Transcript Extensive Form - London School of Economics

Frank Cowell: UB Public Economics
June 2005
Welfare Analysis of Distribution
Public Economics: University of Barcelona
Frank Cowell
http://darp.lse.ac.uk/ub
Frank Cowell:
The role of public economics
UB Public Economics
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What is the motivation for our subject?
What is the reason for intervention by public sector
in private economic activity?
This is a main purpose of this lecture
We will:
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Examine the rationale of the public sector
Analyse alternative philosophical bases for intervention
Develop a simple model of welfare
First: how to characterise the role of the public
sector?
Frank Cowell:
Economic rôle of government...?
UB Public Economics
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Regulator and enforcer
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Spender
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Public goods
Public provision of private goods
Revenue raiser
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Enforcement of property rights
Prices, quantity, quality standards
Taxes, user charges...
Redistributor
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Taxes, and spending...
A brief
agenda....
Frank Cowell:
Agenda
UB Public Economics
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Previous classification is ad hoc.
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We seek a reasoned basis for the rôle of public
sector.
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Use the standard microeconomic model as context.
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Find the rôle for the public sector in this context.
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Examine “Equity-efficiency trade-off”.
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Incorporate social values.
Frank Cowell:
Overview...
Welfare Analysis of
Public Economics
UB Public Economics
A model of
intervention
Roots in basic
microeconomics
Income,
welfare, utility
The basis for
redistribution
Risk and welfare
Frank Cowell:
Finding room for public
economics
UB Public Economics
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We want to ground the public sector within
conventional economics.
The public sector should not be seen as a
kind of alien invader.
It should follow naturally from the model of
the economic system.
In effect we “find room” for the public
sector within microeconomics .
We begin with a standard paradigm.
Frank Cowell:
A simple model of the economy
UB Public Economics
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The basics:
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Private ownership:
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A collection of persons
A collection of resources
A collection of firms
A complete
description of the
economy?
Determines incomes
in market allocation
Entitlement to the resources
Shares in the firms
A market allocation:
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Consumption basket for each person
Output/input programme for each firm
A set of prices
Competitive if everyone is maximising
A complete
description of a
social state?
Frank Cowell:
Market economy: operation
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Assumptions:
UB Public Economics
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Implications:
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Given property distribution
informed optimisation
Free contracting
Known prices
Incomes are automatically generated
Equilibrium (CE) under fairly general conditions
Equilibrium system: a fundamental mapping
property
distribution
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goods
allocation
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individual
welfare
Is this a means of steering the economy?
Frank Cowell:
Market economy: basic results
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UB Public Economics
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Using the mapping seems a powerful argument.
It is strengthened by appeal to welfare theorems:
1.
2.
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Implications:
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Any CE is Pareto Efficient (PE)
Any PE allocation can be “supported” by a CE
Decide on the type of efficient outcome you want.
Use political system to get resource distribution right
Use the competitive system as a delivery vehicle
But could there be trouble in this competitive
paradise?
Frank Cowell:
Problems with the market ?
UB Public Economics
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Why might the delivery system not work?
Classic issues in market failure:
 externalities
 public goods
 non-existence of equilibrium
Informational problems in redistribution
 unobservable resources
 uncertainty about prices
Opens up natural discussion of role for public sector
Frank Cowell:
Rôle for government?
UB Public Economics
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Facilitate the economic system
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Correct “market failure”
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Externalities
Public goods
Information problems
Change the resource distribution
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Enforce property rights
But may not be possible without excessive cost
Change the relationship between resources and
allocations
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A policy trade-off…?
Frank Cowell:
Policy options
UB Public Economics
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Often depicted as a trade-off.
But what kind of trade-off?
Is a trade-off actually necessary?
And how to make the choice from the trade-off
options?
 A classic trade-off
 Social values
 An optimum?
UB Public Economics
efficiency
Frank Cowell:
An standard approach?
Need to define
terms...
What is
“efficiency”?
What is “equity”?
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equity
Frank Cowell:
Efficiency-equity trade-off
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Is there necessarily a trade-off?
UB Public Economics
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What is efficiency?
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PE provides a criterion for the goal of efficiency itself.
Pareto criterion gives no guidance away from efficient point.
Standard approach to efficiency gains and losses:
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Not if we can redistribute resources without transactions cost.
A criterion for Public Economics applications such as tax design.
What is equity?
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Raises issues of definition.
Also of the case for egalitarianism (Putterman et al. - JEL98).
Frank Cowell:
Components of the policy
problem
UB Public Economics
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Specification of the technology
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A definition of equity
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Also related concepts such as inequality
See later lectures
An analysis of the nature of the trade-off
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Production of private and public goods
Enables precise definition of efficiency
Informational problems
See lecture on design issues
A statement of social preferences
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What is the basis for concern with distribution?
We deal with this in the current lecture
Frank Cowell:
Welfare approaches
UB Public Economics
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Ordinal approaches to welfare
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Welfarism
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These are of little use
Run into the Arrow (1953) problem
Hence are hopelessly indecisive
Uses a cardinally measurable and interpersonally
comparable approach to welfare.
Usually based on individualism
Provides the basis for a coherent model
Need to examine the basic building blocks…
Frank Cowell:
Overview...
Welfare Analysis of
Public Economics
UB Public Economics
A model of
intervention
The basic units
of analysis
Income,
welfare, utility
The basis for
redistribution
Risk and welfare
Frank Cowell:
Ingredients of an approach
UB Public Economics
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A model of individual resources
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A measure of individual welfare
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A basis for interpersonal comparisons
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An intellectual base for state intervention
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We will deal with the first three of these now.
Frank Cowell:
Individual resources and
distribution
UB Public Economics
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We adopt two simple paradigms concerning
Fixed total
resources:
income
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The cake-sharing problem
The general case with production
Incorporates
incentive effects
Irene and
Janet
Often distributional analysis can be conducted
in terms of typical individuals i and j. The F-form
approach
In some cases one needs a more general
distributional notation
 Two persons
 The feasible set
 The interesting distributions
UB Public Economics
 The basic cake-sharing
income-distribution
problem
Janet’s income
Frank Cowell:
A simple model for the
distributional problem
Income distributions
with given total
45°
0
Irene’s income
Frank Cowell:
Limitations of this basic model
UB Public Economics
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Just 2 persons
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Fixed-size cake
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Economic growth?
Waste through distortion?
Essential to first-best
welfare economics
Costlessly transferable incomes
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n  3 persons for the inequality problem
The “leaky bucket” problem
Analysed further in discussion of incentives
Incomes or utilities?
Example 1
Frank Cowell:
Example 2
For welfare purposes we are
concerned with utility...
UB Public Economics
Comparability without measurability :
Imagine a world where access to public
services determines utility and the
following ordering is recognised:
•Gas+Electricity
•Electricity only
•Gas only
•Neither
It makes no sense to say “U(G+E)
=2U(E)”, but you could still compare
individuals.
Measurability without comparability:
Imagine a world where utility is
proportional to income, but the constant of
proportionality is known to depend on
family characteristics which may be
unobservable.
Double a family’s income and you double
each member’s utility; but you cannot
compare utilities of persons from different
families.
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What is the relationship of utility to
income?
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What properties does utility have?
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Is it measurable?
Is it comparable?
These properties are independent
We usually need both
We need a
simple model
of utility....
Frank Cowell:
Ingredients
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a: personal attributes
UB Public Economics
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y: income
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Could be exogenous
Or you can model as a function of attributes: y=y(a)
u: individual utility
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Identity
Needs
Abilities
Special “merit” or “desert”
Several ways of modelling this…
…see below
x: “equivalised” income
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Dollar/Pound/Euro units…
Can be treated as a version of “utility”
Frank Cowell:
Ingredients (2)
UB Public Economics
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F : distribution function
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U : utility function
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Standard tool borrowed from statistics
A variety of specifications – see below
Gives indicator of how “well-off” a person of given
attributes is
c : equivalisation function
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A simple way of accounting for differences in needs
Perhaps too simple?
We will try something different in the next lecture
Frank Cowell:
Basic questions about income
UB Public Economics
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Is it unique?
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How comprehensive should it be?
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What is the relevant receiving unit?
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Is it comparable between persons?
Frank Cowell:
Income: Uniqueness?
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Should we use univariate or multivariate analysis?
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UB Public Economics
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A relationship between different types of “income”?
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income and expenditure?
income and wealth?
income over time?
covariance of earnings and asset income?
conditional transfers?
Several definitions may be relevant?
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gross income?
disposable income?
other concepts?
Frank Cowell:
Income: comprehensiveness?
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Is income “full income”?
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UB Public Economics
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Is income a proxy for economic welfare?
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discount for risk?
valuation over time?..
Can income be zero?
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final income +
value of leisure +...?
rental income?
... or less than zero?
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business losses?
Frank Cowell:
Income: Comparability?
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Price adjustment
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UB Public Economics
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Adjustment for needs and household size
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Normalise by price indices
Usual approach is to introduce equivalence scales
The equivalence transformation is
x = c ( y, a )
Equivalised
income
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personal attributes
nominal income
Usually a simplifying assumption is made.
Write transformation as an income-independent
Number of
equivalence scale:
equivalent adults
x = y / n (a)
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Where does the function c come from?
Frank Cowell:
Equivalence Scales
UB Public Economics
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We will assume that there is an agreed
method of determining equivalence scales.
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But there is a variety of possible sources of
information for equivalence scales:
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From official government sources
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From international bodies such as OECD
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From econometric models of household budgets.
Consider an example of the last of these:
 Plot share of food in
budget against household
income
A reference household type...
sfood
 Engel Equivalence Scale
UB Public Economics
proxy for “need”
Frank Cowell:
A model of income and need
childless
couple
couple with
children
xr  yr
From budget
studies
x, y
0
xi
yi
income
Frank Cowell:
Alternative models of utility
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u = U (y)
UB Public Economics
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u = U (y; a)
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Inter-personally comparable utility
Individualistic utility
May not be comparable, depending on information about a.
u = U (y, F)
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Concern for distribution as a kind of externality
Need not be benevolent concern
Evidence that people are
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Concerned about relative incomes
“upward looking” in their comparisons.
Ferrar-i-Carbonell (2005)
x = c(y ; a) = y / n(a)
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A comparable money-metric utility?
Frank Cowell:
The relationship between utility
and income:
UB Public Economics
u
Increase
concavity
u = U(y)
^
u = U(y)
y
Frank Cowell:
A simple model
UB Public Economics
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As an example take the iso-elastic form:
y1 – d – 1
U(y) = ———— , d  
1 –d
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We can think of d as risk aversion
But it may take on an additional welfare significance
Frank Cowell:
What to do with this information?
UB Public Economics
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We need a method of appraising either the
distribution of utilities…
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…or, the system by which they were
produced
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This involves fundamentally different
approaches to welfare judgments.
Frank Cowell:
Overview...
Welfare Analysis of
Public Economics
UB Public Economics
A model of
intervention
Philosophies,
social welfare
and the basis for
intervention
Income,
welfare, utility
The basis for
redistribution
Risk and welfare
Frank Cowell:
Five intellectual bases for public
action
UB Public Economics
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…and five social philosophers
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Entitlement theories
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Unanimity
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Bentham
Concern with the least advantaged
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Pareto
Utilitarianism
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Nozick
Rawls
Egalitarianism
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Plato
 Standard cake-sharing model
 N stands for “Nozick”
UB Public Economics
Janet’s income
Frank Cowell:
A distributional outcome
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N
implications for
utility
possibilities
45°
0
Irene’s income
Frank Cowell:
Utility-possibility set
 Plot utility on the axes
 Simple cake-sharing
uj
 The effect of utility
interdependence
UB Public Economics
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N
N
Assuming that U is strictly
concave...
…and that U is the same
function for both Irene
and Janet.
45°
0
ui
Frank Cowell:
Should we move from N?
UB Public Economics
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What is the case for shifting from the status-quo
point?
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Answer differs dramatically according to social
philosophy:
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Entitlement approach is concerned with process
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Other approaches concerned with end-states
Frank Cowell:
Entitlement approach
UB Public Economics
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Focus on Nozick (Anarchy, State and Utopia, 1974).
Answer depends crucially on how N came about
Distinguish three key issues:
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fairness in original acquisition
fair transfers
rectification of past injustice
Presumption is that there will be little or no role
for the State
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“Night watchman”
Frank Cowell:
Pareto Criterion
UB Public Economics
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Pareto unanimity criterion is an end-state principle
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Individualistic
Based on utilities
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Approve the move from N to another point…
…if at least one person gains
…and no-one loses
But utility may have a complicated relationship with
income
May depend on the income of others
See how Pareto applies in the simple example
Frank Cowell:
Pareto improvement: simple case
 The utility-possibility
set again
 The initial point
uj
 Pareto superior points
UB Public Economics
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N
No case for
intervention?
0
45°
ui
Frank Cowell:
End-state approaches: beyond Pareto
UB Public Economics
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Pareto criterion can be indecisive
Alternative end state approaches use a social
welfare function
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What principles should this embody?
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Typically get unique solution
Individualism?
The Pareto principle?
Additivity?
Take a simple example that combines them all...
Frank Cowell:
Benthamite approach
UB Public Economics
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General principle is “Seek the greatest good of the
greatest number”
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This is typically interpreted as maximising the
sum of individual welfare.
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In Irene-Janet terms: u1 + u2 + ...+ un
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More generally the SWF is:
WB =  u dF(u)
Frank Cowell:
Distributional implications of
utilitarianism
UB Public Economics
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Much of public economics uses utilitarianism.
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But does utilitarianism provide a basis for
egalitarian transfers?
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Efficiency criteria
Sacrifice theories in taxation
Sen has argued that this is a common fallacy
Sen and Foster (1997)
Again look at this within the simple model
Frank Cowell:
Benthamite redistribution?
 Take a symmetric utilitypossibility set
uj
 The initial distribution
UB Public Economics
 Benthamite welfare
contour
 Maximise welfare
 Optimum in this case
 Implied tax/transfer
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N
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B
ui+uj = constant
0
45°
ui
The general
case?
Frank Cowell:
The general case...
uj
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N
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C
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 Incorporates differential
incentive effects etc.
 N. The status quo
B
 Pareto improvements
UB Public Economics
 Points that Paretodominate N
 C The voluntary solution?
 Anywhere above C
might be a candidate
 B. Benthamite solution
ui
0
Paretianism leads to
multiple solutions
Benthamite
utilitarianism leads to
a unique, possibly
different, solution.
Frank Cowell:
General case: discussion
UB Public Economics
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A motive for changing distribution?
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Nozickians might still insist that no move from N is justified
unless it came through private voluntary action
Applies even to C
Implementation:
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Private voluntary action might not be able to implement C
Could rise if there were many individuals
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Case for egalitarianism?
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Clearly Bentham approach does not usually imply egalitarian
outcome.
Consider two further alternative approaches:
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Concern for the least advantaged (Rawls)
Egalitarianism
Frank Cowell:
Rawls (1971)
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UB Public Economics
Rawls’ distributional philosophy is based on two
fundamental principles:
1.
2.
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Economic focus has usually been on 2
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each person has equal right to the most extensive scheme of
equal basic liberties compatible with a similar scheme of
liberties for all
society should so order its decisions as to secure the best
outcome for the least advantaged
Argument based on reasoning behind a “veil of ignorance”
I do not know which position in society I have when making
social judgment
Needs careful interpretation
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Avoid confusion with a probabilistic approach we consider
later
Frank Cowell:
The Rawls approach…?
UB Public Economics
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What is meant by the difference principle?
This is typically interpreted as maximising the welfare of
the worst-off person.
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Based on simplistic interpretation of veil of ignorance argument
Rawls interpreted it differently
But rather vaguely
In Irene-Janet terms: min {u1 , u2 , ..., un}
So the suggested SWF is:
WR = {min u: F(u)>0}
Frank Cowell:
Egalitarianism?
UB Public Economics
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Origin goes back to Plato…
…but reinterpreted by Meade (1974).
 “Superegalitarianism”
Welfare is perceived in terms of pairwise differences:
[ui - uj]...
Welfare might not be expressible as a neat additive
expression involving individual utilities.
 Finds an echo in more recent welfare developments
 Covered in a later lecture
Frank Cowell:
General case (2)
uj

N
 A 'Rawlsian' solution
 Superegalitarianism
UB Public Economics
Contours of max
min function
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R
Maxi-min does not
imply equality
E
Superegalitaranism
implies equality
ui
0
Frank Cowell:
Bergson-Samuelson approach
UB Public Economics
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But why an additive form of the SWF?

We could just use a weaker individualistic form.

This is the basis of the Bergson-Samuelson
formulation

A generalisation

Subsumes several welfare concepts

In Irene-Janet terms: W(u1 , u2 , ..., un)

More generally the SWF is: WBS = W(F)
Frank Cowell:
General individualistic welfare
UB Public Economics
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The specific welfare functions are special cases of
Bergson-Samuelson.

Most satisfy the principle of additivity

Except for the last one (utility differences)

In Irene-Janet terms this means we can write:
u(u1) + u(u2) + ... + u(un)

More generally the SWF is:
WBSa =  u(u) dF(u)
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This is clear for Bentham where u(u)= u. But…
Frank Cowell:
General individualistic welfare (2)
UB Public Economics
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…we can say more

Again take the iso-elastic form, this time of the
(social) u-function:

u 1–e – 1
u(u) = ————, e  
1–e
Bentham corresponds to the case e = .

Max-min (“Rawls”) corresponds to the case e=.

Intermediate cases (0<e<) are interesting too
Frank Cowell:
General case (closeup)
 B. Benthamite (e = 0)
 W. Intermediate (e = 1)
UB Public Economics
 R. 'Rawlsian' ( e =)
B
 ‘E. Superegalitarianism'
(no e value)


W


E
R
Frank Cowell:
A brief summary
UB Public Economics
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Entitlement theories

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Unanimity


A basis for egalitarianism?
Concern with the least advantaged


Blairism?
Utilitarianism


Thatcherism?
How to be interpreted?
(Super)-egalitarianism


Out of fashion in UK.
In Spain...?
Frank Cowell:
Overview...
Welfare Analysis of
Public Economics
UB Public Economics
A model of
intervention
A reinvention of
utilitarianism?
Income,
welfare, utility
The basis for
redistribution
Risk and welfare
Frank Cowell:
But where do the values in the SWF
come from...?
UB Public Economics

Consensus


High-minded idealism


Social and private values...?
The PLUM principle

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Runs into the “Arrow Theorem...”
“People Like Us Matter” – a cynical approach
The Harsanyi approach

Based on individual rationality under uncertainty
take another
look...
Frank Cowell:
High-minded idealism?

UB Public Economics

Do people care about inequality or other distributional
issues?
Multiple values argument

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Externality argument



Suppose that people are “schizophrenic”
They have two sets of values, private and public.
People treat the income distribution as a “public good”
Hochman and Rodgers (AER 1969)
Motivates the formulation u = U (y, F)

Individuals care about the income distribution F
Frank Cowell:
The PLUM principle
UB Public Economics

Interest groups may determine what the SWF is




Champernowne and Cowell (1998)
No reason to suppose that it has a direct
connection with individual utilities
However we may still be able to say something
about how values are/should be determined
For example they should at least be consistent
Frank Cowell:
An approach based on risk analysis
UB Public Economics

Social welfare is based individual utility



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Each citizen ranks social states on the basis of expected
utility
These utilities concern life prospects
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Utility is of a representative person
Harsanyi (Journal of Political Economy 1953, 1955)
made behind a “veil of ignorance” similar to Rawls
Ignorance concerns income, wealth, social position etc
But what of personal values?
We need to reconsider and reinterpret the sum-of-utilities
approach.
Frank Cowell:
Reinterpret sum-of-utilities
UB Public Economics
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The Irene-Janet version: u1 + u2 + ...+ un
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This is equivalent to:
(1/n)u1 + (1/n)u2
+ ...+ (1/n)un
 Reinterpreted as:
p1u1 + p2u2 + ...+ pnun , where pi := 1/n
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Which is simply E ui
Frank Cowell:
Reinterpret sum-of-utilities (2)
UB Public Economics
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The formal utility function:  u dF(u)
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This is equivalent to:  U(y) f(y)dy
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Reinterpreted as: U(y(a)) p(a) da
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Which is simply E U(y(a))
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How do we reach this conclusion…?
Frank Cowell:
Welfare and Risk?
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Expect links between welfare and risk analysis
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UB Public Economics
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Argument by analogy
Atkinson (JET 1970) on inequality
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The Harsanyi paradigm (J.Pol.E. 1953, 1955)
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Harsanyi’s contribution is fundamental
Consists of two strands.
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See Amiel et al (2005)
Frank Cowell:
Harsanyi 1
UB Public Economics
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Aggregation theorem
Consider preferences over set of lotteries L
Individuals’ preferences Vi satisfy EU axioms
i=1,…,n
Social preference V satisfies EU axioms
Assume Pareto indifference is satisfied
Then there are numbers ai and b such that, for all
pL
Frank Cowell:
Harsanyi 1 (contd)
UB Public Economics
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Powerful result
Does not assume interpersonal utility comparisons.
If such comparisons ruled out, the ai are based on
the evaluator’s value judgments (Harsanyi 1978, p. 227)
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personal?
arbitrary?
the evaluator?
“Judges and other public officials” (1978, p. 226)
Need not be a member of the society
Must satisfy some consistency requirements
Frank Cowell:
Harsanyi 2
UB Public Economics
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Impartial observer theorem.
Basic idea already in Vickrey (1945).
Assumes interpersonal comparisons of utility.
An impartial observer sympathetic to the interests of
each member of society makes value judgments.
The observer is to imagine himself being person i.
 i’s objective circumstances
 i’s preferences
Frank Cowell:
Harsanyi 2 (contd)
UB Public Economics
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How to get a representative person?
Thought experiment
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Evaluator imagines he has an equal chance of being any
person in society
Equal consideration to each person’s interests.
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Impartial observer calculates average expected
utility of each lottery in L:
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I.e. person j’s expected utility
Frank Cowell:
Implications of Harsanyi approach
UB Public Economics
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The aggregation theorem gives an argument for
additivity
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The “representative person” induces a
probabilistic approach
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Then social welfare is found to be inherited from
individual expected utility
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But on what basis do we get the probabilities
here?
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And is “expectations” an appropriate basis for
social choice?
Frank Cowell:
Harsanyi: Some difficulties
UB Public Economics
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Are preferences known behind the “Veil of ignorance”?
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Not in the Rawls approach
But Harsanyi assumes that representative person knows others
utilities
Is it useful to suppose equal ignorance?
Subjective probabilities may be inconsistent
Should we be concerned only with expected utility?
It is not clear that individuals view risk-choices and
distributional choices in the same way
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Cowell and Schokkaert (EER 2001).
Carlsson et al (Economica 2005)
UB Public Economics
the veil of ignorance
the cynical approach
a general view
probability
Frank Cowell:
Identity
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1 2 3
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i
n
identity
Frank Cowell:
A difficulty with expected utility?
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Suppose the outcomes depend on uncertain events
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UB Public Economics
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probabilities of Events 1,2 are (p, 1- p)
Payoffs for persons (i,j) under two policies are
Policy
a
b
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Event 2
(1,)
(,1)
Consider choice between policies a and b
Expected payoffs are:
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Event 1
(1,)
(1,)
Under a: (1,0)
Under b: (p, 1- p)
Should society be indifferent between a and b?
Mobility may be important as well as expected outcome
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See Diamond (Journal of Political Economy 1967).
Frank Cowell:
Views on redistribution
UB Public Economics
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Source: Ravallion and Lokshin (JPubE 2000)
Clearly views on distribution depend on (i) your current
position and (ii) your expectations
Frank Cowell:
Concluding remarks
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UB Public Economics
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We can construct a model with an individualistic base for
welfare comparisons.
The alternative social philosophies may give support to
redistributive arguments,
But it raises some awkward questions...
Should the social basis for redistribution rest on private
tastes for equality or aversion to misery?
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Should it rest on individual attitudes to risk?
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What if people like seeing the poor..?
What if people are not risk-averse?
We will come back to consider the implications of these
questions