Frank Cowell: Design-Taxation DESIGN: TAXATION MICROECONOMICS Principles and Analysis Frank Cowell July Almost essential: Design Contract Almost essential: Design Contract Prerequisites

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

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

Frank Cowell: Design-Taxation Overview July Design basics Simple model Generalisations Interpretations Design: Taxation Preferences, incomes, ability and the government

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

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

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 A closer look at utility

Frank Cowell: Design-Taxation The utility function (1) July increasing preference y 1– z  Preferences over leisure and income  Indifference curves   =  (z) + y   z (z) < 0  Reservation utility   ≥  

Frank Cowell: Design-Taxation The utility function (2) July increasing preference y q  Preferences over leisure and output  Indifference curves   =  (q/  ) + y   z (q/  ) < 0  Reservation utility   ≥  

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

Frank Cowell: Design-Taxation The single-crossing condition July 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

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

Frank Cowell: Design-Taxation Overview July Design basics Simple model Generalisations Interpretations Design: Taxation Analogy with contract theory

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

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

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

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

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

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

Frank Cowell: Design-Taxation Two ability types: tax design July 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

Frank Cowell: Design-Taxation Overview July Design basics Simple model Generalisations Interpretations Design: Taxation Moving beyond the two-ability model

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

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

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

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

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

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

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

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

Frank Cowell: Design-Taxation Output and disposable income under the optimal tax July 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

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

Frank Cowell: Design-Taxation Overview July Design basics Simple model Generalisations Interpretations Design: Taxation Applying design rules to practical policy

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

Frank Cowell: Design-Taxation 1t1t A simple tax-benefit system July 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

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

Frank Cowell: Design-Taxation A badly designed tax-benefit system July 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

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