Principles and Analysis Frank Cowell July 2017 1 Almost essential Design Contract Prerequisites The design problem The government needs to raise revenue and it may want to redistribute resources ID: 657443
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Slide1
Design: Taxation
MICROECONOMICSPrinciples and Analysis Frank Cowell
April 2018
1
Almost essential:Design Contract
PrerequisitesSlide2
The design problem
The government needs to raise revenueand it may want to redistribute resourcesTo do this it uses the tax system
personal income taxand income-based subsidiesBase it on “ability to pay”
income rather than wealthability reflected in productivityTax authority may have limited informationwho have the high ability to pay?
what impact on individuals’ willingness to produce output?What’s the right way to construct the tax schedule?
April 2018
2Slide3
A link with contract theory
Base approach on the analysis of contractsclose analogy with case of hidden characteristics
owner hires managerbut manager’s ability is unknown at time of hiring Ability here plays the role of unobservable type
ability may not be directly observablebut distribution of ability in the population is knownA progressive treatment:
outline model componentsuse analogy with contracts to solve two-type caseproceed to large (finite) number of typesthen extend to general continuous distribution
April 2018
3Slide4
Overview
April 2018
4
Design basics
Simple model
Generalisations
Interpretations
Design: Taxation
Preferences, incomes, ability and the governmentSlide5
Model elements
A two-commodity modelleisure (i.e. the opposite of effort) consumption – a basket of all other goods
Income comes only from workindividuals are paid according to their marginal productworkers differ according to their ability
Individuals derive utility from:their leisuretheir disposable income (consumption)Government / tax agency
has to raise a fixed amount of revenue Kseeks to maximise social welfarewhere social welfare is a function of individual utilities
April 2018
5Slide6
Modelling preferences
Individual’s preferencesu =
y(z) + y
u : utility level z : efforty : income received
y() : decreasing, strictly concave, function Special shape of utility 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
April 2018
6Slide7
Ability and income
Individuals work (give up leisure) to provide consumptionIndividuals differ in talent (ability) t
higher ability people produce more and may thus earn moreindividual of type t
works an amount zproduces output q = t
z,but does not necessarily get to keep this outputDisposable income determined by tax authorityintervention 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
and
y
for type
t
utility is given by
y(
z
)
+
yequivalently:
y(q / t) +
yApril 2018
7
A closer look at utilitySlide8
The utility function (1)
April 2018
8
increasing
preference
y
1
–
z
Preferences over leisure and income
Indifference curves
u
=
y
(
z
) +
y
y
z
(
z
) < 0
Reservation utility
u
≥
u
uSlide9
The utility function (2)
April 2018
9
increasing
preference
y
q
Preferences over leisure and output
Indifference curves
u
=
y
(
q/
t
) +
y
y
z
(
q/
t
) < 0
Reservation utility
u
≥
u
uSlide10
Indifference curves: pattern
All types have the same preferencesFunction y(
) is common knowledgeutility level
u of type t depends on effort z and payment y
but value of t may be information that is private to individualTake indifference curves in (q, y) spaceu
=
y(
q
/t)
+
y
slope of given type’s indifference curve depends on value of
t
indifference curves of different types cross once only
April 2018
10Slide11
The single-crossing condition
April 2018
11
increasing
preference
y
q
type
b
type
a
Preferences over leisure and output
High talent
q
a
=
t
a
z
a
Low talent
q
b
=
t
b
z
b
Those with different talent (ability) will have different sloped indifference curves in this diagramSlide12
Similarity with contract model
The position of the Agentnot a single Agent with known ex-ante probability distribution of talentsbut a population of workers with known distribution of abilities
The position of the Principal (designer)designer is the government acting as Principal knows distribution of ability (common knowledge)
the objective function is a standard SWFOne extra constraintthe community has to raise a fixed amount K ≥ 0
the government imposes a tax drives a wedge between market income generated by worker and the amount available to spend on other goodsApril 2018
12Slide13
Overview
April 2018
13
Design basics
Simple model
Generalisations
Interpretations
Design: Taxation
Analogy with contract theorySlide14
A full-information solution?
Consider argument based on the analysis of contractsGiven full information owner can fully exploit any managerpays the minimum amount necessary
“chooses” their effortSame basic story here can impose lump-sum tax
“chooses” agents’ effort — no distortion But the full-information solution may be unattractiveinformational requirements are demanding
perhaps violation of individuals’ privacy?so look at second-best caseApril 2018
14Slide15
Two types
Start with the case closest to optimal contract modelExactly two skill types
ta > t
b proportion of a-types is p
values of ta, tb and p are common knowledge
From contract design we can write down the outcome
essentially all we need to do is rework notation
But let us examine the model in detail:
April 2018
15Slide16
Second-best: two types
The government’s budget constraintp[qa
- ya] + [1-p][q
b - yb ] ≥ Kwhere
qh - yh is the amount raised in tax from agent hParticipation constraint for the b type:
y
b
+
y(
z
b
)
≥
u
b
have to offer at least as much as available elsewhere
Incentive-compatibility constraint for the
a
type:
y
a
+
y(
q
a/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) + y
b) where z is increasing and concave
April 2018
16Slide17
Two types: model
We can use a standard Lagrangian approachgovernment chooses (
q, y) pairs for each typesubject to three constraints
Constraints are:government budget constraintparticipation constraint (for b-types)
incentive-compatibility constraint (for a-types)Choose qa, q
b
,
y
a
,
y
b
to max
p z(y(
q
a
/
t
a
)
+
y
a) + [1
-p] z(y(qb/
tb) + y
b) + k [p[q
a - ya] + [1
-p][qb - yb ]
- K]
+ l [yb + y(qb/
tb)
- ub] + m
[ya + y(qa/t
a) -
yb -
y(
q
b
/
t
a
)
]
where
k, l, m
are Lagrange multipliers for the constraints
April 2018
17Slide18
Two types: method
Differentiate with respect to qa, q
b, ya, yb
to get FOCs:pzu(ua
)yz(za)/ta +
kp
+
my
z
(
z
a
)
/
t
a
≤
0
[1
-p
]
z
u
(
ub)yz
(zb)/
tb + k [1-p] + lyz
(zb)/
tb - myz(q
b/ta)/
ta ≤ 0pzu(
ua) - kp
+ m ≤ 0[1-p]
zu(ub) - k[1
-p] + l - m ≤
0For an interior solution, where qa
,
q
b
,
y
a
,
y
b
are all positive
pz
u
(
u
a
)
y
z
(
z
a
)
/ta + kp +
myz(za)/ta = 0[1-p]
zu(ub)yz
(zb)/tb + k [1-p] + ly
z(zb)/tb
- myz(qb/t
a)/ta = 0
pzu(ua) - kp + m = 0[1-p]zu(ub) - k[1-p] + l - m = 0
From first and third conditions:[kp - m ] yz(za)/ta + kp +
myz(za)/
ta = 0kp yz(z
a)/ta + kp
=
0
April 2018
18Slide19
Two types: solution
Solving the FOC we get:- y
z(qa/ta
) = ta- yz
(qb/tb) = tb
+
k
p
/
[1
-p
],
where
k
:=
y
z
(
q
b
/
t
b
) - [
tb/ta
] yz(q
b/ta) < 0Also, all the Lagrange multipliers are positiveso the associated constraints are binding
follows from standard adverse selection modelResults are as for optimum-contracts model:
MRSa = MRTaMRSb <
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
April 2018
19Slide20
Two ability types: tax design
April 2018
20
y
q
q
a
q
b
y
a
y
b
a-type’s reservation utility
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
Analysis determines
(
q,y
)
combinations at two points
If a tax schedule
T
(
∙
)
is to be designed where
y = q −T
(
q
)
then it must be consistent with these two pointsSlide21
Overview
April 2018
21
Design basics
Simple model
Generalisations
Interpretations
Design: Taxation
Moving beyond the two-ability modelSlide22
A small generalisation
With three types problem becomes a bit more interestingsimilar structure to previous case
ta > tb >
tcproportions of each type in the population are pa,
pb, pcWe now have one more constraint to worry about
participation constraint for
c
type:
y
c
+
y(
q
c
/
t
c
)
≥
u
c
IC constraint for
b
type:
y
b + y(qb/
tb) ≥ y
c + y(qc/tb)
IC constraint for a type: y
a + y(qa/ta) ≥
yb + y(qb
/ta)But this is enough to complete the model specification
the two IC constraints also imply ya + y(q
a/ta) ≥ yc +
y(qc/tb)
so no-one has incentive to misrepresent as lower ability
April 2018
22Slide23
Three types
Methodology is same as two-ability modelset up Lagrangian
Lagrange multipliers for budget constraint, participation constraint and two IC constraintsmaximise with respect to (q
a,ya), (qb,yb), (q
c,yc)Outcome essentially as before :MRSa =
MRT
a
MRS
b
<
MRT
b
MRS
c
<
MRT
c
Again, no distortion at the top and the participation constraint binding at the bottom
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
April 2018
23Slide24
A richer model: N
+ 1 typesThe multi-type case follows immediately from three typesTake
N + l typest0
< t1 < t2
< … < tN (note the required change in notation)proportion of type j is
p
j
this distribution is common knowledge
Budget constraint and SWF are now
S
j
p
j
[
q
j
-
y
j
] ≥
K
S
j
pj
z(y(zj) + yj)
where sum is from 0 to N
April 2018
24Slide25
N + 1 types: behavioural constraints
Participation constraintis relevant for lowest type
j = 0 form is as before: y0 +
y(z0) ≥ u0
Incentive-compatibility constraint applies where j > 0j must be no worse off than if it behaved like the type below (j-
1)
y
j
+
y(
q
j
/
t
j
)
≥
y
j
-
1
+
y(
qj-1
/tj)
implies (qj
-1, yj-1) < (q
j, yj
) and u(tj) ≥ u(tj
-1)
From previous cases we know the methodology(and can probably guess the outcome)
April 2018
25Slide26
N+1 types: solution
Lagrangian is only slightly modified from beforeChoose {(
qj, yj )} to max
Sj=0 pj
z (y(qj / tj)
+
y
j
)
+
k
[
S
j
p
j
[
q
j
-
y
j
] -
K] + l
[y0 + y(z
0) - u0] +
Sj=1 m
j [yj + y(qj/t
j) - y
j-1 - y(qj-
1 /tj
)]where there are now N incentive-compatibility Lagrange multipliersAnd we get the result, as before
MRSN = MRTNMRSN−1
< MRTN−1 …
MRS1 < MRT1
MRS
0
< MRT
0
Now the tax schedule is determined at
N
+1 points
April 2018
26Slide27
A continuum of types
One more step is required in generalisationSuppose the tax agency is faced with a continuum of taxpayers
frequently used assumptionallows for general specification of ability distributionThis case can be reasoned from the case with
N + 1 typesallow N
From previous cases we know form of the participation constraintform that IC constraint must takean outline of the outcome
Can proceed by analogy with previous analysis
April 2018
27Slide28
The continuum model
Continuous ability bounded support [t
,`t ]density f(t)
Utility for talent t as beforeu(t) = y(
t) + y( q(t) / t) Participation constraint is
u(
t
) ≥
u
Incentive compatibility requires
d
u(t
) /d
t
≥
0
SWF is
-
z
(
u(t)
)
f(t) d
tt
April 2018
28Slide29
Continuum model: optimisation
Lagrangian is
`t
z (u(t)) f
(t) dt t
`t
+
k
q
(t)
−
y
(t)
−
K
]
f
(t)
dt
t + l [ u(
t) − u
] `t+
m(t) [d
u(t) / dt ] f(t) d
t
t where u(t) =
y(t) + y( q(t)
/ t) Lagrange multipliers are
k : government budget constraintl
: participation constraint
m(t) :
incentive-compatibility for type
t
Maximise
Lagrangian
with respect to
q
(t)
and
y
(t)
for all
t
[
t
,`t
]
April 2018
29Slide30
Output and disposable income under the optimal tax
April 2018
30
y
q
q
_
q
_
t
_
t
_
45°
Lowest type’s indifference curve
Lowest type’s output and income
Intermediate type’s indifference curve, output and income
Highest type’s indifference curve
Highest type’s output and income
Menu offered by tax authoritySlide31
Continuum model: results
Incentive compatibility implies dy /dq > 0
optimal marginal tax rate < 100%No distortion at top impliesdy /dq
= 1zero optimal marginal tax rate!But explicit form for the optimal income tax requiresspecification of distribution f(∙)
specification of individual preferences y(∙)specification of social preferences z (∙)specification of required revenue K
April 2018
31Slide32
Overview
April 2018
32
Design basics
Simple model
Generalisations
Interpretations
Design: Taxation
Applying design rules to practical policySlide33
Application of design principles
The second-best method provides some pointersbut is not a prescriptive formulamodel is necessarily over-simplified
exact second-best formula might be administratively complexSimple schemes may be worth considering
roughly correspond to actual practiceillustrate good/bad designConsider affine (linear) tax systembenefit B payable to all (guaranteed minimum income)
all gross income (output) taxable at the same marginal rate tconstant 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
April 2018
33Slide34
1
t
A simple tax-benefit system
April 2018
34
y
q
Low-income type’s indiff curve
Low-income type’s output, income
High-income type’s indiff curve
Highest type’s output and income
Constant marginal retention rate
Guaranteed minimum income B
B
Implied attainable set
“Linear” income tax system ensures that incentive-compatibility constraint is satisfied Slide35
Violations of design principles?
Sometimes the IC condition be violated in actual designThis can happen by accident:interaction between income support and income tax
generated by the desire to “target” support more effectivelya well-meant inefficiency?
Commonly known asthe “notch problem” (US)the “poverty trap” (UK)Simple example
suppose some of the benefit is intended for lowest types onlyan amount B0 is withdrawn after a given output levelrelationship between y and q no longer continuous and monotonic
April 2018
35Slide36
A badly designed tax-benefit system
April 2018
36
y
q
Low-income type’s indiff curve
Low type’s output and income
High-income type’s indiff curve
High type’s
intended
output and income
Menu offered to low income groups
Withdrawal of benefit B
0
q
a
q
b
y
a
y
b
Implied attainable set
High type’s utility-maximising choice
B
0
The notch violates IC
causes
a-
types to masquerade as
b-
types Slide37
Summary
Optimal income tax is a standard second-best problemElementary version a reworking of the contract modelCan be extended to general ability distribution
Provides simple rules of thumb for good designIn practice these may be violated by well-meaning policies
April 2018
37