Transcript No Slide Title

17 October 2011
Frank Cowell: EC426 Public Economics
MSc Public Economics
2011/12
http://darp.lse.ac.uk/ec426/
Policy Design: Social Insurance
Frank A. Cowell
Frank Cowell: EC426 Public Economics
Role of social insurance

Insurance and social insurance




Redistribution



A role for society
Why do it through an insurance mechanism?
Paternalist management



Why should a social approach be necessary?
Is this just classic market failure?
Will society succeed where the market fails?
Merit good argument: health insurance?
Failure of perception: life insurance?
Overview of subject:


Barr (1992)
Diamond (2003)
Frank Cowell: EC426 Public Economics
Design issues

Game theoretic approach:



1+n players



Designs a menu of opportunities
Citizen then makes choices



1 Government
n Citizens
Government makes first move


Incomplete information
Asymmetric information
in the light of the “menu” offered
Could be work, saving, etc
More on design theory next week
Frank Cowell: EC426 Public Economics
Overview...
Policy Design:
Social Insurance
Insurance
model
Choice in the
face of risk
Adverse
selection
Moral hazard
Distributional
objectives
Frank Cowell: EC426 Public Economics
Insurance problem: outline

You are endowed with a risky prospect




You can purchase insurance against this risk of loss



Value of wealth ex-ante is y0
There is a risk of loss L
If loss occurs then wealth is y0 – L
Cost of insurance is k
In both states of the world ex-post wealth is y0 – k
Use a standard state-space diagram
Frank Cowell: EC426 Public Economics
Attainable set: insurance
 Endowment
 Full insurance at premium k
xBLUE
 All these points belong to A
 Can you overinsure?
 Can you bet on your loss?
unlikely to be
points here
_
_
y
partial
insurance
 P
L–k
 P0
y0 – L
k
A
_
y
unlikely to be
points here
xRED
y0
Frank Cowell: EC426 Public Economics
Insurance problem: preferences


Also from standard model of risk-taking
Satisfy von-Neumann-Morgenstern axioms




Competitive market means actuarially fair insurance



Can use concept of expected utility
If p is probability of loss…
…slope of indifference curve where it crosses the 45º line is
– [1– p]/p
Slope of budget line given by –[1–p]/p...
Strong implication for equilibrium, but…
Outcome will depend on individual perceptions
Frank Cowell: EC426 Public Economics
Equilibrium and perceptions
Attainable set, insurance
problem
xBLUE
 Equilibrium – correct perception
 Equilibrium – wild over-optimism
 Equilibrium - mild over-optimism
_
_
y
 P
P*

P0

A
_
y
xRED
Frank Cowell: EC426 Public Economics
Overview...
Policy Design:
Social Insurance
Insurance
model
Problems with
different risk
classes
Adverse
selection
Moral hazard
Distributional
objectives
Frank Cowell: EC426 Public Economics
Adverse selection: market model?


Adverse selection models use one of 2 paradigms:
Monopoly provision



Competitive model with free entry into the market




Can draw up menu of contracts
Limited only by possibility of non-participation or
misrepresentation
Numbers determined by zero-profit condition
What type of equilibrium will emerge?
Will there be an equilibrium?
Take multiple-agent (competitive) version of model
Graphical
representation
Frank Cowell: EC426 Public Economics
A single risk type
xBLUE
 a-type indifference map
 income and possible loss
 actuarially expected income
 actuarially fair insurance
with premium k
 a-type attainable set
_
ya
•
 Slope is same on 45
line
 Gets “flatter” as p
increases
L−k
1- pa
- 
pa .
A
•P0
y–L
k
0
 Endowment point P0
has coordinates (y, y – L)
_
ya
xRED
y
 Full insurance
guarantees expected
income
Frank Cowell: EC426 Public Economics
The insurance problem: types


An information problem can arise if there is
heterogeneity of the insured persons
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 prospect P0
Begin with full-information case
Frank Cowell: EC426 Public Economics
Efficient risk allocation
xBLUE
Endowment point
 a-type indifference curves
 b-type indifference curves
 Attainable set and
equilibrium, a-types
 Attainable set and
equilibrium, b-types
• P*a
 p b > pa
•
P*b
y–L
 A b-type would prefer
to get an a-type
contract if it were
possible
1- pa
- 
pa .
b
1- pP
0
- 
b
p .
•
xRED
0
y
Frank Cowell: EC426 Public Economics
Possibility of adverse selection
xBLUE
 Indifference curves
 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...
•
•
• (y, y - L)
xRED
0
Frank Cowell: EC426 Public Economics
Pooling




Suppose the firm “pools” all customers
Same price offered for insurance to all
Proportions of a-types and b-types are (g, 1 - g)
Pooled probability of loss is therefore



`p := gpa + [1 - g] pb
pa <`p < pb
Can this be an equilibrium?
Frank Cowell: EC426 Public Economics
Pooling equilibrium?
 Endowment & indiff curves
 Pure a-type, b-type contracts
xBLUE
 Pooling contract, low g
 Pooling contract, high g
 Pooling contract, intermediate g
 a-type’s choice with pooling
 b-type’s unrestricted choice
 b-type mimics an a-type
•
 A profitable contract preferred
by a-types but not by b-types
•`P
•
 A proposed pooling
contract is always
dominated by a
separating contract
• P0
xRED
0
Frank Cowell: EC426 Public Economics
Separating equilibrium?
 Endowment & indiff curves
 Pure a-type, b-type contracts
xBLUE
 b-type would like a pure atype contract
 Restrict a-types in their coverage
 Then b-types take efficient
contract
 a-type’s and b-type’s preferred
~
prospects to (Pa ,P*b)
•
P*b•
P^
 A pooled contract preferred
by both a-types and b-type
•
•
~
Pa
 Proposed separating
contract might be
dominated by a pooling
contract
• P0
xRED
0
This could happen if g
were large enough
Frank Cowell: EC426 Public Economics
Insurance model: assessment

Insurance case difficult because of role of pa , pb



The population composition affects profitability



Directly: expected profit on each contract written
Indirectly: through the masquerading process
Equilibrium?



As "type indicators" – shift the indifference curves
As weights in the evaluation of profits
Pooling: No (As in monopoly adverse selection models)
Separating: Maybe not (problem of free entry)
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...
Policy Design:
Social Insurance
Insurance
model
Responsible
behaviour?
Adverse
selection
Moral hazard
Distributional
objectives
Frank Cowell: EC426 Public Economics
The moral hazard problem



A key aspect of hidden information
Information relates to actions
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



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:




Payment schedule:





An agreement to provide specified service…
…in exchange for specified payment
Type of contract will depend on information available
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:



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


chance element
effort put in by A

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:
 can set the terms of the contract
 constrained by A’s option to refuse

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
Amount of output q is a random variable
Two possible outcomes



Probability of failure (loss) is common knowledge:



Failure q
–_
Success q
given by p(z)
z is the effort supplied by the agent
Agent chooses either


Low effort z
_
High effort z
Frank Cowell: EC426 Public Economics
Motivation

A's utility derives from




Agent is risk averse


xb = q – xa
Principal is risk neutral



ua(•, •) is strictly concave in its first argument
Principal consumes all output not consumed by Agent


consumption of the single good xa ()
the effort put in, z ()
Given vNM preferences utility is Eua(xa, z)
(In the simple model)
Utility is Eq – xa
Interpret this in an Edgeworth Box trading diagram
Frank Cowell: EC426 Public Economics
Low effort
b
xRED
a
xBLUE
pRED
– ____
pBLUE
 Certainty line for Agent
b
 A's
curves
Obindifference
O
 Certainty line for Boss
 B's indifference curves
 Endowment point
 A's reservation utility
If B exploits A
then outcome is on
reservation IC, ua
If B is risk-neutral
and A is risk
averse then
outcome is on A's
certainty line
ua
b
xBLUE
Oa
a
xRED
Switch to high
effort
Frank Cowell: EC426 Public Economics
High effort
b
xRED
a
xBLUE
pRED
– ____
pBLUE
 Certainty line and indifference
bA
curves
for
O
Obb O
 Certainty line and
indifference curves for B
 Endowment point
 A's reservation utility
High effort tilts
the ICs, shifts
the equilibrium
outcome
Contrast with
low effort
b
xBLUE
Oa
a
xRED
Combine to get
menu of
contracts
Frank Cowell: EC426 Public Economics
Full information: max problem

Schedule of contracts for high and low effort


Contract specifies payment in each state-of-the-world



can be conditioned on effort: w(z)
Agent's consumption is determined by the payment
The Principal chooses a payment schedule...


Effort is verifiable
w = w(z)
...subject to the participation constraint:

Eua(w,z)  ua
So, problem is choose w(•) to maximise
 Eq – w + l[Eua(w,z) – ua]

Equivalently




_
Find w(z) to max [1 – p(z)] q + p(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
–
q
b
xRED
a
xBLUE
Ob
q
–
–
w(z)
w(z)
–
b
xBLUE
Oa
–
w(z)
w(z)
–
a
xRED
 A's low-effort ICs
 B’s 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


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



As before
Essentially as before
(but we'll have another look)
Maths
formulation
New incentive-compatibility constraint


Cannot observe effort
Agent must get the utility level attainable under low effort
Frank Cowell: EC426 Public Economics
Second best: constraints

Principal can condition payment on observed output:



Agent will choose high or low effort


_
_ _
_
_
[1 – p(z)] ua(w, z) + p(z) ua(w, z)  ua
Incentive Compatibility: To ensure high effort:


determines the probability of getting high output, high payment
Participation: must get the utility available elsewhere:


Pay `w if output is `q
Pay w if output is q
_
_ _
_
_
[1 – p(z)] ua(w, z) + p(z) ua(w, z)  [1 – p(z)] ua(w, z) + p(z) ua(w, z)
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
b
xRED
 A's low-effort ICs
Ob B’s ICs
 A's high-effort ICs
 B’s ICs
 Full-information contracts
 Participation constraint
 Incentivecompatibility constraint
 B’s second-best feasible
set
 Second-best contract
a
xBLUE
ua
–w
b
xBLUE
Oa
–
w
a
xRED
 Contract
maximises B’s
utility over
second-best
feasible set
Frank Cowell: EC426 Public Economics
P&A: (social) insurance

“Translate” from wage-contract model


Insurer does not provide full cover




Unobservable states…
Unobservable action…
…moral hazard
Standard application



Replace wages with payments if employed/unemployed
Unemployment: whose fault?
Search effort
Should the state provide full cover?


Traditional welfare state – yes?
Modern approach is to respect incentive-compatibility
Frank Cowell: EC426 Public Economics
Overview...
Policy Design:
Social Insurance
Insurance
model
A model of
multiple
objectives
Adverse
selection
Moral hazard
Distributional
objectives
Frank Cowell: EC426 Public Economics
Alternative SI systems


Two main approaches to social insurance
The Bismarck tradition




The Beveridge tradition



full coverage
fortunate compensate the unfortunate
Elements of both in most countries’ SI



individuals insure themselves
no explicit redistribution element
you get what you pay for
more equal societies have more Bismarckian
schemes? (Conde-Ruiz and Profeta 2007 )
B-index is correlation (wage, pension)
Compare these within a unified model?

Casamatta et al. (2000)
Country B-index
Austria 0.527
Belgium 0.435
Denmark 0.490
France 0.652
Germany 0.555
Greece 0.730
Ireland 0.491
Italy
0.557
Spain
0.710
UK
0.268
US
0.208
Gini
23.1
25.0
24.7
32.7
28.3
32.7
35.9
27.3
32.5
36.1
40.8
Frank Cowell: EC426 Public Economics
Casamatta et al. model (1)

Type: the mix between B & B



Scale: how much insurance



Logically prior?
Model this as being decided first
Decided after the type is determined
Constrained by the tax etc. required to pay for it
Model SI as a two-stage game


Stages corresponding to two features of SI
1 Constitutional stage



2 Voting stage



Take B&B as two polar cases
Decide on what mix of the two is appropriate
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



Two possible states for each person



U(ci, bi) = u(ci) +u(bi) where u is concave
Total benefits bi = bip + bis



Employed – get wi with probability ½
Unemployed – get bi with probability ½
Preferences


Exogenous income levels: w1, w2, w3
Equal numbers, so average income: `w := [w1 + w2 + w3 ] /3
(private) bip = rpqiwi
(social) bis = t[(1 −a)`w + awi]
Rates of return to private and social insurance:


rp determined by the market; assume rp < 1
ris = [(1 −a)`w / wi + a]; ris depends on wi; average = 1
Frank Cowell: EC426 Public Economics
The second stage: t

Consumption and benefits for type i:



ci = wi [1 −qi − t]
bi = wi [ris t + rpqi ]
Linearity in qi 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 qi = 0, t > 0
person with w would be indifferent if rates of return equal
requires a = `a(w) := [rpw −`w ] / [w −`w]

Choose optimal scale given a:

Implies t*(w,1) = ½ and
Frank Cowell: EC426 Public Economics
The first stage: a


ui(a), utility received by person i, given a:

case 1, no private insurance

case 2, there is private insurance
In either case optimal a requires


either Rawlsian objective: max u1(a)
or Utilitarian objective: max Si ui(a)
Frank Cowell: EC426 Public Economics
Which type of scheme?


Results depend on SWF and the distribution of w
No private insurance



Low w2 : only Beveridge (a = 0) under Rawls; t* > 0
High w2 : even under Rawls may need a > 0 to get t* >0 in
second stage
Private insurance


Low w2 : only Beveridge (a = 0) under Rawls; t* > 0
High w2 : under Rawls need critical value of a = `a 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
Private insurance undermines support for SI
Private insurance may increase welfare of the poor
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,
741-803
Casamatta, G. and Cremer, H. and Pestieau, P. (2000) “Political
Sustainability and the Design of Social Insurance”, Journal of
Public Economics, 75, 341-364.
Conde-Ruiz, J. I. and Profeta, P. (2007) “The Redistributive
Design of Social Security Systems,” Economic Journal, 117,
686-712
Diamond, P. A. (2003) Taxation, Incomplete Markets and Social
Security, MIT Press.