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Frank Cowell: UB Public Economics June 2005 Optimal Tax Design Public Economics: University of Barcelona Frank Cowell http://darp.lse.ac.uk/ub Frank Cowell: Purpose of tax design UB Public Economics The issue of design is fundamental to public economics Move from what we would like to achieve… …to what we can actually implement 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. A brief review of the elements of the problem. … Frank Cowell: Components of the problem UB Public Economics Objectives Scope for policy 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. Methods of intervention Constraints Informational problems Available tools The tax base Direct and indirect taxation Frank Cowell: Overview... Optimal Income Taxation Design Issues UB Public Economics Why this kind of problem is set up General labour model “linear” labour model Education model Generalisations •Objectives •Scope for policy •Informational issues •Available tools Frank Cowell: Specific objectives? UB Public Economics Could be a class of The objectives of the tax design could include: functions Could be incorporated in 1. Bergson-Samuelson welfare maximisation objective #1 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. Frank Cowell: Implementation of objectives UB Public Economics What is domain of the SWF? Need a model of cardinal, comparable utility Incomes? Individual utilities? Welfarist approach usually founded on this basis What social Data entities? is often on this basis… Individuals …or this Families Household units? Frank Cowell: Overview... Optimal Income Taxation Design Issues UB Public Economics Types of intervention. The tax base General labour model “linear” labour model Education model Generalisations •Objectives •Scope for policy •Informational issues •Available tools Frank Cowell: Scope for policy UB Public Economics What is potentially achievable? 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 Do this within the context of an elementary microeconomic model. Frank Cowell: The Composition of Income UB Public Economics Take the standard microeconomic model of a person’s total income in a market economy Composed of resources valued at their market prices: endowment income of good i Non-market income price of good i Does this mean public policy has to be limited to . redistributing resources, or 2. manipulating prices? There could be other forms of income Problems with 1 and 2 above are also important And there may be other types of intervention 1. Frank Cowell: Problems with redistributing resources: UB Public Economics The lump-sum tax issue: Non-transferability Special information – such as personal characteristics Political problems of implementation Fixed resources Inalienability of certain rights – No slavery Ways of getting round these problems? Could redistribute the purchasing power generated by the resource? Or modify the supply of “co-operant factors”? Frank Cowell: Problems with price manipulation UB Public Economics Identification of commodities Complexity Proliferation of implied pricing structures Informational problems The boundary problem Artificial definition of a good or service on which a tax is to be levied. Uncertainty leads to wrong price signals? Misinformation leads to wrong price signals? May even be missing markets Need to focus on economics of information Frank Cowell: Overview... Optimal Income Taxation Design Issues UB Public Economics Fundamental theoretical issues in design problem General labour model “linear” labour model Education model Generalisations •Objectives •Scope for policy •Informational issues •Available tools Frank Cowell: Informational issues in microeconomics UB Public Economics There are two key types of informational problem: Both can be relevant to policy design. Hidden action: Hidden information: Regulation and optimal contracts. Moral hazard in social insurance Compliance issues. Problems of “tailoring” tax rates. Adverse selection in social insurance. Focus on this issue here But the “information issue” is quite deep: There is connection with discussion of social welfare A fundamental relationship with the “Arrow” problem Frank Cowell: Social values: the Arrow problem UB Public Economics Uses weak assumptions about preferences/values 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 Well-defined individual orderings over social states Well-defined social ordering over social states Universal Domain Pareto Unanimity Independence of Irrelevant Alternatives Non-Dictatorship There’s no constitution that does all four Except in cases where there are less than three social states Frank Cowell: Social-choice function UB Public Economics Similar to the concept of constitution But maps from set of preference profiles to set of social states Not surprising to find result similar to Arrow Given a particular set of preferences for the population Picks out the preferred social state Introduce weak conditions on the Social-choice function There’s no SCF that satisfies all of them But key point concerns the implementation issue Frank Cowell: Implementation UB Public Economics Is the social-choice function consistent with private economic behaviour? Yes if the social state picked out by the SCF corresponds to an equilibrium 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 Focus on a type that is useful for expositional purposes... Frank Cowell: Direct mechanisms UB Public Economics Map from collection of preferences to states Here the SCF is the mechanism itself An SCF that encourages misrepresentation may be of limited use Is truthful implementation possible? Involves a very simple game. The game is “show me your utility function” Enables us to focus directly on the informational aspects of implementation Will people announce their true attributes? Will it be a dominant strategy to do so? Introduce another key result Frank Cowell: Gibbard-Satterthwaite result UB Public Economics Can be stated in a variety of ways. 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 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 Frank Cowell: Overview... Optimal Income Taxation Design Issues UB Public Economics What practical options available to achieve the objectives? General labour model “linear” labour model Education model Generalisations •Objectives •Scope for policy •Informational issues •Available tools Frank Cowell: Informational issues in taxation UB Public Economics What distinguishes taxation from highway robbery? Taxation principles Appropriate information What information is/should be available? Attributes Behaviour Frank Cowell: Available tools UB Public Economics Availability determined by a variety of considerations. Fundamental economic constraints Institutional constraints. These may come from: Legal restrictions Administrative considerations Historical precedent But each of these institutional aspects may really follow from the economics. Frank Cowell: The Tax Base UB Public Economics We focus here on the taxation of individuals rather than corporations or other entities. An approach to the individual tax-base might begin with number an examination of the individual’s budget constraint: of goods expenditure consumption of good i income 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 First take a closer look at income: Frank Cowell: A fundamental difference? UB Public Economics It is tempting to think of the distinction between different types of tax in terms of the budget Indirect constraint: taxes here? Direct taxes here? This misses the point . Any tax on RHS can be converted to tax on LHS Real question is about information Frank Cowell: Information again UB Public Economics The government and its agencies must work with imperfect information. To model taxes appropriately need to take this into account. Information imposes specific constraints on tax design Income In a typical market economy there are two main Total expenditure types of information:Age, marital status? About individuals About transactions Expenditure by product category Expenditure by industry Input and output quantities Frank Cowell: Fundamental constraints... Public budget constraints UB Public Economics Participation constraints Example: In simple redistribution sum of net receipts (taxes cash subsidies) must be zero Example: Labour supply Incentive-compatibility (self-selection) constraints Example: Differential subsidies for specific commodities Frank Cowell: Design basics: summary UB Public Economics Objectives follow on logically from our discussion in previous lectures. Beware of oversimplifying assumptions about the tax base. Information plays a key role. Frank Cowell: Overview... Optimal Income Taxation Design Issues UB Public Economics What types of tax formula used in theory and practice? General labour model “Linear” labour model Education model Generalisations •Tax schedules •Outline of problem •The solution Frank Cowell: Income tax – example of design problem UB Public Economics Standard types of tax simple examples integration with income support General issues of how to set up an optimisation 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 Frank Cowell: Income tax – notation UB Public Economics y – taxable income c – disposable income (“consumption”) c(y) = y – T(y) T(·) – tax schedule c(·) – disposable income 0 t schedule 1 t – marginal tax rate y0 – exemption-level income B – lumpsum benefit/guaranteed income c=c(y) UB Public Economics disposable income Frank Cowell: Income space pre-tax income y Frank Cowell: The simple income tax c=c(y) UB Public Economics Marginal retention rate 1-t Exemption level y0 y Frank Cowell: ...extended to Negative Income Tax c=c(y) UB Public Economics Incomes subsidised through NIT 1-t B = t y0 B y0 y Frank Cowell: How to generalise this approach…? UB Public Economics Other functional forms of the income tax Administrative complexity of IT Interaction with other contingent taxes and benefits. Frank Cowell: Increasing marginal tax rate t c(y) UB Public Economics y Frank Cowell: Example 1 UB Public Economics UK: piecewise linear tax stepwise jumps in MTR compare with contingent tax/benefit model Germany: linearly increasing marginal tax rate quadratic tax and disposable income schedules Frank Cowell: Example 2 UB Public Economics Germany 1981-1985, single person (§32a Einkommensteuergesetz): up to 4,212: T = 0 4,213 to 18,000: T = 0.22y – 926 18,001 to 59,999: T = 3.05 z4 – 73.76 z3 + 695 z2 + 2,200 z + 3,034 z = y/10,000 - 18,000; 60,000 to 129,999: T = 0.09z4 – 5.45z3 + 88.13 z2 + 5,040 z + 20,018 z = y/10,000 - 60,000; from 130,000: T = 0.56 y – 14,837 (units: DM) 80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000 0 0 20000 40000 60000 80000 100000 120000 140000 Frank Cowell: Interaction with income support UB Public Economics c(y) Straight income tax at constant marginal rate “Clawback” of support Tax-payments kick in with benefits Untaxed income support B 0 y1 y0 y2 y Frank Cowell: The approach to IT – summary UB Public Economics The “linear” form may be a reasonable approximation to some practical cases We may also see an appealing intuitive argument for linearity as simplification “Income tax” may need to be interpreted fairly broadly Interaction amongst various forms of government intervention is important for an appropriate model This may lead to nonlinearity in the effective schedule Frank Cowell: Overview... Optimal Income Taxation Design Issues UB Public Economics Basic ingredients of OIT analysis. General labour model “Linear” labour model Education model Generalisations •Tax schedules •Outline of problem •The solution Frank Cowell: Basic Ingredients of An Optimal Income Tax model UB Public Economics A distribution of abilities Individuals’ behaviour Social-welfare function Feasibility Constraint Restriction on types of functional form What resources are potentially available for redistribution? Frank Cowell: Distribution of Ability... Assume... UB Public Economics a single source of earning power – “ability” ability is fully reflected in the (potential) wage w So ability is effectively measured by w. the distribution F of w is observable individual values of w are not observable by the tax authority Frank Cowell: Can we infer the distribution of ability? UB Public Economics Practical approach Select relevant group or groups in population. Choose appropriate earnings concept. male manual workers? Full time earnings? Divide earnings by hours to get wages. Use parametric model to capture shape of distribution. Lognormal? Frank Cowell: Distribution: example Example from UK 2000 Gives distribution of y=wh for full-time male manual workers UB Public Economics f (y ) NES 2000 Lognormal (5.7,0.13) y £5 0 £1 00 £1 50 £2 00 £2 50 £3 00 £3 50 £4 00 £4 50 £5 00 £5 50 £6 00 £6 50 £7 00 £7 50 £8 00 £8 50 £9 00 £9 5 £1 0 ,0 00 £0 0 Frank Cowell: Basic Ingredients of An Optimal Income Tax model (2) UB Public Economics A distribution of abilities Individuals’ behaviour Social-welfare function Feasibility Constraint Restriction on types of functional form In what ways do we assume that people will respond to the tax authority’s instruments ? Frank Cowell: The individual's problem UB Public Economics Individual’s utility is determined by disposable income (consumption) c and leisure. So the optimisation problem can be written maxh U(c,h) subject to c = y – T(y) and y = wh This yields maximised utility as a function of ability (wage): u(w) Frank Cowell: A Characterisation of Tastes UB Public Economics Introduce a definition to capture the shape of individual preferences Normalised MRS The following restriction on “regularity” of The way slope of indifference curve preferences is important forability clean-cut results changes with Frank Cowell: A representation of preferences UB Public Economics (leisure) Frank Cowell: Indifference curve in (h,c)-space c UB Public Economics (hours worked) h Frank Cowell: Contour translated to (y,c)-space c UB Public Economics slope = q (gross income) y Frank Cowell: The regularity condition c high w UB Public Economics low w Illustrates the qw < 0 property Ensures “single-crossing” of ICs for different ability groups y Frank Cowell: Individual's problem: points to note UB Public Economics Incorporates standard assumptions Same basic model as in earlier lectures Consistent with the model y = wh or with the model y = wh + I Frank Cowell: Basic Ingredients of An Optimal Income Tax model (3) UB Public Economics A distribution of abilities Individuals’ behaviour Social-welfare function Feasibility Constraint Restriction on types of functional form How to represent the objectives of the optimisation problem? Frank Cowell: The Government’s Objective... Take a standard version of the SWF UB Public Economics Assume additive separability Take “weighted average” over types Maximised utility of a w-type person Social evaluation function Proportion of w-type persons in the population Frank Cowell: Basic Ingredients of An Optimal Income Tax model (4) UB Public Economics A distribution of abilities Individuals’ behaviour Social-welfare function Feasibility Constraint Restriction on types of functional form Real-world restrictions on government and the design problem Frank Cowell: Main types of constraint UB Public Economics The Government’s budget constraint Incentive compatibility Frank Cowell: Government’s Budget Constraint disposable income in population Net revenue requirement Earnings in population UB Public Economics Frank Cowell: Two Ability Levels c high w UB Public Economics c(y) low w Incentive compatibility problem: Original and disposable income must increase with ability y Frank Cowell: Basic Ingredients of An Optimal Income Tax model (5) UB Public Economics A distribution of abilities Individuals’ behaviour Social-welfare function Feasibility Constraint Restriction on types of functional form Frank Cowell: Restrictions on form... UB Public Economics It may make sense to consider cases where marginal tax-rate is everywhere constant Pre-1985 Germany? (like the NIT model earlier): Lack of detail about tails Administrative costs of general model Cf the US “flat tax” discussion informational problems “fairness” arguments [?] Frank Cowell: Review: ingredients of OIT model UB Public Economics A distribution of abilities Individuals’ behaviour Assume individualistic additively separable SWF W = u( u(w) f(w) dw Feasibility constraints maxh U(c,h), subject to c = y – T(y) and y = wh Gives utility as function of ability u(w) Social-welfare function Ability is measured by potential wage w. Distribution F of w is observable Individual values of w are not observable The Government’s budget constraint Incentive compatibility Restriction on types of functional form (Piecewise) linear schedule? Frank Cowell: Overview... Optimal Income Taxation Design Issues UB Public Economics Characterising a general optimal income tax. General labour model “Linear” labour model Education model Generalisations •Tax schedules •Outline of problem •The solution Frank Cowell: The general OIT model UB Public Economics No preconditions on form of income tax Use results from economics of information Use a general variational approach to give the solution IC condition required for “sensible” results However, no consideration of administrative complexity Has similarities with techniques used for optimal growth Terminal conditions can be important Illustrate the variational approach diagrammatically Use the disposable income schedule c(•) Frank Cowell: The general tax-design problem UB Public Economics c(y) Variation in general tax schedule y Frank Cowell: Individual optimisation UB Public Economics Begin with the way each tax-payer is assumed to act. Disposable The optimisation problem can be written income work The function c(·) is chosen by the government Define normalised MRS Slope of disposable income function The first-order condition is The solution is of the form u(w) := maxh U(c(wh), h) Frank Cowell: Contour in (y,c)-space c UB Public Economics Disposable income schedule c(•) optimised income Proportional to work y Frank Cowell: A regularity condition c high w UB Public Economics low w If the property qw < 0 holds this ensures “single-crossing” of ICs for different ability groups y Frank Cowell: Incentive compatibility condition c high w UB Public Economics c(y) low w Design of c must ensure that utility and income increase with ability y Frank Cowell: Incentive compatibility UB Public Economics The IC condition means that high ability people should not have an incentive to “masquerade” as low ability. This requires maximised utility u(w) to increase in w. By differentiation of the solution function we have Optimised value of h If c(·) is monotonic and differentiable everywhere then this becomes But if these conditions are violated, problems arise… Frank Cowell: Fundamental design problem UB Public Economics It may seem odd that the IC condition be violated in actual design 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 The “notch problem” (US) The “poverty trap” (UK) Frank Cowell: “Notch problem” / “poverty trap” UB Public Economics c(y) Discontinuous nonmonotonic c(·) Withdrawal of benefit here y0 y Frank Cowell: Violation of IC condition UB Public Economics c(y) Where high w would choose Where high w “should” be low w y0 y Frank Cowell: Government: maximisation UB Public Economics Uses a constrained maximum method But there is a constraint at each ability level from wmin to wmax. Similar to maximisation over time. Choose c(·) to max subject to and, at each ability level: disposable income in population Net revenue requirement Earnings in population Frank Cowell: Government: maximisation UB Public Economics Introduce a Lagrange multiplier l for the budget constraint and a multiplier m(w) for the incentive compatibility constraint at each ability level. Then, on rearranging, the Lagrangean is Frank Cowell: General Model: Characterisation of Marginal Tax Rate UB Public Economics Lagrange multiplier for incentive-compatibility constraint Lagrange multiplier for Government budget constraint Frank Cowell: Interpreting the FOC UB Public Economics 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 Some key results First for the overall shape Second for what happens at each end of the ability range… Frank Cowell: Main result 1 UB Public Economics Mirrlees 1971: The optimal marginal tax rate must be greater than or equal to 0 and less than 1 http://darp.lse.ac.uk/papersdb/Mirrlees_(REStud_71).pdf 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 Agent monotonicity implies So it is immediate that T'(y) < 1. For the lower extreme of the distribution need to look at “bunching”… Frank Cowell: Bunching: w'<w''<w''' UB Public Economics w''' w'' c w' y Frank Cowell: No bunching: w'<w''<w''' UB Public Economics c w''' w' w'' y Frank Cowell: Main result 2 UB Public Economics Seade 1977,Ebert 1992. 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 top of distribution note the FOC: At wmax IC constraint becomes irrelevant; so m(wmax) = 0. Therefore T'(ymax) = T'(wmax h(wmax)) = 0 Frank Cowell: Problems of the general model (1) UB Public Economics There appear to be commonsense general results And clear-cut results for the extremes, But little guidance on the structure for the majority of the workforce. Some broad principles can be adduced from the first order conditions. But you cannot get further without an explicit model On the distribution of w On individual preferences On the SWF Frank Cowell: Problems of the general model (2) UB Public Economics The results for the extremes are not robust 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 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” Frank Cowell: Problems of the general model (3) UB Public Economics 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. Diamond (1998) uses Pareto. Gets high marginal tax rates where ability follows a Pareto distribution http://darp.lse.ac.uk/papersdb/Diamond_(AER98).pdf Saez (2001) is a general extension of the Mirrlees results http://darp.lse.ac.uk/papersdb/Saez_(REStud01).pdf Frank Cowell: Problems of the general model (4) UB Public Economics Cannot get stronger results on tax rates analytically. But to do this you need to implement a specific model which can be: Can do this for special cases Example Salanie model with quasi-linear preferences Or could use simulation in a numerical model 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 Frank Cowell: Overview... Optimal Income Taxation Design Issues UB Public Economics A “cut-down” version of the labour-leisure problem General labour model “linear” labour model Education model Generalisations Frank Cowell: Approach 2: The linear model UB Public Economics Same behavioural assumptions as before Same objectives Restriction to linear (affine) tax functions: two parameters First analysed by Sheshinski (1972) http://darp.lse.ac.uk/papersdb/Sheshinski_(REStud_72).pdf Frank Cowell: Linear Model: outline UB Public Economics No longer choosing a general tax/disposable income schedule c(•) Instead, just a two-parameter model. Disposable income is Marginal tax rate Pre-tax income Minimum disposable income Simplified version is much more tractable analytically Frank Cowell: Arguments for “linear” model UB Public Economics Relatively easy to interpret parameters Pragmatic: t as uniform marginal tax rate B as minimum income, or… B / t as exemption rate Approximates several countries’ tax systems Example – piecewise linear tax in UK Sidesteps the incentive compatibility constraint… Frank Cowell: Incentive compatibility resolved c high w UB Public Economics c(y) low w Original and disposable income will increase with ability y Frank Cowell: Linear Model: Constraint UB Public Economics Given that the IC condition vanishes, there is only one constraint Government TaxThe raised on working Budget constraint: Extra revenue population Minimum guaranteed income for all requirement In effect this makes the issue a one-variable problem… Frank Cowell: Take the linear income tax... c(y) UB Public Economics 1-t B Note Marginal tax-rate is constant: t If B>0 average tax-rate t–B/y is everywhere rising with income y Frank Cowell: Higher B needs higher t c(y) UB Public Economics 1-tDt B+DB y Frank Cowell: Linear Model: Lagrangean UB Public Economics The constrained optimisation problem can be set up as Lagrange Social-welfare the Lagrangean: multiplier function Government budget constraint Maximise Lagrangean by choice of tax instruments t and B This can be done using classical optimisation methods. Frank Cowell: Linear Model: FOC (1) Maximised utility $1 lump-sum UB Public Economics Consider the social value of This is defined as: Average social value Differentiating the Lagrangean with respect to B:of $1 should be 1 income. Frank Cowell: Linear Model: FOC (2) UB Public Economics Differentiating the Lagrangean with respect to t and Covariance of social marginal rearranging we get: valuation and income Optimal marginal tax rate Compensated laboursupply elasticity Again the formula can be used to give guidance on policy… Frank Cowell: Outcomes from the linear model UB Public Economics If R = 0 then B > 0 Implies progressive taxation. FOC cannot be solved to give an explicit formula The covariance and the elasticities will themselves be functions of t. However the “natural” restriction imposed by linearity makes construction of simulation easier Better behaved at special points of the distribution Frank Cowell: Components of simulation UB Public Economics Structure of ability (wage) distribution Individual preferences Determines labour supply responses Social welfare function Empirically determined? Use evidence from social surveys etc? Government budget constraint Experiment with alternative assumptions Frank Cowell: Broome’s revelation UB Public Economics John Broome suggested a great simplification for OIT. t* = 58.6% !! http://darp.lse.ac.uk/papersdb/Broome_(REStud_75).pdf The basis for this astounding claim? 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 Frank Cowell: Broome’s model… UB Public Economics But in UK 2000: Make the (empirically relevant?) assumption that£10.53 no-one 1 Average wage was / has ability less than 0.7071 times thehour average: 2 Min wage was £4.10! Take standard Cobb-Douglas preferences: “Rawlsian” max-min social welfare: Balanced budget: Frank Cowell: The Stern simulation model UB Public Economics Stern’s (1976) model is less tongue-in-cheek. But can be taken as a generalisation of Broome. Also based on a linear OIT Ingredients are: Lognormal ability Isoelastic utility Isoelastic social welfare A variety of assumptions about the government budget constraint Frank Cowell: Representation of ability distribution UB Public Economics Simple two parameter distribution L(w; m, s2 ) First parameter m is log of the median The second parameter s2 is itself an inequality index – the variance of log income. Support is [0, ) 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 0.7071 × mean (Broome) Frank Cowell: The lognormal distribution f(w) UB Public Economics —L(w; 0, 0.25 ) …L(w; 0, 1.0 ) 0 0 1 2 3 4 w Frank Cowell: Isoelastic utility UB Public Economics Take an empirically relevant version of household elasticity of utility: Substitution ( 0) hours worked consumption Becomes the Broome model in the case s=1 Frank Cowell: Labour Supply and Income... Define q := [1t] w UB Public Economics Could have backward-bending labour supply if s<1 Frank Cowell: Resulting Labour Supply and Income… (Broome case) UB Public Economics Frank Cowell: Standard SWF Take additive form of Bergson-Samuelson SWF: UB Public Economics W = u(u) dF(u) = u(u) f(u) du Use the iso-elastic form of the (social) u-function: u 1–e – 1 u(u) = ————, e 1–e Bentham corresponds to the case e=. Max-min (“Rawls”) corresponds to the case e=. Frank Cowell: Stern's Optimal Income Tax Rates UB Public Economics s e= e= 1 e= 0.2 0.4 0.6 0.8 1.0 36.2 22.3 17.0 14.1 12.7 62.7 47.7 38.9 33.1 29.1 92.6 83.9 75.6 68.2 62.1 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. Frank Cowell: “Linear” model: assessment UB Public Economics Solution to problem becomes much more transparent But exact tax formulas are still elusive. Optimal tax rates are very sensitive to precise assumptions about labour-supply elasticity. Distribution of ability Inequality aversion Frank Cowell: Overview... Optimal Income Taxation Design Issues UB Public Economics An alternative focus on human capital General labour model “linear” labour model Education model Generalisations Frank Cowell: Approach 3: Alternative Income Determination UB Public Economics Most OIT models focus on just one area of personal decision making Casual discussion of policy suggest that other economic incentives may be relevant What about the long-run determination of earning power? Need a model of investment. Frank Cowell: Components of Atkinson’s human capital model UB Public Economics Given structure of ability distribution Individuals maximise lifetime disposable income Essentially investment model Based on Becker (and Mincer) human capital model Schooling only, not experience Conventional social welfare function Government budget constraint of zero net revenue Frank Cowell: Notation in Atkinson’s human capital model UB Public Economics w - exogenously given ability y - pretax income S - years of schooling L - length of working life r - interest (discount) rate c - disposable income UB Public Economics earnings Frank Cowell: Life Cycle in the Atkinson Model y=wS t S L+S age Frank Cowell: Atkinson’s Becker-typePareto approach distribution of ability UB Public Economics pretax income determined by Becker schooling model choose schooling to maximise discounted lifetime consumption Frank Cowell: Atkinson’s human capital model: optimised schooling UB Public Economics Disposable income is c = B + [1-t] y Define a critical ability level in terms of tax parameters Ability type w chooses optimal schooling as For medium/high ability schooling increases with ability For low ability it’s not worth investing in education Frank Cowell: Atkinson’s human capital model: optimised utility UB Public Economics Substitute optimal S into formula for discounted lifetime consumption to get: Gives relationship between ability and utility Frank Cowell: Atkinson model: social objectives and constraints UB Public Economics Maximise additively separable SWF as before. Government budget constraint becomes Frank Cowell: Atkinson’s human-capital model: income will become pretax and disposabletaxable income more unequal the more progressive is the tax UB Public Economics disposable income will have the same inequality as ability! Frank Cowell: Ability-Schooling Relationship for values of w0 = rB/[1 – t] 0.12 schooling UB Public Economics 0.1 0.08 0 (No tax) 0.05 (low prog) 0.10 (medium) 0.15 (high) 0.06 0.04 0.02 0 150 200 250 300 350 400 ability In a high-progression model the able invest a lot in education This pays for the income supplements for the less able Frank Cowell: Atkinson’s “Becker” model: optimal marginal tax rates UB Public Economics Frank Cowell: Education model: assessment UB Public Economics Key to model is investment response to anticipated tax In simple model schooling chosen increases when tax progression is increased. Result can appear to offset effect on current income But target is distribution of lifetime utility. Result of low optimal marginal rates depends crucially on appropriateness of the precise investment model Frank Cowell: Overview... Optimal Income Taxation Design Issues UB Public Economics What if we combine insights from the two main branches of optimal taxation? General labour model “linear” labour model Education model Generalisations Frank Cowell: More general tax issues UB Public Economics Should we rely on direct or indirect taxation? Is there much to be gained by combining the two branches of theory? Can a unified optimising model be developed? Frank Cowell: Direct versus Indirect Taxation Issues UB Public Economics 1. Nonlinear commodity taxation? 2. Informational requirements. 3. Participation and incentive compatibility constraints. 4. Direct versus indirect tax progressivity. Frank Cowell: 1 Nonlinear commodity taxation? UB Public Economics Should consider the issue of proportional versus nonlinear taxation of commodities. “Nonlinear” includes affine functions (like the socalled linear income tax function). 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 Frank Cowell: 2 Informational requirements UB Public Economics Recall the main differences between the two types of tax: 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. To see this consider problem of nonlinear pricing of consumer goods. Can work for water, gas, electricity But for food? Clothes? Frank Cowell: 3 Participation and Incentive Compatibility Constraints UB Public Economics ICC issues are central to nonlinear income tax design 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 Supports the strong case for considering linear commodity taxes Frank Cowell: 4 Direct versus Indirect Tax Progressivity UB Public Economics Can measure progressivity in a number of ways 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 In practice indirect taxes are more regressive than direct taxes. Frank Cowell: Implied average tax rates in Economic Trends. UK 1994 0.35 UB Public Economics 0.3 Direct Indirect 0.25 0.2 0.15 0.1 0.05 0 Bottom 10th 2nd 3nd 4th 5th 6th 7th 8th 9th Top 10th Frank Cowell: Integrating direct and indirect taxation: consumer’s problem Total disposable income is given by UB Public Economics . so the budget constraint is: Assume there is no lump sum income (I=0) Frank Cowell: Integrating direct and indirect taxation: government’s problem UB Public Economics Government budget constraint is otherwise you’ll get lump sum taxation again! . Given the generality of the problem we should reduce the number of degrees of freedom Use this to give general guidance on tax First order conditions yield structure. Frank Cowell: Policy rules UB Public Economics 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. Focus tax on goods for which the most able have the strongest preference. Frank Cowell: Conclusions UB Public Economics Direct versus indirect Distinction between the two is essentially an issue of information. Big differences in terms of distributional effect. 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” Argument as for uniform commodity taxation