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Frank Cowell: UB Public Economics Optimal Tax Design June 2005 Public Economics: University of Barcelona Frank Cowell

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1 Frank Cowell: UB Public Economics Optimal Tax Design June 2005 Public Economics: University of Barcelona Frank Cowell

2 Frank Cowell: UB Public Economics Purpose of tax design The issue of design is fundamental to public economics The issue of design is fundamental to public economics Move from what we would like to achieve… Move from what we would like to achieve… …to what we can actually implement …to what we can actually implement Plenty of examples of this issue: Plenty of examples of this issue:  Public-good provision  Regulation  Social insurance  Optimal taxation – see below. Important to be clear what the purpose of the tax design problem is. Important to be clear what the purpose of the tax design problem is. A brief review of the elements of the problem. … A brief review of the elements of the problem. …

3 Frank Cowell: UB Public Economics Components of the problem Objectives Objectives  Could be an attempt to satisfy a particular objective function or class of functions  Could be a characterisation of policies that achieve some broad objectives. Scope for policy Scope for policy  Methods of intervention  Constraints  Informational problems Available tools Available tools  The tax base  Direct and indirect taxation

4 Frank Cowell: UB Public Economics Overview... Design Issues General labour model Generalisations Optimal Income Taxation Why this kind of problem is set up “linear” labour model Education model Objectives Scope for policy Informational issues Available tools

5 Frank Cowell: UB Public Economics Specific objectives? The objectives of the tax design could include: The objectives of the tax design could include: 1. Bergson-Samuelson welfare maximisation 2. Overall concern for efficiency 3. Overall concern for reduction of inequality of outcome. 4. Inequality of opportunity 5. Poverty, horizontal inequity... More than one of the above may be relevant. More than one of the above may be relevant. Could be a class of functions Could be incorporated in objective #1

6 Frank Cowell: UB Public Economics Implementation of objectives What is domain of the SWF? What is domain of the SWF?  Incomes?  Individual utilities? What social entities? What social entities?  Individuals  Families  Household units? Need a model of cardinal, comparable utility Welfarist approach usually founded on this basis Data is often on this basis… …or this

7 Frank Cowell: UB Public Economics Overview... Design Issues General labour model Generalisations Optimal Income Taxation Types of intervention. The tax base “linear” labour model Education model Objectives Scope for policy Informational issues Available tools

8 Frank Cowell: UB Public Economics Scope for policy What is potentially achievable? What is potentially achievable? We need to do this before we can examine specific policy tools and their associated constraints. We need to do this before we can examine specific policy tools and their associated constraints. If we have in mind income redistribution it is appropriate to look at the determinants of income If we have in mind income redistribution it is appropriate to look at the determinants of income Do this within the context of an elementary microeconomic model. Do this within the context of an elementary microeconomic model.

9 Frank Cowell: UB Public Economics Take the standard microeconomic model of a person’s total income in a market economy Take the standard microeconomic model of a person’s total income in a market economy Composed of resources valued at their market prices: Composed of resources valued at their market prices: income endowment of good i Non-market income Does this mean public policy has to be limited to Does this mean public policy has to be limited to 1. redistributing resources, or 2. manipulating prices? There could be other forms of income There could be other forms of income The Composition of Income price of good i. Problems with 1 and 2 above are also important Problems with 1 and 2 above are also important And there may be other types of intervention And there may be other types of intervention

10 Frank Cowell: UB Public Economics Problems with redistributing resources: The lump-sum tax issue: The lump-sum tax issue:  Special information – such as personal characteristics  Political problems of implementation Non-transferability Non-transferability  Fixed resources  Inalienability of certain rights – No slavery Ways of getting round these problems? Ways of getting round these problems?  Could redistribute the purchasing power generated by the resource?  Or modify the supply of “co-operant factors”?

11 Frank Cowell: UB Public Economics Problems with price manipulation Identification of commodities Identification of commodities  The boundary problem  Artificial definition of a good or service on which a tax is to be levied. Complexity Complexity  Proliferation of implied pricing structures Informational problems Informational problems  Uncertainty leads to wrong price signals?  Misinformation leads to wrong price signals?  May even be missing markets Need to focus on economics of information Need to focus on economics of information

12 Frank Cowell: UB Public Economics Overview... Design Issues General labour model Generalisations Optimal Income Taxation Fundamental theoretical issues in design problem “linear” labour model Education model Objectives Scope for policy Informational issues Available tools

13 Frank Cowell: UB Public Economics Informational issues in microeconomics There are two key types of informational problem: There are two key types of informational problem: Both can be relevant to policy design. Both can be relevant to policy design. Hidden action: Hidden action:  Regulation and optimal contracts.  Moral hazard in social insurance  Compliance issues. Hidden information: Hidden information:  Problems of “tailoring” tax rates.  Adverse selection in social insurance.  Focus on this issue here But the “information issue” is quite deep: But the “information issue” is quite deep:  There is connection with discussion of social welfare  A fundamental relationship with the “Arrow” problem

14 Frank Cowell: UB 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

15 Frank Cowell: UB Public Economics Social-choice function Similar to the concept of constitution Similar to the concept of constitution But maps from set of preference profiles to set of social states But maps from set of preference profiles to set of social states  Given a particular set of preferences for the population  Picks out the preferred social state Not surprising to find result similar to Arrow Not surprising to find result similar to Arrow  Introduce weak conditions on the Social-choice function  There’s no SCF that satisfies all of them But key point concerns the implementation issue But key point concerns the implementation issue

16 Frank Cowell: UB Public Economics Implementation Is the social-choice function consistent with private economic behaviour? Is the social-choice function consistent with private economic behaviour? Yes if the social state picked out by the SCF corresponds to an equilibrium Yes if the social state picked out by the SCF corresponds to an equilibrium Problem becomes finding an appropriate mechanism Problem becomes finding an appropriate mechanism  mechanism can be thought of as a kind of cut-down game  to be interesting the game is one of imperfect information  is the desired social state an equilibrium of the game? There is a wide range of possible mechanisms There is a wide range of possible mechanisms Focus on a type that is useful for expositional purposes... Focus on a type that is useful for expositional purposes...

17 Frank Cowell: UB Public Economics Direct mechanisms Map from collection of preferences to states Map from collection of preferences to states  Involves a very simple game.  The game is “show me your utility function”  Enables us to focus directly on the informational aspects of implementation Here the SCF is the mechanism itself Here the SCF is the mechanism itself An SCF that encourages misrepresentation may be of limited use An SCF that encourages misrepresentation may be of limited use Is truthful implementation possible? Is truthful implementation possible?  Will people announce their true attributes?  Will it be a dominant strategy to do so? Introduce another key result Introduce another key result

18 Frank Cowell: UB Public Economics Gibbard-Satterthwaite result Can be stated in a variety of ways. Can be stated in a variety of ways. A standard versions is: A standard versions is:  If the set of social states contains at least three elements; ...and the social choice function is defined for the all logically possible preference profiles... ...and the SCF is truthfully implementable in dominant strategies... ...then the SCF 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  Underlies issues of public-good provision, regulation, tax design

19 Frank Cowell: UB Public Economics Overview... Design Issues General labour model Generalisations Optimal Income Taxation What practical options available to achieve the objectives? “linear” labour model Education model Objectives Scope for policy Informational issues Available tools

20 Frank Cowell: UB Public Economics Informational issues in taxation What distinguishes taxation from highway robbery? What distinguishes taxation from highway robbery?  Taxation principles  Appropriate information What information is/should be available? What information is/should be available?  Attributes  Behaviour

21 Frank Cowell: UB Public Economics Available tools Availability determined by a variety of considerations. Availability determined by a variety of considerations. Fundamental economic constraints Fundamental economic constraints Institutional constraints. These may come from: Institutional constraints. These may come from:  Legal restrictions  Administrative considerations  Historical precedent But each of these institutional aspects may really follow from the economics. But each of these institutional aspects may really follow from the economics.

22 Frank Cowell: UB Public Economics We focus here on the taxation of individuals rather than corporations or other entities. We focus here on the taxation of individuals rather than corporations or other entities. An approach to the individual tax-base might begin with an examination of the individual’s budget constraint: An approach to the individual tax-base might begin with an examination of the individual’s budget constraint: The Tax Base expenditure consumption of good i number of goods So taxation might be based on consumption of specific goods or on some concept of income or expenditure So taxation might be based on consumption of specific goods or on some concept of income or expenditure We will see that using the above as an elementary method of classifying taxes can be misleading We will see that using the above as an elementary method of classifying taxes can be misleading First take a closer look at income: First take a closer look at income: income

23 Frank Cowell: UB Public Economics It is tempting to think of the distinction between different types of tax in terms of the budget constraint: It is tempting to think of the distinction between different types of tax in terms of the budget constraint: A fundamental difference? Indirect taxes here? Direct taxes here? This misses the point This misses the point Any tax on RHS can be converted to tax on LHS Any tax on RHS can be converted to tax on LHS Real question is about information Real question is about information.

24 Frank Cowell: UB Public Economics Information again The government and its agencies must work with imperfect information. The government and its agencies must work with imperfect information. To model taxes appropriately need to take this into account. To model taxes appropriately need to take this into account. Information imposes specific constraints on tax design Information imposes specific constraints on tax design In a typical market economy there are two main types of information: In a typical market economy there are two main types of information:  About individuals  About transactions Income Total expenditure Age, marital status? Income Total expenditure Age, marital status? Expenditure by product category Expenditure by industry Input and output quantities Expenditure by product category Expenditure by industry Input and output quantities

25 Frank Cowell: UB Public Economics Fundamental constraints... Public budget constraints Public budget constraints  Example: In simple redistribution sum of net receipts (taxes  cash subsidies) must be zero Participation constraints Participation constraints  Example: Labour supply Incentive-compatibility (self-selection) constraints Incentive-compatibility (self-selection) constraints  Example: Differential subsidies for specific commodities

26 Frank Cowell: UB Public Economics Design basics: summary Objectives follow on logically from our discussion in previous lectures. Objectives follow on logically from our discussion in previous lectures. Beware of oversimplifying assumptions about the tax base. Beware of oversimplifying assumptions about the tax base. Information plays a key role. Information plays a key role.

27 Frank Cowell: UB Public Economics Overview... Design Issues General labour model Generalisations Optimal Income Taxation What types of tax formula used in theory and practice? “Linear” labour model Education model Tax schedules Outline of problem The solution

28 Frank Cowell: UB Public Economics Income tax – example of design problem Standard types of tax Standard types of tax  simple examples  integration with income support General issues of how to set up an optimisation problem General issues of how to set up an optimisation problem Solution of optimal tax problem: Solution of optimal tax problem:  Solution of the general tax design problem  Solution of the special “linear” case  Alternative models of optimal income tax

29 Frank Cowell: UB Public Economics Income tax – notation y – taxable income y – taxable income c – disposable income (“consumption”) c – disposable income (“consumption”) T(·) – tax schedule T(·) – tax schedule c(·) – disposable income schedule c(·) – disposable income schedule  – marginal tax rate  – marginal tax rate y 0 – exemption-level income y 0 – exemption-level income B – lumpsum benefit/guaranteed income B – lumpsum benefit/guaranteed income 0   1 c(y) = y – T(y)

30 Frank Cowell: UB Public Economics Income space y c=c(y) pre-tax income disposable income no-intervention line

31 Frank Cowell: UB Public Economics The simple income tax y c=c(y) 1-  y0y0 Exemption level Marginal retention rate

32 Frank Cowell: UB Public Economics...extended to Negative Income Tax y c=c(y) 1-  B y0y0 B =  y 0 Incomes subsidised through NIT

33 Frank Cowell: UB Public Economics How to generalise this approach…? Other functional forms of the income tax Other functional forms of the income tax Administrative complexity of IT Administrative complexity of IT Interaction with other contingent taxes and benefits. Interaction with other contingent taxes and benefits.

34 Frank Cowell: UB Public Economics Increasing marginal tax rate  y c(y)c(y)

35 Frank Cowell: UB Public Economics Example 1 UK: UK:  piecewise linear tax  stepwise jumps in MTR  compare with contingent tax/benefit model Germany: Germany:  linearly increasing marginal tax rate  quadratic tax and disposable income schedules

36 Frank Cowell: UB Public Economics Germany , single person (§32a Einkommensteuergesetz): up to 4,212: T = 0 up to 4,212: T = 0 4,213 to 18,000: T = 0.22y – 926 4,213 to 18,000: T = 0.22y – ,001 to 59,999: T = 3.05 z 4 – z z 2 + 2,200 z + 3,034 18,001 to 59,999: T = 3.05 z 4 – z z 2 + 2,200 z + 3,034 z = y/10, ,000; z = y/10, ,000; 60,000 to 129,999: T = 0.09z 4 – 5.45z z 2 + 5,040 z + 20,018 60,000 to 129,999: T = 0.09z 4 – 5.45z z 2 + 5,040 z + 20,018 z = y/10, ,000; from 130,000: T = 0.56 y – 14,837 from 130,000: T = 0.56 y – 14,837 (units: DM) (units: DM) Example ,000 20,000 30,000 40,000 50,000 60,000 70,000 80,

37 Frank Cowell: UB Public Economics Interaction with income support y c(y)c(y) B y1y1 y2y2 y0y0 0 Untaxed income support “Clawback” of support Tax-payments kick in with benefits Straight income tax at constant marginal rate

38 Frank Cowell: UB Public Economics The approach to IT – summary The “linear” form may be a reasonable approximation to some practical cases The “linear” form may be a reasonable approximation to some practical cases We may also see an appealing intuitive argument for linearity as simplification We may also see an appealing intuitive argument for linearity as simplification “Income tax” may need to be interpreted fairly broadly “Income tax” may need to be interpreted fairly broadly Interaction amongst various forms of government intervention is important for an appropriate model Interaction amongst various forms of government intervention is important for an appropriate model This may lead to nonlinearity in the effective schedule This may lead to nonlinearity in the effective schedule

39 Frank Cowell: UB Public Economics Overview... Design Issues General labour model Generalisations Optimal Income Taxation Basic ingredients of OIT analysis. “Linear” labour model Education model Tax schedules Outline of problem The solution

40 Frank Cowell: UB Public Economics Basic Ingredients of An Optimal Income Tax model A distribution of abilities A distribution of abilities Individuals’ behaviour Individuals’ behaviour Social-welfare function Social-welfare function Feasibility Constraint Feasibility Constraint Restriction on types of functional form Restriction on types of functional form What resources are potentially available for redistribution?

41 Frank Cowell: UB Public Economics Distribution of Ability... Assume... a single source of earning power – “ability” a single source of earning power – “ability” ability is fully reflected in the (potential) wage w ability is fully reflected in the (potential) wage w So ability is effectively measured by w. So ability is effectively measured by w. the distribution F of w is observable the distribution F of w is observable individual values of w are not observable by the tax authority individual values of w are not observable by the tax authority

42 Frank Cowell: UB Public Economics Can we infer the distribution of ability? Practical approach Practical approach Select relevant group or groups in population. Select relevant group or groups in population.  male manual workers? Choose appropriate earnings concept. Choose appropriate earnings concept.  Full time earnings? Divide earnings by hours to get wages. Divide earnings by hours to get wages. Use parametric model to capture shape of distribution. Use parametric model to capture shape of distribution.  Lognormal?

43 Frank Cowell: UB Public Economics Distribution: example Example from UK 2000 Example from UK 2000 Gives distribution of y=wh for full-time male manual workers Gives distribution of y=wh for full-time male manual workers

44 Frank Cowell: UB Public Economics Basic Ingredients of An Optimal Income Tax model (2) A distribution of abilities A distribution of abilities Individuals’ behaviour Individuals’ behaviour Social-welfare function Social-welfare function Feasibility Constraint Feasibility Constraint Restriction on types of functional form Restriction on types of functional form In what ways do we assume that people will respond to the tax authority’s instruments ?

45 Frank Cowell: UB Public Economics The individual's problem Individual’s utility is determined by disposable income (consumption) c and leisure. Individual’s utility is determined by disposable income (consumption) c and leisure. So the optimisation problem can be written So the optimisation problem can be written  max h U(c,h)  subject to c = y – T(y)  and y = wh This yields maximised utility as a function of ability (wage): This yields maximised utility as a function of ability (wage): (w)(w)(w)(w)

46 Frank Cowell: UB Public Economics A Characterisation of Tastes Introduce a definition to capture the shape of individual preferences Introduce a definition to capture the shape of individual preferences Normalised MRS The following restriction on “regularity” of preferences is important for clean-cut results The following restriction on “regularity” of preferences is important for clean-cut results The way slope of indifference curve changes with ability

47 Frank Cowell: UB Public Economics A representation of preferences (consumption) (leisure)

48 Frank Cowell: UB Public Economics Indifference curve in (h,c)-space h c (consumption) (hours worked)

49 Frank Cowell: UB Public Economics Contour translated to (y,c)-space c y (consumption = net income) (gross income) slope = q

50 Frank Cowell: UB Public Economics The regularity condition y c low w high w ll ll ll ll Illustrates the q w < 0 property Ensures “single-crossing” of ICs for different ability groups Illustrates the q w < 0 property Ensures “single-crossing” of ICs for different ability groups

51 Frank Cowell: UB Public Economics Individual's problem: points to note Incorporates standard assumptions Incorporates standard assumptions Same basic model as in earlier lectures Same basic model as in earlier lectures Consistent with the model Consistent with the model  y = wh or with the model or with the model  y = wh + I

52 Frank Cowell: UB Public Economics Basic Ingredients of An Optimal Income Tax model (3) A distribution of abilities A distribution of abilities Individuals’ behaviour Individuals’ behaviour Social-welfare function Social-welfare function Feasibility Constraint Feasibility Constraint Restriction on types of functional form Restriction on types of functional form How to represent the objectives of the optimisation problem?

53 Frank Cowell: UB Public Economics The Government’s Objective... Social evaluation function Take a standard version of the SWF Take a standard version of the SWF Assume additive separability Assume additive separability Take “weighted average” over types Take “weighted average” over types Maximised utility of a w-type person Proportion of w-type persons in the population

54 Frank Cowell: UB Public Economics Basic Ingredients of An Optimal Income Tax model (4) A distribution of abilities A distribution of abilities Individuals’ behaviour Individuals’ behaviour Social-welfare function Social-welfare function Feasibility Constraint Feasibility Constraint Restriction on types of functional form Restriction on types of functional form Real-world restrictions on government and the design problem

55 Frank Cowell: UB Public Economics Main types of constraint The Government’s budget constraint The Government’s budget constraint Incentive compatibility Incentive compatibility

56 Frank Cowell: UB Public Economics Government’s Budget Constraint disposable income in population Earnings in population Net revenue requirement

57 Frank Cowell: UB Public Economics Two Ability Levels y c low w high w ll ll ll ll Incentive compatibility problem: Original and disposable income must increase with ability Incentive compatibility problem: Original and disposable income must increase with ability c(y)c(y)

58 Frank Cowell: UB Public Economics Basic Ingredients of An Optimal Income Tax model (5) A distribution of abilities A distribution of abilities Individuals’ behaviour Individuals’ behaviour Social-welfare function Social-welfare function Feasibility Constraint Feasibility Constraint Restriction on types of functional form Restriction on types of functional form

59 Frank Cowell: UB Public Economics Restrictions on form... It may make sense to consider cases where marginal tax-rate is everywhere constant (like the NIT model earlier): It may make sense to consider cases where marginal tax-rate is everywhere constant (like the NIT model earlier): Administrative costs of general model Administrative costs of general model informational problems informational problems “fairness” arguments [?] “fairness” arguments [?] Pre-1985 Germany? Cf the US “flat tax” discussion Lack of detail about tails

60 Frank Cowell: UB Public Economics Review: ingredients of OIT model A distribution of abilities A distribution of abilities  Ability is measured by potential wage w.  Distribution F of w is observable  Individual values of w are not observable Individuals’ behaviour Individuals’ behaviour  max h U(c,h), subject to c = y – T(y) and y = wh  Gives utility as function of ability  (w) Social-welfare function Social-welfare function  Assume individualistic additively separable SWF  W =  u(  (w) f(w) dw Feasibility constraints Feasibility constraints  The Government’s budget constraint  Incentive compatibility Restriction on types of functional form Restriction on types of functional form  (Piecewise) linear schedule?

61 Frank Cowell: UB Public Economics Overview... Design Issues General labour model Generalisations Optimal Income Taxation Characterising a general optimal income tax. “Linear” labour model Education model Tax schedules Outline of problem The solution

62 Frank Cowell: UB Public Economics The general OIT model No preconditions on form of income tax No preconditions on form of income tax Use results from economics of information Use results from economics of information  IC condition required for “sensible” results  However, no consideration of administrative complexity Use a general variational approach to give the solution Use a general variational approach to give the solution  Has similarities with techniques used for optimal growth  Terminal conditions can be important Illustrate the variational approach diagrammatically Illustrate the variational approach diagrammatically  Use the disposable income schedule c()

63 Frank Cowell: UB Public Economics The general tax-design problem c(y)c(y) y Variation in general tax schedule

64 Frank Cowell: UB Public Economics Individual optimisation Begin with the way each tax-payer is assumed to act. Begin with the way each tax-payer is assumed to act. The optimisation problem can be written The optimisation problem can be written Disposable income work The function c(·) is chosen by the government The function c(·) is chosen by the government Define normalised MRS Define normalised MRS Slope of disposable income function The solution is of the form  (w) := max h U(c(wh), h) The solution is of the form  (w) := max h U(c(wh), h) The first-order condition is The first-order condition is

65 Frank Cowell: UB Public Economics Contour in (y,c)-space c y slope = q Proportional to work Disposable income schedule c() optimised income

66 Frank Cowell: UB Public Economics A regularity condition y c low w high w ll ll ll ll If the property q w < 0 holds this ensures “single-crossing” of ICs for different ability groups

67 Frank Cowell: UB Public Economics Incentive compatibility condition y c low w high w ll ll ll ll Design of c must ensure that utility and income increase with ability c(y)c(y)

68 Frank Cowell: UB Public Economics Incentive compatibility The IC condition means that high ability people should not have an incentive to “masquerade” as low ability. The IC condition means that high ability people should not have an incentive to “masquerade” as low ability. This requires maximised utility  (w) to increase in w. This requires maximised utility  (w) to increase in w. By differentiation of the solution function we have By differentiation of the solution function we have Optimised value of h If c(·) is monotonic and differentiable everywhere then this becomes If c(·) is monotonic and differentiable everywhere then this becomes But if these conditions are violated, problems arise… But if these conditions are violated, problems arise…

69 Frank Cowell: UB Public Economics Fundamental design problem It may seem odd that the IC condition be violated in actual design It may seem odd that the IC condition be violated in actual design But it can happen by accident: But it can happen by accident:  interaction between income support and income tax.  generated by the desire to “target” support more effectively.  A well-meant gross inefficiency? Commonly known as Commonly known as  The “notch problem” (US)  The “poverty trap” (UK)

70 Frank Cowell: UB Public Economics “Notch problem” / “poverty trap” c(y)c(y) y y0y0 Withdrawal of benefit here Discontinuous non- monotonic c(·)

71 Frank Cowell: UB Public Economics Violation of IC condition c(y)c(y) y y0y0 low w Where high w “should” be Where high w would choose

72 Frank Cowell: UB Public Economics Government: maximisation Uses a constrained maximum method Uses a constrained maximum method But there is a constraint at each ability level from w min to w max. But there is a constraint at each ability level from w min to w max. Similar to maximisation over time. Similar to maximisation over time. Choose c( · ) to max Choose c( · ) to max subject to subject to and, at each ability level: and, at each ability level: disposable income in population Earnings in population Net revenue requirement

73 Frank Cowell: UB Public Economics Government: maximisation Introduce a Lagrange multiplier for the budget constraint Introduce a Lagrange multiplier for the budget constraint and a multiplier  (w) for the incentive compatibility constraint at each ability level. and a multiplier  (w) for the incentive compatibility constraint at each ability level. Then, on rearranging, the Lagrangean is Then, on rearranging, the Lagrangean is

74 Frank Cowell: UB Public Economics General Model: Characterisation of Marginal Tax Rate Lagrange multiplier for incentive-compatibility constraint Lagrange multiplier for Government budget constraint

75 Frank Cowell: UB Public Economics Interpreting the FOC Can be used to give us an impression of the shape of the solution Can be used to give us an impression of the shape of the solution But an explicit form for the OIT is usually not possible But an explicit form for the OIT is usually not possible Some key results Some key results  First for the overall shape  Second for what happens at each end of the ability range…

76 Frank Cowell: UB Public Economics Main result 1 Mirrlees 1971: The optimal marginal tax rate must be greater than or equal to 0 and less than 1 Mirrlees 1971: The optimal marginal tax rate must be greater than or equal to 0 and less than 1 The condition “  0” means that in trying to raise tax it never makes sense to introduce a distortionary labour subsidy − see Tuomala (1990) The condition “  0” means that in trying to raise tax it never makes sense to introduce a distortionary labour subsidy − see Tuomala (1990) The condition “<1” follows from The condition “<1” follows from Agent monotonicity implies Agent monotonicity implies So So it is immediate that T'(y) < 1. For the lower extreme of the distribution need to look

77 Frank Cowell: UB Public Economics Bunching: w'

78 Frank Cowell: UB Public Economics No bunching: w'

79 Frank Cowell: UB Public Economics Main result 2 Seade 1977,Ebert The optimal marginal tax rate: Seade 1977,Ebert The optimal marginal tax rate:  is 0 on the highest income  is 0 on the lowest income if there is no bunching  is positive on the lowest income if there is bunching For bottom of distribution see Tuomala (1990) For bottom of distribution see Tuomala (1990) For top of distribution note the FOC: For top of distribution note the FOC: At w max IC constraint becomes irrelevant; so  (w max ) = 0. At w max IC constraint becomes irrelevant; so  (w max ) = 0. Therefore y max ) = w max h(w max )) = 0 Therefore T'(y max ) = T'(w max h(w max )) = 0

80 Frank Cowell: UB Public Economics Problems of the general model (1) There appear to be commonsense general results There appear to be commonsense general results And clear-cut results for the extremes, And clear-cut results for the extremes, But little guidance on the structure for the majority of the workforce. But little guidance on the structure for the majority of the workforce. Some broad principles can be adduced from the first order conditions. Some broad principles can be adduced from the first order conditions. But you cannot get further without an explicit model But you cannot get further without an explicit model  On the distribution of w  On individual preferences  On the SWF

81 Frank Cowell: UB Public Economics Problems of the general model (2) The results for the extremes are not robust The results for the extremes are not robust Should have low or decreasing tax rates close to the top of the income distribution? Should have low or decreasing tax rates close to the top of the income distribution? This does not seem to be the case from simulation study Tuomala: J. Pub. Econ 1984 This does not seem to be the case from simulation study Tuomala: J. Pub. Econ 1984 Part of the problem arises from assumed F() of w Part of the problem arises from assumed F() of w  convenient to assume that support of the distribution F is finite  But this means an artificial assumption about known “endpoints”

82 Frank Cowell: UB Public Economics Problems of the general model (3) Most applied models assume something like lognormal or Pareto Most applied models assume something like lognormal or Pareto  Support is unbounded above.  No “maximum” income If you rework the model with a distribution that is “open-ended” at the top things appear very different. If you rework the model with a distribution that is “open-ended” at the top things appear very different.  Diamond (1998) uses Pareto.  Gets high marginal tax rates where ability follows a Pareto distribution  Saez (2001) is a general extension of the Mirrlees results

83 Frank Cowell: UB Public Economics Problems of the general model (4) Cannot get stronger results on tax rates analytically. Cannot get stronger results on tax rates analytically.  Can do this for special cases  Example Salanie model with quasi-linear preferences  Or could use simulation in a numerical model But to do this you need to implement a specific model which can be: But to do this you need to implement a specific model which can be:  Computationally messy  Sensitive to specific assumptions made about labour supply and ability It may make sense to impose more structure a priori on the tax function It may make sense to impose more structure a priori on the tax function

84 Frank Cowell: UB Public Economics Overview... Design Issues General labour model Generalisations Optimal Income Taxation A “cut-down” version of the labour-leisure problem “linear” labour model Education model

85 Frank Cowell: UB Public Economics Approach 2: The linear model Same behavioural assumptions as before Same behavioural assumptions as before Same objectives Same objectives Restriction to linear (affine) tax functions: two parameters Restriction to linear (affine) tax functions: two parameters First analysed by Sheshinski (1972) First analysed by Sheshinski (1972)

86 Frank Cowell: UB Public Economics Simplified version is much more tractable analytically Simplified version is much more tractable analytically No longer choosing a general tax/disposable income schedule c() No longer choosing a general tax/disposable income schedule c() Instead, just a two-parameter model. Instead, just a two-parameter model. Disposable income is Disposable income is Linear Model: outline Marginal tax rate Pre-tax income Minimum disposable income

87 Frank Cowell: UB Public Economics Arguments for “linear” model Relatively easy to interpret parameters Relatively easy to interpret parameters   as uniform marginal tax rate  B as minimum income, or…  B /  as exemption rate Pragmatic: Pragmatic:  Approximates several countries’ tax systems  Example – piecewise linear tax in UK Sidesteps the incentive compatibility constraint… Sidesteps the incentive compatibility constraint…

88 Frank Cowell: UB Public Economics Incentive compatibility resolved y c low w high w ll ll ll ll Original and disposable income will increase with ability c(y)c(y)

89 Frank Cowell: UB Public Economics In effect this makes the issue a one-variable problem… In effect this makes the issue a one-variable problem… Given that the IC condition vanishes, there is only one constraint Given that the IC condition vanishes, there is only one constraint The Government Budget constraint: The Government Budget constraint: Linear Model: Constraint Tax raised on working population Minimum guaranteed income for all Extra revenue requirement

90 Frank Cowell: UB Public Economics Take the linear income tax... y c(y)c(y) 1-  B Note Marginal tax-rate is constant:  If B>0 average tax-rate  –B/y is everywhere rising with income

91 Frank Cowell: UB Public Economics Higher B needs higher  y c(y)c(y) B+BB+B 1- 

92 Frank Cowell: UB Public Economics The constrained optimisation problem can be set up as the Lagrangean: The constrained optimisation problem can be set up as the Lagrangean: Linear Model: Lagrangean Social-welfare function Lagrange multiplier Maximise Lagrangean by choice of tax instruments  and B Maximise Lagrangean by choice of tax instruments  and B This can be done using classical optimisation methods. This can be done using classical optimisation methods. Government budget constraint

93 Frank Cowell: UB Public Economics Linear Model: FOC (1) Consider the social value of $1 lump-sum income. Consider the social value of $1 lump-sum income. This is defined as: This is defined as: Maximised utility Differentiating the Lagrangean with respect to B: Differentiating the Lagrangean with respect to B: Average social value of $1 should be 1

94 Frank Cowell: UB Public Economics Again the formula can be used to give guidance on policy… Again the formula can be used to give guidance on policy… Differentiating the Lagrangean with respect to  and rearranging we get: Differentiating the Lagrangean with respect to  and rearranging we get: Linear Model: FOC (2) Compensated labour- supply elasticity Optimal marginal tax rate Covariance of social marginal valuation and income

95 Frank Cowell: UB Public Economics Outcomes from the linear model If R = 0 then B > 0 If R = 0 then B > 0  Implies progressive taxation. FOC cannot be solved to give an explicit formula FOC cannot be solved to give an explicit formula  The covariance and the elasticities will themselves be functions of . However the “natural” restriction imposed by linearity makes construction of simulation easier However the “natural” restriction imposed by linearity makes construction of simulation easier Better behaved at special points of the distribution Better behaved at special points of the distribution

96 Frank Cowell: UB Public Economics Components of simulation Structure of ability (wage) distribution Structure of ability (wage) distribution  Empirically determined? Individual preferences Individual preferences  Determines labour supply responses Social welfare function Social welfare function  Use evidence from social surveys etc? Government budget constraint Government budget constraint  Experiment with alternative assumptions

97 Frank Cowell: UB Public Economics John Broome suggested a great simplification for OIT. John Broome suggested a great simplification for OIT.  * = 58.6% !!  * = 58.6% !!  The basis for this astounding claim? The basis for this astounding claim? When we spot that the tax rate is in fact 2 –  2 the remark is not so outlandish When we spot that the tax rate is in fact 2 –  2 the remark is not so outlandish Rather it serves as a useful lesson in applied modelling Rather it serves as a useful lesson in applied modelling Broome’s revelation

98 Frank Cowell: UB Public Economics Take standard Cobb-Douglas preferences: Take standard Cobb-Douglas preferences: Broome’s model… Make the (empirically relevant?) assumption that no-one has ability less than times the average: Make the (empirically relevant?) assumption that no-one has ability less than times the average: “Rawlsian” max-min social welfare: “Rawlsian” max-min social welfare: Balanced budget: Balanced budget: But in UK 2000: 1 1Average wage was £10.53 / hour 2 2Min wage was £4.10! But in UK 2000: 1 1Average wage was £10.53 / hour 2 2Min wage was £4.10!

99 Frank Cowell: UB Public Economics The Stern simulation model Stern’s (1976) model is less tongue-in-cheek. Stern’s (1976) model is less tongue-in-cheek. But can be taken as a generalisation of Broome. But can be taken as a generalisation of Broome. Also based on a linear OIT Also based on a linear OIT Ingredients are: Ingredients are:  Lognormal ability  Isoelastic utility  Isoelastic social welfare  A variety of assumptions about the government budget constraint

100 Frank Cowell: UB Public Economics Representation of ability distribution Simple two parameter distribution Simple two parameter distribution   (w; m, s 2 )  First parameter m is log of the median  The second parameter s 2 is itself an inequality index – the variance of log income. Support is [0,  ) Support is [0,  ) Not a bad approximation to empirical distributions Not a bad approximation to empirical distributions  Particularly for manual workers  Stern assumed s = 0.39 (same as Mirrlees)  In this case less than 2% of the population have less than × mean (Broome)

101 Frank Cowell: UB Public Economics The lognormal distribution f(w)f(w) w — —  (w; 0, 0.25 ) … …  (w; 0, 1.0 )

102 Frank Cowell: UB Public Economics Becomes the Broome model in the case  =1 Becomes the Broome model in the case  =1 Take an empirically relevant version of household utility: Take an empirically relevant version of household utility: Isoelastic utility hours worked consumption elasticity of Substitution (  0) elasticity of Substitution (  0)

103 Frank Cowell: UB Public Economics Labour Supply and Income... Define Define  w Could have backward-bending labour supply if  <1

104 Frank Cowell: UB Public Economics Resulting Labour Supply and Income… (Broome case)

105 Frank Cowell: UB Public Economics Standard SWF Take additive form of Bergson-Samuelson SWF: Take additive form of Bergson-Samuelson SWF: u()W = u() dF()u()W = u() dF() u()f()= u() f() du()f()= u() f() d Use the iso-elastic form of the (social) u-function: Use the iso-elastic form of the (social) u-function:  1 –  – 1 u(  ) = ————,   1 –  Bentham corresponds to the case Bentham corresponds to the case  Max-min (“Rawls”) corresponds to the case Max-min (“Rawls”) corresponds to the case 

106 Frank Cowell: UB Public Economics Stern's Optimal Income Tax Rates  Notes: Calculations are for a purely redistributive tax: i.e. R = 0 Broome case corresponds to bottom right corner. But he assumed that there was no-one below 70.71% of the median.  

107 Frank Cowell: UB Public Economics “Linear” model: assessment Solution to problem becomes much more transparent Solution to problem becomes much more transparent But exact tax formulas are still elusive. But exact tax formulas are still elusive. Optimal tax rates are very sensitive to precise assumptions about Optimal tax rates are very sensitive to precise assumptions about  labour-supply elasticity.  Distribution of ability  Inequality aversion

108 Frank Cowell: UB Public Economics Overview... Design Issues General labour model Generalisations Optimal Income Taxation An alternative focus on human capital “linear” labour model Education model

109 Frank Cowell: UB Public Economics Approach 3: Alternative Income Determination Most OIT models focus on just one area of personal decision making Most OIT models focus on just one area of personal decision making Casual discussion of policy suggest that other economic incentives may be relevant Casual discussion of policy suggest that other economic incentives may be relevant What about the long-run determination of earning power? What about the long-run determination of earning power? Need a model of investment. Need a model of investment.

110 Frank Cowell: UB Public Economics Components of Atkinson’s human capital model Given structure of ability distribution Given structure of ability distribution Individuals maximise lifetime disposable income Individuals maximise lifetime disposable income Essentially investment model Essentially investment model  Based on Becker (and Mincer) human capital model  Schooling only, not experience Conventional social welfare function Conventional social welfare function Government budget constraint of zero net revenue Government budget constraint of zero net revenue

111 Frank Cowell: UB Public Economics Notation in Atkinson’s human capital model w - exogenously given ability w - exogenously given ability y - pretax income y - pretax income S - years of schooling S - years of schooling L - length of working life L - length of working life r - interest (discount) rate r - interest (discount) rate c - disposable income c - disposable income

112 Frank Cowell: UB Public Economics Life Cycle in the Atkinson Model age earnings SL+S y=wS t

113 Frank Cowell: UB Public Economics Atkinson’s Becker-type approach Pareto distribution of ability pretax income determined by Becker schooling model choose schooling to maximise discounted lifetime consumption

114 Frank Cowell: UB Public Economics Atkinson’s human capital model: optimised schooling Disposable income is c = B + [1-  ] y Disposable income is c = B + [1-  ] y Define a critical ability level in terms of tax parameters Define a critical ability level in terms of tax parameters For medium/high ability schooling increases with ability For medium/high ability schooling increases with ability For low ability it’s not worth investing in education For low ability it’s not worth investing in education Ability type w chooses optimal schooling as Ability type w chooses optimal schooling as

115 Frank Cowell: UB Public Economics Atkinson’s human capital model: optimised utility Substitute optimal S into formula for discounted lifetime consumption to get: Substitute optimal S into formula for discounted lifetime consumption to get: Gives relationship between ability and utility Gives relationship between ability and utility

116 Frank Cowell: UB Public Economics Atkinson model: social objectives and constraints Maximise additively separable SWF as before. Maximise additively separable SWF as before. Government budget constraint becomes Government budget constraint becomes

117 Frank Cowell: UB Public Economics Atkinson’s human-capital model: pretax and disposable income taxable income will become more unequal the more progressive is the tax disposable income will have the same inequality as ability!

118 Frank Cowell: UB Public Economics Ability-Schooling Relationship for values of w 0 = rB/[1 –  ] In a high-progression model the able invest a lot in education In a high-progression model the able invest a lot in education This pays for the income supplements for the less able This pays for the income supplements for the less able

119 Frank Cowell: UB Public Economics Atkinson’s “Becker” model: optimal marginal tax rates

120 Frank Cowell: UB Public Economics Education model: assessment Key to model is investment response to anticipated tax Key to model is investment response to anticipated tax In simple model schooling chosen increases when tax progression is increased. In simple model schooling chosen increases when tax progression is increased. Result can appear to offset effect on current income Result can appear to offset effect on current income But target is distribution of lifetime utility. But target is distribution of lifetime utility. Result of low optimal marginal rates depends crucially on appropriateness of the precise investment model Result of low optimal marginal rates depends crucially on appropriateness of the precise investment model

121 Frank Cowell: UB Public Economics Overview... Design Issues General labour model Generalisations Optimal Income Taxation What if we combine insights from the two main branches of optimal taxation? “linear” labour model Education model

122 Frank Cowell: UB Public Economics More general tax issues Should we rely on direct or indirect taxation? Should we rely on direct or indirect taxation? Is there much to be gained by combining the two branches of theory? Is there much to be gained by combining the two branches of theory? Can a unified optimising model be developed? Can a unified optimising model be developed?

123 Frank Cowell: UB Public Economics Direct versus Indirect Taxation Issues 1. Nonlinear commodity taxation? 2. Informational requirements. 3. Participation and incentive compatibility constraints. 4. Direct versus indirect tax progressivity.

124 Frank Cowell: UB Public Economics 1 Nonlinear commodity taxation? Should consider the issue of proportional versus nonlinear taxation of commodities. Should consider the issue of proportional versus nonlinear taxation of commodities. “Nonlinear” includes affine functions (like the so- called linear income tax function). “Nonlinear” includes affine functions (like the so- called linear income tax function). The argument is whether each commodity should be “repriced”, perhaps not in a proportional fashion. The argument is whether each commodity should be “repriced”, perhaps not in a proportional fashion. Similar argument is applied in other areas: tariffs for output of state-owned industries, price support schemes Similar argument is applied in other areas: tariffs for output of state-owned industries, price support schemes

125 Frank Cowell: UB Public Economics 2 Informational requirements Recall the main differences between the two types of tax: Recall the main differences between the two types of tax: Not the formal tax base (income versus expenditure) but the informational base. Not the formal tax base (income versus expenditure) but the informational base.  Direct tax authority can know details of personal resources.  Indirect tax authority can know structure of production and transactions Informational requirements may preclude extensive application of nonlinear commodity taxes. Informational requirements may preclude extensive application of nonlinear commodity taxes. To see this consider problem of nonlinear pricing of consumer goods. To see this consider problem of nonlinear pricing of consumer goods.  Can work for water, gas, electricity  But for food? Clothes?

126 Frank Cowell: UB Public Economics 3 Participation and Incentive Compatibility Constraints ICC issues are central to nonlinear income tax design ICC issues are central to nonlinear income tax design Same difficulty can arise with nonlinear pricing schemes: Same difficulty can arise with nonlinear pricing schemes:  Some groups may choose the “wrong contract”  Arises both in private and public sector Difficulties usually disappear if you impose the regularity conditions implied by linearity Difficulties usually disappear if you impose the regularity conditions implied by linearity Supports the strong case for considering linear commodity taxes Supports the strong case for considering linear commodity taxes

127 Frank Cowell: UB Public Economics 4 Direct versus Indirect Tax Progressivity Can measure progressivity in a number of ways Can measure progressivity in a number of ways A standard method is to compute the implied tax rates that emerge from actual expenditure decisions A standard method is to compute the implied tax rates that emerge from actual expenditure decisions Can do this for the definitions of “direct” and “indirect” taxes in the UK Can do this for the definitions of “direct” and “indirect” taxes in the UK In practice indirect taxes are more regressive than direct taxes. In practice indirect taxes are more regressive than direct taxes.

128 Frank Cowell: UB Public Economics Implied average tax rates in Economic Trends. UK 1994

129 Frank Cowell: UB Public Economics Integrating direct and indirect taxation: consumer’s problem so the budget constraint is: so the budget constraint is: Total disposable income is given by Total disposable income is given by. Assume there is no lump sum income (I=0) Assume there is no lump sum income (I=0)

130 Frank Cowell: UB Public Economics Integrating direct and indirect taxation: government’s problem First order conditions yield First order conditions yield Government budget constraint is Government budget constraint is. Given the generality of the problem we should reduce the number of degrees of freedom Given the generality of the problem we should reduce the number of degrees of freedom otherwise you’ll get lump sum taxation again! Use this to give general guidance on tax structure.

131 Frank Cowell: UB Public Economics Policy rules Commodity taxes should be zero if preferences are weakly separable in leisure and other goods Commodity taxes should be zero if preferences are weakly separable in leisure and other goods Tax on good i should be higher if the MRS between good i and labour increases. Tax on good i should be higher if the MRS between good i and labour increases. Focus tax on goods for which the most able have the strongest preference. Focus tax on goods for which the most able have the strongest preference.

132 Frank Cowell: UB Public Economics Conclusions Direct versus indirect Direct versus indirect  Distinction between the two is essentially an issue of information.  Big differences in terms of distributional effect. Uniform commodity taxation Uniform commodity taxation  No compelling case within the context of the model  There may be a case if you appeal to other factors “Flat tax” “Flat tax”  Argument as for uniform commodity taxation


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