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Frank Cowell: Design-Taxation DESIGN: TAXATION MICROECONOMICS Principles and Analysis Frank Cowell July 2015 1 Almost essential: Design Contract Almost essential: Design Contract Prerequisites
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Frank Cowell: Design-Taxation 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 2
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Frank Cowell: Design-Taxation 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 3
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Frank Cowell: Design-Taxation Overview July 2015 4 Design basics Simple model Generalisations Interpretations Design: Taxation Preferences, incomes, ability and the government
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Frank Cowell: Design-Taxation 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 July 2015 5
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Frank Cowell: Design-Taxation Modelling preferences Individual’s preferences = z + y : utility level z : effort y : income received : decreasing, strictly concave, function Special shape of utility function quasi-linear form zero-income effect z gives the disutility of effort in monetary units Individual does not have to work reservation utility level requires z + y ≥ July 2015 6
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Frank Cowell: Design-Taxation Ability and income Individuals work (give up leisure) to provide consumption Individuals differ in talent (ability) higher ability people produce more and may thus earn more individual of type works an amount z produces output q = z 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 is implicitly given by q − y Preferences can be expressed in terms of q and y for type utility is given by z + y equivalently: q / + y July 2015 7 A closer look at utility
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Frank Cowell: Design-Taxation The utility function (1) July 2015 8 increasing preference y 1– z Preferences over leisure and income Indifference curves = (z) + y z (z) < 0 Reservation utility ≥
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Frank Cowell: Design-Taxation The utility function (2) July 2015 9 increasing preference y q Preferences over leisure and output Indifference curves = (q/ ) + y z (q/ ) < 0 Reservation utility ≥
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Frank Cowell: Design-Taxation Indifference curves: pattern All types have the same preferences Function is common knowledge utility level of type depends on effort z and payment y but value of may be information that is private to individual Take indifference curves in (q, y) space = q + y slope of given type’s indifference curve depends on value of indifference curves of different types cross once only July 2015 10
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Frank Cowell: Design-Taxation The single-crossing condition July 2015 11 increasing preference y q type b type a Preferences over leisure and output High talent q a = a z a Low talent q b = b z b Those with different talent (ability) will have different sloped indifference curves in this diagram
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Frank Cowell: Design-Taxation 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 12
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Frank Cowell: Design-Taxation Overview July 2015 13 Design basics Simple model Generalisations Interpretations Design: Taxation Analogy with contract theory
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Frank Cowell: Design-Taxation 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 July 2015 14
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Frank Cowell: Design-Taxation Two types Start with the case closest to optimal contract model Exactly two skill types a > b proportion of a-types is values of a, b and 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: July 2015 15
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Frank Cowell: Design-Taxation Second-best: two types The government’s budget constraint [q a y a ] + [1 ][q b y b ] ≥ K where q h y h is the amount raised in tax from agent h Participation constraint for the b type: y b + z b ≥ b have to offer at least as much as available elsewhere Incentive-compatibility constraint for the a type: y a + q a / a ≥ y b + q b / a must be no worse off than if it behaved like a b-type implies q b, y b q a, y a The government seeks to maximise standard SWF z a + y a ) + [1 ] z b + y b ) where is increasing and concave July 2015 16
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Frank Cowell: Design-Taxation 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 q a q b y a y b to max q a / a + y a ) + [1 ] q b / b + y b ) + [ [q a y a ] + [1 ][q b y b ] K] + [y b + q b / b b ] + [y a + q a / a y b q b / a ] where are Lagrange multipliers for the constraints July 2015 17
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Frank Cowell: Design-Taxation Two types: method Differentiate with respect to q a q b y a y b to get FOCs: a z z a / a + + z z a / a ≤ 0 [1 ] b z z b / b + [1 ] + z z b / b z q b / a / a ≤ 0 a + ≤ 0 [1 ] b [1 ] + ≤ 0 For an interior solution, where q a q b y a y b are all positive a z z a / a + + z z a / a = 0 [1 ] b z z b / b + [1 ] + z z b / b z q b / a / a = 0 a + = 0 [1 ] b [1 ] + = 0 Manipulating these gives the main results for example, from first and third condition: [ ] z z a / a + + z z a / a = 0 z z a / a + = 0 July 2015 18
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Frank Cowell: Design-Taxation Two types: solution Solving the FOC we get: z q a / a = a z q b / b = b + k [1 ], where k := z q b / b [ b / a ] z q b / a 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: MRS a = MRT a MRS b < MRT b Interpretation no distortion at the top (for type a ) no surplus at the bottom (for type b ) determine the “menu” of (q,y)-choices offered by tax agency July 2015 19
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Frank Cowell: Design-Taxation Two ability types: tax design July 2015 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 points
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Frank Cowell: Design-Taxation Overview July 2015 21 Design basics Simple model Generalisations Interpretations Design: Taxation Moving beyond the two-ability model
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Frank Cowell: Design-Taxation A small generalisation With three types problem becomes a bit more interesting similar structure to previous case a > b > c proportions of each type in the population are a, b, c We now have one more constraint to worry about 1. participation constraint for c type: y c + q c / c ≥ c 2. IC constraint for b type: y b + q b / b ≥ y c + q c / b 3. IC constraint for a type: y a + q a / a ≥ y b + q b / a But this is enough to complete the model specification the two IC constraints also imply y a + q a / a ≥ y c + q c / b so no-one has incentive to misrepresent as lower ability July 2015 22
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Frank Cowell: Design-Taxation 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 q a,y a q b,y b q c,y c Outcome essentially as before : MRS a = 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 July 2015 23
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Frank Cowell: Design-Taxation A richer model: N + 1 types The multi-type case follows immediately from three types Take N + l types 0 < 1 < 2 < … < N (note the required change in notation) proportion of type j is j this distribution is common knowledge Budget constraint and SWF are now j j [q j y j ] ≥ K j j z j + y j ) where sum is from 0 to N July 2015 24
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Frank Cowell: Design-Taxation N + 1 types: behavioural constraints Participation constraint is relevant for lowest type j = 0 form is as before: y 0 + z 0 ≥ 0 Incentive-compatibility constraint applies where j > 0 j must be no worse off than if it behaved like the type below (j 1) y j + q j / j ≥ y j 1 + q j 1 / j implies q j 1, y j 1 q j, y j and j ≥ j 1 From previous cases we know the methodology (and can probably guess the outcome) July 2015 25
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Frank Cowell: Design-Taxation N+1 types: solution Lagrangian is only slightly modified from before Choose {(q j y j )} to max j=0 j q j j + y j ) + [ j j [q j y j ] K] + [y 0 + z 0 0 ] + j=1 j [y j + q j / j y j 1 q j 1 / j ] where there are now N incentive-compatibility Lagrange multipliers And we get the result, as before MRS N = MRT N MRS N−1 < MRT N−1 … MRS 1 < MRT 1 MRS 0 < MRT 0 Now the tax schedule is determined at N+1 points July 2015 26
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Frank Cowell: Design-Taxation 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 July 2015 27
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Frank Cowell: Design-Taxation The continuum model Continuous ability bounded support [ density f( ) Utility for talent as before y( ) + q( ) Participation constraint is ) ≥ Incentive compatibility requires d ) /d ≥ SWF is ( ) f d July 2015 28
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Frank Cowell: Design-Taxation Continuum model: optimisation Lagrangian is ( ) f d + q − y − f d + [ − + d d f d where y( ) + q( ) Lagrange multipliers are : government budget constraint : participation constraint incentive-compatibility for type Maximise Lagrangian with respect to q and y for all [ July 2015 29
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Frank Cowell: Design-Taxation Output and disposable income under the optimal tax July 2015 30 y q q _ q _ 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 authority
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Frank Cowell: Design-Taxation 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 (∙) specification of social preferences (∙) specification of required revenue K July 2015 31
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Frank Cowell: Design-Taxation Overview July 2015 32 Design basics Simple model Generalisations Interpretations Design: Taxation Applying design rules to practical policy
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Frank Cowell: Design-Taxation 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 July 2015 33
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Frank Cowell: Design-Taxation 1t1t A simple tax-benefit system July 2015 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
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Frank Cowell: Design-Taxation 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 B 0 is withdrawn after a given output level relationship between y and q no longer continuous and monotonic July 2015 35
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Frank Cowell: Design-Taxation A badly designed tax-benefit system July 2015 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 B0B0 The notch violates IC causes a- types to masquerade as b- types
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Frank Cowell: Design-Taxation 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 July 2015 37
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