Prerequisites Almost essential Design Contract DESIGN TAXATION MICROECONOMICS

Prerequisites Almost essential: Design Contract DESIGN: TAXATION MICROECONOMICS Principles and Analysis Frank Cowell April 2018 Frank Cowell: Design-Taxation 1

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? April 2018 Frank Cowell: Design-Taxation 2

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 April 2018 Frank Cowell: Design-Taxation 3

Overview Design: Taxation Design basics Preferences, incomes, ability and the government Simple model Generalisations Interpretations April 2018 Frank Cowell: Design-Taxation 4

Model elements § A two-commodity model • leisure (i. e. the opposite of effort) • consumption – a basket of all other goods § Incomes 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 April 2018 Frank Cowell: Design-Taxation 5

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 April 2018 Frank Cowell: Design-Taxation 6

Ability and income § Individuals work (give up leisure) to provide consumption § Individuals differ in talent (ability) • higher ability people produce more and may thus earn more • individual of type works an amount z • produces output q = z, but 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 is implicitly given by q − y § Preferences can be expressed in terms of q and y • for type utility is given by y(z) + y • equivalently: y(q / ) + y April 2018 Frank Cowell: Design-Taxation A closer look at utility 7

The utility function (1) § Preferences over leisure and income § Indifference curves § Reservation utility y sing a e r e inc erenc f pre § u = y(z) + y § yz(z) < 0 § u≥u u 1– z April 2018 Frank Cowell: Design-Taxation 8

The utility function (2) y § Preferences over leisure and output § Indifference curves § Reservation utility inc r pre easing fere nce § u = y(q/ ) + y § yz(q/ ) < 0 § u≥u u q April 2018 Frank Cowell: Design-Taxation 9

Indifference curves: pattern § All types have the same preferences § Function y( ) is common knowledge • utility level u of type depends on effort z and payment y • but value of may be information that is private to individual § Take indifference curves in (q, y) space • u = y(q/ ) + y • slope of given type’s indifference curve depends on value of • indifference curves of different types cross once only April 2018 Frank Cowell: Design-Taxation 10

The single-crossing condition § Preferences over leisure and output y § High talent inc r pre easing fere nce § Low talent § Those with different talent (ability) will have different sloped indifference curves in this diagram type b § qa = aza § qb = bzb type a q April 2018 Frank Cowell: Design-Taxation 11

Similarity with contract model § The position of the Agent • not a single Agent with known ex-ante probability distribution of talents • but 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 April 2018 Frank Cowell: Design-Taxation 12

Overview Design: Taxation Design basics Analogy with contract theory Simple model Generalisations Interpretations April 2018 Frank Cowell: Design-Taxation 13

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 April 2018 Frank Cowell: Design-Taxation 14

Two types § Start with the case closest to optimal contract model § Exactly two skill types • a > b • proportion of a-types is p • values of a, b 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: April 2018 Frank Cowell: Design-Taxation 15

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/ a) ≥ yb + y(qb/ a) • 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 April 2018 Frank Cowell: Design-Taxation 16

Two types: model § We can use a standard Lagrangian 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/ a) + ya) + [1 p] z(y(qb/ b) + yb) + k [p[qa ya] + [1 p][qb yb ] K] + l [yb + y(qb/ b) ub] + m [ya + y(qa/ a) yb y(qb/ a)] where k, l, m are Lagrange multipliers for the constraints April 2018 Frank Cowell: Design-Taxation 17

Two types: method § Differentiate with respect to qa, qb, ya, yb to get FOCs: • pzu(ua)yz(za)/ a + kp + myz(za)/ a ≤ 0 • [1 p]zu(ub)yz(zb)/ b + k [1 p] + lyz(zb)/ b myz(qb/ a)/ a ≤ 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)/ a + kp + myz(za)/ a = 0 • [1 p]zu(ub)yz(zb)/ b + k [1 p] + lyz(zb)/ b myz(qb/ a)/ a = 0 • pzu(ua) kp + m = 0 • [1 p]zu(ub) k[1 p] + l m = 0 § From first and third conditions: • [kp m ] yz(za)/ a + kp + myz(za)/ a = 0 • kp yz(za)/ a + kp = 0 April 2018 Frank Cowell: Design-Taxation 18

Two types: solution § Solving the FOC we get: • yz(qa/ a) = a • yz(qb/ b) = b + kp/[1 p], • where k : = yz(qb/ b) [ b/ a] yz(qb/ a) < 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 a) • no surplus at the bottom (for type b) • determine the “menu” of (q, y)-choices offered by tax agency April 2018 Frank Cowell: Design-Taxation 19

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) y §menu of (q, y) offered by tax authority a y §Analysis determines (q, y) combinations at two points b §If a tax schedule T(∙) is to be designed where y = q −T(q) q q April 2018 b q §then it must be consistent with these two points a Frank Cowell: Design-Taxation 20

Overview Design: Taxation Design basics Moving beyond the two-ability model Simple model Generalisations Interpretations April 2018 Frank Cowell: Design-Taxation 21

A small generalisation § With three types problem becomes a bit more interesting • similar structure to previous case • a > b > c • proportions of each type in the population are pa, pb, pc § We now have one more constraint to worry about 1. participation constraint for c type: yc + y(qc/ c) ≥ uc 2. IC constraint for b type: yb + y(qb/ b) ≥ yc + y(qc/ b) 3. IC constraint for a type: ya + y(qa/ a) ≥ yb + y(qb/ a) § But this is enough to complete the model specification April 2018 • the two IC constraints also imply ya + y(qa/ a) ≥ yc + y(qc/ b) • so no-one has incentive to misrepresent as lower ability Frank Cowell: Design-Taxation 22

Three types § Methodology is same as two-ability model • set up Lagrangian • 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 April 2018 Frank Cowell: Design-Taxation 23

A richer model: N + 1 types § The multi-type case follows immediately from three types § Take N + l types • 0 < 1 < 2 < … < N • (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 April 2018 Frank Cowell: Design-Taxation 24

N + 1 types: behavioural constraints § Participation constraint • is relevant for lowest type j = 0 • form is as before: • y 0 + y(z 0) ≥ u 0 § 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/ j) ≥ yj 1 + y(qj 1 / j) • implies (qj 1, yj 1) < (qj, yj) and u( j) ≥ u( j 1) § From previous cases we know the methodology • (and can probably guess the outcome) April 2018 Frank Cowell: Design-Taxation 25

N+1 types: solution § Lagrangian is only slightly modified from before § Choose {(qj, yj )} to max Sj=0 pj z (y(qj / j) + yj) + k [Sj pj [qj yj] K] + l [y 0 + y(z 0) u 0] + Sj=1 mj [yj + y(qj/ j) yj 1 y(qj 1 / j)] where there are now N incentive-compatibility Lagrange multipliers § And we get the result, as before • MRSN = MRTN • MRSN− 1 < MRTN− 1 • … • MRS 1 < MRT 1 • MRS 0 < MRT 0 • Now the tax schedule is determined at N+1 points April 2018 Frank Cowell: Design-Taxation 26

A continuum of types § One more step is required in generalisation § Suppose the tax agency is faced with a continuum of taxpayers • frequently used 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 April 2018 Frank Cowell: Design-Taxation 27

The continuum model § Continuous ability • bounded support [ , ` ] • density f( ) § Utility for talent as before • u( ) = y( ) + y( q( ) / ) § Participation constraint is • u( ) ≥ u § Incentive compatibility requires • du( ) /d ≥ 0 § SWF is z (u( )) f( ) d April 2018 Frank Cowell: Design-Taxation 28

Continuum model: optimisation § Lagrangian is ` z (u( )) f( ) d ` +k q( ) − y( ) − K] f( ) d + l [ u( ) − u] ` + m( ) [du( ) / d ] f( ) d where u( ) = y( ) + y( q( ) / ) § Lagrange multipliers are • k : government budget constraint • l : participation constraint • m( ) : incentive-compatibility for type § Maximise Lagrangian with respect to q( ) and y( ) for all [ , ` ] April 2018 Frank Cowell: Design-Taxation 29

Output and disposable income under the optimal tax y _ §Lowest type’s indifference curve §Lowest type’s output and income §Intermediate type’s indifference curve, output and income §Highest type’s indifference curve _ 45° §Highest type’s output and income §Menu offered by tax authority _ q_ April 2018 q q Frank Cowell: Design-Taxation 30

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 April 2018 Frank Cowell: Design-Taxation 31

Overview Design: Taxation Design basics Applying design rules to practical policy Simple model Generalisations Interpretations April 2018 Frank Cowell: Design-Taxation 32

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 if q < B / t and vice versa April 2018 Frank Cowell: Design-Taxation 33

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 B q April 2018 Frank Cowell: Design-Taxation 34

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 B 0 is withdrawn after a given output level • relationship between y and q no longer continuous and monotonic April 2018 Frank Cowell: Design-Taxation 35

A badly designed tax-benefit system §Menu offered to low income groups y y §Withdrawal of benefit B 0 §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 a y §High type’s utility-maximising choice b § The notch violates IC B 0 §causes a-types to masquerade as b-types q q April 2018 b q a Frank Cowell: Design-Taxation 36

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 April 2018 Frank Cowell: Design-Taxation 37