Prerequisites
Almost essential:
Design Contract
DESIGN: TAXATION
MICROECONOMICS
Principles and Analysis
Frank Cowell
July 2015
Frank Cowell: Design-Taxation
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The design problem
 The government needs to raise revenue
• and it may want to redistribute resources
 To do this it uses the tax system
• personal income tax
• and income-based subsidies
 Base it on “ability to pay”
• income rather than wealth
• ability reflected in productivity
 Tax authority may have limited information
• 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?
July 2015
Frank Cowell: Design-Taxation
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A link with contract theory
 Base approach on the analysis of contracts
• close analogy with case of hidden characteristics
• owner hires manager
• but manager’s ability is unknown at time of hiring
 Ability here plays the role of unobservable type
• ability may not be directly observable
• but distribution of ability in the population is known
 A progressive treatment:
• outline model components
• use analogy with contracts to solve two-type case
• proceed to large (finite) number of types
• then extend to general continuous distribution
July 2015
Frank Cowell: Design-Taxation
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Overview
Design: Taxation
Design basics
Preferences, incomes,
ability and the government
Simple model
Generalisations
Interpretations
July 2015
Frank Cowell: Design-Taxation
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Model elements
 A two-commodity model
• leisure (i.e. the opposite of effort)
• consumption – a basket of all other goods
 Income comes only from work
• individuals are paid according to their marginal product
• workers differ according to their ability
 Individuals derive utility from:
• their leisure
• their disposable income (consumption)
• Government / tax agency
• has to raise a fixed amount of revenue K
• seeks to maximise social welfare
• where social welfare is a function of individual utilities
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Frank Cowell: Design-Taxation
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Modelling preferences
 Individual’s preferences
• u = y(z) + y
• u : utility level
• z : effort
• y : 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
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Ability and income
 Individuals work (give up leisure) to provide consumption
 Individuals differ in talent (ability) t
• 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?
 Disposable income determined by tax authority
• 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 and y
• for type t utility is given by y(z) + y
• equivalently: y(q / t) + y
July 2015
Frank Cowell: Design-Taxation
A closer look
at utility
7
The utility function (1)
 Preferences over leisure and income
 Indifference curves
 Reservation utility
y
 u = y(z) + y
 yz(z) < 0
 u≥u
u
1– z
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Frank Cowell: Design-Taxation
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The utility function (2)
 Preferences over leisure and output
 Indifference curves
 Reservation utility
y
 u = y(q/t) + y
 yz(q/t) < 0
 u≥u
u
q
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Indifference curves: pattern
 All types have the same preferences
 Function y() is common knowledge
• utility level u of type t depends on effort z and payment y
• but value of t may be information that is private to individual
 Take indifference curves in (q, y) space
• u = y(q/t) + y
• slope of given type’s indifference curve depends on value of t
• indifference curves of different types cross once only
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Frank Cowell: Design-Taxation
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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
 qa = taza
 qb = tbzb
type a
q
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Frank Cowell: Design-Taxation
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Similarity with contract model
 The position of the Agent
• not a single Agent with known ex-ante probability distribution of talents
• but 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 SWF
 One extra constraint
• the 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 goods
July 2015
Frank Cowell: Design-Taxation
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Overview
Design: Taxation
Design basics
Analogy with
contract theory
Simple model
Generalisations
Interpretations
July 2015
Frank Cowell: Design-Taxation
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A full-information solution?
 Consider argument based on the analysis of contracts
 Given full information owner can fully exploit any manager
• pays the minimum amount necessary
• “chooses” their effort
 Same basic story here
• can impose lump-sum tax
• “chooses” agents’ effort — no distortion
 But the full-information solution may be unattractive
• informational requirements are demanding
• perhaps violation of individuals’ privacy?
• so look at second-best case
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Two types
 Start with the case closest to optimal contract model
 Exactly two skill types
• ta > tb
• 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:
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Second-best: two types
 The government’s budget constraint
• p[qa - ya] + [1-p][qb - yb ] ≥ K
• where qh - yh is the amount raised in tax from agent h
 Participation constraint for the b type:
• yb + y(zb) ≥ ub
• have to offer at least as much as available elsewhere
 Incentive-compatibility constraint for the a type:
• 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
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Two types: model
 We can use a standard Lagrangian approach
• government chooses (q, y) pairs for each type
• subject to three constraints
 Constraints are:
• 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
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Two types: method
 Differentiate with respect to qa, qb, ya, yb to get FOCs:
• pzu(ua)yz(za)/ta + kp + myz(za)/ta ≤ 0
• [1-p]zu(ub)yz(zb)/tb + k [1-p] + lyz(zb)/tb - myz(qb/ta)/ta ≤ 0
• pzu(ua) - kp + m ≤ 0
• [1-p]zu(ub) - k[1-p] + l - m ≤ 0
 For an interior solution, where qa, qb, ya, yb are all positive
• pzu(ua)yz(za)/ta + kp + myz(za)/ta = 0
• [1-p]zu(ub)yz(zb)/tb + k [1-p] + lyz(zb)/tb - myz(qb/ta)/ta = 0
• pzu(ua) - kp + m = 0
• [1-p]zu(ub) - k[1-p] + l - m = 0
 Manipulating these gives the main results
• for example, from first and third condition:
• [kp - m ] yz(za)/ta + kp + myz(za)/ta = 0
• kp yz(za)/ta + kp = 0
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Frank Cowell: Design-Taxation
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Two types: solution
 Solving the FOC we get:
• - yz(qa/ta) = ta
• - yz(qb/tb) = tb + kp/[1-p],
• where k := yz(qb/tb) - [tb/ta] yz(qb/ta) < 0
 Also, all the Lagrange multipliers are positive
• so the associated constraints are binding
• follows from standard adverse selection model
 Results are as for optimum-contracts model:
• 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
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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
July 2015
then it must be consistent with
these two points
qa
Frank Cowell: Design-Taxation
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Overview
Design: Taxation
Design basics
Moving beyond the
two-ability model
Simple model
Generalisations
Interpretations
July 2015
Frank Cowell: Design-Taxation
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A small generalisation
 With three types problem becomes a bit more interesting
• similar structure to previous case
• ta > tb > tc
• proportions of each type in the population are pa, pb, pc
 We now have one more constraint to worry about
1. participation constraint for c type: yc + y(qc/tc) ≥ uc
2. IC constraint for b type: yb + y(qb/tb) ≥ yc + y(qc/tb)
3. 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
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Three types
 Methodology is same as two-ability model
• set up Lagrangian
• Lagrange multipliers for budget constraint, participation constraint and
two IC constraints
• maximise with respect to (qa,ya), (qb,yb), (qc,yc)
 Outcome essentially as before :
• MRSa = MRTa
• MRSb < MRTb
• MRSc < MRTc
 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
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A richer model: N + 1 types
 The multi-type case follows immediately from three types
 Take N + l types
• 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
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Frank Cowell: Design-Taxation
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N + 1 types: behavioural constraints
 Participation constraint
• is relevant for lowest type j = 0
• form is as before:
• y0 + y(z0) ≥ u0
 Incentive-compatibility constraint
• applies where j > 0
• j must be no worse off than if it behaved like 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)
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N+1 types: solution
 Lagrangian 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
• MRSN = MRTN
• MRSN−1 < MRTN−1
• …
• MRS1 < MRT1
• MRS0 < MRT0
• Now the tax schedule is determined at N+1 points
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A continuum of types
 One more step is required in generalisation
 Suppose the tax agency is faced with a continuum of taxpayers
• frequently used assumption
• allows for general specification of ability distribution
 This case can be reasoned from the case with N + 1 types
• allow N  
 From previous cases we know
• form of the participation constraint
• form that IC constraint must take
• an outline of the outcome
 Can proceed by analogy with previous analysis
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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
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Continuum model: optimisation
 Lagrangian is
`t

t
z (u(t)) f(t) dt
`t
+k

q(t) − y(t) − K] f(t) dt
t
+ l [ u(t) − u]
`t
+

t
m(t) [du(t) / dt ] f(t) dt
where u(t) = y(t) + y( q(t) / t)
 Lagrange multipliers are
• k : government budget constraint
• l : participation constraint
• m(t) : incentive-compatibility for type t
 Maximise Lagrangian with respect to q(t) and y(t) for all t  [t,`t ]
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Frank Cowell: Design-Taxation
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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_
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q
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Continuum model: results
 Incentive compatibility implies
• dy /dq > 0
• optimal marginal tax rate < 100%
 No distortion at top implies
• dy /dq = 1
• zero optimal marginal tax rate!
 But explicit form for the optimal income tax requires
• specification of distribution f(∙)
• specification of individual preferences y(∙)
• specification of social preferences z (∙)
• specification of required revenue K
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Overview
Design: Taxation
Design basics
Applying design
rules to practical
policy
Simple model
Generalisations
Interpretations
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Application of design principles
 The second-best method provides some pointers
• but is not a prescriptive formula
• model is necessarily over-simplified
• exact second-best formula might be administratively complex
 Simple schemes may be worth considering
• roughly correspond to actual practice
• illustrate good/bad design
 Consider affine (linear) tax system
• 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
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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
B
q
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Violations of design principles?
 Sometimes the IC condition be violated in actual design
 This can happen by accident:
• interaction between income support and income tax
• generated by the desire to “target” support more effectively
• a well-meant inefficiency?
 Commonly known as
• 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
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A badly designed tax-benefit system
Menu offered to low income groups
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
y
High type’s intended output and income
ya
High type’s utility-maximising choice
yb
 The notch violates IC
B0
causes a-types to masquerade as
b-types
q
qb
July 2015
qa
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Summary
 Optimal income tax 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
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