Transcript No Slide Title

24 October 2011
Frank Cowell: EC426 Public Economics
MSc Public Economics
2011/12
http://darp.lse.ac.uk/ec426/
Policy Design: Income Tax
Frank A. Cowell
Frank Cowell: EC426 Public Economics
Overview...
Policy Design:
Income Tax
Design
principles
Roots in social
choice and
asymmetric
information
Simple model
Generalisations
Interpretations
Frank Cowell: EC426 Public Economics
Social values: the Arrow problem

Uses weak assumptions about preferences/values


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Uses a general notion of social preferences

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The constitution
A map from set of preference profiles to social preference
Also weak assumptions about the constitution
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Well-defined individual orderings over social states
Well-defined social ordering over social states
Universal Domain
Pareto Unanimity
Independence of Irrelevant Alternatives
Non-Dictatorship
There’s no constitution that does all four

Except in cases where there are less than three social states
Frank Cowell: EC426 Public Economics
Social choice function
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A social state: q  Q
Individual h’s evaluation of the state vh(q)

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A profile: [v1, v2, …, vh, … ]

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A given population is indexed by h = 1,2,…, nh
A “reduced-form” utility function vh().
An ordered list of utility functions
Set of all profiles: V
A social choice function G: V→Q


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For a particular profile q = G(v1, v2, …, vh, … )
Argument is a utility function not a utility level
Picks exactly one chosen element from Q
Frank Cowell: EC426 Public Economics
Implementation
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Is the SCF consistent with private economic behaviour?
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Implementation problem: find/design an appropriate mechanism
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Mechanism is a partially specified game of imperfect information…
rules of game are fixed
strategy sets are specified
preferences for the game are not yet specified
Plug preferences into the mechanism:
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Yes if the q picked out by G is also…
… the equilibrium of an appropriate economic game
Does the mechanism have an equilibrium?
Does the equilibrium correspond to the desired social state q?
If so, the social state is implementable
There is a wide range of possible mechanisms

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Example: the market as a mechanism
Given the distribution of resources and the technology…
…the market maps preferences into prices.
The prices then determine the allocation
Frank Cowell: EC426 Public Economics
Manipulation

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Consider outcomes from a “direct” mechanism in two cases:
If all, including h, tell the truth about preferences:
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If h misrepresents his preferences but others tell the truth:
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q = G(v1,…, vh, …, )
q = G(v1,…, vh, …, )
How does the person “really” feel about q and q?
If vh(q) > vh(q) then there is an incentive to misrepresent
information
If h realises this we say that G is manipulable.
Frank Cowell: EC426 Public Economics
Gibbard-Satterthwaite result

Result on SCF G can be stated in several ways
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A standard version is:
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(Gibbard 1973, Satterthwaite, 1975 )
If the set of social states Q contains at least three elements;
and G is defined for all logically possible preference profiles
and G is truthfully implementable in dominant strategies...
then G must be dictatorial
Closely related to the Arrow theorem
Has profound implications for public economics

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Misinformation may be endemic to the design problem
May only get truth-telling mechanisms in special cases
Frank Cowell: EC426 Public Economics
Overview...
Policy Design:
Income Tax
Design
principles
Preferences,
incomes, ability
and the
government
Simple model
Generalisations
Analogy with
contract theory
Interpretations
Frank Cowell: EC426 Public Economics
The design problem

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The government needs to raise revenue…
…and it may want to redistribute resources
To do this it uses the tax system
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Base it on “ability to pay”

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income rather than wealth
ability reflected in productivity
Tax authority may have limited information

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personal income tax…
…and income-based subsidies
who have the high ability to pay?
what impact on individuals’ willingness to produce output?
What’s the right way to construct the tax schedule?
Frank Cowell: EC426 Public Economics
Model elements
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A two-commodity model
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Income comes only from work
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individuals are paid according to their marginal product
workers differ according to their ability
Individuals derive utility from:
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leisure (i.e. the opposite of effort)
consumption – a basket of all other goods
similar to optimal contracts (Bolton and Dewatripont 2005)
their leisure
their disposable income (consumption)
Government / tax agency


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has to raise a fixed amount of revenue K
seeks to maximise social welfare…
…where social welfare is a function of individual utilities
Frank Cowell: EC426 Public Economics
Modelling preferences
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Individual’s preferences
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Special shape of utility function
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u = y(z) + y
u : utility level
z : effort
y : income received
y() : decreasing, strictly concave, function
quasi-linear form
zero-income effect
y(z) gives the disutility of effort in monetary units
Individual does not have to work


reservation utility level u
requires y(z) + y ≥ u
Frank Cowell: EC426 Public Economics
Ability and income
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Individuals work (give up leisure) to get consumption
Individuals differ in talent (ability) t
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Disposable income determined by tax authority

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higher ability people produce more and may thus earn more
individual of type t works an amount z
produces output q = tz
but individual does not necessarily get to keep this output?
intervention via taxes and transfers
fixes a relationship between individual’s output and income
(net) income tax on type t is implicitly given by q − y
Preferences can be expressed in terms of q, y


for type t utility is given by y(z) + y
equivalently: y(q / t) + y
A closer look
at utility
Frank Cowell: EC426 Public Economics
The utility function
 Preferences over leisure and income
 Indifference curves
y
 Reservation utility
Transform into (leisure, output) space

u = y(z) + y
yz(z) < 0

u≥u

u
u
1– zq


u = y(q/t) + y
yz(q/t) < 0
Frank Cowell: EC426 Public Economics
The single-crossing condition
Preferences over leisure and
output
y
 High talent
 Low talent
 Those with different
talent (ability) will
have different sloped
indifference curves in
this diagram
type b
type a

q

qa = taza
qb = tbzb
Frank Cowell: EC426 Public Economics
A full-information solution?
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Consider argument based on the analysis of contracts
Full information: owner can fully exploit any manager
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Same basic story here

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Pays the minimum amount necessary
“Chooses” their effort
Can impose lump-sum tax
“Chooses” agents’ effort — no distortion
But the full-information solution may be unattractive

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Informational requirements are demanding
Perhaps violation of individuals’ privacy?
So look at second-best case…
Frank Cowell: EC426 Public Economics
Two types
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Start with the case closest to the optimal contract model
Exactly two skill types

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From contract design we can write down the outcome

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ta > tb
proportion of a-types is p
values of ta , tb and p are common knowledge
essentially all we need to do is rework notation
But let us examine the model in detail:
Frank Cowell: EC426 Public Economics
Second-best: two types
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The government’s budget constraint
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Participation constraint for the b type:
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yb + y(zb) ≥ ub
have to offer at least as much as available elsewhere
Incentive-compatibility constraint for the a type:
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p[qa - ya] + [1-p][qb - yb ] ≥ K
where qh - yh is the amount raised in tax from agent h
ya + y(qa/ta) ≥ yb + y(qb/ta)
must be no worse off than if it behaved like a b-type
implies (qb, yb) < (qa, ya)
The government seeks to maximise standard SWF


p z(y(za) + ya) + [1-p] z(y(zb) + yb)
where z is increasing and concave
Frank Cowell: EC426 Public Economics
Two types: model
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We can use a standard Lagrangean approach
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Constraints are:
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government chooses (q, y) pairs for each type
…subject to three constraints
government budget constraint
participation constraint (for b-types)
incentive-compatibility constraint (for a-types)
Choose qa, qb, ya, yb to max
p z(y(qa/ta) + ya) + [1-p] z(y(qb/tb) + yb)
+ k [p[qa - ya] + [1-p][qb - yb ] - K]
+ l [yb + y(qb/tb) - ub]
+ m [ya + y(qa/ta) - yb - y(qb/ta)]
where k, l, m are Lagrange multipliers for the constraints
Frank Cowell: EC426 Public Economics
Two types: solution
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From first-order conditions we get:
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Also, all the Lagrange multipliers are positive
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so the associated constraints are binding
follows from standard adverse selection model
Results are as for optimum-contracts model:
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- yz(qa/ta) = ta
- yz(qb/tb) = tb + kp/[1-p],
where k := yz(qb/tb) - [tb/ta] yz(qb/ta) < 0
MRSa = MRTa
MRSb < MRTb
Interpretation



no distortion at the top (for type ta)
no surplus at the bottom (for type tb)
determine the “menu” of (q,y)-choices offered by tax agency
Frank Cowell: EC426 Public Economics
Two ability types: tax design
a type’s reservation utility
y
b type’s reservation utility
b type’s (q,y)
incentive-compatibility constraint
a type’s (q,y)
menu of (q,y) offered by tax
authority
ya
 Analysis determines (q,y)
combinations at two points
yb
If a tax schedule T(∙) is to
be designed where
y = q −T(q) …
q
qb
qa
…then it must be consistent
with these two points
Frank Cowell: EC426 Public Economics
Overview...
Policy Design:
Income Tax
Design
principles
Moving beyond
the two-ability
model
Simple model
Generalisations
Interpretations
Frank Cowell: EC426 Public Economics
A small generalisation

With three types problem becomes a bit more
interesting

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We now have one more constraint to worry about
1.
2.
3.

Similar structure to previous case
ta > tb > tc
proportions of each type in the population are pa, pb, pc
Participation constraint for c type: yc + y(qc/tc) ≥ uc
IC constraint for b type: yb + y(qb/tb) ≥ yc + y(qc/tb)
IC constraint for a type: ya + y(qa/ta) ≥ yb + y(qb/ta)
But this is enough to complete the model specification


the two IC constraints also imply ya + y(qa/ta) ≥ yc + y(qc/tb)
so no-one has incentive to misrepresent as lower ability
Frank Cowell: EC426 Public Economics
Three types

Methodology is same as two-ability model

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Outcome essentially as before :

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MRSa = MRTa
MRSb < MRTb
MRSc < MRTc
Again, no distortion at the top and the participation constraint
binding at the bottom



set up Lagrangean
Lagrange multipliers for budget constraint, participation constraint and
two IC constraints
maximise with respect to (qa,ya), (qb,yb), (qc,yc)
determines (q,y)-combinations at exactly three points
tax schedule must be consistent with these points
A stepping stone to a much more interesting model…
Frank Cowell: EC426 Public Economics
A richer model: N+1 types


The multi-type case follows immediately from the
three-type case
Take N + l types

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t0 < t1 < t2 < … < tN
(note the required change in notation)
proportion of type j is pj
this distribution is common knowledge
Budget constraint and SWF are now



Sj pj [qj - yj] ≥ K
Sj pj z(y(zj) + yj)
where sum is from 0 to N
Frank Cowell: EC426 Public Economics
N+1 types: behavioural constraints

Participation constraint

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Incentive-compatibility constraint

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is relevant for lowest type j = 0
form is as before:
y0 + y(z0) ≥ u0
applies where j > 0
j must be no worse off than if it behaved as the type below (j-1)
yj + y(qj/tj) ≥ yj-1 + y(qj-1 /tj).
implies (qj-1, yj-1) < (qj, yj)
and u(tj) ≥ u(tj-1)
From previous cases we know the methodology

(and can probably guess the outcome)
Frank Cowell: EC426 Public Economics
N+1 types: solution


Lagrangean is only slightly modified from before
Choose {(qj, yj )} to max
Sj=0 pj z (y(qj / tj) + yj)
+ k [Sj pj [qj - yj] - K]
+ l [y0 + y(z0) - u0]
+ Sj=1 mj [yj + y(qj/tj) - yj-1 - y(qj-1 /tj)]
where there are now N incentive-compatibility Lagrange multipliers

And we get the result, as before
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MRSN = MRTN
MRSN−1 < MRTN−1
…
MRS1 < MRT1
MRS0 < MRT0
Now the tax schedule is determined at N+1 points
Frank Cowell: EC426 Public Economics
A continuum of types

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One more step is required in generalisation
Tax agency is faced with a continuum of taxpayers

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This can be reasoned from the case with N + 1 types

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allow N  
From previous cases we know



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common assumption
allows for general specification of ability distribution
form of the participation constraint
form that IC constraint must take
an outline of the outcome
Can proceed by analogy with previous analysis…
Frank Cowell: EC426 Public Economics
The continuum model

Continuous ability
bounded support [t,`t ]
density f(t)



Utility for talent t as before
u(t) = y(t) + y( q(t) / t)

Participation constraint is
u(t) ≥ u

Incentive compatibility requires
du(t) /dt ≥ 0

SWF is
`t
∫
z (u(t)) f(t) dt
t
Frank Cowell: EC426 Public Economics
Output and disposable income
under the optimal tax
y
t_
Lowest type’s indifference curve
Lowest type’s output and income
Intermediate type’s indifference
curve, output and income
_
t
Highest type’s indifference curve
45°
Highest type’s output and income
Menu offered by tax authority
q
_
q
_
q
Frank Cowell: EC426 Public Economics
Continuum model: results

Incentive compatibility implies dy /dq > 0

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No distortion at top implies dy /dq = 1

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
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zero optimal marginal tax rate! (Seade 1977)
but does not generalise to incomes close to top (Tuomala 1984)
does not hold if there is no “topmost income” (Diamond 1998 )
May be 0 on the lowest income


optimal marginal tax rate < 100% (Mirrlees 1971)
depends on distribution of ability there (Ebert 1992)
Explicit form for the optimal income tax requires




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specification of distribution f(∙)
specification of individual preferences y(∙)
specification of social preferences z (∙)
specification of required revenue K
(Saez 2001, Brewer et al. 2010, Mankiw 2009)
Frank Cowell: EC426 Public Economics
Overview...
Design: Taxation
Design basics
Apply design
rules to practical
policy….
Plus a “cutdown” version of
the OIT problem
Simple model
Generalisations
Interpretations
Frank Cowell: EC426 Public Economics
Application of design principles

The second-best method provides some pointers

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Simple schemes may be worth considering

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roughly correspond to actual practice
illustrate good/bad design
Consider affine (linear) tax system

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

but is not a prescriptive formula
explicit form of OIT usually not possible (Salanié 2003)
model is necessarily over-simplified
exact second-best formula might be administratively complex
benefit B payable to all (guaranteed minimum income)
all gross income (output) taxable at the same marginal rate t…
…constant marginal retention rate: dy /dq = 1 - t
Effectively a negative income tax scheme:


(net) income related to output thus: y = B + [1 - t] q
so y > q if q < B / t … and vice versa
Frank Cowell: EC426 Public Economics
A simple tax-benefit system
Guaranteed minimum income B
y
Constant marginal retention rate
Implied attainable set
Low-income type’s indiff curve
Low-income type’s output, income
1-t
High-income type’s indiff curve
Highest type’s output and income
 “Linear” income tax
system ensures that
incentive-compatibility
constraint is satisfied
Analysed by
Sheshinski (1972)
B
q
Frank Cowell: EC426 Public Economics
Violations of design principles?


The IC condition be violated in actual design
This can happen by accident:

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Commonly known as



interaction between income support and income tax.
generated by the desire to “target” support more effectively
a well-meant inefficiency?
the “notch problem” (US)
the “poverty trap” (UK)
Simple example



suppose some of the benefit is intended for lowest types only
an amount B0 is withdrawn after a given output level
relationship between y and q no longer continuous and
monotonic
Frank Cowell: EC426 Public Economics
A badly designed tax-benefit system
Menu offered to low income groups
y
Withdrawal of benefit B0
Implied attainable set
Low-income type’s indiff curve
Low type’s output and income
High-income type’s indiff curve
ya
High type’s intended output and
income
High type’s utility-maximising
choice
yb
 The notch violates IC…
B0
…causes a-types to
masquerade as b-types
q
qb
qa
Frank Cowell: EC426 Public Economics
Neglected design issues?



Administrative complexity
Example 1. UK today (Mirrlees et al 2011)
Example 2. Germany 1981-1985:


linearly increasing marginal tax rate
quadratic tax and disposable income schedules
rates
for single person (§32a Einkommensteuergesetz);
units DM:
income x up to 4,212: T = 0
4,213 to 18,000: T = 0.22x – 926
4
3
2
18,001 to 59,999: T = 3.05 X – 73.76 X + 695 X +
2,200 X + 3,034
 where X = x/10,000 – 18,000;
60,000 to 129,999: T = 0.09X4 – 5.45X3 + 88.13 X2 +
5,040 X + 20,018
 where X = x/10,000 – 60,000;
from 130,000:
T = 0.56 x – 14,837
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
0
20000
40000
60000
80000
100000
120000
140000
Frank Cowell: EC426 Public Economics
Arguments for “linear” model


Relatively easy to interpret parameters
Pragmatic:




Approximates several countries’ tax systems
Example – piecewise linear tax in UK
Sidesteps the incentive compatibility constraint
Simplified version is more tractable analytically




Not choosing a general tax/disposable income schedule
Given t, B and the government budget constraint…
…in effect we have a single-parameter problem
See Kaplow (2008), pp 58-63
Frank Cowell: EC426 Public Economics
Linear model: Lagrangean


Social welfare is a function of individual utility
Individual utility is maximised subject to budget constraint



Optimisation problem: choose B and t to max social welfare




the covariance of social marginal valuation and income
the compensated labour-supply elasticity
If K = 0 then B > 0
No explicit general formula?



Subject to government budget constraint
From maximised Lagrangean get a messy result involving


Determined by individual ability
Tax parameters B and t
FOC cannot be solved to give t
covariance and elasticities will themselves be functions of t
And in some cases you get a clear-cut result…
Frank Cowell: EC426 Public Economics
John Broome’s revelation


Broome (1975) suggested a great simplification.
Optimal income tax rate should be 58.6% !!




The basis for this astounding claim?
Tax rate is in fact 2 – 2; follows from a simple model
Rather it is a useful lesson in applied modelling
He makes conventional assumptions




no-one has ability less than 0.707 times the average
Cobb-Douglas preferences:
“Rawlsian” max-min social welfare
Balanced budget: pure redistribution
Frank Cowell: EC426 Public Economics
A simulation model

Stern’s (1976) model of linear OIT



Lognormal ability


…more on this below
Isoelastic individual utility


can be taken as a generalisation of Broome
simulation uses standard ingredients:
elasticity of substitution s
Isoelastic social welfare

W =  z (u) dF(u)
u 1–e – 1
z(u) = ———— , e  0
1–e
 inequality aversion e

Variety of assumptions about government budget constraint
Frank Cowell: EC426 Public Economics
Lognormal ability

f(w)
0
0
Two parameter distribution
L(w; m, s2 )

—L(w; 0, 0.25 )
…L(w; 0, 1.0 )



Approximation to empirical
distributions

1
2
3
4
w
m is log of the median
s2 is the variance of log
income
support is [0, )


Particularly manual workers
Stern took s = 0.39 (same as
Mirrlees 1971)
In this case less than 2% of the
population have less than
0.707 × mean (Broome 1975 )
Frank Cowell: EC426 Public Economics
Stern's Optimal Tax Rates
s
e=0
e=1
e=
0.2
0.4
0.6
0.8
1.0
36.2
22.3
17.0
14.1
12.7
62.7
47.7
38.9
33.1
29.1
92.6
83.9
75.6
68.2
62.1
• Calculations are for a purely redistributive tax: i.e. K = 0
• Broome case corresponds to bottom right corner. But he
assumed that there was no-one below 70.71% of the median.
Frank Cowell: EC426 Public Economics
Summary






Could we have “full information” taxation?
OIT is a standard second-best problem
Elementary version a reworking of the contract model
Can be extended to general ability distribution
Provides simple rules of thumb for good design
In practice these may be violated by well-meaning
policies
Frank Cowell: EC426 Public Economics
References (1)
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Bolton, P. and Dewatripont, M. (2005) Contract Theory, The MIT Press, pp
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*Brewer, M., Saez, E. and Shephard, A. (2010) “Means-testing and Tax
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Oxford University Press, Chapter 2, pp 90-164
Broome, J. (1975) “An important theorem on income tax,” Review of
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Diamond, P.A. (1998) “Optimal Income taxation: an example with a UShaped pattern of optimal marginal tax rates,” American Economic Review,
88, 83-95
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Journal of Public Economics, 49, 47-73
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Gibbard, A. (1973) “Manipulation of voting schemes: a general result,”
Econometrica, 41, 587-60
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*Kaplow, L. (2008) The Theory of Taxation and Public Economics,
Princeton University Press
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*Mankiw, N.G., Weinzierl, M. and Yagan, D. (2009) “Optimal Taxation in
Theory and Practice,” Journal of Economic Perspectives, 23, 147-174
Frank Cowell: EC426 Public Economics
References (2)
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Mirrlees, J. A. (1971) “An exploration in the theory of the optimal income
tax,” Review of Economic Studies, 38, 135-208
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Mirrlees, J. A. et al (2011) “The Mirrlees Review: Conclusions and
Recommendations for Reform,” Fiscal Studies, 32, 331–359
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Saez, E. (2001) “Using elasticities to derive optimal income tax rates,”
Review of Economic Studies, 68,205-22
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*Salanié, B. (2003) The Economics of Taxation, MIT Press, pp 59-61, 79-109
Satterthwaite, M. A. (1975) “Strategy-proofness and Arrow's conditions,
Journal of Economic Theory, 10, 187-217
Seade, J. (1977) “On the shape of optimal tax schedules,” Journal of Public
Economics, 7, 203-23
Sheshinski, E. (1972) “The optimal linear income tax,” Review of Economic
Studies, 39, 297-302
Stern, N. (1976) “On the specification of models of optimum income
taxation” Journal of Public Economics, 6,123-162
Tuomala, M. (1984) “On the Optimal Income Taxation: Some Further
Numerical Results,” Journal of Public Economics, 23, 351-366
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