MSc Public Economics 2011/12 darp.lse.ac.uk/ec426/

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

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

3. 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

4. Social choice function • A social state: q  Q • Individual h’s evaluation of the state vh(q) • A given population is indexed by h = 1,2,…, nh • A “reduced-form” utility function vh(). • A profile: [v1, v2, …, vh, … ] • An ordered list of utility functions • Set of all profiles: V • A social choice function G: V→Q • For a particular profile q = G(v1, v2, …, vh, … ) • Argument is a utility function not a utility level • Picks exactly one chosen element from Q

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

6. Manipulation • Consider outcomes from a “direct” mechanism in two cases: • If all, including h, tell the truth about preferences: • q = G(v1,…, vh, …,) • If h misrepresents his preferences but others tell the truth: • q = G(v1,…, vh, …,) • How does the person “really” feel about q and q? • If vh(q) > vh(q) then there is an incentive to misrepresent information • If h realises this we say that G is manipulable.

7. Gibbard-Satterthwaite result • Result on SCF G can be stated in several ways • (Gibbard 1973, Satterthwaite, 1975 ) • A standard version is: • If the set of social states Q contains at least three elements; • and G is defined for all logically possible preference profiles • and Gis truthfully implementable in dominant strategies... • then G 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

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

9. The design problem • The government needs to raise revenue… • …and it may want to redistribute resources • To do this it uses the tax system • personal income tax… • …and income-based subsidies • Base it on “ability to pay” • income rather than wealth • ability reflected in productivity • Tax authority may have limited information • who have the high ability to pay? • what impact on individuals’ willingness to produce output? • What’s the right way to construct the tax schedule?

10. Model elements • A two-commodity model • leisure (i.e. the opposite of effort) • consumption – a basket of all other goods • similar to optimal contracts (Bolton and Dewatripont 2005) • Income comes only from work • individuals are paid according to their marginal product • workers differ according to their ability • Individuals derive utility from: • their leisure • their disposable income (consumption) • Government / tax agency • has to raise a fixed amount of revenue K • seeks to maximise social welfare… • …where social welfare is a function of individual utilities

11. Modelling preferences • Individual’s preferences • u = y(z) + y • u: utility level • z : effort • y : income received • y(): decreasing, strictly concave, function • Special shape of utility function • quasi-linear form • zero-income effect • y(z) gives the disutility of effort in monetary units • Individual does not have to work • reservation utility level u • requires y(z) + y ≥u

12. Ability and income • Individuals work (give up leisure) to get consumption • Individuals differ in talent (ability) t • higher ability people produce more and may thus earn more • individual of type t works an amount z • produces output q= tz • but individual does not necessarily get to keep this output? • Disposable income determined by tax authority • intervention via taxes and transfers • fixes a relationship between individual’s output and income • (net) income tax on type t is implicitly given by q− y • Preferences can be expressed in terms of q,y • for type t utility is given by y(z) + y • equivalently: y(q /t) + y A closer look at utility

13. The utility function • Preferences over leisure and income y y • Indifference curves increasing preference increasing preference • Reservation utility • Transform into (leisure, output) space • u = y(z) + y • yz(z) < 0 • u = y(q/t) + y • yz(q/t) < 0 • u≥u u u 1– z q

14. The single-crossing condition • Preferences over leisure and output y • High talent increasing preference • Low talent • Those with different talent (ability) will have different sloped indifference curves in this diagram type b type a • qa = taza q • qb = tbzb

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

16. Two types • Start with the case closest to the optimal contract model • Exactly two skill types • ta > tb • proportion of a-types is p • values of ta , tb and p are common knowledge • From contract design we can write down the outcome • essentially all we need to do is rework notation • But let us examine the model in detail:

17. Second-best: two types • The government’s budget constraint • p[qa - ya] + [1-p][qb - yb] ≥ K • where qh - yh is the amount raised in tax from agent h • Participation constraint for the b type: • yb + y(zb)≥ ub • have to offer at least as much as available elsewhere • Incentive-compatibility constraint for the a type: • ya + y(qa/ta)≥ yb + y(qb/ta) • must be no worse off than if it behaved like a b-type • implies (qb,yb) < (qa,ya) • The government seeks to maximise standard SWF • p z(y(za) + ya) + [1-p] z(y(zb) + yb) • where z is increasing and concave

18. Two types: model • We can use a standard Lagrangean approach • government chooses (q, y) pairs for each type • …subject to three constraints • Constraints are: • government budget constraint • participation constraint (for b-types) • incentive-compatibility constraint (for a-types) • Choose qa, qb, ya, yb to max p z(y(qa/ta) + ya) + [1-p] z(y(qb/tb) + yb) + k [p[qa - ya] + [1-p][qb - yb] -K] + l [yb + y(qb/tb)-ub] + m [ya + y(qa/ta)-yb-y(qb/ta)] where k, l, m are Lagrange multipliers for the constraints

19. Two types: solution • From first-order conditions we get: • - yz(qa/ta)= ta • - yz(qb/tb) = tb+ kp/[1-p], • where k :=yz(qb/tb)- [tb/ta] yz(qb/ta) < 0 • Also, all the Lagrange multipliers are positive • so the associated constraints are binding • follows from standard adverse selection model • Results are as for optimum-contracts model: • MRSa = MRTa • MRSb< MRTb • Interpretation • no distortion at the top (for type ta) • no surplus at the bottom (for type tb) • determine the “menu” of (q,y)-choices offered by tax agency

20. a y b y b a q q Two ability types: tax design • a type’s reservation utility y • b type’s reservation utility • b type’s (q,y) • incentive-compatibility constraint • a type’s (q,y) • menu of (q,y) offered by tax authority • Analysis determines (q,y) combinations at two points • If a tax schedule T(∙) is to be designed where y = q −T(q)… • …then it must be consistent with these two points q

21. Overview... Policy Design: Income Tax Design principles Moving beyond the two-ability model Simple model Generalisations Interpretations

22. A small generalisation • With three types problem becomes a bit more interesting • Similar structure to previous case • ta > tb > tc • proportions of each type in the population are pa, pb, pc • We now have one more constraint to worry about • Participation constraint for c type: yc + y(qc/tc) ≥ uc • IC constraint for b type: yb + y(qb/tb)≥ yc + y(qc/tb) • IC constraint for a type: ya + y(qa/ta)≥ yb + y(qb/ta) • But this is enough to complete the model specification • the two IC constraints also imply ya + y(qa/ta)≥ yc + y(qc/tb) • so no-one has incentive to misrepresent as lower ability

23. Three types • Methodology is same as two-ability model • set up Lagrangean • Lagrange multipliers for budget constraint, participation constraint and two IC constraints • maximise with respect to (qa,ya), (qb,yb), (qc,yc) • Outcome essentially as before : • MRSa = MRTa • MRSb< MRTb • MRSc< MRTc • Again, no distortion at the top and the participation constraint binding at the bottom • determines (q,y)-combinations at exactly three points • tax schedule must be consistent with these points • A stepping stone to a much more interesting model…

24. A richer model: N+1 types • The multi-type case follows immediately from the three-type case • Take N + l types • t0< t1< t2< … < tN • (note the required change in notation) • proportion of type j is pj • this distribution is common knowledge • Budget constraint and SWF are now • Sjpj[qj - yj] ≥ K • Sjpj z(y(zj) + yj) • where sum is from 0 to N

25. N+1 types: behavioural constraints • Participation constraint • is relevant for lowesttype j = 0 • form is as before: • y0 + y(z0)≥ u0 • Incentive-compatibility constraint • applies where j > 0 • j must be no worse off than if it behaved as the type below (j-1) • yj + y(qj/tj)≥ yj-1 + y(qj-1/tj). • implies (qj-1,yj-1) < (qj,yj) • and u(tj) ≥ u(tj-1) • From previous cases we know the methodology • (and can probably guess the outcome)

26. N+1 types: solution • Lagrangean is only slightly modified from before • Choose {(qj, yj )} to max Sj=0 pj z(y(qj / tj) + yj) + k [Sjpj[qj - yj] -K] + l [y0 + y(z0)-u0] + Sj=1 mj [yj + y(qj/tj)-yj-1-y(qj-1/tj)] where there are now N incentive-compatibility Lagrange multipliers • And we get the result, as before • MRSN = MRTN • MRSN−1< MRTN−1 • … • MRS1< MRT1 • MRS0< MRT0 • Now the tax schedule is determined at N+1 points

27. A continuum of types • One more step is required in generalisation • Tax agency is faced with a continuum of taxpayers • common assumption • allows for general specification of ability distribution • This can be reasoned from the case with N + 1 types • allow N  • From previous cases we know • form of the participation constraint • form that IC constraint must take • an outline of the outcome • Can proceed by analogy with previous analysis…

28. The continuum model • Continuous ability • bounded support [t,`t ] • density f(t) • Utility for talent t as before u(t) = y(t) + y( q(t) / t) • Participation constraint is u(t) ≥u • Incentive compatibility requires du(t) /dt≥ 0 • SWF is `t ∫z (u(t)) f(t)dt t

29. t _ 45° q _ _ _ t q Output and disposable income under the optimal tax • Lowest type’s indifference curve y • Lowest type’s output and income • Intermediate type’s indifference curve, output and income • Highest type’s indifference curve • Highest type’s output and income • Menu offered by tax authority q

30. Continuum model: results • Incentive compatibility implies dy /dq> 0 • optimal marginal tax rate < 100% (Mirrlees 1971) • No distortion at top implies dy /dq = 1 • zero optimal marginal tax rate! (Seade 1977) • but does not generalise to incomes close to top (Tuomala 1984) • does not hold if there is no “topmost income” (Diamond 1998 ) • May be 0 on the lowest income • depends on distribution of ability there (Ebert 1992) • Explicit form for the optimal income tax requires • specification of distribution f(∙) • specification of individual preferences y(∙) • specification of social preferences z (∙) • specification of required revenue K • (Saez 2001, Brewer et al. 2010, Mankiw 2009)

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

32. Application of design principles • The second-best method provides some pointers • but is not a prescriptive formula • explicit form of OIT usually not possible (Salanié 2003) • model is necessarily over-simplified • exact second-best formula might be administratively complex • Simple schemes may be worth considering • roughly correspond to actual practice • illustrate good/bad design • Consider affine (linear) tax system • benefit B payable to all (guaranteed minimum income) • all gross income (output) taxable at the same marginal rate t… • …constant marginal retention rate: dy/dq= 1  t • Effectively a negative income tax scheme: • (net) income related to output thus: y = B + [1  t] q • so y > q ifq < B / t … and vice versa

33. 1t A simple tax-benefit system • Guaranteed minimum income B y • Constant marginal retention rate • Implied attainable set • Low-income type’s indiff curve • Low-income type’s output, income • High-income type’s indiff curve • Highest type’s output and income • “Linear” income tax system ensures that incentive-compatibility constraint is satisfied • Analysed by Sheshinski (1972) B q

34. Violations of design principles? • The IC condition be violated in actual design • This can happen by accident: • interaction between income support and income tax. • generated by the desire to “target” support more effectively • a well-meant inefficiency? • Commonly known as • the “notch problem” (US) • the “poverty trap” (UK) • Simple example • suppose some of the benefit is intended for lowest types only • an amount B0 is withdrawn after a given output level • relationship between y and q no longer continuous and monotonic

35. a y b y b a q q A badly designed tax-benefit system • Menu offered to low income groups y • Withdrawal of benefit B0 • Implied attainable set • Low-income type’s indiff curve • Low type’s output and income • High-income type’s indiff curve • High type’s intended output and income • High type’s utility-maximising choice • The notch violates IC… • …causes a-types to masquerade as b-types B0 q

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 Neglected design issues? • Administrative complexity • Example 1. UK today (Mirrlees et al 2011) • Example 2. Germany 1981-1985: • linearly increasing marginal tax rate • quadratic tax and disposable income schedules • rates for single person (§32a Einkommensteuergesetz); units DM: • income x up to 4,212: T = 0 • 4,213 to 18,000: T = 0.22x – 926 • 18,001 to 59,999: T =  3.05 X4 – 73.76 X3 + 695 X2 + 2,200 X + 3,034 • where X = x/10,000 – 18,000; • 60,000 to 129,999: T = 0.09X4 – 5.45X3 + 88.13 X2 + 5,040 X + 20,018 • where X = x/10,000 – 60,000; • from 130,000:    T = 0.56 x – 14,837

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

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

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

40. A simulation model • Stern’s (1976) model of linear OIT • can be taken as a generalisation of Broome • simulation uses standard ingredients: • Lognormal ability • …more on this below • Isoelastic individual utility • elasticity of substitution s • Isoelastic social welfare • W=ò z(u)dF(u) u1 – e– 1 z(u) = ———— , e ³ 0 1 – e • inequality aversion e • Variety of assumptions about government budget constraint

41. Two parameter distribution L(w; m, s2 ) m is log of the median s2 is the variance of log income support is [0, ) Approximation to empirical distributions Particularly manual workers Stern took s = 0.39 (same as Mirrlees 1971) In this case less than 2% of the population have less than 0.707 × mean (Broome 1975) Lognormal ability f(w) • L(w; 0, 0.25 ) • L(w; 0, 1.0 ) 0 w 0 1 2 3 4

42. Stern's Optimal Tax Rates s e = 0 0.2 36.2 0.4 22.3 0.6 17.0 0.8 14.1 1.0 12.7 e = 1 62.7 47.7 38.9 33.1 29.1 e =  92.6 83.9 75.6 68.2 62.1 • Calculations are for a purely redistributive tax: i.e. K= 0 • Broome case corresponds to bottom right corner. But he assumed that there was no-one below 70.71% of the median.

43. Summary • Could we have “full information” taxation? • OIT is a standard second-best problem • Elementary version a reworking of the contract model • Can be extended to general ability distribution • Provides simple rules of thumb for good design • In practice these may be violated by well-meaning policies

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

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