Microsimulation

European Commission Joint Research Center, Seville, Spain 2-3 October 2017

Microsimulation and Optimal Taxation Theory Ugo Colombino EST (Turin), CHILD (Turin) and LISER (Luxembourg)

1

Motivation • Tradition: • Optimal Taxation: formula for optimal general tax-transfer rules

• Microsimulation: evaluation of specific tax-transfer rules • More recently: • Use of Microsimulation for implementing OT formulas • Drop formulas, integration of microeconometrics, microsimulation and numerical optimization to derive optimal rules

2

Outline: 1st day, 9.30 – 11.00

Preliminaries • Two models: • Labour supply • Taxable earnings • Elasticity concepts

• Marginal tax reforms • No income effects (quasi-linear utility) • Microsimulation • Non-behavioural • Behavioural, semi-structural • Behavioural, structural

3

Outline: 1st day, 11.30 – 13.00

Optimal Taxation • First-best and Second-best taxation • First best: lump-sum based on exogenous endowments or characteristics • Second best: based on earnings

• Edgeworth (1897) • Inverse elasticity rules: • Revenue maximizing tax rate • Ramsey 1927

4

Outline: 1st day, 14.30 – 16.00, 16.30 – 17.30

Optimal Taxation • Mirrlees (1971) • Diamond (1998), Saez (2001, 2002)

• Limitations of OT theory: Analytical Approach (AA) • New strands

5

Outline: 2nd day, 9.30 – 11.00

• A computational approach (CA): combining microeconometrics, microsimulation and numerical optimization • Steps in implementing CA

6

Outline: 2nd day, 11.30 – 13.00

Examples of CA • • • • • • • • •

Fortin et al. (1993) Aaberge and Colombino (2006) Sefton and de Ven (2009) Colombino et al. (2010) Colombino and Narazani (2011) Aaberge and Colombino (2012, 2013) Blundell and Shephard (2012) Colombino (2015) Islam and Colombino (2017)

7

Outline: 2nd day, 14.30 – 16.00, 16.30 – 17.30

• Islam and Colombino (2017)

8

Two (almost) equivalent models

1) Labour supply

c  consumption l = hours of work w = wage rate maxl,cU(c,l), s.t. c = wl - T(wl) maxl U wl - T(wl) , l  FOC: Uc  w - wT'(wl) + ul = 0 9

Two (almost) equivalent models

2) Taxable income

z  taxable earnings c = z - T(z) max z Uz - T(z) , z  FOC: Uc 1 - T'(z) + uz = 0

10

Two (almost) equivalent models

The two models are essentially equivalent if we interpret z =wl. However, the «taxable income» model is more general

Mirrlees 1971 interprets z as the observable result of the product of two unobservables: z = nl, n= «ability», l = «effort» Feldstein 1999 interprets z as the product of a vector of incomeproducing activities and a vector of specific productivities.

11

Elasticity concepts

l w 1) el   elasticity of l w.r.t. w w l z w 2) e z  = elasticity of z w.r.t. w = w z

l  1  l  w  1  e l   w  l

12

Elasticity conceps

Notice: l w l w(1  T ') l (1  T ') el    w l w(1  T ') l (1  T ') l elast. of l w.r.t. w   elast. of l w.r.t. net w   elast. of l w.r.t. 1-T' z w z w(1  T ') z (1  T ') ez    w z w(1  T ') z (1  T ') z elast. of z w.r.t. w   elast. of z w.r.t. net w   elast. of z w.r.t. 1-T' 13

Marginal tax reforms (Envelope Th.)

U(wl  T (wl), l) Supose l is the optimally chosen value of l given T(.). FOC:Uc w  wT '(z)  Ul  0 Consider a marginal reform dT(wl). dU  Uc  dT (wl)  Uc w  wT '(wl)dl  Ul dl

 Uc  dT (wl)  Uc w  wT '(z)  Ul dl  Uc dT (wl)

since Uc w  wT '(z)  Ul  0! 14

Marginal tax reforms (Envelope Th.)

U(z  T (z), z) Supose z is the optimally chosen value of z given T(.). FOC: Uc 1  T '(z)  Uz  0 Consider a marginal reform dT(z). dU  Uc  dT (z)  Uc 1  T '(z)dz  Uz dz

 Uc  dT (z)  Uc 1  T '(z)  Uz dz  Uc dT (z)

since Uc 1  T '(z)  Uz  0! 15

No income effects (quasi-linear utility)

U(c , l)  c   (l) c  wl  y U  wl  y   (l) Ul  w   '(l*)  0 l *depends only on w (no income effects!)

16

No income effects (quasi-linear utility)

U(c , l)  c   (l) c  wl  y U  wl  y   (l) Ul  w   '(l*)  0 l *depends only on w (no income effects!)

17

No income effects (quasi-linear utility)

U(c , z)   z  y    (z) Uz  1   '(z*)  0 z *does not depend on y (no income effects!)

18

Microsimulation • Orcutt 1957: behavioural, reduced form • 60’ – 90’: non-behavioural • Since 2000: behavioural, reduced form, structural • Euromod conference in Cambridge, 1998: a couple of behavioural papers • Euromod conference in Essex, 2010: all papers dedicated to behavioural microsimulation

19

Microsimulation Non-behavioural The effect of a marginal reform dT(z) is just -UcdT(z). Moreover, with quasi-linear preferences, we can normalize UC=1. This justifies non-behavioural microsimulation: the effect of a (marginal) tax reform is just its budget effect.

20

Microsimulation Behavioural reduced form Non-marginal reform ΔT(z). It turns out that the the effect on C = z – T(z) can be written as

z c  T (z)ez T (z)

where ez is some reduced form estimate of the elasticity of z with respect to (1-T(z)). Close to original Orcutt’s idea. Revived recently as the «sufficient statistics» approach (Chetty 2009) 21

Microsimulation Behavioural structural Non-marginal reform: T0(wl) → T1(wl). Microeconometric model: Estimate Ui(wl –T0(wl), l) Microsimulation: simulate the solution to Maxl Ui(wl –T1(wl), l)

22

Optimal taxation • First Best vs Second Best taxation • Edgewoth (1887): 1° best, complete equalization • Ramsey (1927): 2° best taxation of goods • Laffer curve: revenue maximizing tax rate • Mirrlees (1971): 2° best taxation of earnings, standard labour supply model • Diamond (1998): simplifies Mirrlees • Feldstein (1999): taxable income approach • Saez (2001): merges Mirrlees, Diamond and Feldstein

23

First Best and Second Best Taxation U(c, l) Uc > 0, Ul < 0 (l is a «bad») Compare a lump sum tax S: c = wl – S → Tax paid = S (does not depend on l) with a tax on earnings: c = (1-t)wl → Tax paid = twl (does depend on l)

24

First Best and Second Best Taxation U(c, z) Uc > 0, Uz < 0 (z is a «bad») Compare a lump sum tax S: c = z – S → Tax paid = S (does not depend on z) with a tax on earnings: c = (1-t)z → Tax paid = tz (does depend on z)

25

First Best and Second Best taxation

C

C=Z

Z

26

First Best and Second Best taxation

C

C = (1-t)Z

Z

27

First Best and Second Best taxation

C

C = (1-t)Z

Z

28

First Best and Second Best taxation

C

tZ

C = (1-t)Z

Z

29

First Best and Second Best taxation

C

C=Z-T T = tZ

C = (1-t)Z

Z

30

First Best and Second Best taxation

C

C=Z-T T = tZ

C = (1-t)Z

Z

T generates the same revenue as t but with a higher utility level

31

First Best and Second Best Taxation (Algebra) 2nd Best 2

1 z  u(c , z)  c    , c  z  tz 2 w  z * (t )  w 2 (1  t ) 1 2 w (1  t )2 2 1st Best u * (t ) 

2

1 z  u(c , z)  c    ,c  z  T 2 w  z * (T )  w 2 , c * (T )  w 2  T  w 2  tw 2  w 2 (1  t ) u * (T ) 

1 1 2 w (1  t )2  u * (t ) t 2 32

First Best and Second Best Taxation • As a First Best solution we should adopt Lump Sum taxes.

• A Lump Sum tax by itself is not a problem. We could easily levy an equal Lump Sum tax on all the cityzens. • That would be called a Poll Tax.

33

First Best and Second Best Taxation • However, for redistribution purposes, L.S. taxes should be differentiated and based on exogenous endowments or characteristics. • Problem: • Exogenous endowments are typically inobservable (Talent? Ability?) or not truthfully revealed by citizens • Exogenous characteristics (Age? Gender? Beauty?) are ethically or polically hard to consider. • Therefore we adopt 2nd best: tax endogenous quantities (e.g. earnings) that are related to exogenous endowments or characteristics (e.g. talent, ability).

34

First Best and Second Best Taxation However, what should we do if we could observe and tax an exogenous endowment?

35

First Best Taxation Edgeworth (1897)

zi  exogenous ci  zi  Ti Common preferences : U(ci )

Public Budget constraint :  Ti  E i

36

First Best Taxation Edgeworth (1897) max T1 ,...,TN

 U(z  T ) (utilitarian criterion) s.t.T  E i

i

i

i

  max    U(zi  Ti )    Ti  E   i  i U '(zi  Ti )   (constant)

ci  zi  Ti  cons tant   zi  E (perfect equalization!) i

We get a less extreme results with heterogeneus preferences: Ui' (zi  Ti )   (constant)  equalization of marginal utilities

37

Second Best Taxation • Edgeworth attains equality with no efficiency loss since he redistributes an exogenously given amount.

• If we are constrained to redistribute an endogenously determined amount, in general we will face an efficiency loss: efficiency/equality trade-off

38

Revenue maximizing tax rate • Just as an introduction: maximize tax revenue • Assume a flat tax rate t • R(t) = tz(t) = tax revenue (note that z depends on t) • R(0) = R(1) = 0 (Inverse U profile: Laffer curve)

39

Revenue maximizing tax rate dz(t*) t*  0 maxt tz(t )  FOC : z(t*)  dt * Define: dz(t*) 1  t * dz(t*) dz(t*) , ez   d (1  t*) z(t*) d (1  t*) dt * Rewrite the FOC: 1 t*    0  t*  z(t*) 1  ez  1  ez 1  t *  40

Revenue maximizing tax rate 1 t*  1  ez Clearly no t  t * can be efficient. This is a useful introduction to the crucial role of the elasticity ez . It reminds of Ramsey 1927 ... 41

Ramsey 1927 2nd best tax rate t j on consumption good j: 1 tj  k | j | where k is a constant and

 j  price elasticity of demand for good

42

Mirrlees 1971 Individuals differs only accordind to ability n (unobserved): f (n) = p.d.f. of n Common (separable) utility function u(c , l) l = effort (unobservable) z  nl = earnings, T (z) = Taxes c  z - T (z) E  taxes to be collected f (n),u(.,.) and E are called the "primitives" of the economy

43

Mirrlees 1971 

max    u  nl  T (nl), l   f (n)dn T (.)

0

s.t. 

 T (nl) f (n)dn  E

(Public Budget Constraint)

0

max l u  nl  T (nl), l   n 1  T '(nl) u  u  0 , c

, l

(Incentive-Compatibility Constraint  Utility maximization FOC) 44

Mirrlees 1971 Solution with quasi-linear preferences  no income effects T '(nl) 1   1  F (n)     1   1  G(n)  1  T '(nl)     nf (n) 

  elasticity of l with respect to n G(n) 





z

 '(u(m)) f (m)dm / 

1  F (n) = relative social weight assigned to individuals with ability  n

T (0) 

 T (nl) f (n)dn  E (if T(0) < 0, -T(0) is a transfer)

nl 0

45

Mirrlees 1971 • Very influential approach in theoretical work • Analytical optimization: control theory (hard to work with) • Formula expressed in terms of unobservable n

• Interior solution (only intensive behaviour) • Weak link with empirical research

• Simulation with hypotetical f(n), simple u(c, l) and constant ε

46

Saez 2001 • Taxable income approach (Feldstein 1999) • «Perturbation» approach (Diamond 1998, much simpler than control theory used by Mirrlees 1971)) • Formula expressed in terms of observables • More direct link with empirical research • But: only intensive behaviour (as in Mirrlees 1971) 47

Saez 2001 Common (separable) utility function u(z  T (z), z) z = taxable income h(z) = p.d.f. of z T (z) = taxes

48

Saez 2001 A special case: Optimal Top Flat rate  above z zm  average z above z (N individuals) Assume no income effects (quasi-linear preferences) "Perturbation" approach When  is optimal, a small change d will iply dSWF = 0 Decompose dSWF:

49

Saez 2001 "Mechanical effect": dM = N  z m  z  d m  "Welfare effect": dW = - gN  z  z  d g is a social welfare weight put on earnings above z

dz m "Behavioural effect": dB =  N d (1   )

50

Saez 2001 By setting dM + dW + dB = 0 we get: z   1  g  1  m    z    1  ez

51

Saez 2001 z   1  g  1  m    z    1  ez z For sufficiently high z,empirically one finds m constant 0.5 (US). z a ak z 1 If h(z)  1a (i.e. Pareto distribution) then m  (Diamond 1998). a z z

52

Saez 2001

If z = 0 we get the formula for the optimal Flat Tax

 1 

1  g   ez

53

Saez 2001 If z  z we approach the optimal tax rate for the top z:   0 m

(provided the distribution of z is bounded) Irrelevant, in practice. z   1  g  1  m    z     0 for z  z m 1  ez

54

Saez 2001 General non linear tax rule 

max    u  z  T (z), z  h(z)dz T (.)

0

s.t. 

 T (z)h(z)dz  E

(Public Budget Constraint)

0

max z u  z  T (z), z   1  T '(z) u  u  0 , c

, z

(Incentive-Compatibility Constraint) 55

Saez 2001 "Perturbation" approach Let T '(z) be the optimal marginal tax (yet to be found!) Let’s consider a local perturbation of T ’  z  , i.e. T ’  z   T '(z)  d in a small interval (z , z  dz) Since we are at the optimum T(z), the effect dSWF of the (marginal) "perturbation"on the SWF should be dSWF = 0. Let's decompose dSWF into three effects:

56

Saez 2001 "Mechanical effect": dM  d dz 1  H(z) where 1  H(z) is the proportion of people with earnings above z "Welfare effect": dW  dzd 1  H(z)G(z) where G(z) is the social weight put on people with earnings above z dz "Behavioural effect": dB  h(z)dzT '(z) d (1  T '(z)) Note: dz has no effect on utility since individual optimally choose z! (FOC)

57

Saez 2001 Setting dM + dW + dB = 0 and assuming no income effects (quasi-linear preferences) one can write the

solution as follows:

58

Saez 2001 1 T '(z)  1  T '(z)  ez

  1  H (z)   1  G(z)      zh(z) 

ez  elasticity of z with respect to 1  T '(z) G(z) 





z

 '(u(s))h(s)ds / 

1  H (z) = relative social weight assigned to individuals with earnings  z

T (0) 

 T (z)h(z)dz  E (if T(0) < 0, -T(0) is a transfer)

z 0

59

Saez 2001

Note on social welfare weights: g(s)   '(u(z))uc  where  is the P.B.C. multiplier 

G(z)   g(s)h(s)ds 1  H(z) z

With no income effects: 

G(0)  1 since it turns out that    g(s)h(s)ds 0

60

Saez 2001 • Both Mirrlees 1971 and Saez 2001 rule out T’(z) < 0. • Therefore, policies like tax credits, in-work benefits etc cannot be optimal according to their models.

61

Saez 2001 Recalling the perturbabion effect, suppose at the optimum T(z), T'(z) < 0 for some z. Now consider a marginal change d > 0 dM+dW: d dz 1  H(z)  dzd 1  H(z)G(z)  d dz 1  H(z)1  G(z)  0 dz dB  h(z)dzT '(z)  0 (since T'(z) < 0) d (1  T '(z)) But then dM+dW+dB > 0, and therefore T'(z) cannot be optimal!

62

Saez 2001 However this is due to the assumption that everyone is at an interior solution (no extensive margin). Diamond (1980) and Saez (2002) introduce the extensive margin. Here’s a simplified model.

63

Brewer, Saez & Shephard 2008 Simplified model, extensive margin only. Every individual has potential earning z (exogenous to the individual)

If working, gets z – T(z) – q, q = fixed cost of working If not working, gets –T(0) > 0. Work if z – T(z) – q > – T(0) or z + T(0) – T(z) > q 64

Brewer, Saez & Shephard 2008 P(q \ z)  distribution of q conditional on z P(z  T (z)  T (0) \ z)  P(qz )   proportion of individuals working conditional on z dP(qz ) qz  participation elasticity z  dqz P(qz ) T (z)  T (0)  average tax rate for earnings z t (z)  z

65

Brewer, Saez & Shephard 2007 A "perturbation" argument leads to the solution: t (z) 1  1  g(z) 1  t (z)  z The average of the social weights g(z) is equal to 1. g(z) < 1 for high values of z g(z) > 1 for low values of z For some value of z the optimal t(z) < 0. The model is a possible rationalization of policies like EITC, In-work benefit etc. 66

Saez 2002 • Discrete alternatives (jobs), j = 1, 2, …

• Both intensive and extensive margins • Attractive since it matches the popular microeconometric discrete choice models of labour supply

67

Saez 2002 J

T '( j) 

h(k) 1  g(k)   (k)T '(k)  1 k j



h( j)

where T '( j) = marginal tax rate at j  ( j) = elasticity of labour supply at j  = participation elasticity h( j) = number of people at j g( j)= social weight put on people at j 68

Empirical identification of optimal T( ) Saez 2001 1 T '(z)  1  T '(z)  ez

  1  H (z)   1  G(z)      zh(z) 

It is not an explicit formula since z depends on T '(z). In order to compute T '(z) one has to specify structurally how z depends on T '(z).

69

Saez 2001, Brewer et al. 2007 Simulation procedure 1 e

e  1  z  1 e 1) u(z  T (z), z)   z  T (z)     z  n 1  T '( z )      1  e  n  2) Calibrate F (n) such that h(z(n)) given actual tax rate fits actual h(z) 1 e

3) Compute zn  n

1  T0 '(z) 

e

given an initial T0 '  microsimulation.

4) Compute G0 (n) 5) Use formula (Mirrlees 1971)

T1 '(zn )  1   1  F (n)   1  G0 (zn )    1  T1 '(zn )  1  e   nf (n) 

to get T1 ' 6) Repeat 3 - 5 until convergence: Tx 1 '  Tx ' 70

Analitical Approach Empirical identification of optimal T(θ ) Max T Ω[u(c,z)]f(z)dz s.t. P.B.C. and I.C.C. This is solved analytically

Formula T*(z)

T*(z(θ))

z(θ) Microsimulate z with an imputed utility maximization and «primitives» : Labour supply elasticity Productivity distribution etc. Primitives are imputed by calibration or taken from external estimation

71

72

Main policy prescriptions • Mirrlees: NIT and (approx.) FT or mildly increasing • Saez et al.: NIT and/or IWB + (approx.) FT or mildly increasing • However, very sensitive w.r.t. specification of utility function and productivity distribution (Tuomala 2010)

73

Inverse Optimal taxation • The formulas by Mirrlees 1971, Saez 2001 or Saez 2002 can be used to retrieve the implicit welfare weights, given the observed T’(z), H(z), h(z) and ez: • Blundell et al. 2009 • Bourguignon and Spadaro 2012 • Bargain et al. 2014

74

Outline: 2nd day, 9.30 – 11.00

• Limitations of OT theory: Analytical Approach (AA) • A computational approach (CA): combining microeconmetrics, microsimulation and numerical optimization • Steps in implementing CA

75

Outline: 2nd day, 11.30 – 13.00

Examples of CA • • • • • • • • •

Fortin et al. (1993) Aaberge and Colombino (2006) Sefton and de Ven (2009) Colombino et al. (2010) Colombino and Narazani (2011) Aaberge and Colombino (2012, 2013) Blundell and Shephard (2012) Colombino (2015) Islam and Colombino (2017)

76

Outline: 2nd day, 14.30 – 16.00, 16.30 – 17.30

• Islam and Colombino (2017)

77

Main limitations of the Analytical Approach The approach combines analytical optimization, imputations and microsimulation • Several limitations due to going through a «formula» : • Hard to represent household decisions • Most applications: quasi-linear utility, no income effects • Hard to account for multidimensional productivity or endogenous characteristics) • Too general tax-benefit rule

• Potential inconsistency between the theoretical assumptions and the calibrated or (elsewhere) estimated parameters

78

New strands • Back to Ramsey: parametric representations of the tax-benefit rule (Heathcote and Tsujiyama 2016) • Microeconometric model replaces imputation of primitives • Microsimulation of household decisions replaces FOCs for utility maximization • Numerical Optimization of Social Welfare replaces Analytical Optimization 79

Parametric representation of T(.) • The theoretical literature produces non-parametric representations of T(.) • The result requires many restrictive assumptions • It is overly general: in practice, the resulting profiles of T(.) are purely suggestive in view of practical implementations • Why not starting with realistic - and easier to manage – parametric representations? • Parametric representations of the tax rule in principle allows for a richer representation of the economy: dynamics, general equilibrium etc., e.g. Heathcote and Tsujiyama 2016

80

Microsimulation of Household Decisions • Microsimulation of households’ optimal choices and maximum attained utility replaces the FOCs for utility maximization • More flexibility in representing preferencies, budget constraints, opportunity sets

81

Numerical Optimization of Social Welfare • Numerical optimization of the Social Welfare function replaces the analytical optimization • Allows to drop the regularity assumptions required by standard methods of analytical optimization • In principle allows to use alternative criteria: e.g. non-welfaristic social welfare etc.

82

Computational Approach (CA) • Combine numerical optimization, microeconometric estimates and microsimulation • Parametric (π) representation of T( ) (back to Ramsey)

• Household decisions – given π – represented by simulation of a microeconometric model (replaces I.C.C) • Numerical optimization of S.W.F. w.r.t. π, s.t. P.B.C. 83

Analytical Approach vs Computational Approach

Analytical Approach

Computational Approach

• • • • • • •

• • • • • • •

Individuals Quasi-linear utility Non-parametric tax rule Common preferences Calibrated «primitives» FOCs for max U(c,l) Analytical optimization

Households Flexible microeconometric model Parametric tax rule Heterogenuous preferences Estimated «primitives» Microsimulation of choices Numerical optimization

84

Identifying optimal πs with CA for economy s For a specific economy s (e.g. a country): • π = parameters of the policy • Ui(c, l;π) = utility of household i, given income c, hours of work l and policy π • Vi(π) = maxh,c Ui(c, l; π) = maximized utility given policy π s.t. household’s budget constraint • Transform Vi(π) into comparable money-metric μi(π)

• Identify optimal policy: πs = argmaxπ W(μ1(π), …, μN(π)) s.t. public budget constraint. 85

Identifying the correspondence between θ and π • AA produces a general function from the «primitives» θ (elasticities, productivityty distribution, P.B.C. etc.) to optimal tax-benefit rules for a generic economy • CA identifies the optimal tax-benefit rule for a specific economy (e.g. a country in a certain year, a certain subsample of the population etc.), i.e. specific primitives θ. • Can CA attain a level of generality close to the one attained by AA? 86

Identifying the correspondence θ  π • S «economies» • θ1, θ2, …, θS = «primitives» • π1, π2, …, πS = optimal tax-benefit rules identified by CA • Identify the «general» correspondence between θ and π

87

Identifying optimal πs with CA for economy s

π* at convergence

Maxπ W(μ1(π),…, N(π))

π

s.t. P.B.C. This is solved by numerical optimization

μi(π)

Vi(π) = maxc,l Ui(c, l) s.t. household’s budget constraint given π This is solved by microsimulating a microeconometric model of household decisions V is turned into μ (King 1983)

88

Identifying the correspondence between π and θ

θ1 … … … θS

Regression analysis, Clusters, Correspondence analysis, etc.

π1 … … … πS

89

Analytical Approach vs Computational Approach

• CA can address issues that are (so far) very hard to address with AA e.g. • Simultaneous household decisions • Tagging • Comparing individual vs joint tax-benefit rules • Accounting for administrative features • Non-welfaristic criteria • Adopting alternative modelling of household choices, such as • Agent-based • Behavioural approach 90

Complementarity of AA and CA Lessons from other areas: • Design of tax-transfers can be closely compared to the design of auctions • In both cases, the problem is designing rules that should be based on the unobservable (ability and willingness to pay, respectively), and must therefore be based on observable proxies (earnings and bids, respectively) • In both cases, we have on the one hand elegant theoretical model and on the other hand complicated implementation practices • The recent history of auction design teaches us a virtuous interaction between theoretical models and the practical designs (Milgrom 2004)

91

Complementarity of AA and CA Other areas: • Traditional engineering: theory + «experience» • Modern engineering: theory + simulation (and/or experiments) • Matching mechanisms (Roth 2002)

92

Complementarity of AA and CA • Theory sets basic framework, concepts and promising directions for solution • Simulations, experiments and numerical methods help in designing implementations

93

Complementarity of AA and CA • Optimal taxation theory suggests general promising solutions (e.g. NIT, FT, IWB etc.) • Combining microeconometrics, microsimulation and numerical optimization allows to explore the policy space with more realistic and richer representations of preferences and constraints

94

Steps in applying CA • Microeconometric model • Microsimulation

• Comparable utility • Social Welfare function

• Parametric representation of tax-transfer rules • Numerical Optimization

95

Microeconometric Model max V(c, l; β) + ε s.t. c = wl +y – T(wl +y) lB U( ) = utility function V( ) = «systematic» utility β = prefence parameters (to be estimated) ε = random variable (extreme value distribution) T( ) = tax-transfer rule c = net income l = labour supply w = wage rate y = exogenous income B = opportunity set

96

Microeconometric Model 1) V(c, l) quadratic form in c and l, with coefficients function of socio-demographic characteristics + some alternative-specific dummies, e.g. singles:

V  1c  2c 2   3 (L  l)   4 (L  l)2   5 (L  l)c 2) An alternative in the literature is the so-called Box-Cox form, e.g. singles: 2 4   c 1 (L  l)  1 c  1 (L  l)  1  V  1  3  5    2 4 4  2   2

4

97

Microeconometric Model B contains alternative values of l Some papers allow also for choice of w Number of alternatives (min. 3 – 6) not very important for estimation. More important for simulation Alternatives sampled from empirical distribution (should improve efficiency of the estimates) More common in the literature are fixed alternatives See Aaberge et al. (2009) on the implications of alternative representations of the opportunity set 98

Microeconometric Model Problem: w is not observed for non-working people Many alternative solutions: 1) Estimate w equation on working subsample with Heckman correction, then impute expected w to everyone 2) Estimate w equation on working subsample with Dagsvik 2000 correction, then impute expected w to everyone 3) Variant of (1) or (3): impute w + simulated residuals of w equation 4) Simltaneous estimation of V( ) and w equation (accounting for selectivity) See survey Loeffer et al. 2014

99

Microeconometric Model Choice probability (Conditional Logit)

Pi ,k 

expVi (ci ,k , li ,k ;  )

 expV (c j

i

i,j

, li , j ;  )

Can be extended: random coefficients, multidimentional choice etc.

100

Microsimulation Simulate the value of some function of c and/or l, x(c,l), given a new T0 0 0

ci ,k  wi ,k li ,k  T (wi ,k li ,k ) 0 ˆ) V ( c , l ;  i i ,k i ,k 0 ˆ Pi ,k  0 ˆ) V ( c , l ;   j i i,j i,j x  x(c , li ,k ) 0 i ,k

0 i ,k

xˆi0  simulated value of xi0 Three methods: 101

Microsimulation 0 0 0 ˆ ˆ 1) xi   Pi ,k xi ,k k

0 0 0 ˆ ˆ 2) xi  xi ,k* , k*  argmax k Pi ,k

0 0 0 ˆ 3) xi  xi ,k* , k*  argmax k Vi (ci ,k , li ,k ; ˆ)  ˆk

The ˆk are simulated by drawing from the Type I extreme value distribution 102

Comparable utility Given the expected maximum utility (McFadden 1978) …

Vi *  ln

  expV (c j

i

i,j

, li , j ;  )



103

Comparable utility … define the comparable money-metric utility μi (King 1983):

ln

  expV ( ,l ;  )  V * j

R

i

j

i

Decoster & Haan (2015) present a different implementation inspired by Fleurbeay (2011) An alternative (e.g. Aaberge & Colombino 2013) is to compute V* with common preference parameters (Deaton & Muellbauer 1980)

104

Social Welfare (Atkinson 1970) 1 1  Social welfare    i i  N  1 Efficiency   i i   N

1 1



 Inequality  1  

105

Social Welfare (Sen 1973) 1 Social welfare   i i (1  C ) N 1 Efficiency   i i   N Inequality  C  Gini concentration index of  distribution You get different version with generalizations of C (Aaberge 2007)

106

Social Welfare (Kolm 1976) expk(i   )  1  Social Welfare =   ln   i  k  N 

1 Efficiency =  i i   N expk(i   )  1  Inequality = ln   i  k  N  k  Inequality Aversion parameter

107

Parametric representation of the tax-transfer rule NIT & FT

For example, the NIT & FT is represented by three parameters: The guaranteed minimum income G The tax rate t1 of the first segment The tax rate t2 of the second segment

NIT & FT

Net

G

Gross

108

Numerical Optimization • πs = argmaxπ W(μ1(π), …, μN(π)) s.t. public budget constraint

• Likely many local «maxima» • Explore parameter space with grid-search (some papers stop there) • Use BFGS with different starting points or global optimization derivative-free procedures 109

Some examples of CA • Fortin et al. 1993 • Sefton & de Ven 2009 • Colombino, Locatelli, Narazani & O’Donoghue et al. 2010 • Colombino & Narazani 2011

• Aaberge & Colombino 2012, 2013 • Blundell & Shephard 2012 • Colombino 2015 • Colombino & Islam (work-in-progress) 2017

110

Fortin et al. 1993 • Quebec • Calibrated household labour supply model • Microsimulation of various versions of NIT and IWB tax reform • Comparability with common parameters • Iso-elastic Social Welfare function • Grid-search of optimal policy

111

Sefton & de Ven 2009 UK Calibrated dynamic stochastic utility maximization (common preferences) Microsimulation of pension transfers reform Utilitarian criterion Numerical optimization

112

Colombino, Locatelli, Narazani & O’Donoghue et al. 2010 Various European countries Microeconometric models of labour supply (Conditional logit) Microsimulation of various basic income policies Money-metric utility Gini-type social welfare function Grid-search exploration of policy space

113

Colombino & Narazani 2013 Italy Microeconometric models of labour supply (Conditional logit) Microsimulation (equilibrium procedure, Colombino 2013) of various basic income policies Money-metric utility Gini-type social welfare function Grid-search exploration of policy space, partial optimization

114

Aaberge & Colombino 2006, 2013 Norway Microeconometric model of labour supply (extended Conditional Logit: RURO model, Aaberge & Colombino 2014) Microsimulation of personal income tax rule Common utility for social evaluation Gini-type social welfare function Numerical optimization

115

Aaberge & Colombino 2012 Italy Microeconometric model of labour supply (extended Conditional Logit: RURO model, Aaberge & Colombino 2014) Microsimulation of personal income tax rule Common utility for social evaluation Roemer-Gini-type social welfare function (equal opportunity) Numerical optimization

116

Blundell & Shephard 2012 UK, lonely mothers Microeconometric model of labour supply (Conditional Logit) Microsimulation of personal income tax No comparability Kolm-type social welfare function Numerical optimization

117

Colombino 2015a Italy Microeconometric models of labour supply (Conditional logit) Microsimulation (equilibrium procedure) of various basic income policies Money-metric utility Gini-type corrected for poverty social welfare function Grid-search of exploration of policy space, partial optimization

118

The case for NIT+FT in Europe: An empirical optimal taxation exercise N. Islam (LISER, Luxembourg) & U. Colombino (EST, Turin, Italy)

119

Outline • The debate CBI (= Conditional, or Means-Tested Basic Income) vs • UBI (= Unconditional Basic Income) vs • IWB (= In-Work Benefits) • CBI, UBI and IWB as special cases of NIT (= Negative Income Tax) • Searching for optimal (= Social Welfare maximizing) CBI, UBI, IWB and NIT in European countries • NIT is expected to be no worse – social welfare-wise – than its special cases • However CBI, UBI or IBW might be preferable according to other criteria (poverty, labour supply etc.) • Work-in-progress, preliminary results

120

The debate CBI vs UBI vs IWB • • • •

CBI prevailing income-support mechanisms in most countries until ’70s Problems with CBI: bad incentives, high administration costs Friedman 1962 proposed NIT as an alternative Rather than Friedman’s NIT, many countries progressively introduced some form of IWB • Problems with IWB: distorsions, high admistration costs • More recently, revived interest in UBI (e.g. survey in Colombino 2015b, experiments in Finland, Netherlands etc.) • Possible benefits from UBI: better incentives, low administration costs

121

CBI Net

Welfare- (or poverty-) Trap: there is no incentive to work for a gross income lower than G

Cheating: if your gross income is larger than G, you have an incentive to hide (part of) it in order to get the subsidy

G

Gross G

122

NIT Net

Same problems as with CBI but to a lesser extent

Gross G

123

UBI vs CBI and the poverty (welfare) trap CBI Net

UBI

G

The individual will work under UBI but not under CBI

Gross

124

CBI, UBI and IWB: special cases of NIT CBI

NIT

Net

Net

G

G

Gross

Gross

G

125

CBI, UBI and IWB: special cases of NIT UBI

NIT

Net

Net

G

G

Gross

Gross

126

CBI, UBI and IWB: special cases of NIT IWB

NIT

Net

Net

G

Gross

Gross

127

Different profiles of NIT Concave

Convex

Net

Net

G

G Gross

Gross

128

Policies We consider four types of income-support mechanism matched with FT (Atkinson 1996): NIT, UBI, CBI and IWB The members of each type are defined by a vector of parameters π. πUBI = (GUBI, tUBI) πCBI = (GCBI, tCBI) πNIT = (GNIT, t1,NIT, t2,NIT ) πIWB = (GIWB, t1,IWB, t2,IWB ) G is adjusted according to the household size (square root rule) The policies replace the whole tax-benefit system

129

Identifying Optimal Policies π = parameters of the policy

Ui(c, l;π) = utility of household i, given income c, hours of work l and policy π Vi(π) = Emaxh,c Ui(c, l;π) = expected maximized utility given policy π s.t. household’s budget constraint μi(π) = comparable money-metric Emax utility W(μ1(π), …, μN(π)) = Social Welfare function π* = argmaxπ W(μ1(π), …, μN(π)) s.t. public budget constraint. 130

Model and Data • Microeconometric model of household labour supply • Separate estimation for each country, couples and single • Sampled alternatives, 7 for singles, 49 for couples • Non working, wage employed, self-employed, retired excluded • Age 18 - 6 • EUROMOD datasets based on EU-SILC 2010 • This presentation is limited to 6 countries • The project will cover most of the countries in Euromod

131

Microeconometric Model max V(c, l; β) + ε s.t. c = wl +y – T(wl +y) lB U( ) = utility function V( ) = «systematic» utility β = prefence parameters (to be estimated) ε = random variable (extreme value distribution) T( ) = tax-transfer rule c = net income l = labour supply w = wage rate y = exogenous income B = opportunity set

132

Microeconometric Model V(c, l) is a quadratic form in c and l, with coefficients function of socio-demographic characteristics + some alternative-specific dummies (not working, part-time, full-time) e.g. singles:

V  1c  2c   3 (L  l)   4 (L  l)   5 (L  l)c 2

2

133

Microeconometric Model B contains alternative values of l Alternatives sampled from empirical distribution (should improve efficiency of the estimates) See Aaberge et al. (2009) on the implications of alternative representations of the opportunity set

134

Microeconometric Model Choice probability (Conditional Logit)

Pi ,k 

exp(Vi (ci ,k , li ,k ;  ))

 exp(V (c j

i

i,j

, li , j ;  ))

135

Microsimulation Simulate the value of some function of c and/or l, x(c,l), given a new T0

ci0,k  wi ,k li ,k  T 0 (wi ,k li ,k ) 0 ˆ) V ( c , l ;  Pˆi 0,k  i i ,k 0 i ,k ˆ) V ( c , l ;   j i i ,k i , j x  x(c , li ,k ) 0 i ,k

0 i ,k

ˆxi0  simulated value of xi0   Pˆi 0,k xi0,k k

136

Comparable utility Given the expected maximum utility (McFadden 1978) …

Vi *  ln

  exp(V (c j

i

i,j

, li , j ;  ))



137

Comparable utility … define the comparable money-metric utility μi (King 1983):

ln

  exp(V ( ,l ;  ))  V * j

R

i

j

i

Decoster & Haan (2015) present a different implementation inspired by Fleurbeay (2011) An alternative (e.g. Aaberge & Colombino 2013) is to compute V* with common preference parameters (Deaton & Muellbauer 1980)

138

Social Welfare (Kolm 1976) expk(i   )  1  Social Welfare =   ln   i  k  N 

1 Efficiency =  i i   N expk(i   )  1  Inequality = ln   i  k  N  k  Inequality Aversion parameter

139

Identifying T*( ) with Computational Approach

π* at convergence

Maxπ W(μi(π),…, μN(π))

π

s.t. P.B.C. This is solved by numerical optimization

μi(π)

Vi(π) = maxc,l Ui(c(π), l) s.t. household’s budget constraint given π This is solved by microsimulating a microeconometric model of household decisions V is turned into μ (King 1983)

140

An example: Italy Italy Second Best: Optimal UBI, k = 0.05

Italy Fourth Best: Optimal CBI, k = 0.05

Net

Net

t 2 = 0.31

G =337

t1 = 1

Gross Gross

141

An example: Italy

Italy Third Best: Optimal IWB, k = 0.05

Italy First Best: Optim al NIT, k = 0.05 Net

Net t2 = 0.47

t2 = 0.35

t 1 = 0.37

t1 = -0.04

G =144

Gross

Gross

142

Optimal CBI, NIT, UBI and IWB, k = 0.05 Belgium G

t1

t2

Hours

Poverty (%)

Winners (%)

%Δ Social Welfare

CBI

804

1.00

0.44

1635

0.0

57

-1.07

NIT

522

0.28

0.72

1635

5.0

61

1.51

UBI

905

0.64

0.64

1645

7.9

57

1.29

IWB

441

-0.02

0.52

1672

9.0

65

0.37

Current

----

----

----

1645

15.0

----

---143

16870 NIT

16860

Belgium

16850 UBI 16840

Efficiency

IWB Current

16830 16820

Iso-social welfare lines

16810 16800

CBI 16790

-805

-800

-795

-790

-785

-Inequality

-780

-775

16780 -770

Optimal CBI, NIT, UBI and IWB, k = 0.05 France G

t1

t2

Hours

Poverty (%)

Winners (%)

%Δ Social Welfare

CBI

374

1.00

0.17

1688

18.5

72

1.48

NIT

123

0.36

0.16

1690

18.4

74

1.70

UBI

322

0.26

0.26

1645

7.9

57

0.23

IWB

245

0.26

-0.026

1684

9.0

73

0.06

Current

----

----

----

1645

15

----

---145

10770 NIT 10760 CBI

France

10750

Efficiency

10740 10730 10720 10710 IWB UBI 10700 Current 10690

Iso-social welfare line -110

-109

-108

-107

-106

-105

-104

-103

10680 -102 -101

-Inequality

146

Optimal CBI, NIT, UBI and IWB, k = 0.05 Ireland G

t1

t2

Hours

Poverty (%)

Winners (%)

%Δ Social Welfare

CBI

981

1.00

0.27

1188

26.6

47

-2.81%

NIT

904

0.32

0.89

1191

12.8

61

3.99

UBI

1062

0.57

057

1161

0.0

57

1.48

IWB

494

-0.09

0.31

1274

15.8

48

-0.41

1249

19.6

Current

147

26750

NIT

26700

Ireland

26650

Efficiency

UBI 26600

Current IWB 26550

Iso-social welfare line

CBI

-2375

-2370

-2365

-Inequality

-2360

26500

-2355

26450 -2350

Optimal CBI, NIT, UBI and IWB, k = 0.05 Italy G

t1

t2

Hours

Poverty (%)

Winners (%)

%Δ Social Welfare

CBI

337

1.00

0.31

1530

29.2

65

0.53

NIT

303

0.37

0.47

1510

21.3

72

0.93

UBI

196

0.35

0.35

1535

25.8

73

0.62

IWB

144

-0.04

0.35

1539

16.2

73

0.59

Current

----

----

----

1540

26.6

----

---149

4570 IWB 4565

UBI

Italy CBI

4560

Efficiency

4555

4550 NIT

Iso-social welfare line

4545

4540 Current

-1055

-1050

-1045

-1040

-Inequality

-1035

-1030

4535 -1025

Optimal CBI, NIT, UBI and IWB, k = 0.05 Luxembourg G

t1

t2

Hours

Poverty (%)

Winners (%)

%Δ Social Welfare

CBI

626

1.00

0.19

1664

20.0

62

-1.01

NIT

605

0.19

0.49

1643

11.0

63

0.10

UBI

1297

0.48

0.48

1642

3.7

51

-0.16

IWB

542

-0.01

0.33

1660

7.5

65

-0.24

1648

10.7

Current

151

7860 Current 7840

NIT

Luxembourg

UBI

7820 IWB

Efficiency

7800

Iso-social welfare line

7780

7760

CBI 7740

-2880

-2870

-2860

-2850

-Inequality

-2840

-2830

7720 -2820

Optimal CBI, NIT, UBI and IWB, k = 0.05 United Kingdom G

t1

t2

Hours

Poverty (%)

Winners (%)

%Δ Social Welfare

CBI

641

1.00

0.25

1182

27.3

56

-0.46

NIT

774

0.63

0.64

1157

16.0

76

9.06

UBI

674

0.55

0.55

1175

30.3

74

6.91

IWB

174

-0.03

0.19

1262

23.3

46

-6.60

1196

30.3

Current

153

9050 NIT 9000

United Kingdom

UBI 8950

Efficiency

8900

8850 Current

CBI 8800

8750

Iso-social welfare line

8700 IWB

-1100

-1095

-1090

-1085

-1080

-Inequality

-1075

-1070

-1065

8650 -1060

Comments • NIT is best (welfare-wise) • UBI in most cases better than CBI

• CBI may lead to a significant reduction in labour supply and poverty-trap effects • IWB in most cases dominated by NIT and UBI • In Luxembourg the reforms hardly improve upon the current system

155

Colombino & Islam • AA suggests NIT as best with convex profile • CA confirms good performance of NIT • However, in most cases, with concave profile • UBI (= Unconditional Basic Income) could outperform NIT when taking administrative implementation details into account (feasible with CA but not with AA)

156

NIT Concave

Convex

Net

Net

G

G Gross

Gross

157

Colombino & Islam (in progress, very preliminary) AA

CA

Previous analyses based on AA suggest IWB might dominate in many European countries

According to our exercise, IWB dominated by both NIT and UBI

158

Identifying the correspondence between θ and π • S «economies» • θ1, θ2, …, θS = «primitives» • π1, π2, …, πS = optimal tax-benefit rules

159

Identifying the correspondence between π and θ

θ1 … … … θS

Regression analysis, Clusters, Correspondence analysis, etc.

π1 … … … πS

160

How NIT parameters π depend on «primitives» θ (OLS, very preliminary) G

t1

t2/t1

451.8

0.290

1.910

Inequality aver. (k)

11.8

0.008

-0.009

Productivity (wage)

8.7

-0.008

0.020

P.B.C.

-0.6

0.000

0.000

L.S. Elasticity

-11.8

-0.002

-0.030

Constant

161

References Aaberge R. (2007). Gini Nuclear Family, Journal of Economic Inequality, 5(3), 305–322. Aaberge, R. and U. Colombino (2014) Labour Supply Models, in C. O’Donoghue (Ed.), Handbook of Microsimulation Modelling, Contributions to Economic Analysis vol. 293, Emerald. Aaberge, R. & Colombino, U. (2013). Using a microeconometric model of household labour supply to design optimal income taxes. Scandinavian Journal of Economics 115(2), 449–475. Aaberge, R. & Colombino, U. (2012). Accounting for family background when designing optimal income taxes: a microeconometric simulation analysis. Journal of Population Economics 25(2), 741–761. Aaberge, R. & Colombino, U. (2006). Designing optimal taxes with a microeconometric model of household labour supply. IZA Discussion Papers 2468, Institute for the Study of Labor (IZA).

Aaberge, R., Colombino, U. and T. Wennemo (2009) Evaluating Alternative Representations of the Choice Sets in Models of Labour Supply, Journal of Economic Surveys, 23 (3), 586-612. Atkinson A. B. (1970). On the measurement of inequality. Journal of Economic Theory, 2 (3), 244–263. Atkinson, A. B. (1996). Public Economics in Action: The Basic Income/Flat Tax Proposal. . Bargain. O., Dolls M., Neumann D., Peichl A. and S. Siegloch (2014). Tax-Benefit Revealed Social Preferences in Europe and the US, Annals of Economics and Statistics, GENES, 13-114, 257-289. Blundell, R. and A. Shephard, A. (2012). Employment, hours of work and the optimal taxation of low-income families. Review of Economic Studies 79(2), 481–510. 162

Blundell, R, Brewer M. Haan P. and A. Shephard (2009). Optimal Income Taxation of Lone Mothers: An Empirical Comparison of the UK and Germany, Economic Journal, Royal Economic Society, 19(535), 101-121. Bourguignon F. and A. Spadaro (2012). Tax–benefit revealed social preferences, The Journal of Economic Inequality, 10(1), 75-108. Brewer, M., Saez, E. and A. Shaphard, (2008) Means-testing and tax rates on earnings (Mirrlees Commission Report), Institute for Fiscal Studies, LondonChetty, 2009. Chetty, R. (2009). Sufficient Statistics for Welfare Analysis: A Bridge Between Structural and Reduced-Form Methods, Annual Review of Economics, Annual Reviews 1(1), 451-488. Colombino U. (2013) A new equilibrium simulation procedure with discrete choice models, International Journal of Microsimulation, 6(3), 25-49. Colombino, U. (2015a). Five crossroads on the way to basic income. An Italian tour. Italian Economic Journal 1(3), 353–389. Colombino, U. (2015b). Is unconditional basic income a viable alternative to other social welfare measures? IZA World of Labor, Bloomsbury Publishing Plc., February 2015: http://wol.iza.org/articles/isunconditional-basic-income-viable-alternative-to-other-social-welfare-measures#link

163

Colombino, U. and E. Narazani (2013). Designing a Universal Income Support Mechanism for Italy. An Exploratory Tour, Basic Income Studies, 8(1), 1-17. Colombino, U., Locatelli, M., Narazani, E. and O’Donoghue, C. (2010). Alternative basic income mechanisms: An evaluation exercise with a microeconometric model. Basic Income Studies 5(1), 1–31. Dagsvik, J. (2000) Multinomial Choice and Selectivity, Discussion Papers 264, Research Department of Statistics Norway. Deaton, A. and J. Muellbauer (1980). Economics and Consumer Behavior, Cambridge University Press. Decoster, A. and P. Haan (2015). Empirical welfare analysis with preference heterogeneity, International Tax and Public Finance, 22(2), 224-251. Diamond, P. (1998). Optimal Income Taxation: An Example with a U-Shaped Pattern of Optimal Marginal Tax Rates, American Economic Review, 88(1), 83-95. Edgeworth, F. (1897). The Pure Theory of Taxation, The Economic Journal, 7(25), 46-70. Feldstein, M. (1999). Tax Avoidance And The Deadweight Loss Of The Income Tax, The Review of Economics and Statistics, 81(4), 674-680. Fleurbaey, M. (2011). Willingness-to-pay and the equivalence approach, Revue d'économie politique, 121(1), 35-58. Fortin, B., Truchon, M. and L. Beausejour (1993). On reforming the welfare system: Workfare meets the negative income tax. Journal of Public Economics 51(2), 119–151.

164

Friedman, M. (1962). Capitalism and Freedom. Chicago: University of Chicago Press. Heathcote, J. and H. Tsujiyama (2015). Optimal Income Taxation: Mirrlees Meets Ramsey, CEPR Discussion Papers 10380, https://ideas.repec.org/p/cpr/ceprdp/10380.html Islam, N. and U. Colombino (2017). The case for NIT+FT in Europe. An empirical optimal taxation exercise, mimeo, LISER, Luxembourg. King M. A. (1983). Welfare analysis of tax reforms using household data. Journal of Public Economics, Volume 21, Issue 2, 183-214.

Kolm, S. C. (1976). Inequal Inequalities I, Journal of Economic Theory. 12, 416-442. Loeffler et al. (2014). Structural Labor Supply Models and Wage Exogeneity, IZA Discussion Papers 8281. McFadden, D. (1978). Modelling the Choice of Residential Location. In: A. Karlqvist, F. Snickars and J. Weibull, Eds., Spatial Interaction Theory and Planning Models, North Holland, 1978, pp. 75-96. P. (ed.). Frontiers in Econometrics, Academic Press. Milgrom, P. (2004). Putting Auction Theory to Work, Cambridge University Press. Mirrlees, J. A. (1971). An exploration in the theory of optimum income taxation. Review of Economic Studies 38(2), 175–208. Orcutt, G. (1957). A New Type of Economic System, The Review of Economics and Statistics, 39(2), 116-123. 165

Ramsey, F. (1927). A Contribution to the Theory of Taxation, The Economic Journal, 37(145), 4761. Roth, A. (2002). The economist as engineer: game theory, experimentation and computations as tools for design economics, Econometrica, 78(4), 1341-1378. Saez, E. (2002). Optimal income transfer programs: Intensive versus extensive labor supply responses, Quarterly Journal of Economics 117(3), 1039–1073. Saez, E. (2001). Using elasticities to derive optimal income tax rates, Review of Economic Studies 68(234), 205–229.

Sen A. (1973). On Economic Inequality, Oxford University Press. Sefton J., Van De Ven J. (2009). Optimal Design of Means Tested Retirement Benefits, Economic Journal, 119(541), 461-481. Tuomala, M. (2010), On optimal non-linear income taxation: numerical results revisited, International Tax and Public Finance 17(3), 259–270. 166