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24 October 2011 Frank Cowell: EC426 Public Economics MSc Public Economics 2011/12 http://darp.lse.ac.uk/ec426/ Policy Design: Income Tax Frank A. Cowell Frank Cowell: EC426 Public Economics Overview... Policy Design: Income Tax Design principles Roots in social choice and asymmetric information Simple model Generalisations Interpretations Frank Cowell: EC426 Public Economics Social values: the Arrow problem 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: EC426 Public Economics Social choice function A social state: q Q Individual h’s evaluation of the state vh(q) A profile: [v1, v2, …, vh, … ] A given population is indexed by h = 1,2,…, nh A “reduced-form” utility function 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 Frank Cowell: EC426 Public Economics Implementation Is the SCF consistent with private economic behaviour? 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: Yes if the q picked out by G is also… … the equilibrium of an appropriate economic game 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 Frank Cowell: EC426 Public Economics Manipulation Consider outcomes from a “direct” mechanism in two cases: If all, including h, tell the truth about preferences: If h misrepresents his preferences but others tell the truth: q = G(v1,…, vh, …, ) 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. Frank Cowell: EC426 Public Economics Gibbard-Satterthwaite result Result on SCF G can be stated in several ways A standard version is: (Gibbard 1973, Satterthwaite, 1975 ) If the set of social states Q contains at least three elements; and G is defined for all logically possible preference profiles and G is 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 Frank Cowell: EC426 Public Economics Overview... Policy Design: Income Tax Design principles Preferences, incomes, ability and the government Simple model Generalisations Analogy with contract theory Interpretations Frank Cowell: EC426 Public Economics The design problem The government needs to raise revenue… …and it may want to redistribute resources To do this it uses the tax system Base it on “ability to pay” income rather than wealth ability reflected in productivity Tax authority may have limited information personal income tax… …and income-based subsidies 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? Frank Cowell: EC426 Public Economics Model elements A two-commodity model Income comes only from work individuals are paid according to their marginal product workers differ according to their ability Individuals derive utility from: leisure (i.e. the opposite of effort) consumption – a basket of all other goods similar to optimal contracts (Bolton and Dewatripont 2005) 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 Frank Cowell: EC426 Public Economics Modelling preferences Individual’s preferences Special shape of utility function u = y(z) + y u : utility level z : effort y : income received y() : decreasing, strictly concave, 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 Frank Cowell: EC426 Public Economics Ability and income Individuals work (give up leisure) to get consumption Individuals differ in talent (ability) t Disposable income determined by tax authority 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? 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 Frank Cowell: EC426 Public Economics The utility function Preferences over leisure and income Indifference curves y Reservation utility Transform into (leisure, output) space u = y(z) + y yz(z) < 0 u≥u u u 1– zq u = y(q/t) + y yz(q/t) < 0 Frank Cowell: EC426 Public Economics The single-crossing condition Preferences over leisure and output y High talent Low talent Those with different talent (ability) will have different sloped indifference curves in this diagram type b type a q qa = taza qb = tbzb Frank Cowell: EC426 Public Economics A full-information solution? Consider argument based on the analysis of contracts Full information: owner can fully exploit any manager Same basic story here Pays the minimum amount necessary “Chooses” their effort 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… Frank Cowell: EC426 Public Economics Two types Start with the case closest to the optimal contract model Exactly two skill types From contract design we can write down the outcome ta > tb proportion of a-types is p values of ta , tb and p are common knowledge essentially all we need to do is rework notation But let us examine the model in detail: Frank Cowell: EC426 Public Economics Second-best: two types The government’s budget constraint 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: p[qa - ya] + [1-p][qb - yb ] ≥ K where qh - yh is the amount raised in tax from agent h 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 Frank Cowell: EC426 Public Economics Two types: model We can use a standard Lagrangean approach Constraints are: government chooses (q, y) pairs for each type …subject to three constraints 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 Frank Cowell: EC426 Public Economics Two types: solution From first-order conditions we get: 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: - yz(qa/ta) = ta - yz(qb/tb) = tb + kp/[1-p], where k := yz(qb/tb) - [tb/ta] yz(qb/ta) < 0 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 Frank Cowell: EC426 Public Economics 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 ya Analysis determines (q,y) combinations at two points yb If a tax schedule T(∙) is to be designed where y = q −T(q) … q qb qa …then it must be consistent with these two points Frank Cowell: EC426 Public Economics Overview... Policy Design: Income Tax Design principles Moving beyond the two-ability model Simple model Generalisations Interpretations Frank Cowell: EC426 Public Economics A small generalisation With three types problem becomes a bit more interesting We now have one more constraint to worry about 1. 2. 3. Similar structure to previous case ta > tb > tc proportions of each type in the population are pa, pb, pc 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 Frank Cowell: EC426 Public Economics Three types Methodology is same as two-ability model Outcome essentially as before : MRSa = MRTa MRSb < MRTb MRSc < MRTc Again, no distortion at the top and the participation constraint binding at the bottom set up Lagrangean Lagrange multipliers for budget constraint, participation constraint and two IC constraints maximise with respect to (qa,ya), (qb,yb), (qc,yc) 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… Frank Cowell: EC426 Public Economics 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 Sj pj [qj - yj] ≥ K Sj pj z(y(zj) + yj) where sum is from 0 to N Frank Cowell: EC426 Public Economics N+1 types: behavioural constraints Participation constraint Incentive-compatibility constraint is relevant for lowest type j = 0 form is as before: y0 + y(z0) ≥ u0 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) Frank Cowell: EC426 Public Economics 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 [Sj pj [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 Frank Cowell: EC426 Public Economics A continuum of types One more step is required in generalisation Tax agency is faced with a continuum of taxpayers This can be reasoned from the case with N + 1 types allow N From previous cases we know common assumption allows for general specification of ability distribution form of the participation constraint form that IC constraint must take an outline of the outcome Can proceed by analogy with previous analysis… Frank Cowell: EC426 Public Economics 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 Frank Cowell: EC426 Public Economics Output and disposable income under the optimal tax y t_ Lowest type’s indifference curve Lowest type’s output and income Intermediate type’s indifference curve, output and income _ t Highest type’s indifference curve 45° Highest type’s output and income Menu offered by tax authority q _ q _ q Frank Cowell: EC426 Public Economics Continuum model: results Incentive compatibility implies dy /dq > 0 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 optimal marginal tax rate < 100% (Mirrlees 1971) 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) Frank Cowell: EC426 Public Economics Overview... Design: Taxation Design basics Apply design rules to practical policy…. Plus a “cutdown” version of the OIT problem Simple model Generalisations Interpretations Frank Cowell: EC426 Public Economics Application of design principles The second-best method provides some pointers Simple schemes may be worth considering roughly correspond to actual practice illustrate good/bad design Consider affine (linear) tax system 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 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 if q < B / t … and vice versa Frank Cowell: EC426 Public Economics 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 1-t 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 Frank Cowell: EC426 Public Economics Violations of design principles? The IC condition be violated in actual design This can happen by accident: Commonly known as interaction between income support and income tax. generated by the desire to “target” support more effectively a well-meant inefficiency? 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 Frank Cowell: EC426 Public Economics 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 ya High type’s intended output and income High type’s utility-maximising choice yb The notch violates IC… B0 …causes a-types to masquerade as b-types q qb qa Frank Cowell: EC426 Public Economics 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 4 3 2 18,001 to 59,999: T = 3.05 X – 73.76 X + 695 X + 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 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: EC426 Public Economics 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 Frank Cowell: EC426 Public Economics Linear model: Lagrangean Social welfare is a function of individual utility Individual utility is maximised subject to budget constraint Optimisation problem: choose B and t to max social welfare the covariance of social marginal valuation and income the compensated labour-supply elasticity If K = 0 then B > 0 No explicit general formula? Subject to government budget constraint From maximised Lagrangean get a messy result involving Determined by individual ability Tax parameters B and t 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… Frank Cowell: EC426 Public Economics 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 Frank Cowell: EC426 Public Economics A simulation model Stern’s (1976) model of linear OIT Lognormal ability …more on this below Isoelastic individual utility can be taken as a generalisation of Broome simulation uses standard ingredients: elasticity of substitution s Isoelastic social welfare W = z (u) dF(u) u 1–e – 1 z(u) = ———— , e 0 1–e inequality aversion e Variety of assumptions about government budget constraint Frank Cowell: EC426 Public Economics Lognormal ability f(w) 0 0 Two parameter distribution L(w; m, s2 ) —L(w; 0, 0.25 ) …L(w; 0, 1.0 ) Approximation to empirical distributions 1 2 3 4 w m is log of the median s2 is the variance of log income support is [0, ) 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 ) Frank Cowell: EC426 Public Economics Stern's Optimal Tax Rates s e=0 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 • 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. Frank Cowell: EC426 Public Economics 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 Frank Cowell: EC426 Public Economics References (1) 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 UShaped 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 Frank Cowell: EC426 Public Economics References (2) Mirrlees, J. A. (1971) “An exploration in the theory of the optimal income tax,” Review of Economic Studies, 38, 135-208 Mirrlees, J. A. 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