Design: Taxation

1. Prerequisites Almost essential Contract Design Design: Taxation MICROECONOMICS Principles and Analysis Frank Cowell September 2006

2. 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?

3. A link with contract theory • Base approach on the analysis of contracts • close analogy with case of hidden characteristics • owner hires manager… • …but manager’s ability is unknown at time of hiring • Ability here plays the role of unobservable type • ability may not be directly observable… • …but distribution of ability in the population is known • A progressive treatment: • outline model components • use analogy with contracts to solve two-type case • proceed to large (finite) number of types • then extend to general continuous distribution

4. Overview... Design: Taxation Design basics Preferences, incomes, ability and the government Simple model Generalisations Interpretations

5. Model elements • A two-commodity model • leisure (i.e. the opposite of effort) • consumption – a basket of all other goods • 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

6. 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

7. Ability and income • Individuals work (give up leisure) to provide 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 and y • for type t utility is given by y(z) + y • equivalently: y(q /t) + y A closer look at utility

8. The utility function (1) • Preferences over leisure and income y • Indifference curves increasing preference • Reservation utility • u = y(z) + y • yz(z) < 0 • u≥u u 1– z

9. The utility function (2) • Preferences over leisure and output y • Indifference curves increasing preference • Reservation utility • u = y(q/t) + y • yz(q/t) < 0 • u≥u u q

10. Indifference curves: pattern • All types have the same preferences • Function y() is common knowledge • but utility level uof type t depends on effort z and payment y • value of t may be information that is private to individual • Take indifference curves in (q, y) space • u = y(q/t) + y • slope of a given type’s indifference curve depends on value of t • indifference curves of different types cross once only

11. 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

12. Similarity with contract model • The position of the Agent • instead of a single Agent with known ex-ante probability distribution of talents,… • … a population of workers with known distribution of abilities. • The position of the Principal (designer) • designer is the government acting as Principal. • knows distribution of ability (common knowledge) • the objective function is a standard SWF • One extra constraint • the community has to raise a fixed amount K≥ 0 • the government imposes a tax • drives a wedge between market income generated by worker and the amount available to spend on other goods.

13. Overview... Design: Taxation Design basics Analogy with contract theory Simple model Generalisations Interpretations

14. A full-information solution? • Consider argument based on the analysis of contracts • Given 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…

15. 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:

16. 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

17. 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

18. Two types: method • Differentiate with respect to qa, qb, ya, yb to get FOCs: • pzu(ua)yz(za)/ta + kp + myz(za)/ta≤ 0 • [1-p]zu(ub)yz(zb)/tb + k [1-p] + lyz(zb)/tb - myz(qb/ta)/ta ≤ 0 • pzu(ua) - kp + m ≤ 0 • [1-p]zu(ub) - k[1-p] + l -m ≤ 0 • For an interior solution, where qa, qb, ya, yb are all positive • pzu(ua)yz(za)/ta + kp + myz(za)/ta= 0 • [1-p]zu(ub)yz(zb)/tb + k [1-p] + lyz(zb)/tb - myz(qb/ta)/ta = 0 • pzu(ua) - kp + m = 0 • [1-p]zu(ub) - k[1-p] + l -m = 0 • Manipulating these gives the main results • For example, from first and third condition: • [kp - m ] yz(za)/ta + kp + myz(za)/ta= 0 • kp yz(za)/ta + kp= 0

19. Two types: solution • Solving the FOC 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... Design: Taxation Design basics 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 three types • 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 like 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 pjz(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 • Suppose the tax agency is faced with a continuum of taxpayers • common assumption • allows for general specification of ability distribution • This case 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. Continuum model: optimisation • Lagrangean is `t ⌠ │ z (u(t))f(t) dt ⌡t `t ⌠ +k │ [ q(t) − y(t) −K]f(t) dt ⌡t + l [ u(t) − u] `t ⌠ du(t) + │ m(t) ——f(t) dt ⌡tdt where u(t) = y(t) + y( q(t) / t) • Lagrange multipliers are • k : government budget constraint • l : participation constraint • m(t) : incentive-compatibility for type t • Maximise Lagrangean with respect to q(t) and y(t) for all t  [t,`t ]

30. 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

31. Continuum model: results • Incentive compatibility implies • dy /dq> 0 • optimal marginal tax rate < 100% • No distortion at top implies • dy /dq = 1 • zero optimal marginal tax rate! • But 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

32. Overview... Design: Taxation Design basics Applying design rules to practical policy Simple model Generalisations Interpretations

33. Application of design principles • The second-best method provides some pointers • but is not a prescriptive formula • 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

34. 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 B q

35. Violations of design principles? • Sometimes 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

36. 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

37. Summary • Optimal income tax 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