Optimal Tax Design

1. Optimal Tax Design Public Economics: University of Barcelona Frank Cowell http://darp.lse.ac.uk/ub June 2005

2. Purpose of tax design • 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. …

3. Components of the problem • 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 • Methods of intervention • Constraints • Informational problems • Available tools • The tax base • Direct and indirect taxation

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

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

6. Implementation of objectives • What is domain of the SWF? • Incomes? • Individual utilities? • 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. Optimal Income Taxation Overview... • Objectives • Scope for policy • Informational issues • Available tools Design Issues General labour model Types of intervention. The tax base “linear” labour model Education model Generalisations

8. Scope for policy • 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.

9. The Composition of Income • Take the standard microeconomic model of a person’s total income in a market economy • Composed of resources valued at their market prices: endowment of good i Non-market income income price of good i • Does this mean public policy has to be limited to • redistributing resources, or • 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

10. Problems with redistributing resources: • The lump-sum tax issue: • Special information – such as personal characteristics • Political problems of implementation • Non-transferability • 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”?

11. Problems with price manipulation • Identification of commodities • The boundary problem • Artificial definition of a good or service on which a tax is to be levied. • Complexity • Proliferation of implied pricing structures • 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

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

13. Informational issues in microeconomics • There are two key types of informational problem: • Both can be relevant to policy design. • Hidden action: • Regulation and optimal contracts. • Moral hazard in social insurance • Compliance issues. • Hidden information: • 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

14. Social values: the Arrow problem • 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 • The constitution • A map from set of preference profiles to social preference • Also weak assumptions about the constitution • 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

15. Social-choice function • Similar to the concept of constitution • 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 • Introduce weak conditions on the Social-choice function • There’s no SCF that satisfies all of them • But key point concerns the implementation issue

16. Implementation • 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...

17. Direct mechanisms • 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 • An SCF that encourages misrepresentation may be of limited use • Is truthful implementation possible? • Will people announce their true attributes? • Will it be a dominant strategy to do so? • Introduce another key result

18. Gibbard-Satterthwaite result • 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

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

20. Informational issues in taxation • What distinguishes taxation from highway robbery? • Taxation principles • Appropriate information • What information is/should be available? • Attributes • Behaviour

21. Available tools • 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.

22. The Tax Base • 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: number of goods consumption of good i expenditure 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:

23. A fundamental difference? • It is tempting to think of the distinction between different types of tax in terms of the budget constraint: Indirect 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 .

24. Information again • 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 • In a typical market economy there are two main types of information: • About individuals • About transactions Income Total expenditure Age, marital status? Expenditure by product category Expenditure by industry Input and output quantities

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

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

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

28. Income tax – example of design problem • 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

29. Income tax – notation • y – taxable income • c – disposable income (“consumption”) • T(·) – tax schedule • c(·) – disposable income schedule • t – marginal tax rate • y0 – exemption-level income • B – lumpsum benefit/guaranteed income c(y) = y – T(y) 0  t  1

30. Income space c=c(y) disposable income no-intervention line y pre-tax income

31. The simple income tax c=c(y) Marginal retention rate 1-t Exemption level y0 y

32. 1-t ...extended to Negative Income Tax c=c(y) Incomes subsidised through NIT B = t y0 B y0 y

33. How to generalise this approach…? • Other functional forms of the income tax • Administrative complexity of IT • Interaction with other contingent taxes and benefits.

34. Increasing marginal tax rate t c(y) y

35. Example 1 • 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

36. 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 Example 2 • 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)

37. Interaction with income support c(y) Straight income tax at constant marginal rate Tax-payments kick in with benefits “Clawback” of support Untaxed income support B y2 y y1 y0 0

38. The approach to IT – summary • 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

39. Optimal Income Taxation Overview... Design Issues • Tax schedules • Outline of problem • The solution General labour model Basic ingredients of OIT analysis. “Linear” labour model Education model Generalisations

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

41. Distribution of Ability... • Assume... • 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

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

43. Distribution: example • Example from UK 2000 • Gives distribution of y=wh for full-time male manual workers

44. Basic Ingredients of An Optimal Income Tax model (2) • 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 ?

45. The individual's problem • 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)

46. A Characterisation of Tastes • 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 way slope of indifference curve changes with ability

47. A representation of preferences (consumption) (leisure)

48. Indifference curve in (h,c)-space c (consumption) h (hours worked)

49. Contour translated to (y,c)-space c (consumption = net income) slope = q y (gross income)

50. The regularity condition c high w Illustrates the qw< 0 property Ensures “single-crossing” of ICs for different ability groups low w y