Transcript Extensive Form - London School of Economics

Prerequisites
Almost essential
Contract Design
Frank Cowell: Microeconomics
September 2006
Design: Taxation
MICROECONOMICS
Principles and Analysis
Frank Cowell
The design problem
Frank Cowell: Microeconomics
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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?
A link with contract theory
Frank Cowell: Microeconomics
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Base approach on the analysis of contracts
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Ability here plays the role of unobservable type
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close analogy with case of hidden characteristics
owner hires manager…
…but manager’s ability is unknown at time of hiring
ability may not be directly observable…
…but distribution of ability in the population is known
A progressive treatment:
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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
Overview...
Design: Taxation
Frank Cowell: Microeconomics
Design basics
Preferences,
incomes, ability
and the
government
Simple model
Generalisations
Interpretations
Model elements
Frank Cowell: Microeconomics
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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
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
Modelling preferences
Frank Cowell: Microeconomics
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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
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reservation utility level u
requires y(z) + y ≥ u
Ability and income
Frank Cowell: Microeconomics
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Individuals work (give up leisure) to provide 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 and y
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for type t utility is given by y(z) + y
equivalently: y(q / t) + y
A closer look
at utility
The utility function (1)
Frank Cowell: Microeconomics
Preferences over leisure and
income
y
 Indifference curves
 Reservation utility

u = y(z) + y
yz(z) < 0

u≥u

u
1– z
The utility function (2)
Frank Cowell: Microeconomics
Preferences over leisure and
output
y
 Indifference curves
 Reservation utility
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u = y(q/t) + y
yz(q/t) < 0
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u≥u
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u
q
Indifference curves: pattern
Frank Cowell: Microeconomics
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All types have the same preferences
Function y() is common knowledge
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but utility level u of type t depends on effort z and
payment y
value of t may be information that is private to
individual
Take indifference curves in (q, y) space
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u = y(q/t) + y
slope of a given type’s indifference curve depends on
value of t
indifference curves of different types cross once only
The single-crossing condition
Frank Cowell: Microeconomics
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
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q

qa = taza
qb = tbzb
Similarity with contract model
Frank Cowell: Microeconomics
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The position of the Agent
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The position of the Principal (designer)
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instead of a single Agent with known ex-ante probability
distribution of talents,…
… a population of workers with known distribution of abilities.
designer is the government acting as Principal.
knows distribution of ability (common knowledge)
the objective function is a standard SWF
One extra constraint
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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.
Overview...
Design: Taxation
Frank Cowell: Microeconomics
Design basics
Analogy with
contract theory
Simple model
Generalisations
Interpretations
A full-information solution?
Frank Cowell: Microeconomics
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Consider argument based on the analysis of contracts
Given 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…
Two types
Frank Cowell: Microeconomics
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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:
Second-best: two types
Frank Cowell: Microeconomics
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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
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p z(y(za) + ya) + [1-p] z(y(zb) + yb)
where z is increasing and concave
Two types: model
Frank Cowell: Microeconomics
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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
Two types: method
Frank Cowell: Microeconomics
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Differentiate with respect to qa, qb, ya, yb to get FOCs:
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For an interior solution, where qa, qb, ya, yb are all positive
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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
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
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For example, from first and third condition:
[kp - m ] yz(za)/ta + kp + myz(za)/ta = 0
kp yz(za)/ta + kp = 0
Two types: solution
Frank Cowell: Microeconomics
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Solving the FOC 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
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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….
Two ability types: tax design
Frank Cowell: Microeconomics
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
Overview...
Design: Taxation
Frank Cowell: Microeconomics
Design basics
Moving beyond
the two-ability
model
Simple model
Generalisations
Interpretations
A small generalisation
Frank Cowell: Microeconomics
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With three types problem becomes a bit more interesting
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We now have one more constraint to worry about
1.
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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
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the two IC constraints also imply ya + y(qa/ta) ≥ yc + y(qc/tb) …
… so no-one has incentive to misrepresent as lower ability
Three types
Frank Cowell: Microeconomics
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Methodology is same as two-ability model
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Outcome essentially as before :
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set up Lagrangean
Lagrange multipliers for budget constraint, participation constraint and
two IC constraints
maximise with respect to (qa,ya), (qb,yb), (qc,yc)
MRSa = MRTa
MRSb < MRTb
MRSc < MRTc
Again, no distortion at the top and the participation constraint binding
at the bottom
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determines (q,y)-combinations at exactly three points
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tax schedule must be consistent with these points
A stepping stone to a much more interesting model…
A richer model: N+1 types
Frank Cowell: Microeconomics
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The multi-type case follows immediately from three types
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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
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Sj pj [qj - yj] ≥ K
Sj pj z(y(zj) + yj)
where sum is from 0 to N
N+1 types: behavioural constraints
Frank Cowell: Microeconomics
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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 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
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(and can probably guess the outcome)
N+1 types: solution
Frank Cowell: Microeconomics
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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
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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
A continuum of types
Frank Cowell: Microeconomics
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One more step is required in generalisation
Suppose the tax agency is faced with a continuum of
taxpayers
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This case 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…
The continuum model
Frank Cowell: Microeconomics
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Continuous ability
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bounded support [t,`t ]
density f(t)
Utility for talent t as before
u(t) = y(t) + y( q(t) / t)
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Participation constraint is
u(t) ≥ u
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Incentive compatibility requires
du(t) /dt ≥ 0
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SWF is
`t
│ z (u(t)) f(t) dt
⌡t
Continuum model: optimisation
Frank Cowell: Microeconomics
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Lagrangean 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
du(t)
+ │ m(t) —— f(t) dt
⌡t
dt
where u(t) = y(t) + y( q(t) / t)
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Lagrange multipliers are
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k : government budget constraint
l : participation constraint
m(t) : incentive-compatibility for type t
Maximise Lagrangean with respect to q(t) and y(t) for all t  [t,`t ]
Output and disposable income under
the optimal tax
Frank Cowell: Microeconomics
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
Continuum model: results
Frank Cowell: Microeconomics
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Incentive compatibility implies
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No distortion at top implies
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dy /dq > 0
optimal marginal tax rate < 100%
dy /dq = 1
zero optimal marginal tax rate!
But 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
Overview...
Design: Taxation
Frank Cowell: Microeconomics
Design basics
Applying design
rules to practical
policy
Simple model
Generalisations
Interpretations
Application of design principles
Frank Cowell: Microeconomics
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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
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:
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(net) income related to output thus: y = B + [1 - t] q
so y > q if q < B / t … and vice versa
A simple tax-benefit system
Frank Cowell: Microeconomics
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
Violations of design principles?
Frank Cowell: Microeconomics
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Sometimes the IC condition be violated in actual design
This can happen by accident:
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Commonly known as
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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
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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
A badly designed tax-benefit system
Frank Cowell: Microeconomics
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
Summary
Frank Cowell: Microeconomics
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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