1. Frank Cowell: EC426 Public Economics MSc Public Economics 2011/12 http://darp.lse.ac.uk/ec426/ Policy Design: Income Tax Frank A. Cowell 24 October 2011

2. Frank Cowell: EC426 Public Economics Overview... Design principles Simple model Generalisations Interpretations Policy Design: Income Tax Roots in social choice and asymmetric information

3. Frank Cowell: EC426 Public Economics Social values: the Arrow problem Uses weak assumptions about preferences/values Uses weak assumptions about preferences/values  Well-defined individual orderings over social states  Well-defined social ordering over social states Uses a general notion of social preferences Uses a general notion of social preferences  The constitution  A map from set of preference profiles to social preference Also weak assumptions about the constitution Also weak assumptions about the constitution  Universal Domain  Pareto Unanimity  Independence of Irrelevant Alternatives  Non-Dictatorship There’s no constitution that does all four There’s no constitution that does all four  Except in cases where there are less than three social states

4. Frank Cowell: EC426 Public Economics Social choice function A social state:  A social state:  Individual h’s evaluation of the state v h (  ) Individual h’s evaluation of the state v h (  )  A given population is indexed by h = 1,2,…, n h  A “reduced-form” utility function v h (  ). A profile: [v 1, v 2, …, v h, … ] A profile: [v 1, v 2, …, v h, … ]  An ordered list of utility functions  Set of all profiles: V A social choice function  : V →  A social choice function  : V →   For a particular profile  (v 1, v 2, …, v h, … )  Argument is a utility function not a utility level  Picks exactly one chosen element from 

5. Frank Cowell: EC426 Public Economics Implementation Is the SCF consistent with private economic behaviour? Is the SCF consistent with private economic behaviour?  Yes if the  picked out by  is also…  … the equilibrium of an appropriate economic game Implementation problem: find/design an appropriate mechanism Implementation problem: find/design an appropriate mechanism  Mechanism is a partially specified game of imperfect information…  rules of game are fixed  strategy sets are specified  preferences for the game are not yet specified Plug preferences into the mechanism: Plug preferences into the mechanism:  Does the mechanism have an equilibrium?  Does the equilibrium correspond to the desired social state  ?  If so, the social state is implementable There is a wide range of possible mechanisms There is a wide range of possible mechanisms  Example: the market as a mechanism  Given the distribution of resources and the technology…  …the market maps preferences into prices.  The prices then determine the allocation

6. Frank Cowell: EC426 Public Economics Manipulation Consider outcomes from a “direct” mechanism in two cases: Consider outcomes from a “direct” mechanism in two cases: If all, including h, tell the truth about preferences: If all, including h, tell the truth about preferences:   (v 1,…, v h, …, ) If h misrepresents his preferences but others tell the truth: If h misrepresents his preferences but others tell the truth:   (v 1,…, v h, …, ) How does the person “really” feel about  and  ? How does the person “really” feel about  and  ? If v h (  ) > v h (  ) then there is an incentive to misrepresent information If v h (  ) > v h (  ) then there is an incentive to misrepresent information If h realises this we say that  is manipulable. If h realises this we say that  is manipulable.

7. Frank Cowell: EC426 Public Economics Gibbard-Satterthwaite result Result on SCF  can be stated in several ways Result on SCF  can be stated in several ways  (Gibbard 1973, Satterthwaite, 1975 ) Gibbard 1973Satterthwaite, 1975 Gibbard 1973Satterthwaite, 1975 A standard version is: A standard version is:  If the set of social states  contains at least three elements;  and  is defined for all logically possible preference profiles  and  is truthfully implementable in dominant strategies...  then  must be dictatorial Closely related to the Arrow theorem Closely related to the Arrow theorem Has profound implications for public economics Has profound implications for public economics  Misinformation may be endemic to the design problem  May only get truth-telling mechanisms in special cases

8. Frank Cowell: EC426 Public Economics Overview... Design principles Simple model Generalisations Interpretations Policy Design: Income Tax Preferences, incomes, ability and the government Analogy with contract theory

9. Frank Cowell: EC426 Public Economics 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?

10. Frank Cowell: EC426 Public Economics Model elements A two-commodity model A two-commodity model  leisure (i.e. the opposite of effort)  consumption – a basket of all other goods  similar to optimal contracts (Bolton and Dewatripont 2005) 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

11. Frank Cowell: EC426 Public Economics 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 ≥ 

12. Frank Cowell: EC426 Public Economics Ability and income Individuals work (give up leisure) to get consumption Individuals work (give up leisure) to get 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, y Preferences can be expressed in terms of q, y  for type  utility is given by  z  + y  equivalently:  q /  + y A closer look at utility

13. Frank Cowell: EC426 Public Economics The utility function increasing preference y 1– z   =  z  + y   z  z  < 0  ≥  ≥  ≥  ≥     =  q/  + y   z  q/  < 0 increasing preference y q    Preferences over leisure and income   Indifference curves   Reservation utility   Transform into (leisure, output) space

14. Frank Cowell: EC426 Public Economics 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

15. Frank Cowell: EC426 Public Economics A full-information solution? Consider argument based on the analysis of contracts Consider argument based on the analysis of contracts Full information: owner can fully exploit any manager 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…

16. Frank Cowell: EC426 Public Economics 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:

17. Frank Cowell: EC426 Public Economics 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

18. Frank Cowell: EC426 Public Economics 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

19. Frank Cowell: EC426 Public Economics 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 positive For an interior solution, where q a  q b  y a  y b are 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

20. Frank Cowell: EC426 Public Economics Two types: solution From first-order conditions we get: From first-order conditions 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

21. Frank Cowell: EC426 Public Economics 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

22. Frank Cowell: EC426 Public Economics Overview... Design principles Simple model Generalisations Interpretations Policy Design: Income Tax Moving beyond the two-ability model

23. Frank Cowell: EC426 Public Economics 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

24. Frank Cowell: EC426 Public Economics 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…

25. Frank Cowell: EC426 Public Economics A richer model: N+1 types The multi-type case follows immediately from the three-type case The multi-type case follows immediately from the three-type case 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

26. Frank Cowell: EC426 Public Economics 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 as 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)

27. Frank Cowell: EC426 Public Economics 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

28. Frank Cowell: EC426 Public Economics A continuum of types One more step is required in generalisation One more step is required in generalisation Tax agency is faced with a continuum of taxpayers Tax agency is faced with a continuum of taxpayers  common assumption  allows for general specification of ability distribution This can be reasoned from the case with N + 1 types This 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…

29. Frank Cowell: EC426 Public Economics 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  

30. Frank Cowell: EC426 Public Economics 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  Max Lagrangean with respect to q  and y  for all  [  Max Lagrangean with respect to q  and y  for all  [ 

31. Frank Cowell: EC426 Public Economics 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

32. Frank Cowell: EC426 Public Economics Continuum model: results Incentive compatibility implies dy /dq > 0 Incentive compatibility implies dy /dq > 0  optimal marginal tax rate < 100% (Mirrlees 1971) Mirrlees 1971Mirrlees 1971 No distortion at top implies dy /dq = 1 No distortion at top implies dy /dq = 1  zero optimal marginal tax rate! (Seade 1977) Seade 1977Seade 1977  but does not generalise to incomes close to top (Tuomala 1984) Tuomala 1984Tuomala 1984  does not hold if there is no “topmost income” (Diamond 1998 ) Diamond 1998Diamond 1998 May be 0 on the lowest income May be 0 on the lowest income  depends on distribution of ability there (Ebert 1992) Ebert 1992Ebert 1992 Explicit form for the optimal income tax requires 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  (Saez 2001, Brewer et al. 2010, Mankiw 2009) Saez 2001Brewer et al. 2010Mankiw 2009Saez 2001Brewer et al. 2010Mankiw 2009

33. Frank Cowell: EC426 Public Economics Overview... Design basics Simple model Generalisations Interpretations Design: Taxation Apply design rules to practical policy…. Plus a “cut- down” version of the OIT problem

34. Frank Cowell: EC426 Public Economics Application of design principles The second-best method provides some pointers The second-best method provides some pointers  but is not a prescriptive formula  explicit form of OIT usually not possible (Salanié 2003)  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

35. Frank Cowell: EC426 Public Economics 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  Sheshinski (1972)  Analysed by Sheshinski (1972) Sheshinski (1972) Sheshinski (1972)

36. Frank Cowell: EC426 Public Economics Violations of design principles? The IC condition be violated in actual design 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

37. Frank Cowell: EC426 Public Economics 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

38. Frank Cowell: EC426 Public Economics Neglected design issues? Administrative complexity Administrative complexity Example 1. UK today (Mirrlees et al 2011) Example 1. UK today (Mirrlees et al 2011)Mirrlees et al 2011Mirrlees et al 2011 Example 2. Germany 1981-1985 : Example 2. Germany 1981-1985 :  linearly increasing marginal tax rate  quadratic tax and disposable income schedules 0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 020000400006000080000100000120000140000 rates for single person (§32a Einkommensteuergesetz); units DM:   income x up to 4,212: T = 0   4,213 to 18,000: T = 0.22x – 926   18,001 to 59,999: T = 3.05 X 4 – 73.76 X 3 + 695 X 2 + 2,200 X + 3,034   where X = x/10,000 – 18,000;   60,000 to 129,999: T = 0.09X 4 – 5.45X 3 + 88.13 X 2 + 5,040 X + 20,018   where X = x/10,000 – 60,000;   from 130,000: T = 0.56 x – 14,837

39. Frank Cowell: EC426 Public Economics Arguments for “linear” model Relatively easy to interpret parameters Relatively easy to interpret parameters Pragmatic: Pragmatic:  Approximates several countries’ tax systems  Example – piecewise linear tax in UK Sidesteps the incentive compatibility constraint Sidesteps the incentive compatibility constraint Simplified version is more tractable analytically Simplified version is more tractable analytically  Not choosing a general tax/disposable income schedule  Given t, B and the government budget constraint…  …in effect we have a single-parameter problem  See Kaplow (2008), pp 58-63

40. Frank Cowell: EC426 Public Economics Linear model: Lagrangean Social welfare is a function of individual utility Social welfare is a function of individual utility Individual utility is maximised subject to budget constraint Individual utility is maximised subject to budget constraint  Determined by individual ability  Tax parameters B and t Optimisation problem: choose B and t to max social welfare Optimisation problem: choose B and t to max social welfare  Subject to government budget constraint From maximised Lagrangean get a messy result involving From maximised Lagrangean get a messy result involving  the covariance of social marginal valuation and income  the compensated labour-supply elasticity If K = 0 then B > 0 If K = 0 then B > 0 No explicit general formula? No explicit general formula?  FOC cannot be solved to give t  covariance and elasticities will themselves be functions of t And in some cases you get a clear-cut result… And in some cases you get a clear-cut result…

41. Frank Cowell: EC426 Public Economics John Broome’s revelation Broome (1975) suggested a great simplification. Broome (1975) suggested a great simplification. Broome (1975) Broome (1975) Optimal income tax rate should be 58.6% !! Optimal income tax rate should be 58.6% !!  The basis for this astounding claim?  Tax rate is in fact 2 –  2; follows from a simple model Rather it is a useful lesson in applied modelling Rather it is a useful lesson in applied modelling He makes conventional assumptions He makes conventional assumptions  no-one has ability less than 0.707 times the average  Cobb-Douglas preferences:  “Rawlsian” max-min social welfare  Balanced budget: pure redistribution

42. Frank Cowell: EC426 Public Economics A simulation model Stern’s (1976) model of linear OIT Stern’s (1976) model of linear OIT Stern’s (1976) Stern’s (1976)  can be taken as a generalisation of Broome  simulation uses standard ingredients: Lognormal ability Lognormal ability  …more on this below Isoelastic individual utility Isoelastic individual utility  elasticity of substitution  Isoelastic social welfare Isoelastic social welfare  ()  W =  (  ) dF(  )  1 –  – 1  (  ) = ————,   1 –   inequality aversion  Variety of assumptions about government budget constraint Variety of assumptions about government budget constraint

43. Frank Cowell: EC426 Public Economics Lognormal ability 01234 0 f(w)f(w) w — —  (w; 0, 0.25 ) … …  (w; 0, 1.0 ) Two parameter distribution  (w; m, s 2 ) Two parameter distribution  (w; m, s 2 )  m is log of the median  s 2 is the variance of log income  support is [0,  ) Approximation to empirical distributions Approximation to empirical distributions  Particularly manual workers  Stern took s = 0.39 (same as Mirrlees 1971) Mirrlees 1971 Mirrlees 1971  In this case less than 2% of the population have less than 0.707 × mean (Broome 1975 ) Broome 1975Broome 1975

44. Frank Cowell: EC426 Public Economics Stern's Optimal Tax Rates  0.2 36.2 0.4 22.3 0.6 17.0 0.8 14.1 1.0 12.7 Calculations are for a purely redistributive tax: i.e. K = 0 Broome case corresponds to bottom right corner. But he assumed that there was no-one below 70.71% of the median.  62.7 47.7 38.9 33.1 29.1  92.6 83.9 75.6 68.2 62.1

45. Frank Cowell: EC426 Public Economics Summary Could we have “full information” taxation? Could we have “full information” taxation? OIT is a standard second-best problem OIT 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

46. Frank Cowell: EC426 Public Economics References (1) Bolton, P. and Dewatripont, M. (2005) Contract Theory, The MIT Press, pp 62-67. Bolton, P. and Dewatripont, M. (2005) Contract Theory, The MIT Press, pp 62-67. *Brewer, M., Saez, E. and Shephard, A. (2010) “Means-testing and Tax Rates on Earnings,” in Dimensions of Tax Design: The Mirrlees Review, Oxford University Press, Chapter 2, pp 90-164 *Brewer, M., Saez, E. and Shephard, A. (2010) “Means-testing and Tax Rates on Earnings,” in Dimensions of Tax Design: The Mirrlees Review, Oxford University Press, Chapter 2, pp 90-164Brewer, M., Saez, E. and Shephard, A. (2010)Brewer, M., Saez, E. and Shephard, A. (2010) Broome, J. (1975) “An important theorem on income tax,” Review of Economic Studies, 42, 649-652 Broome, J. (1975) “An important theorem on income tax,” Review of Economic Studies, 42, 649-652 Broome, J. (1975) Broome, J. (1975) Diamond, P.A. (1998) “Optimal Income taxation: an example with a U- Shaped pattern of optimal marginal tax rates,” American Economic Review, 88, 83-95 Diamond, P.A. (1998) “Optimal Income taxation: an example with a U- Shaped pattern of optimal marginal tax rates,” American Economic Review, 88, 83-95 Diamond, P.A. (1998) Diamond, P.A. (1998) Ebert, U. (1992) “A re-examination of the optimal non-linear income tax,” Journal of Public Economics, 49, 47-73 Ebert, U. (1992) “A re-examination of the optimal non-linear income tax,” Journal of Public Economics, 49, 47-73 Ebert, U. (1992) Ebert, U. (1992) Gibbard, A. (1973) “Manipulation of voting schemes: a general result,” Econometrica, 41, 587-60 Gibbard, A. (1973) “Manipulation of voting schemes: a general result,” Econometrica, 41, 587-60 Gibbard, A. (1973) Gibbard, A. (1973) *Kaplow, L. (2008) The Theory of Taxation and Public Economics, Princeton University Press *Kaplow, L. (2008) The Theory of Taxation and Public Economics, Princeton University Press *Mankiw, N.G., Weinzierl, M. and Yagan, D. (2009) “Optimal Taxation in Theory and Practice,” Journal of Economic Perspectives, 23, 147-174 *Mankiw, N.G., Weinzierl, M. and Yagan, D. (2009) “Optimal Taxation in Theory and Practice,” Journal of Economic Perspectives, 23, 147-174Mankiw, N.G., Weinzierl, M. and Yagan, D. (2009)Mankiw, N.G., Weinzierl, M. and Yagan, D. (2009)

47. Frank Cowell: EC426 Public Economics References (2) Mirrlees, J. A. (1971) “An exploration in the theory of the optimal income tax,” Review of Economic Studies, 38, 135-208 Mirrlees, J. A. (1971) “An exploration in the theory of the optimal income tax,” Review of Economic Studies, 38, 135-208 Mirrlees, J. A. (1971) Mirrlees, J. A. (1971) Mirrlees, J. A. et al (2011) “The Mirrlees Review: Conclusions and Recommendations for Reform,” Fiscal Studies, 32, 331–359 Mirrlees, J. A. et al (2011) “The Mirrlees Review: Conclusions and Recommendations for Reform,” Fiscal Studies, 32, 331–359 Mirrlees, J. A. et al (2011) Mirrlees, J. A. et al (2011) Saez, E. (2001) “Using elasticities to derive optimal income tax rates,” Review of Economic Studies, 68,205-22 Saez, E. (2001) “Using elasticities to derive optimal income tax rates,” Review of Economic Studies, 68,205-22 Saez, E. (2001) Saez, E. (2001) *Salanié, B. (2003) The Economics of Taxation, MIT Press, pp 59-61, 79-109 *Salanié, B. (2003) The Economics of Taxation, MIT Press, pp 59-61, 79-109 Satterthwaite, M. A. (1975) “Strategy-proofness and Arrow's conditions, Journal of Economic Theory, 10, 187-217 Satterthwaite, M. A. (1975) “Strategy-proofness and Arrow's conditions, Journal of Economic Theory, 10, 187-217 Satterthwaite, M. A. (1975) Satterthwaite, M. A. (1975) Seade, J. (1977) “On the shape of optimal tax schedules,” Journal of Public Economics, 7, 203-23 Seade, J. (1977) “On the shape of optimal tax schedules,” Journal of Public Economics, 7, 203-23 Seade, J. (1977) Seade, J. (1977) Sheshinski, E. (1972) “The optimal linear income tax,” Review of Economic Studies, 39, 297-302 Sheshinski, E. (1972) “The optimal linear income tax,” Review of Economic Studies, 39, 297-302 Sheshinski, E. (1972) Sheshinski, E. (1972) Stern, N. (1976) “On the specification of models of optimum income taxation” Journal of Public Economics, 6,123-162 Stern, N. (1976) “On the specification of models of optimum income taxation” Journal of Public Economics, 6,123-162 Stern, N. (1976) Stern, N. (1976) Tuomala, M. (1984) “On the Optimal Income Taxation: Some Further Numerical Results,” Journal of Public Economics, 23, 351-366 Tuomala, M. (1984) “On the Optimal Income Taxation: Some Further Numerical Results,” Journal of Public Economics, 23, 351-366 Tuomala, M. (1984) Tuomala, M. (1984)