1. Frank Cowell: Microeconomics Design: Taxation MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Contract Design Almost essential Contract Design Prerequisites September 2006

2. Frank Cowell: Microeconomics The design problem The government needs to raise revenue… The government needs to raise revenue… …and it may want to redistribute resources …and it may want to redistribute resources To do this it uses the tax system To do this it uses the tax system  personal income tax…  …and income-based subsidies Base it on “ability to pay” Base it on “ability to pay”  income rather than wealth  ability reflected in productivity Tax authority may have limited information 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? What’s the right way to construct the tax schedule?

3. Frank Cowell: Microeconomics A link with contract theory Base approach on the analysis of contracts 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 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: 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

4. Frank Cowell: Microeconomics Overview... Design basics Simple model Generalisations Interpretations Design: Taxation Preferences, incomes, ability and the government

5. Frank Cowell: Microeconomics Model elements A two-commodity model A two-commodity model  leisure (i.e. the opposite of effort)  consumption – a basket of all other goods Income comes only from work Income comes only from work  individuals are paid according to their marginal product  workers differ according to their ability Individuals derive utility from: Individuals derive utility from:  their leisure  their disposable income (consumption) Government / tax agency 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

6. Frank Cowell: Microeconomics Modelling preferences Individual’s preferences Individual’s preferences   =  z  + y   : utility level  z : effort  y : income received   : decreasing, strictly concave, function Special shape of utility 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 Individual does not have to work  reservation utility level   requires  z  + y ≥ 

7. Frank Cowell: Microeconomics Ability and income Individuals work (give up leisure) to provide consumption Individuals work (give up leisure) to provide consumption Individuals differ in talent (ability)  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 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 Preferences can be expressed in terms of q and y  for type  utility is given by  z  + y  equivalently:  q /  + y A closer look at utility

8. Frank Cowell: Microeconomics The utility function (1) increasing preference y 1– z   Preferences over leisure and income   Indifference curves   =  z  + y   z  z  < 0   Reservation utility  ≥  ≥  ≥  ≥  

9. Frank Cowell: Microeconomics The utility function (2) increasing preference y q   Preferences over leisure and output   Indifference curves   =  q/  + y   z  q/  < 0   Reservation utility  ≥  ≥  ≥  ≥  

10. Frank Cowell: Microeconomics Indifference curves: pattern All types have the same preferences All types have the same preferences Function  is common knowledge Function  is common knowledge  but utility level  of type  depends on effort z and payment y  value of  may be information that is private to individual Take indifference curves in (q, y) space Take indifference curves in (q, y) space   =  q  + y  slope of a given type’s indifference curve depends on value of   indifference curves of different types cross once only

11. Frank Cowell: Microeconomics The single-crossing condition 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

12. Frank Cowell: Microeconomics Similarity with contract model The position of the Agent   instead of a single Agent with known ex-ante probability distribution of talents,…   … 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.

13. Frank Cowell: Microeconomics Overview... Design basics Simple model Generalisations Interpretations Design: Taxation Analogy with contract theory

14. Frank Cowell: Microeconomics A full-information solution? Consider argument based on the analysis of contracts Consider argument based on the analysis of contracts Given full information owner can fully exploit any manager Given full information owner can fully exploit any manager  Pays the minimum amount necessary  “Chooses” their effort Same basic story here Same basic story here  Can impose lump-sum tax  “Chooses” agents’ effort — no distortion But the full-information solution may be unattractive But the full-information solution may be unattractive  Informational requirements are demanding  Perhaps violation of individuals’ privacy?  So look at second-best case…

15. Frank Cowell: Microeconomics Two types Start with the case closest to the optimal contract model Start with the case closest to the optimal contract model Exactly two skill types 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 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: But let us examine the model in detail:

16. Frank Cowell: Microeconomics Second-best: two types The government’s budget constraint 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: 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: 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 The government seeks to maximise standard SWF   z a  + y a ) + [1  ]  z b  + y b )  where  is increasing and concave

17. Frank Cowell: Microeconomics Two types: model We can use a standard Lagrangean approach We can use a standard Lagrangean approach  government chooses (q, y) pairs for each type  …subject to three constraints Constraints are: 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 Choose q a  q b  y a  y b  to max  q a /  a  + y a ) + [1  ]  q b /  b  + y b )  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

18. Frank Cowell: Microeconomics Two types: method Differentiate with respect to q a  q b  y a  y b to get FOCs: 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 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 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

19. Frank Cowell: Microeconomics Two types: solution Solving the FOC we get: 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 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: Results are as for optimum-contracts model:  MRS a = MRT a  MRS b < MRT b Interpretation 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….

20. Frank Cowell: Microeconomics Two ability types: tax design 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

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

22. Frank Cowell: Microeconomics A small generalisation With three types problem becomes a bit more interesting 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 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 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

23. Frank Cowell: Microeconomics Three types Methodology is same as two-ability model Methodology is same as two-ability model  set up Lagrangean  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 : 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 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… A stepping stone to a much more interesting model…

24. Frank Cowell: Microeconomics A richer model: N+1 types The multi-type case follows immediately from three types The multi-type case follows immediately from three types Take N + l 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 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

25. Frank Cowell: Microeconomics N+1 types: behavioural constraints Participation constraint Participation constraint  is relevant for lowest type j = 0  form is as before:  y 0 +  z 0  ≥  0 Incentive-compatibility constraint 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 From previous cases we know the methodology  (and can probably guess the outcome)

26. Frank Cowell: Microeconomics N+1 types: solution Lagrangean is only slightly modified from before Lagrangean is only slightly modified from before Choose {(q j  y j )} to max Choose {(q j  y j )} to max  j=0  j    q j  j  + y j )  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 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 Now the tax schedule is determined at N+1 points

27. Frank Cowell: Microeconomics A continuum of types One more step is required in generalisation One more step is required in generalisation Suppose the tax agency is faced with a continuum of taxpayers Suppose the tax agency is faced with a continuum of taxpayers  common assumption  allows for general specification of ability distribution This case can be reasoned from the case with N + 1 types This case can be reasoned from the case with N + 1 types  allow N   From previous cases we know 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… Can proceed by analogy with previous analysis…

28. Frank Cowell: Microeconomics The continuum model Continuous ability Continuous ability  bounded support [   density f(  ) Utility for talent  as before Utility for talent  as before  y(  ) +  q(  )  Participation constraint is Participation constraint is  ) ≥  Incentive compatibility requires Incentive compatibility requires d  ) /d  ≥  SWF is SWF is   │  (  ) f  d  ⌡⌡⌡⌡

29. Frank Cowell: Microeconomics Continuum model: optimisation Lagrangean is Lagrangean is   │  (  )  f  d  │  (  )  f  d  ⌡  ⌡     +   │ [ q  − y  −  f  d  ⌡  ⌡  +  [  −    d  + │  ——  f  d  ⌡  d  ⌡  d  where  y(  ) +  q(  )  Lagrange multipliers are Lagrange multipliers are   : government budget constraint   : participation constraint   incentive-compatibility for type  Maximise Lagrangean with respect to q  and y  for all  [  Maximise Lagrangean with respect to q  and y  for all  [ 

30. Frank Cowell: Microeconomics Output and disposable income under the optimal tax 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

31. Frank Cowell: Microeconomics Continuum model: results Incentive compatibility implies Incentive compatibility implies  dy /dq > 0  optimal marginal tax rate < 100% No distortion at top implies No distortion at top implies  dy /dq = 1  zero optimal marginal tax rate! But explicit form for the optimal income tax requires 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

32. Frank Cowell: Microeconomics Overview... Design basics Simple model Generalisations Interpretations Design: Taxation Applying design rules to practical policy

33. Frank Cowell: Microeconomics Application of design principles The second-best method provides some pointers 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 Simple schemes may be worth considering  roughly correspond to actual practice  illustrate good/bad design Consider affine (linear) tax system 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: Effectively a negative income tax scheme:  (net) income related to output thus: y = B + [1  t] q  so y > q if q q if q < B / t … and vice versa

34. Frank Cowell: Microeconomics 1t1t A simple tax-benefit system 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

35. Frank Cowell: Microeconomics Violations of design principles? Sometimes the IC condition be violated in actual design Sometimes the IC condition be violated in actual design This can happen by accident: 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 Commonly known as  the “notch problem” (US)  the “poverty trap” (UK) Simple example 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

36. Frank Cowell: Microeconomics A badly designed tax-benefit system 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

37. Frank Cowell: Microeconomics Summary Optimal income tax is a standard second-best problem Optimal income tax is a standard second-best problem Elementary version a reworking of the contract model Elementary version a reworking of the contract model Can be extended to general ability distribution Can be extended to general ability distribution Provides simple rules of thumb for good design Provides simple rules of thumb for good design In practice these may be violated by well-meaning policies In practice these may be violated by well-meaning policies