by b-types   A proposed pooling contract is always dominated by a separating contract P P

Frank Cowell: EC426 Public Economics Separating equilibrium? x BLUE x RED   Endowment & indiff curves 0   Pure a-type, b-type contracts   b-type would like a pure a- type contract   Then b-types take efficient contract   Restrict a-types in their coverage P0 P0   a-type’s and b-type’s preferred prospects to (P a,P* b )   A pooled contract preferred by both a-types and b-type P *b PaPa ~   Proposed separating contract might be dominated by a pooling contract   This could happen if  were large enough ~ P ^

Frank Cowell: EC426 Public Economics Insurance model: assessment Insurance case difficult because of role of  a,  b Insurance case difficult because of role of  a,  b  As "type indicators" – shift the indifference curves  As weights in the evaluation of profits The population composition affects profitability The population composition affects profitability  Directly: expected profit on each contract written  Indirectly: through the masquerading process Equilibrium? Equilibrium?  Pooling: No (As in monopoly adverse selection models)  Separating: Maybe not (problem of free entry) Role of government? Role of government?  regulatory – to offset market failure  as monopoly supplier – to avoid problem of new entrants  concern for high-risk people– subsidies to individuals?

Frank Cowell: EC426 Public Economics Overview... Insurance model Adverse selection Distributional objectives Policy Design: Social Insurance Responsible behaviour? Moral hazard

Frank Cowell: EC426 Public Economics The moral hazard problem A key aspect of hidden information A key aspect of hidden information Information relates to actions Information relates to actions Hidden action by one party affects probability of favourable/unfavourable outcomes Hidden action by one party affects probability of favourable/unfavourable outcomes  Hidden information about personal characteristics is different... ... “adverse selection”  but similar issues in setting up the economic problem Set-up is principal-and-agent analysis Set-up is principal-and-agent analysis  based on model of trade under uncertainty  interpret as management contract  or as an insurance contract

Frank Cowell: EC426 Public Economics Principal and agent: concepts Contract: Contract:  An agreement to provide specified service…  …in exchange for specified payment  Type of contract will depend on information available Payment schedule: Payment schedule:  Set-up involving a menu of contracts  The Principal draws up the menu  Allows selection by the Agent  Payment schedule will depend on information available Events: Events:  Assume that events consist of single states-of-the-world  Distribution of these is common knowledge  But distribution may be conditional on the Agent’s effort

Frank Cowell: EC426 Public Economics Outline of the problem Output depends on Output depends on  chance element  effort put in by A A's effort affects probability of chance element A's effort affects probability of chance element  High effort – high probability of favourable outcome  Low effort – low probability of favourable outcome Because B moves first: Because B moves first:  can set the terms of the contract  constrained by A’s option to refuse The issues are: The issues are:  Does B find it worthwhile to pay A for high effort?  Is it possible to monitor whether high effort is provided?  If not, how can B best construct the contract?

Frank Cowell: EC426 Public Economics Model: basics A single good A single good Amount of output q is a random variable Amount of output q is a random variable Two possible outcomes Two possible outcomes  Failure q – _  Success q Probability of failure (loss) is common knowledge: Probability of failure (loss) is common knowledge:  given by  (z)  z is the effort supplied by the agent Agent chooses either Agent chooses either  Low effort z _  High effort z

Frank Cowell: EC426 Public Economics Motivation A's utility derives from A's utility derives from  consumption of the single good x a (  )  the effort put in, z (  )  Given vNM preferences utility is E u a (x a, z) Agent is risk averse Agent is risk averse  u a (, ) is strictly concave in its first argument Principal consumes all output not consumed by Agent Principal consumes all output not consumed by Agent  x b = q – x a Principal is risk neutral Principal is risk neutral  (In the simple model)  Utility is E q – x a Interpret this in an Edgeworth Box trading diagram Interpret this in an Edgeworth Box trading diagram

Frank Cowell: EC426 Public Economics Low effort OaOa ObOb x RED a b x BLUE b a  RED – ____  BLUE  RED – ____  BLUE ObOb   Certainty line for Agent   A's indifference curves   Certainty line for Boss   B's indifference curves   Endowment point   A's reservation utility   If B exploits A then outcome is on reservation IC,  a   If B is risk-neutral and A is risk averse then outcome is on A's certainty line aa Switch to high effort

Frank Cowell: EC426 Public Economics High effort OaOa ObOb x RED a b x BLUE b a ObOb ObOb   Certainty line and indifference curves for A   Certainty line and indifference curves for B   Endowment point   A's reservation utility  RED – ____  BLUE  RED – ____  BLUE   High effort tilts the ICs, shifts the equilibrium outcome   Contrast with low effort Combine to get menu of contracts

Frank Cowell: EC426 Public Economics Full information: max problem Schedule of contracts for high and low effort Schedule of contracts for high and low effort  Effort is verifiable Contract specifies payment in each state-of-the-world Contract specifies payment in each state-of-the-world  can be conditioned on effort: w(z) Agent's consumption is determined by the payment Agent's consumption is determined by the payment The Principal chooses a payment schedule... The Principal chooses a payment schedule...  w = w(z)...subject to the participation constraint:...subject to the participation constraint:  E u a (w,z)   a So, problem is choose w() to maximise So, problem is choose w() to maximise  E q – w + [ E u a (w,z) –  a ] Equivalently Equivalently _  Find w(z) to max [1 –  (z)] q +  (z) q – w(z)... _ ... for the two cases z = z and z = z  Choose the one with higher expected payoff to Principal

Frank Cowell: EC426 Public Economics Full-information contracts OaOa ObOb x RED a b x BLUE b a – w(z)w(z) q – – w(z)w(z) – w(z)w(z) – w(z)w(z) – q   A's low-effort ICs   A's high-effort ICs   B’s ICs   Low-effort contract   High-effort contract

Frank Cowell: EC426 Public Economics Second best: principles Utility functions Utility functions  As before Payment schedule Payment schedule  Because effort is unobservable… ...can’t condition payment on effort or state-of-the-world  But resulting output is observable... ... so you can condition payment on output w(q) Participation constraint Participation constraint  Essentially as before  (but we'll have another look) New incentive-compatibility constraint New incentive-compatibility constraint  Cannot observe effort  Agent must get the utility level attainable under low effort Maths formulation

Frank Cowell: EC426 Public Economics Second best: constraints Principal can condition payment on observed output: Principal can condition payment on observed output:  Pay  w if output is  q  Pay w if output is q Agent will choose high or low effort Agent will choose high or low effort  determines the probability of getting high output, high payment Participation: must get the utility available elsewhere: Participation: must get the utility available elsewhere: _ _ _ _ _ _ _ _ _ _  [1 –  (z)] u a (w, z) +  (z) u a (w, z)   a Incentive Compatibility: To ensure high effort: Incentive Compatibility: To ensure high effort: _ _ _ _ _ _ _ _ _ _  [1 –  (z)] u a (w, z) +  (z) u a (w, z)  [1 –  (z)] u a (w, z) +  (z) u a (w, z) This condition determines a set of w-pairs This condition determines a set of w-pairs  a set of contingent consumptions for A  must not reward A too highly if failure is observed

Frank Cowell: EC426 Public Economics Second-best contracts OaOa ObOb w – – w x RED a b x BLUE b a   A's low-effort ICs   B’s ICs   A's high-effort ICs   B’s ICs   Full-information contracts   Participation constraint   Incentive- compatibility constraint aa   Second-best contract   B’s second-best feasible set   Contract maximises B’s utility over second-best feasible set

Frank Cowell: EC426 Public Economics P&A: (social) insurance “Translate” from wage-contract model “Translate” from wage-contract model  Replace wages with payments if employed/unemployed Insurer does not provide full cover Insurer does not provide full cover  Unobservable states…  Unobservable action…  …moral hazard Standard application Standard application  Unemployment: whose fault?  Search effort Should the state provide full cover? Should the state provide full cover?  Traditional welfare state – yes?  Modern approach is to respect incentive-compatibility

Frank Cowell: EC426 Public Economics Overview... Insurance model Adverse selection Distributional objectives Policy Design: Social Insurance A model of multiple objectives Moral hazard

Frank Cowell: EC426 Public Economics Alternative SI systems Two main approaches to social insurance Two main approaches to social insurance The Bismarck tradition The Bismarck tradition  individuals insure themselves  no explicit redistribution element  you get what you pay for The Beveridge tradition The Beveridge tradition  full coverage  fortunate compensate the unfortunate Elements of both in most countries’ SI Elements of both in most countries’ SI  more equal societies have more Bismarckian schemes? ()  more equal societies have more Bismarckian schemes? (Conde-Ruiz and Profeta 2007 )Conde-Ruiz and Profeta 2007  B-index is correlation (wage, pension) Compare these within a unified model? Compare these within a unified model?   Casamatta et al. (2000)Casamatta et al. (2000) CountryB-indexGini Austria Belgium Denmark France Germany Greece Ireland Italy Spain UK US CountryB-indexGini Austria Belgium Denmark France Germany Greece Ireland Italy Spain UK US

Frank Cowell: EC426 Public Economics Casamatta et al. model (1) Type: the mix between B & B Type: the mix between B & B  Logically prior?  Model this as being decided first Scale: how much insurance Scale: how much insurance  Decided after the type is determined  Constrained by the tax etc. required to pay for it Model SI as a two-stage game Model SI as a two-stage game  Stages corresponding to two features of SI  1 Constitutional stage  Take B&B as two polar cases  Decide on what mix of the two is appropriate  2 Voting stage  Determine the amount of coverage  Decide on amount of a “payroll tax”  Work backwards through stages

Frank Cowell: EC426 Public Economics The Casamatta et al. model (2) Three types of individual Three types of individual  Exogenous income levels: w 1, w 2, w 3  Equal numbers, so average income:  w := [w 1 + w 2 + w 3 ] /3 Two possible states for each person Two possible states for each person  Employed – get w i with probability ½  Unemployed – get b i with probability ½ Preferences Preferences  U(c i, b i ) = u(c i ) +u(b i ) where u is concave Total benefits b i = b i p + b i s Total benefits b i = b i p + b i s  (private) b i p =  p  i w i  (social) b i s = t[(1 −  )  w +  w i ] Rates of return to private and social insurance: Rates of return to private and social insurance:   p determined by the market; assume  p < 1   i s = [(1 −  )  w / w i +  ];  i s depends on w i ; average = 1

Frank Cowell: EC426 Public Economics The second stage: t Consumption and benefits for type i: Consumption and benefits for type i:  c i = w i [1 −  i − t]  b i = w i [  i s t +  p  i ] Linearity in  i implies choose only one type of insurance Linearity in  i implies choose only one type of insurance  go private if social rate of return low (i.e. where w is high)  low-wage people choose  i = 0, t > 0  person with w would be indifferent if rates of return equal  requires  (w) := [  p w −  w ] / [w −  w] Choose optimal scale given  : Choose optimal scale given  : Implies t*(w,1) = ½ and Implies t*(w,1) = ½ and

Frank Cowell: EC426 Public Economics The first stage:   i (  ), utility received by person i, given  :  i (  ), utility received by person i, given  :  case 1, no private insurance  case 2, there is private insurance In either case optimal  requires In either case optimal  requires  either Rawlsian objective: max  1 (  )  or Utilitarian objective: max  i  i (  )

Frank Cowell: EC426 Public Economics Which type of scheme? Results depend on SWF and the distribution of w Results depend on SWF and the distribution of w No private insurance No private insurance  Low w 2 : only Beveridge (  = 0) under Rawls; t * > 0  High w 2 : even under Rawls may need  > 0 to get t * >0 in second stage Private insurance Private insurance  Low w 2 : only Beveridge (  = 0) under Rawls; t * > 0  High w 2 : under Rawls need critical value of   so that median voter chooses t * >0

Frank Cowell: EC426 Public Economics Casamatta et al.: conclusions May need a less redistributive scheme in order to get support for tax in second stage May need a less redistributive scheme in order to get support for tax in second stage Private insurance undermines support for SI Private insurance undermines support for SI Private insurance may increase welfare of the poor Private insurance may increase welfare of the poor Case for prohibiting private insurance stronger when private market more efficient Case for prohibiting private insurance stronger when private market more efficient

Frank Cowell: EC426 Public Economics References Barr, N. A. (1992) “Economic theory and the welfare state: A survey and interpretation,” Journal of Economic Literature, 30, Barr, N. A. (1992) Casamatta, G. and Cremer, H. and Pestieau, P. (2000) “Political Sustainability and the Design of Social Insurance”, Journal of Public Economics, 75, Casamatta, G. and Cremer, H. and Pestieau, P. (2000) Conde-Ruiz, J. I. and Profeta, P. (2007) “The Redistributive Design of Social Security Systems,” Economic Journal, 117, Conde-Ruiz, J. I. and Profeta, P. (2007) Diamond, P. A. (2003) Taxation, Incomplete Markets and Social Security, MIT Press.