earned income response to tax changes in denmark

Earned Income Response to Tax Changes in Denmark

Hans Bækgaard1

Ministry of Finance

Copenhagen, April 2010

Abstract This paper presents new Danish estimates for the response of taxable earned income to income tax changes. Such estimates are often referred to as the elasticity of taxable income. While previous Danish estimates of behavioural response to tax changes have focussed on worked hours, inter-national studies have increasingly turned the attention to income responses. The attraction is that income responses ostensibly cover a broader range of behavioural adjustments and hence cap-ture a larger share of the deadweight loss of taxation. The paper presents results from two differ-ent approaches to measuring the elasticity of taxable income namely, differences-in-differences and error correction. The main results are obtained by a three-year differences-in-differences approach. The uncompensated elasticity is 0.11 for males and 0.06 for females. The results from error correction model suggest that the elasticities could be substantially higher in the long run (and lower in the short run). The income elasticity is -0.017 for males and -0.013 for females. These results are at the lower end of the range produced by international studies though consistent with economic theory. However, the methodological challenges associated with measuring the elasticity of tax-able income are many and the policy implications of the results are profound. Accordingly, a main focus of the paper is to screen the results for sensitivity to the methodology assumptions applied by previous studies. The analysis shows that the results are sensitive to model assump-tions such as the dynamic specification and the method for dealing with tax variable endogeneity.

1 The views expressed in the working paper are those of the author, and not necessarily those of the Ministry of Finance.

2

Introduction

Recent years have seen a plethora of new studies into the estimation of taxable income response to tax changes. The traditional approach to labour supply response to taxation has focussed on the quantitative labour supply that is, worked hours (the intensive margin) and labour market participation (the extensive margin). The attraction of the alternative approach is that the elastic-ity of taxable income ostensibly covers a broader range of behavioural adjustments including the qualitative labour supply (productivity) and the effect of DIY work and tax compliance. The elasticity of taxable income thus captures a larger share of the deadweight loss of taxation.

This paper focuses on earned income response to tax changes in Denmark. The variable of analysis is labour income as measured by the tax base for the Danish labour market contribution. The main motivation for looking at earned income is a desire to analyse how tax changes affect labour supply. Analysing labour income also alleviates the problem that different income types are treated differently by the Danish tax system and, as result, have different marginal tax rates.

Results from two different approaches to measuring the elasticity of taxable income are pre-sented. The two approaches are differences-in-differences and error correction. The main results are ob-tained by a three-year differences-in-differences approach. The uncompensated elasticity is 0.11 for males and 0.06 for females. The income elasticity is -0.017 for males and -0.013 for females. The error-correction model identifies short and long term elasticities separately and the results are indica-tive of an adjustment process with smaller short term and larger long term elasticities. The elas-ticities from the error-correction model are nevertheless less well-identified than those from the differences-in-differences model.

The seminal work by Fieldstein (1995) produced very large substitution elasticities in the range from 0.75 to 3.05 depending on income concept and model specification. More recent consensus has, however, rejected these results as flawed as the study failed to account for external effects on taxable income developments, cf. Slemrod (1996) og Goolsbee (2000).

The more recent estimates of substitution elasticities are generally non-negative and cover a range of values between 0 and 0.5, cf. Table 1. The studies in Table 1 all correct for mean income reversion by including lagged income in the regressions. All studies apply a 2SLS estimation method where the first step is IV-regressions to account for endogeneity of the tax variables and the second step is an OLS regression of the main behavioural equation.

The methodologies applied by these studies vary although the similarities are more prevalent than the differences. The differences-in-differences approach is a common feature of all except one study. The exception is Holmlund and Söderström (2007) who obtain their main results from an error correction model that identifies separate short and long run effects for assessed income, broad income and earned income. They find that the short run elasticities are insignificant for assessed and broad income and indistinguishable, at 0.2, from the long run elasticity for earned income. The long run elasticities for assessed and broad income are around 0.3. Holmlund and Söder-ström (2007) also estimate a differences-in-differences model and find either very small or insignificant elasticities.

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Gruber and Saez (2002) analyse US data and obtain their main results from a model with splines of income deciles, which they show leads to slightly lower elasticities. They find an elasticity of 0.1 for broad income and 0.4 for taxable income.

In a recent study by Blomquist and Selin (2009), the income effect was estimated by using virtual income, which makes it more in line with standard labour supply theory. In contrast, previous attempt to measure the income effect use net income rate or the (equivalent) average tax rate, cf. Gruber and Saez (2002) and Ljunge and Ragan (2005).

A few studies find small negative income elasticities, albeit these estimates are generally not well-defined.

Table 1

Selected studies of the elasticity of taxable income

Elasticity

Study Country Income Substitution Income

Ljunge and Ragan, 2005 Sweden Taxable labour income 0.3 – 0.4 -0.14 - 0.022

Hansson, 2007 Sweden Taxable labour income 0.4 – 0.5 -

Holmlund and Söderström, 20073 Sweden Assessed income

Broad income

Taxable labour income

0.27 0.02

0.32 0.03

0.22 0.012

-

Blomquist and Selin, 2009 Sweden Taxable labour income

- males

- females

0.2

1.0 – 1.41

-0.072

-0.042

Aarbu and Thoresen, 2001 Norway Taxable income 0.0 – 0.2 -

Gruber and Saez, 2002 USA Broad income

Taxable income

0.1

0.4

-0.072

-0.132

1 The standard deviation on these estimates is very large in part due to the small sample size. 2 Insignificant estimates. 3 This study uses an error correction model (ECM) to distinguish between short and long run elasticities. The estimates

in the 1st column are the long run ECM-based elasticities; the 2nd column has differences-in-differences estimates.

It is not possible to reconcile the results in Table 1 into a single universal elasticity of taxable in-come. The main reason is that different methods lead to different results but also, that different economic systems and different types of tax changes targeted at different population groups are likely to produce different outcomes. In spite of these difficulties, all existing studies present positive substitution elasticities that concord with economic theory. The key questions are then, it seems, where in the 0 - ½ interval a reliable estimate of the elasticity of taxable income lies, and how the elasticity varies across population groups and types of income. The answers to these questions clearly have profound implications for public finance.

The present paper makes the following contributions to the existing literature into the elasticity of taxable income. Firstly, it provides new estimates for Denmark where research in this area has been non-existing until recently. Estimates are obtained for both a (traditional) differences-in-

4

differences and an error-correction approach, and the results are compared. Secondly, the paper proposes an alternative method for model estimation. Unlike the 2SLS method applied by previ-ous studies, the method accounts for error term autocorrelation and, more generally, encom-passes a more elaborate dynamic model specification. Third, and importantly, the paper demon-strates that the results depend crucially on the sample selection and model specification more generally, in particular the method for handling tax variable endogeneity.

Methodology

The starting point is standard micro economic theory. A person maximizes utility year by year subject to the local budget constraint itititit Ryc +−= )1( τ . TP

2PT itc is consumption, ity is pre-tax

income, itτ is the (local) marginal tax rate (given ity ), and itR is virtual income. The budget con-straint for the average Danish tax payer in 2008 is illustrated by Figure 1.

Figure 1

Budget constraint and virtual income in the Danish tax system 2008

0

50

100

150

200

250

300

0 50 100

150

200

250

300

350

400

450

500

Annual income

Net income

0

50

100

150

200

250

300Net income

D

C

B

A

V2

V1

V3

Slope = 1-τB

Slope = 1-τC

Slope = 1-τD

Slope = 1-τA

Small changes to the marginal tax rate and virtual income affect pre-tax income by differentia-tion:

(1) itit

itit

it

itit dR

Ry

dy

dy∂∂

+−∂∂

−= ττ )1(

TP

2PT The notation and the first part of the derivation is from Gruber and Saez (2002).

5

The uncompensated elasticity of pre-tax income with respect to the marginal net income (1- itτ ) is

)1(

)1(

)1()1( it

it

it

it

it

it

it

it

u zz

zz

ττ

ττ

ξ−∂∂−=

−−∂

∂

=

The income elasticity is

it

itit

it

it

it

Rz

Rz

∂∂

−=

−∂

∂−= )1(

)1(

τ

τ

η

Substitute uξ and η in (1)

it

it

it

itit

uit

dRdydy

τη

ττ

ξ−

+−−

=11

)1(

it

it

itit

it

it

itu

it

it

RdR

yRd

ydy

)1(1)1(

τη

ττ

ξ−

+−−

=

(2) it

itR

it

itu

it

it

RdRd

ydy

ηττ

ξ +−−

=1

)1(

The following log-differences version approximates (2) for ‘small’ changes to itτ and itR

(3) )log()1log()log( itdRitdu

itd Ry ∆+−∆=∆ ητξ

The operator d∆ takes differences across d years. The relationship between the income elasticity,

η , and the elasticity of virtual income, )1( itit

itR y

Rτ

ηη−

= , is person specific and can be evalu-

ated as a population average. (3) is used in the present study for estimating uξ and η as measures of the income response to tax changes.

An alternative specification uses the compensated elasticity, cξ , and is derived from the Slutsky equation ηξξ += cu

(4) )1(1

)1(

itit

ititit

it

itc

it

it

ydydRd

ydy

ττ

ηττ

ξ−

−+

−−

=

The measure )( ititit dydR τ− is the change in net (after tax) income as a result of a change in the marginal tax rate given a fixed level of income. Gruber and Saez (2002) apply (4) in the following log-specification where )( ititit dydR τ− replaces the change in net income, ))(( ittitd yTy −∆ , and )1( itity τ− approximated by )( ittit yTy −

(5) )log()1log()log( itditdc

itd y υητξ υ∆+−∆=∆

6

Here, the income effect is represented by the elasticity υη of net income )( itititit yTy −=υ . As noted by Gruber and Saez (2002), )()1( ittititit yTyy −≅−τ is only a good approximation for small tax changes.

Empirical specification The general empirical model formulation is a dynamic panel model with lagged explanatory vari-ables, an autoregressive error process and random (individual) effects for the constant term ( iα ) and the tax parameters ),,,( '

2'1

'2

'1 iiii ηηξξ .

(6) itdititditditiitiditiitiiit XXyRRy εββγηητξτξα ++++++−+−+= −−−−'2

'1

''2

'1

'2

'1 )1()1(

ititit e+= −1ρεε dineit ..~

),(~),,,,( '2

'1

'2

'1 Σµηηξξα Niiiii

Background variables, including their lagged values, are included to control for external (non-tax related) factors that may have affected income changes differently across the income distribu-tion.

The differences-in-differences and the error correction models are both special cases of (6).

The differences-in-differences model’s parameters are estimated with the following empirical versions of (3) and (5):

(7) ititditdititdRitdu

itd XXyRy εββγητξ +∆+++∆+−∆=∆ − 21)1(

(8) ititditdititditdc

itd XXyy εββγυητξ υ +∆+++∆+−∆=∆ − 21)1(

These specifications have been derived from microeconomic theory, cf. above. The model corre-sponds to (6) with the restrictions '

2'

1 ξξξ −== and '2

'1 ηηη −== , and the re-

parameterisation: '1 γγ −= , 21'

1 βββ += and 2'2 ββ −= . As in most other studies, three-year

differences are applied. The random effects are differenced out.

The lagged income (t-d) is included in the differences-in-differences model based on a mean-income-reversion argument, though it is also part of the generel dynamic specification. As noted by Holm-lund and Söderström (2007), some previous studies suffer from the problem that estimation of the differences-in-differences model with OLS while including lagged income that typically is corre-lated with the error term. This could result in biased parameter estimates.

Also, most previous studies fail to test for error term autocorrelationTP

3PT. In other words, it is Uas-

sumedU that itε is uncorrelated over time (i.e. 0=ρ ). In contrast, the present study, accounts for first-order error term autocorrelation, that is ititit e+= −1ρεε , as well as lagged income thereby resulting in consistent and efficient parameter estimates.

TP

3PT This critique is off course only relevant for panel studies.

7

The estimated error correction model is a generalisation of the model analysed by Holmlund and Söderström (2007):TP

4PT

(9) +∆+∆+−∆=∆ 111 )1( βητξ ititiitiit XRy

itititiitiiit XRy εβητξαγ +++−+−− −−−− )])1(()[1( 2112121

),0(~),( 111 ΣNii ηξ and ),0(~),,( 222 ΣNiii ηξα

This specification is a rewrite of (6) without error term autocorrelation, that is 0=ρ . TP

5PT Error cor-

rection is a more general specification than differences-in-differences. Importantly, it identifies short and long term effects separately. 1ξ and 2ξ are the short and long run substitution elasticity. 1η and 2η are the short and long run income elasticities. TP

6PT

Both models are estimated by a Bayesian estimation technique, cf. Appendix 1.

Endogeneity In a non-linear tax system, a change in a person’s taxable income may cause a change in the cen-tral tax parameters, i.e. the person’s marginal tax rate and virtual income. As a result, the tax vari-ables itτ and itR are endogenous. With a progressive tax system, failure to account for the en-dogeneity problem will typically lead to elasticity parameters with a ‘wrong’ sign.

The endogeneity problem has been discussed intensively in the literature. A common approach to dealing with the problem is to apply instrumental variables (IV) for the tax variables in conjunc-tion with a two-stage least square (2SLS) estimation method. In the first-stage, the endogenous vari-ables are regressed on a ‘suitable’ instrument and the model’s other background variables. In the second-stage, the predictions from the first-stage equations replace the endogenous variables in the original equation. Applying ordinary least squares to the second-stage estimation make the method a 2SLS. More generally, applying a maximum likelihood procedure makes the method a two-stage maximum likelihood (2SML) estimation.

Predicted marginal tax rates and virtual incomes are used as instruments for the tax variables. The predictions are based on ’updated’ taxable incomes from previous years and current year tax rules. More formally, let tz be the tax rules in year t, and ),( tit zyτ and ),( tit zyR the marginal tax rate and virtual income given taxable income and tax rules.

TP

4PT The model analysed by Holmlund and Söderström (2007) is an error correction model, which is estimated in a version, that looks like (6), but in

first-differences. The individual random effect iα is differenced out, but unlike (9), it does not allow random tax parameters. The sensitivity

analysis in the present study shows that allowing random effects change the tax parameters. Holmlund and Söderström (2007) assume no error

term autocorrelation in the undifferenced model although by use of an indirect argument, they make it plausible that this does not constitute a

problem. Error term autocorrelation in the model’s differentiated version means that ∆yBt-1B is correlated with error term. This is handled by using

yBt-2B as instrument for ∆yBt-1B.

TP

5PT The error-correction model is less prone to error-term autocorrelation than the differences-in-differences model, which has overlapping differ-

ences, where two consecutive three-year differences cover the same year-on-year changes in two out of three years.

TP

6PT For detailed account of the error correction model see Hendry, Pagan and Sargan (1984).

8

The following variables are then used as instruments for itτ and itω in the differences-in-differences model:TP

7PT

(10) ),( ,t

tpdit

pit zy −= ττ and ),( ,

ttpdit

pit zyRR −=

tpdity ,

− is predicted income in period t given income in period t-d. The method for calculating pre-dicted incomes and the definition of the instruments is described in more details below.

The estimation variables are calculated as 1−−=∆ itp

itp

it τττ and 1−−=∆ itp

itp

it RRR . This method is labelled IV-method I. An alternative, IV-method II, is to instrument directly on the differences

itτ∆ and itR∆ . IV-method II is examined in the subsequent sensitivity analysis.

The following IV-variables are applied for the error correction model: TP

8PT

(11) ),( ,1 ttp

itp

it zy −= ττ and ),( ,1 ttp

itp

it zyRR −=

A similar endogeneity problem would result if variables for income tested social transfers were used as explanatory variables (Ljunge and Ragan, 2007). Such variables should therefore not be included among the regressors.

Data

The models are estimated with a 1994-2006 random panel sample of 3.3 per cent of the Danish population. The data are merged from several administrative registers and include income tax records, education attainment and attendance and demographic information.

Danish tax law implies that income from different sources are taxed differently and with differ-ent marginal tax rates for different income levels. It therefore makes sense to look at the tax re-sponse of the different income components separately albeit it is acknowledged that, in doing so, potential cross elasticities are ignored.

The present study focuses on earned income that is, wage and salaries, and wage-like income from self-employment. Since the introduction of the so-called labour market contribution (LMC) as part of the 1994-tax reform, earned income has been the only form of income to which all the tax rates apply. The tax base for the LMC is a good measure for earned income and will be used as such in the following.

The tax variables The tax records provide detailed information about reported incomes from taxable sources. The tax records for the 13 years from 1994 to 2006 were run through the Ministry of Finance’s tax simulator to simulate taxes for the tax year and up to three years out. The simulated taxes for the out-years are used to calculate the instrumental variables in (10) and (11). The incomes for future

TP

7PT The notation is due to Holmlund and Söderström (2007))

TP

8PT Holmlund and Söderström (2007) instrument itτ and 1−itτ using predicted incomes based on (t-2) incomes ),( 1,

2,2

−−−tp

ittp

it yy which is

necessary because they estimate a double differenced model. The estimation method applied in the present study makes this unnecessary.

9

years were projected on the basis of average year-to-year income development for the major income components such as wages and salaries, self-employment income, interest, dividends and social transfers.

The tax simulator creates the tax law’s main income components and the three relevant tax vari-ables are calculated from total income and total taxes:

total tax = total income tax + labour market contribution

total income = personal income + labour market contribution

The marginal tax rateτ is calculated by comparing taxes in the base case with taxes calculated by increasing the earned income by DKR 100. The virtual income is derived from the formula:

virtual income = other taxable income – total tax + LMC*τ

The average tax is simply total tax divided by total income.

The background variables Over time, the distribution of taxable income is influenced by exogenous factors that cannot be attributed to changes in the tax system. A number of background variables are included in the regressions to adjust for such external, non-tax, influences on the development in taxable in-come. The background variables have information about macro economic conditions and per-sonal characteristics that may influence income developments, including age, marital status, chil-dren, educational attainment and attendance, income deciles and splines, unemployment rates and measures of economic activity (GDP). Interaction effects are included for some variables, for example, education specific age profiles and unemployment rates.

The sample The full sample is restricted to persons aged between 22 and 56 years (in the first year) and ex-cludes observations where a person completes an education between the first and the second year across the differencing period for the differences-in-differences model (equivalent for the error-correction model). Only persons with a minimum labour income of DKR 10,000 (1994 level) in both years are included to avoid extreme percentage changes in labour income.

The main results are obtained with a sample that excludes persons who received income trans-fers in either year t or year t-d or in both years. We refer to this as the ‘no unemployed’ sample. This is roughly equivalent to restricting the sample to only cover persons who were in either part-time or full-time employment the full year.

The reason for omitting transfer recipients is that the marginal tax rate does not account for withdrawal of income transfers – for example unemployment benefits and social security – when a person works a day or a week more or less during the year. The marginal tax rate is therefore an incomplete employment incentive measure where, indeed, the net replacement rate is more relevant – i.e. the relationship between net income (after taxes and transfers) when a person is employed or not.

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Identification Measuring income responses to tax changes requires a period with tax changes that are both sufficiently large and have differed sufficiently across income levels. Variations in the changes to marginal tax rates are particularly important. The period analysed here, 1994 to 2006, has not seen ‘large’ tax reforms that match, for example, the Swedish 1991 tax reform. In Denmark, however, a series of smaller tax changes add up to quite substantial changes to both marginal and average taxes for large income groups. The next section will explore these changes in more de-tail.

Danish income tax 1994 to 2006

The period from 1994 to 2006 has seen a general decrease in the average tax paid by Danish tax payers. The percentage decrease in the average tax rate is increasing with income up to around 10 per cent at an annual income around DKR 250,000 (1994 level), cf. Figure 2. For incomes above that level, the decrease in the average tax rate is slightly smaller.

Figure 2

Average tax and marginal tax rate and labour income (1994-level), change 1994 - 2006

-25

-20

-15

-10

-5

0

100,000

150,000

200,000

250,000

300,000

350,000

400,000

450,000

Per cent

-25

-20

-15

-10

-5

0Per cent

Average tax (change) Marginal tax rate (change)

Top tax rate reduced

Six per cent tax phased-out

Middle bracket increased

Bottom tax rate reduced

Source: Ministry of Finance’s Family Type Model.

The marginal tax rates have also been reduced over the period and the decrease exhibits a larger variation across income levels than the average tax rate. The decrease is largest, at 24 per cent, for incomes around DKR 200,000 (1994 level). The marginal tax rate has decreased by around 7 per cent for incomes above the top tax bracket.

11

Identification of the substitution effect requires sufficient variation in the change to the marginal tax rate across income levels. Judging from Figure 2, such variation seems to be present. Never-theless, as indicated on Figure 2, a large part of the changes to the marginal tax rates are a result of increasing the tax brackets rather than lowering the tax rates. Looking at the development in marginal tax rates for incomes within the three main tax bands (bottom, middle and top tax) confirms this picture, cf. Figure 3.9

Figure 3

Marginal tax rates for selected income levels (1994-level), 1994 – 2006

40

45

50

55

60

65

70

19941995

19961997

19981999

20002001

20022003

20042005

2006

Per cent

40

45

50

55

60

65

70

Per cent

DKR 120,000 DKR 160,000 DKR 200,000DKR 240,000 DKR 280,000


For persons with incomes above the top tax bracket (above ca. DKR 247,000 in 1994)10, the marginal tax rate decreased by 5 percentage points from 1994 to 1998, but has stabilised thereaf-ter. Tax payers with incomes in the bottom bracket (less than ca. DKR 137,000 in 1994) experi-enced a gradual decrease in the marginal tax rate totalling 4.7 percentage points by 2004 and un-changed thereafter. The marginal tax rate for the middle bracket (ca. DKR 137,000 – 182,000 in 1994) decreased by a modest 2½ percentage points from 1997 to 2004, but due to a string of bracket increases from 1999 to 2003 culminating with a DKR 56,000 one-off increase in 2004, a large number of tax payers no longer pay the middle tax. The marginal tax rate for tax payers who in 1994 had incomes in the now abolished ‘6 per cent tax’ bracket (ca. DKR 182,000 –

9 The short lived 6 per cent tax was phased-out between 1994 and 1996. From 1996 onwards there has only been three tax brackets that is, the

bottom tax, the middle tax and the top tax. 10 The tax brackets have been converted to reflect that the tax rates apply to taxable income after labour market contribution, introduced in 1994

at 5 per cent and subsequently increased to the current 8 per cent by 1996.

12

247,000 in 1994) dropped sharply by 4.5 percentage points from 1994 to 1996 as the six per cent tax was phased-out.

Across the period from 1994 to 2006, the marginal tax rates decreased by 9.9 per cent for the bottom bracket, 4.9 per cent for middle bracket, 12.9 per cent for the ‘6 per cent tax’ bracket and 6.7 per cent for the top bracket, cf. Table 2. The largest tax changes were, however, experienced by tax payers who jumped between tax brackets either because of changes to the tax brackets delimiters or to a person’s income. Indeed, the tax payers with incomes around DKR 200,000 in 1994 have benefited from both the phase-out of the 6 per cent tax and the increase in the middle tax bracket, cf. Figure 2.

Table 2

Marginal tax rates for the 1994 tax brackets I n1994 and 2006

Marginal tax rate (per cent) Change

1994 2006 %-point Per cent

Bottom 47,4 42,7 -4,7 -9,9

Middle 51,7 49,2 -2,5 -4,9

6 per cent tax 56,5 49,2 -7,3 -12,9

Top 67,4 62,9 -4,5 -6,7

Note: average municipal tax rates.


It is quite plausible that tax payers, who have experienced large marginal tax changes as a result of tax bracket or (relatively small) income changes, could weigh disproportionately in the estima-tions. As discussed later, it is questionable whether the model specification is geared to fully ac-commodate bracket changes.

Results

The differences-in-differences and error-correction models have been estimated for males and females separately. Table 3 shows the results from the differences-in-differences model for the restricted sample (‘no unemployed’) and the full sample respectively. The substitution elasticity is positive and significant for both males and females. The uncompensated is estimated at 0,109 for males and 0,056 for females. Including transfer recipients leads to much higher elasticities (0.278 for males and 0,287 for females).

Consistent with the theoretical outset, the income elasticity is negative and significant. In the baseline results – without transfer recipients – the income elasticity is slightly higher for males (-0.017) than for females (-0.013). Based on the full sample, the income elasticity is lower for males (-0.008) than for females (-0.017).

The lower substitution elasticity for females is a new result and several other studies indicate that the reverse relationship exists, cf. Ljunge og Ragan (2005), Blomquist og Selin (2009) and Frederiksen et

13

al. (2009)TP

11PT. A higher elasticity for females is normally explained by females often being the sec-

ondary income-earner in the family and therefore they are more flexible than males. The lower female elasticity in the present study can possibly be explained by the highly segmented Danish labour market with a large share of women employed in the public sector, where wages and working hours are less flexible. In addition to that, and unlike previous studies, the results are based on a sample of persons who are generally employed throughout the year.

Table 3

Estimation results for the differences-in-differences model

All 22-56 year olds (year one) and earned income above DKR 10,000 (1994 level)

Males Females

Full-year

employed Full sample

Full-year

employed Full sample

Substitution elasticity (ξ) P

1P 0.109 (0.013) 0. 278 (0.016) 0. 056 (0.012) 0.287 (0.021)

Income elasticity (η) -0.017 (0.002) -0.008 (0.003) -0.013 (0.002) -0.017 (0.003)

Gamma (γ)

- primary schooling -0.395 (0.002) -0.510 (0.003) -0.380 (0.003) -0.590 (0.002)

- secondary schooling -0.448 (0.006) -0.556 (0.004) -0.407 (0.003) -0.598 (0.003)

- vocational -0.459 (0.002) -0.608 (0.001) -0.391 (0.002) -0.632 (0.001)

- short tertiary -0.430 (0.003) -0.658 (0.005) -0.406 (0.005) -0.642 (0.004)

- medium tertiary -0.329 (0.002) -0.513 (0.002) -0.380 (0.006) -0.639 (0.002)

- long tertiary -0.329 (0.003) -0.526 (0.002) -0.381 (0.007) -0.629 (0.008)

Rho (ρ) 0.418 (0.001) 0.488 (0.001) 0.423 (0.001) 0.488 (0.001)

2σ 0.458 0.642 0.315 0.623

Number of persons 39,729 49,504 33,604 45,295

Number of observations 230,957 344,664 183,527 304,883

Year (1994/7 – 2003/6) 10 10 10 10

P

1P The parameter estimates for the substitution elasticity are uncompensated elasticities

A note of caution is warranted regarding the higher elasticities obtained by using the full sample. These results are potentially heavily biased and have only been included here for comparison with previous studies that mostly do not exclude transfer recipients. The problem is that the de-cision regarding how many days or weeks to work during the year is influenced by incentives that are not fully captured by a marginal tax rate. Indeed, the extensive labour supply margin is influ-enced by the effect of income transfer withdrawal and therefore, the net replacement rate is the more appropriate incentive measure for persons who are out of employment in either year t or year t-d. However, an annual differences-in-differences model is not suitable for representing the combined effect of income transfer withdrawal and a non-linear tax schedule because the net replacement rate depends on a person’s actual duration of unemployment in year t and year t-d.

The large negative γ-parameters are indicative of a noticeable presence of mean reversion. This re-sult is in line with previous studies and is no different to including a lagged dependant variable in

TP

11PT Frederiksen et al. (2009) analyses labour supply in terms of worked hours.

14

a dynamic panel model (4). The estimations show that there is a significant error term autocorre-lation with a ρ -value around 0.42. This is hardly surprising in a model with overlapping differ-ences, where two consecutive three-year differences cover the same year-on-year changes in two out of three years. It is important that the estimation procedure accounts for the considerable autocorrelation since, if it fails to do so, it could lead to unreliable conclusion regarding the tax parameters. It is a particular concern, that unrecognised autocorrelation can cause problems for the endogeneity of the instrumental variables, since these are based on lagged income and hence will be correlated with the error term, cf. Holmlund and Söderström (2007).

The error-correction model identifies separate short and long term elasticities and thereby provides insight into the dynamics of behavioural response to tax changes. The results are based on the restricted sample without transfer recipients. The uncompensated elasticity for males is estimated at 0.073 short run and 0.265 long run and the income elasticity is -0.001 (insignificant) short run and -0.042 long run, cf. Table 4. The results for males are thus consistent with those from the differences-in-differences model in the sense that the uncompensated elasticity (0.109) and the income elasticity (-0.017) fall between the short and the long run elasticities from the error-correction model.

The uncompensated elasticity for females is small (insignificant) both short run (0.014) and long run (0.023), cf. Table 4. The income elasticity is significant both short run (-0.012) and long run (-0.037). Considering the uncertainty attached to these estimates the results are nevertheless con-sistent with those from the differences-in-differences model. Indeed, excluding the income effect in-creases the substitution elasticity for both males and females – in particular the long run elastic-ities.

Table 4

Estimation results for the error correction model

All 23-56 year olds (year one) and earned income above DKR 10,000 (1994 level) who did not receive income transfers

Males Females

Income effect No income effect Income effect No income effect

Substitution elasticity (ξ)

- short run effect 0.073 (0.0023) 0.076 (0.020) 0.014 (0.021) 0.043 (0.021)

- long run effect 0.265 (0.047) 0.388 (0.046) 0.023 (0.041) 0.132 (0.043)

Income elasticity (η)

- short run effect -0.001 (0.002) - -0.012 (0.003) -

- long run effect -0.042 (0.003) - -0.037 (0.005)

Gamma (γ) 0.642 (0.002) 0.661 (0.003) 0.647 (0.003) 0.671 (0.002)

2ασ 0.018 (0.001) 0.014 (0.001) 0.020 (0.001) 0.015 (0.001)

2σ 0.060 0.061 0.040 0.040



Year (1996 – 2006) 11 11 11 11

Note: Tests show a small but significant error term autocorrelation (ρ < 0.07), with only marginal effects on the results.

15

The error-correction results indicate the differences-in-differences model may overestimate the elasticities in the short run while under estimating them in the long run.

The individual random effect parameter 2ασ is significant in all cases thereby indicating that unob-

served personal characteristics explain a relatively large proportion of the income dispersion.

Sensitivity analysis The results presented above are based on a range of choices and assumptions regarding the model specifications. The importance of these assumptions is now explored.

Several previous studies have used net income to measure the income effect, cf. Gruber and Saez (2002) and Ljunge and Ragan (2005). As shown above, using net income means that the substitu-tion elasticity is approximately equivalent to the compensated elasticity and following the Slutsky equation, the compensated elasticity is larger than uncompensated elasticity by the absolute size of the income elasticity. According to this relationship, the estimated substitution elasticities are at par with the baseline results, cf. Table 5.

Using the baseline sample that excludes income transfer recipients, the compensated elasticity is estimated at 0.133 for males and 0.079 for females. Considering the parameter uncertainties, these estimates are not significantly different from the compensated elasticities obtained by using virtual income (0.116 for males and 0.069 for females).

Table 5

Differences-in-differences sensitivity analysis – income elasticity measured by net income

All 22-56 year olds (year one) and earned income above DKR 10,000 (1994 level)

Males Females

Full-year

employed All

Full-year

employed All

Substitution elasticity (ξ) P

1P 0.133 (0.013) 0.267 (0.014) 0.079 (0.011) 0.308 (0.017)


Gamma (γ)

- primary schooling -0.390 (0.002) -0.507 (0.002) -0.382 (0.001) -0.597 (0.001)

- secondary schooling -0.461 (0.002) -0.556 (0.003) -0.406 (0.002) -0.610 (0.003)

- vocational -0.458 (0.000) -0.602 (0.001) -0.392 (0.001) -0.643 (0.001)

- short tertiary -0.423 (0.002) -0.635 (0.002) -0.401 (0.004) -0.658 (0.002)

- medium tertiary -0.331 (0.001) -0.502 (0.001) -0.364 (0.001) -0.658 (0.002)

- long tertiary -0.325 (0.001) -0.530 (0.002) -0.388 (0.002) -0.648 (0.002)

Rho (ρ) 0.416 (0.001) 0.483 (0.001) 0.420 (0.001) 0.483 (0.001)

2σ 0.458 0.642 0.315 0.623



Year (1994/7 – 2003/6) 10 10 10 10

The parameter estimates for the substitution elasticity are compensated elasticities

16

The estimated income elasticities, -0.009 for males and -0.017 for females, differ somewhat from those obtained with virtual income (-0.017 for males and -0.023 for females). A likely explanation for this discrepancy is that measuring the income elasticity using net income is only an approxi-mation, which is only good for small tax changes. The theoretical attractiveness of the virtual income based income elasticity therefore gives more credence to this estimate.

Like most previous studies into the elasticity of taxable income, the baseline results presented above account for mean reversion by including lagged income in the regressions. Gruber and Saez (2002) generalized this approach by including income decile splines in the regressions. The moti-vation is to account for mean reversion in a more detailed manner.TP

12PT Accordingly, the effect of

including income splines in the differences-in-differences model has been tested, cf. Table 6.

Table 6

Differences-in-differences sensitivity analysis – income spline and IV-method

All 22-56 year olds (year one) and earned income above DKR 10,000 (1994 level) who did not receive income transfers

Males Females

Decile splines IV-method II Decile splines IV-method II

Substitution elasticity (ξ)P

1P -0.012 (0.014) 0.022 (0.026) -0.039 (0.015) -0.067 (0.018)


Gamma (γ)

- 1 P

stP decile / primary schooling -0.637 (0.002) -0.389 (0.002) -0.624 (0.002) -0.384 (0.001)

- 2 P

ndP decile / sec. schooling -0.610 (0.001) -0.458 (0.004) -0.598 (0.001) -0.405 (0.001)

- 3 P

rdP decile / vocational -0.599 (0.002) -0.460 (0.001) -0.588 (0.001) -0.390 (0.000)

- 4 P

thP decile / short tertiary -0.591 (0.002) -0.429 (0.003) -0.581 (0.002) -0.409 (0.001)

- 5 P

thP decile / medium tertiary -0.584 (0.002) -0.323 (0.003) -0.573 (0.002) -0.380 (0.001)

- 6 P

thP decile / long tertiary -0.576 (0.002) -0.325 (0.002) -0.566 (0.001) -0.381 (0.001)

- 7 P

thP decile -0.568 (0.002) -0.557 (0.001)

- 8 P

thP decile -0.556 (0.002) -0.549 (0.001)

- 9 P

thP decile -0.539 (0.002) -0.539 (0.001)

- 10 P

thP decile -0.507 (0.002) -0.517 (0.001)

Rho (ρ) 0.390 (0.001) 0.423 (0.000) 0.396 (0.001) 0.427 (0.000)

2σ 0.452 0.458 0.310 0.315



Year (1994/7 – 2003/6) 10 10 10 10

The parameter estimates for the substitution elasticity are uncompensated elasticities

The substitution elasticity disappears for males (-0.012) and becomes significantly negative – and therefore counterintuitive – for females (-0.039). The estimates for the income elasticity are lar- TP

12PT ’Income decile splines’ split lagged income into separate variables for the 10 income deciles of lagged income. This approach is not without

pitfalls. Indeed, a more detailed account of mean reversion places tougher requirements on identification through the number of years analysed

and the cross-income variation in marginal tax rates. It is questionable, whether the tax changes through the period from 1994 to 2006 satisfy this

criteria.

17

ger than the baseline results for both males (-0.031) and females (-0.022). The parameters for lagged income ( 101 γγ − ) are decreasing with income, which indicates that income mobility is decreasing with income.

A plausible explanation for the counterintuitive results is that the deciles splines eliminate the identification of the substitution elasticities because they ‘absorb’ the differences in income changes across the income levels. In other words, the analysed period does not encompass a sufficient variety of tax changes across income levels to allow identification income dependent mean reversion as well as substitution elasticities.

As previously mentioned, the endogeneity of the tax variables is handled by instrumenting the year-two tax variables by simulated tax variables based on year-one incomes. This is the IV-method I referred to above. As an alternative, the instrumentation can be applied directly to the tax variable differences in (7) and (8). This method (IV-method II), has been used by several previous studies. IV-method II leads to results that are markedly different from the results ob-tained by IV-method I, cf. Table 6.

For males, the substitution elasticity disappears almost entirely and turns negative for females (0.056 for IV-method I to -0.067 for IV-method II). The income elasticities are similar if not slightly higher than the baseline results at -0.021 for males and -0.019 for females. It is noted that the IV-method I provides a substantially better prediction than IV-method II. Accordingly, the IV-method I instrument is considered to be superior to IV-method II instrument.

Table 7

ECM sensitivity analysis – Random effects for the tax variables

23-56 year olds (year one) with earned income above DKR 10,000 (1994 level) who did not receive income transfers

Random Effects

Males Females

Substitution elasticity (ξ)

- short run 0.036 (0.026) -0.059 (0.020)

- long run -0.068 (0.044) -0.042 (0.034)

Income elasticity (η)

- short run -0.005 (0.006) -0.065 (0.009)

- long run -0.032 (0.007) 0.127 (0.010)

Gamma (γ) 0.563 (0.002) 0.584 (0.0035)

1Σ 0.170 (0.009) 0.539 (0.023)

0.024 (0.002) 0.010 (0.000) 0.181 (0.007) 0.070 (0.002)

2Σ 1.379 (0.044) 8.347 (0.275)

0.559 (0.023) 0.955 (0.026) -0.887 (0.071) 1.432 (0.047)

-0.081 (0.004) 0.027 (0.002) 0.010 (0.000) -0.849 (0.029) 0.190 (0.008) 0.094 (0.003)

2σ 0.047 0.029

18

Finally, the error-correction model has been estimated with random effects for the tax parameters, cf. Table 7. As a result, the elasticities are no longer meaningful, i.e. negative substitution and posi-tive income elasticities occur. Most likely, this is because the identification of the parameters is being stretched beyond capacity. As an indication of this, the estimated variation around the mean for the random effect parameters is generally very large and there is also quite substantial correlation between the parameters.

As a result of the identification problems, no interpretation should be attached to estimated ran-dom effect parameters. The parameters do, however, reveal an interesting property of the nature of the behavioural response to tax changes namely, that the causality cannot be observed for individuals, but only as an aggregated response by many individuals. Indeed, at the individual level there is a huge variation in the ‘relationship’ between changed incentives (i.e. tax changes) and income response. A large proportion of this ‘relationship’ is obviously unrelated to tax changes. Nonetheless, it contributes to illustrate that taxpayers respond very differently to tax changes and, likely, many not at all, but the sum of those that do can make a substantial differ-ence to the fiscal implications of tax reform.

Discussion

The baseline results from the differences-in-differences model presented in this paper are broadly in line with previous research into the elasticity of taxable income. The compensated elasticity for males and females are small but positive (0.109 and 0.056) and the income elasticity is -0.017 for males and -0.013 for females. The error-correction model suggests that the elasticities are smaller in the short run and somewhat larger in the long run. This is consistent with economic theory. Nevertheless, the results are somewhat sensitive to the choice of assumptions and model specifi-cation.

The purpose of the study is twofold. A primary interest is to provide Danish estimates for the elasticity of taxable income. Secondly, was an important quest to shed light on the robustness and scope of the research in this space and, through that, to assess the economic and political implications of existing results. Indeed, the analysis points to a number of methodology issues associated with the traditional approach to measuring the elasticity of taxable income. Some of these issues are addressed below.

Mean income reversion It is well-known from a number of differences-in-differences studies that the estimated elasticities strongly depend on whether and how mean reversion is accounted for (Gruber and Saez, 2002). These results are confirmed by the present study. Indeed, the analysis shows that it is problem-atic to use income decile splines as this may lead to negative, albeit small, substitution elasticities. A likely explanation is that decile splines eliminate the identification of the substitution elastic-ities because they ‘absorb’ the differences in income changes across the income levels.

The method for handling endogeneity of the tax variables The results show that the estimated elasticities are very sensitive to the choice of instrumentation method. Although the preferred IV-method I is arguably a better choice than IV-method II, the large differences in the estimated substitution elasticities suggest that this is an area where more

19

work is warranted. A particular issue of interest is to what extent different types of incomes are interrelated and, leading on from that, the question of how best to account for these interde-pendencies.

Estimation method The results from the differences-in-differences model show that estimation that accounts for error-term autocorrelation and a lagged dependent variable leads to results that are different from a simple OLS-estimation.

The behaviour of taxpayers is affected by the marginal tax rate of neighbouring budget segments The relationship between taxable income and marginal tax rates is ’myopic’ in the sense that it only accounts for the tax payer’s actual budget segment. This has several implications. First, there is no account of neighbouring budget segments and their slope. A more realistic assump-tion is that tax payer behaviour is affected by the entire budget curve. For example, a person who in year 1 has an income below the top tax bracket – to avoid paying top tax – may respond to a reduction in the top tax even though based on current income the marginal tax rate is un-changed. In other words, it is likely that many tax payers have an incentive to up their efforts to increase taxable income that is not measured by the myopic representation of the budget curve. It is equally likely that any behavioural effects in response to these incentives are not measured.

Secondly, the models do not allow for utility optimisation error as a result of tax payers not hav-ing full control over their taxable income (see Graversen and Smith (2001), and Frederiksen et al. (2009)). This means that a tax payer’s budget segment of choice is not necessarily the observed budget segment.

Finally, a potential source or error could be introduced by the predicted incomes used for the tax variable instruments because they may lead to predicted taxable income that inadvertently pushes a tax payer into a neighbouring budget segment.

Literature

Aarbu K. O. and T. O. Thoresen (2001) Income Responses to Tax Changes – Evidence from the Norwe-gian Tax Reform, National Tax Journal 54, pp. 319-335.

Blomquist, S., A. Kumar and C. Liang (2009) Estimation of Taxable Income Elasticity in a Nonlinear Budget Set Framework, Working Paper, Department of Economics, Uppsala University, Sweden. Blomquist, S. and H. Selin (2009) Hourly Wage Rate and Taxable Labor Income Responsiveness to Changes in Marginal Tax Rates, Working Paper 2009:1, Uppsala Centre for Fiscal Studies, Depart-ment of Economics, Uppsala University. Frederiksen, A., E.K. Graversen, and N. Smith (2009) Dual Job Holding and Taxation, Research in Labor Economics, vol. 28, s. 25-55.

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Goolsbee, A. (2000) It’s not about the money: why natural experiments don’t work on the rich, Working Paper, Graduate School of Business, University of Chicago, American Bar Foundation, and N.B.E.R..

Graversen, E.K. and N. Smith (1998) Labour Supply, Overtime Work and Taxation in Denmark, CLS Woorking Paper 98-06, Centre for Labour Market and Social Research, University of Aarhus and Aarhus School of Business.

Gruber J. and E. Saez (2002) The Elasticity of Taxable Income: Evidence and Implications, Journal of Public Economics 84, pp 1-32.

Hansson Å. (2004) Taxpayer Responsiveness to Tax Rate Changes and Implications for the Cost of Taxa-tion, Working Paper, Department of Economics, Lund University, February 2004.

Hendry, D. F., A. R. Pagan and J. D. Sargan (1984) Dynamic Specification, in Z. Griliches and M. D. Intriligator (Eds.), Handbook of Econometrics, Vol. 2, Chap. 18, North-Holland, Amsterdam, pp. 1023-1100.

Holmund B. and M. Söderström (2007) Estimating Income Responses to Tax Changes: A Dynamic Panel Data Approach, CESIFO Working Paper No. 2121.

Ljunge M. and K. Ragan (2005) Labour supply and the Tax Reform of the Century, unpublished work-ing paper, University of Chicago.

Slemrod (2006) High income families and the tax changes of the 1980s: the anatomy of behavioral response. In: Feldstein, M., Poterba, J. (Eds.), Empirical Foundations of Household Taxation. University of Chi-cago.

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Appendix 1 Parameter estimation

The 2nd step in the two-stage maximum likelihood estimation of the models’ parameters is per-formed by a Bayesian estimation technique whereby a model’s parameter space is simulated itera-tively conditional on prior and posterior distributions for the parameters.

The priors and posteriors for the differences-in-differences and the error correction models are now de-scribed in turn.

The differences-in-differences model The prior and posterior distributions for the differences-in-differences model’s parameters are specified in the following.

Prior distributions The parameters are grouped in blocks with the prior distributions for the DiD model’s parame-ters specified as:

(i) ),(~ [1,1] γγγ VTN − where 11 <<− γ

(ii) ),(~ βββ VN

(iii) ),(~ [1,1] ρρρ VTN − where 11 <<− ρ

(iv) )(~ 02

2

20 fsf χσ

σ where 00 >f and 02 >σs

),( ΣµN is the multinomial normal distribution with meanµ and variance Σ . ),( 2[,] σµbaTN is

the truncated normal distribution on the interval ]a,b[. )(2 fχ is a 2χ -distribution with f degrees of freedom. The parameters of the prior distributions are chosen to ensure that the conditional posterior distributions are well-defined.

Posterior distributions The conditional posterior distributions for the parameter blocks are iterated through the Gipps sampler:

(i) ),,,,|(~ 0001 YXp σρβγγ

(ii) ),,,,|(~ 0001 YXp σργγβ

(iii) ),,,,|(~ 0001 YXp σβγγρ

(iv) ),,,,|(~ 0001 YXp ρβγγσ

22

Parameter index ’0’ and ’1’ are the previous and current draws from the Gipps sam-pler. { } TtNiitXX ,..,1;,..,1 === and { } TtNiitYY ,..,1;,..,1 === . The conditional posterior distributions are derived from the joint posterior density of the model parameters.

The error correction model The prior and posterior distributions for the error correction model’s parameters are specified in the following.

Prior distributions The error correction model’s prior distributions are specified as:

(i) ),(~ [1,1] γγγ VTN − where 11 <<− γ

(ii) ),(~111111 βββ VN

(iii) ),(~212121 βββ VN

(iv) ),(~ 1101

1 sSW−Σ where 010 >S and 01 >s

(v) ),(~ 2201

2 sSW−Σ where 020 >S and 02 >s

(vi) )(~ 02

2

20 fsf χσ

σ where 00 >f and 02 >σs

),( ΣµN is the multinomial normal distribution with meanµ and variance Σ . ),( 2[,] σµbaTN is

the truncated normal distribution on the interval ]a,b[. )(2 fχ is a 2χ -distribution with f degrees of freedom. ),( sSW is a Wishart distribution with mean matrix S and s degrees of freedom. The parameters of the prior distributions are chosen to ensure that the conditional posterior distribu-tions are well-defined.

Posterior distributions The conditional posterior distributions for the parameter blocks are iterated through the Gipps sampler:

(i) ),,,,,,,,,|(~ 1020101

021

0120101 YXXp ii ∆∆ΣΣ −σββββγγ

(ii) ),,,,,,,,,|(~ 1020101

021

0120011 YXXp ii ∆∆ΣΣ −σβββγγβ

(iii) ),,,,,,,,,|(~ 1020101

021

0110021 YXXp ii ∆∆ΣΣ −σβββγγβ

(iv) ),,,,,,,,,|(~ 1020101

0220100111 YXXp ii ∆∆ΣΣ −σβββγγβ i=1,…,N

23

(v) ),,,,,,,,,|(~ 1020101

01201001

12 YXXp ii ∆∆ΣΣ −σβββγγβ i=1,…,N

(vi) ),,,,,,,,,|(~ 12020

102


(vii) ),,,,,,,,,|(~ 12010

102


(viii) ),,,,,,,,,|(~ 120101

021

01201001 YXXp ii ∆∆ΣΣ −ββββγγσ

Parameter index ’0’ and ’1’ are the previous and current draws from the Gipps sam-pler. { } TtNiitXX ,..,1;,..,1 === and { } TtNiitYY ,..,1;,..,1 === . The conditional posterior distributions are derived from the joint posterior density of the model parameters.

earned income response to tax changes in denmark

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