Analytical Studies Branch Research Paper Series
Intra-household Labour Income Responses to Changes in Tax Rates Among Older Workers

by Derek Messacar, Social Analysis and Modelling Division

Release date: November 23, 2017

Abstract

Despite a large literature estimating the effects of income taxation on the labour decisions of young and middle-aged workers, little is known about the extent to which older workers respond to changes in their income taxes. This paper explores this unresolved empirical issue, using longitudinal administrative data on more than one million individuals from Canada and exploiting a recent tax reform in the empirical identification strategy that explicitly targeted older couples. Specifically, this paper makes two contributions. First, older workers are shown to respond meaningfully to changes in their income taxes; however, in contrast with standard predictions, they are responsive to the average—not the marginal—tax rate. The compensated elasticities of labour income with respect to the average and marginal net-of-tax shares are estimated to be 0.421 and -0.112 conditional on being employed, respectively, of which only the former is statistically significant. In addition, each 1% increase in total after-tax income reduces the likelihood of being employed by 1.6 to 2.3 percentage points. Second, changes in individuals’ taxes have spillover effects on the employment and labour income decisions of their spouses, although the income pooling prediction of the unitary model is rejected empirically. These findings offer new insight into the “black box” of intra-household labour supply and inform the optimal designs of income tax and retirement income systems.

Keywords: labour income, pension, effective tax rate, income splitting, unitary model, instrumental variables.

Executive summary

Using longitudinal administrative tax data on more than one million individuals and their spouses in Canada, this paper assesses the extent to which older workers respond to changes in their income taxes. In particular, the primary findings from this paper are as follows:

  1. The compensated elasticity of taxable labour income with respect to the average net-of-tax share (one minus the tax rate) is 0.421, conditional on being employed. Therefore, a 10% increase in the average net-of-tax share leads to a 4.21% increase in the labour income of older workers.
  2. Changes in tax liabilities also have large effects on individuals’ labour market participation decisions. The analysis suggests that each 1% increase in total after-tax income reduces the likelihood that individuals are employed by 1.6 to 2.3 percentage points.

In addition, changes in individuals’ tax rates and tax liabilities can have spillover effects on the employment and labour income decisions of their spouses. The findings from this intra-household analysis indicate the following:

  1. Each 10% change in individuals’ total after-tax income is estimated to induce spouses to reduce their labour earnings by 0.76% to 1.17% conditional on being employed, and to become 1.7 to 1.8 percentage points less likely to be employed.
  2. Each 10% change in individuals’ labour income following a tax reform is estimated to induce spouses to reduce their labour earnings by approximately 15%.

Taken together, the results of this paper suggest that the employment and labour income decisions of older workers respond meaningfully to changes in both their own income taxes and the income taxes of their spouses. These findings offer new insight into intra-household labour supply, and inform the optimal designs of income tax and retirement income systems.

1 Introduction

The aging of workforces and increasing life expectancies pose significant challenges for economic growth, national savings, and the solvency of public pension systems in many countries (Organisation for Economic Co-operation and Development [OECD] 2014). To address these concerns, governments have been increasing retirement ages and strengthening work incentives in order to raise employment rates among older workers and improve the sustainability of their pension systems (OECD 2012). These initiatives are motivated by a large literature that finds individuals’ pension receipts or decisions to retire are responsive to the retirement incentives that public pension plans create (Baker and Benjamin 1999; Feldstein and Liebman 2002; Baker, Gruber and Milligan 2003; French and Jones 2012). In contrast to these pension reforms, the tax codes of many countries offer age-related deductions and reduced taxation of public pension income (OECD 2011), which lower the effective tax rates of older workers and create ambiguous labour supply incentives. Yet the extent to which older workers respond to income taxes, so that the tax code is a viable policy lever for influencing labour supply, has not received considerable attention in empirical research (Schmidt and Sevak 2009; Alpert and Powell 2014). A better understanding of the extent to which older workers respond to changes in their income taxes would inform the optimal designs of income tax and retirement income systems.

Using administrative data on more than one million taxfilers from Canada, this paper provides new insight into this unresolved issue by estimating employment and labour income responses to policy-induced changes in income tax rates among older workers. Specifically, this paper makes two key contributions. First, it credibly estimates whether older workers respond to exogenous changes in their effective net-of-tax shares.Note  The empirical identification strategy exploits a recent tax reform that targeted individuals near or of retirement age, making Canada an ideal setting to study this issue empirically. On January 1, 2007, the federal government implemented pension income splitting, which permits couples to notionally “split” private pension income for tax purposes. The unit of taxation in Canada is the individual, but the income tax system recognizes individuals’ reduced ability to pay taxes when they have dependent spouses. Due to the reform, pension recipients can now transfer up to half of this income to their spouses to reduce household tax liabilities. When income is split, the effective tax rate of each individual may either decrease or increase depending on whether that individual sends or receives income, respectively. The reform also had different effects on tax liabilities depending on the pensioners’ age and whether the pension income derives from an employer-sponsored pension plan (EPP), which creates several margins of variation in effective income tax rates to be exploited empirically. Instrumental variables (IV) are constructed for individuals’ net-of-tax shares that vary over time due solely to this reform, following Auten and Carroll (1999) and Gruber and Saez (2002).

The results show that compensated changes in income tax rates induce large labour income responses conditional on being employed (the intensive margin). However, in contrast with standard predictions, older workers are responsive to the average—not the marginal—tax rate. The compensated elasticities of labour income with respect to the average and marginal net-of-tax shares are 0.421 and -0.112, respectively, of which only the former is statistically significant.Note  These findings support the “schmeduling” hypothesis of Liebman and Zeckhauser (2004), which posits that individuals use the average price as a proxy for the marginal price when nonlinear price schedules are so complex that knowing what marginal price is actually faced is difficult.Note  In addition, individuals exhibit large participation responses to changes in total tax liabilities (the extensive margin). For each 1% exogenous change in total after-tax income, individuals are estimated to become 1.6 to 2.3 percentage points less likely to be employed, on average.

The second contribution of this paper is to test for spillover effects of changes in tax rates across spouses—an important issue among older couples whose retirement decisions could be codependent (Gustman and Steinmeier 2004, 2009; Banks, Blundell and Rivas 2010). In a model of taxable labour income augmented to include intra-household decisions, it is shown that the effect of a tax reform across spouses can be decomposed into 1) a cross-spouse income effect, owing to the change in household disposable income; and 2) a cross-spouse labour effect. Then, the extent to which each of these effects matters for determining employment and labour income decisions is examined separately. These results indicate that each 1% change in individuals’ total after-tax income induces spouses to reduce their labour earnings by 0.076% to 0.117% conditional on being employed, and also reduces the likelihood of being employed by an average of 1.7 to 1.8 percentage points. Hence, there are large cross-spouse effects of changes in income taxes along both the intensive and extensive margins. However, in contrast with the income-pooling prediction of the unitary model of labour supply, individuals appear to be slightly more responsive along the extensive margin to changes in their own after-tax income than to the after-tax income of their spouses.

Overall, the results indicate that the tax code is a viable policy lever for influencing the employment and labour income decisions of older workers. These findings relate to several interconnected literatures. While many studies estimate the elasticity of taxable income with respect to the marginal net-of-tax share, that research has produced a range of empirical results owing to the diversity of tax reforms analyzed and model specifications used (Gruber and Saez 2002).Note  Of direct relevance to this paper are previous studies that estimate the elasticity of taxable labour or wage income to the marginal net-of-tax share (Moffitt and Wilhelm 1998; Saez 2003; Bloomquist and Selin 2010; Bosch and van der Klaauw 2012; Kleven and Schultz 2014). This work tends to find lower elasticities than the taxable income literature, perhaps owing to inflexible labour contracts or work preferences.Note 

Another explanation for the low tax responsiveness typically observed in this literature is that individuals use the average tax rate as a proxy for the marginal tax rate given the complexity of nonlinear income tax schedules. While this schmeduling hypothesis has been tested in other contexts—for example, Ito (2014) finds evidence of this behaviour in electricity consumption—this paper is the first to show that schmeduling occurs with respect to income taxes using a quasi-experimental design. The only other study that addresses this issue is that of de Bartolome (1995), who conducts a controlled experiment and shows that there are as many individuals who use the average tax rate as if it were the marginal tax rate as there are individuals who correctly use the marginal tax rate. Hence, this paper's findings contribute to the wider issue of tax salience (Feldman and Katuščák 2006; Chetty, Looney and Kroft 2009; Finkelstein 2009; Chetty and Saez 2013; Taubinsky and Rees-Jones 2015; Feldman, Katuščák and Kawano 2016).

This paper is also among the first to estimate intra-household labour supply decisions using a reduced-form experimental design that exploits a tax reform in the identification. Previous studies tend to estimate cross-spouse effects structurally (Gustman and Steinmeier 2004, 2009; Banks, Blundell and Rivas 2010; Laitner and Silverman 2012; Michaud and Vermeulen 2011; Van Soest and Vonkova 2014). Two notable exceptions are the following: 1) Yamada (2011), who shows that the hours worked of married women responded significantly to a series of Japanese tax reforms during the 1990s, although that analysis assumed labour supply decisions within couples are determined sequentially for identification; and 2) Kabátek, Van Soest, and Stancanelli (2014), who estimate the effects of a tax reform on intra-household labour supply and domestic work decisions among couples in France, where the unit of taxation is the couple. Using a random coefficients model, the study concludes that spouses’ labour market and housework hours are both responsive to changes in the tax code.Note  In contrast, the intra-household model of taxable labour income developed in Section 2 of this paper shows that, when the unit of taxation is the individual, a tax reform can be exploited to credibly estimate intra-household labour supply decisions using an IV approach, which offers a novel contribution to this literature.

The paper proceeds as follows. The next section presents a unitary model of taxable labour income to provide a framework for the empirical analysis. Section 3 describes the data, sample selection, and empirical methodology, including key features of the tax reform to be exploited in the identification and how the predicted tax measures are constructed. Sections 4 and 5 assess the effects of income tax changes on individuals’ labour income and cross-spouse effects, respectively. Section 6 concludes.

2 Theoretical framework

This section presents a stylized model of taxable labour income (Gruber and Saez 2002; Kleven and Schultz 2014) and derives comparative statics to provide a framework for interpreting the empirical results. More precisely, the model is extended to a setting with intra-household labour supply (Chiappori 1988). This yields predictions of how individuals should respond to changes in their own tax rates relative to changes in the tax rates of their spouses, and validates an IV approach used in Section 5 to estimate cross-spouse labour effects.

2.1 Setup

In the intra-household model, a single economic agent maximizes a weighted average of utilities for individual i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaaaa@36FA@  and spouse s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Caaaa@3704@ , U( c, z i , z s ; ψ i , ψ s )λ u i ( c, z i , z s ; ψ i )+( 1λ ) u s ( c, z i , z s ; ψ s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamyvamaabmaapaqaa8qacaWGJbGaaiilaiaadQhapaWaaSbaaSqa a8qacaWGPbaapaqabaGcpeGaaiilaiaadQhapaWaaSbaaSqaa8qaca WGZbaapaqabaGcpeGaai4oaiabeI8a59aadaWgaaWcbaWdbiaadMga a8aabeaak8qacaGGSaGaeqiYdK3damaaBaaaleaapeGaam4CaaWdae qaaaGcpeGaayjkaiaawMcaaiabggMi6kabeU7aSjaadwhapaWaaWba aSqabeaapeGaamyAaaaakmaabmaapaqaa8qacaWGJbGaaiilaiaadQ hapaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeGaaiilaiaadQhapaWa aSbaaSqaa8qacaWGZbaapaqabaGcpeGaai4oaiabeI8a59aadaWgaa WcbaWdbiaadMgaa8aabeaaaOWdbiaawIcacaGLPaaacqGHRaWkdaqa daWdaeaapeGaaGymaiabgkHiTiabeU7aSbGaayjkaiaawMcaaiaadw hapaWaaWbaaSqabeaapeGaam4Caaaakmaabmaapaqaa8qacaWGJbGa aiilaiaadQhapaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeGaaiilai aadQhapaWaaSbaaSqaa8qacaWGZbaapaqabaGcpeGaai4oaiabeI8a 59aadaWgaaWcbaWdbiaadohaa8aabeaaaOWdbiaawIcacaGLPaaaaa a@6EC4@ , by choosing consumption c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4yaaaa@36F4@  and taxable labour incomes z i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaaa@3853@  and z s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaWgaaWcbaWdbiaadohaa8aabeaaaaa@385D@  conditional on vectors of personal traits ψ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiYdK3damaaBaaaleaapeGaamyAaaWdaeqaaaaa@3922@  and ψ s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiYdK3damaaBaaaleaapeGaam4CaaWdaeqaaaaa@392C@ . The parameter λ[ 0,1 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4UdWMaeyicI48aamWaa8aabaWdbiaaicdacaGGSaGaaGymaaGa ay5waiaaw2faaaaa@3D7A@  is the weight factor.Note  This setup implicitly assumes that z i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaaa@3853@  and z s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaWgaaWcbaWdbiaadohaa8aabeaaaaa@385D@  depend on such factors as hours worked, effort, or tax sheltering, and that these activities are separable from consumption in the utility function (Kleven and Schultz 2014). The agent chooses the bundle { c, z i , z s } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaiWaa8aabaWdbiaadogacaGGSaGaamOEa8aadaWgaaWcbaWdbiaa dMgaa8aabeaak8qacaGGSaGaamOEa8aadaWgaaWcbaWdbiaadohaa8 aabeaaaOWdbiaawUhacaGL9baaaaa@3F70@  subject to the budget constraint c= z i ( 1 τ i )+ z s ( 1 τ s )+ R i + R s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4yaiabg2da9iaadQhapaWaaSbaaSqaa8qacaWGPbaapaqabaGc peWaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaSbaaSqaa8 qacaWGPbaapaqabaaak8qacaGLOaGaayzkaaGaey4kaSIaamOEa8aa daWgaaWcbaWdbiaadohaa8aabeaak8qadaqadaWdaeaapeGaaGymai abgkHiTiabes8a09aadaWgaaWcbaWdbiaadohaa8aabeaaaOWdbiaa wIcacaGLPaaacqGHRaWkcaWGsbWdamaaBaaaleaapeGaamyAaaWdae qaaOWdbiabgUcaRiaadkfapaWaaSbaaSqaa8qacaWGZbaapaqabaaa aa@50C6@ , where τ j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdq3damaaBaaaleaapeGaamOAaaWdaeqaaaaa@391A@  is the marginal tax rate and R j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaWgaaWcbaWdbiaadQgaa8aabeaaaaa@382C@  is virtual income for each j{ i,s } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiabgIGiopaacmaapaqaa8qacaWGPbGaaiilaiaadohaaiaa wUhacaGL9baaaaa@3D65@ .

The following standard assumptions on the agent’s utility function are imposed. For each  j{ i,s } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiabgIGiopaacmaapaqaa8qacaWGPbGaaiilaiaadohaaiaa wUhacaGL9baaaaa@3D65@ : 1)  u c j >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaqhaaWcbaWdbiaadogaa8aabaWdbiaadQgaaaGccqGH +aGpcaaIWaaaaa@3B14@  and u z j j <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaqhaaWcbaWdbiaadQhapaWaaSbaaWqaa8qacaWGQbaa paqabaaaleaapeGaamOAaaaakiabgYda8iaaicdaaaa@3C6D@ , more consumption but less work is desired; 2)  u cc j <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaqhaaWcbaWdbiaadogacaWGJbaapaqaa8qacaWGQbaa aOGaeyipaWJaaGimaaaa@3BF8@  and u z j z j j <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaqhaaWcbaWdbiaadQhapaWaaSbaaWqaa8qacaWGQbaa paqabaWcpeGaamOEa8aadaWgaaadbaWdbiaadQgaa8aabeaaaSqaa8 qacaWGQbaaaOGaeyipaWJaaGimaaaa@3ED1@ , there are diminishing returns to consumption and increasing marginal costs of labour supply; and 3) u c z j j = u z j c j <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaqhaaWcbaWdbiaadogacaWG6bWdamaaBaaameaapeGa amOAaaWdaeqaaaWcbaWdbiaadQgaaaGccqGH9aqpcaWG1bWdamaaDa aaleaapeGaamOEa8aadaWgaaadbaWdbiaadQgaa8aabeaal8qacaWG Jbaapaqaa8qacaWGQbaaaOGaeyipaWJaaGimaaaa@4405@ , the marginal utility of consumption is decreasing in the labour income requirements to achieve this consumption. On the assumption that leisure is (weakly) complementary across spouses, for each j,k{ i,s } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiaacYcacaWGRbGaeyicI48aaiWaa8aabaWdbiaadMgacaGG SaGaam4CaaGaay5Eaiaaw2haaaaa@3F05@  subject to jk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiabgcMi5kaadUgaaaa@39B2@ , the following conditions are also assumed: 4) u z k j 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaqhaaWcbaWdbiaadQhapaWaaSbaaWqaa8qacaWGRbaa paqabaaaleaapeGaamOAaaaakiabgsMiJkaaicdaaaa@3D1F@ ; 5) u c z k j = u z k c j 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaqhaaWcbaWdbiaadogacaWG6bWdamaaBaaameaapeGa am4AaaWdaeqaaaWcbaWdbiaadQgaaaGccqGH9aqpcaWG1bWdamaaDa aaleaapeGaamOEa8aadaWgaaadbaWdbiaadUgaa8aabeaal8qacaWG Jbaapaqaa8qacaWGQbaaaOGaeyizImQaaGimaaaa@44B8@ ; and 6) u z j z k j = u z k z j j 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyDa8aadaqhaaWcbaWdbiaadQhapaWaaSbaaWqaa8qacaWGQbaa paqabaWcpeGaamOEa8aadaWgaaadbaWdbiaadUgaa8aabeaaaSqaa8 qacaWGQbaaaOGaeyypa0JaamyDa8aadaqhaaWcbaWdbiaadQhapaWa aSbaaWqaa8qacaWGRbaapaqabaWcpeGaamOEa8aadaWgaaadbaWdbi aadQgaa8aabeaaaSqaa8qacaWGQbaaaOGaeyizImQaaGimaaaa@4791@ .

2.2 The intensive margin

Conditional on participating in the labour market, the optimal taxable labour income z i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaqhaaWcbaWdbiaadMgaa8aabaWdbiaacQcaaaaaaa@3912@  (for a given z s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaWgaaWcbaWdbiaadohaa8aabeaaaaa@385D@  ) is implicitly solved for by the first-order condition:

λ( u c i ( 1 τ i )+ u z i i )+( 1λ )( u c s ( 1 τ i )+ u z i s )=0   (1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4UdW2aaeWaa8aabaWdbiaadwhapaWaa0baaSqaa8qacaWGJbaa paqaa8qacaWGPbaaaOWaaeWaa8aabaWdbiaaigdacqGHsislcqaHep aDpaWaaSbaaSqaa8qacaWGPbaapaqabaaak8qacaGLOaGaayzkaaGa ey4kaSIaamyDa8aadaqhaaWcbaWdbiaadQhapaWaaSbaaWqaa8qaca WGPbaapaqabaaaleaapeGaamyAaaaaaOGaayjkaiaawMcaaiabgUca Rmaabmaapaqaa8qacaaIXaGaeyOeI0Iaeq4UdWgacaGLOaGaayzkaa WaaeWaa8aabaWdbiaadwhapaWaa0baaSqaa8qacaWGJbaapaqaa8qa caWGZbaaaOWaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaS baaSqaa8qacaWGPbaapaqabaaak8qacaGLOaGaayzkaaGaey4kaSIa amyDa8aadaqhaaWcbaWdbiaadQhapaWaaSbaaWqaa8qacaWGPbaapa qabaaaleaapeGaam4CaaaaaOGaayjkaiaawMcaaiabg2da9iaaicda aaa@6148@

To find the individual’s labour income responses to changes in the own net-of-tax share and the net-of-tax share of the spouse, totally differentiate Equation (1) with respect to ( 1 τ i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaSbaaSqaa8qa caWGPbaapaqabaaak8qacaGLOaGaayzkaaaaaa@3C83@  and ( 1 τ s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaSbaaSqaa8qa caWGZbaapaqabaaak8qacaGLOaGaayzkaaaaaa@3C8D@  and evaluate at { z i * , z s * } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaiWaa8aabaWdbiaadQhapaWaa0baaSqaa8qacaWGPbaapaqaa8qa caGGQaaaaOGaaiilaiaadQhapaWaa0baaSqaa8qacaWGZbaapaqaa8 qacaGGQaaaaaGccaGL7bGaayzFaaaaaa@3F36@  to obtain:

d z i * =Ω( λ u c i +( 1λ ) u c s )d( 1 τ i )Γ( z i * d( 1 τ i )+d R i )Θd z s *    (2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadQhapaWaa0baaSqaa8qacaWGPbaapaqaa8qacaGGQaaa aOGaeyypa0JaaeyQdmaabmaapaqaa8qacqaH7oaBcaWG1bWdamaaDa aaleaapeGaam4yaaWdaeaapeGaamyAaaaakiabgUcaRmaabmaapaqa a8qacaaIXaGaeyOeI0Iaeq4UdWgacaGLOaGaayzkaaGaamyDa8aada qhaaWcbaWdbiaadogaa8aabaWdbiaadohaaaaakiaawIcacaGLPaaa caWGKbWaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaSbaaS qaa8qacaWGPbaapaqabaaak8qacaGLOaGaayzkaaGaeyOeI0Iaae4K dmaabmaapaqaa8qacaWG6bWdamaaDaaaleaapeGaamyAaaWdaeaape GaaiOkaaaakiaadsgadaqadaWdaeaapeGaaGymaiabgkHiTiabes8a 09aadaWgaaWcbaWdbiaadMgaa8aabeaaaOWdbiaawIcacaGLPaaacq GHRaWkcaWGKbGaamOuamaaBaaaleaacaaMc8UaamyAaaqabaaakiaa wIcacaGLPaaacqGHsislcaqGyoGaamizaiaadQhapaWaa0baaSqaa8 qacaWGZbaapaqaa8qacaGGQaaaaaaa@6CE3@

d z i * =Γ( z s * d( 1 τ s )+d R s )Θd z s *    (3) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadQhapaWaa0baaSqaa8qacaWGPbaapaqaa8qacaGGQaaa aOGaeyypa0JaeyOeI0Iaae4Kdmaabmaapaqaa8qacaWG6bWdamaaDa aaleaapeGaam4CaaWdaeaapeGaaiOkaaaakiaadsgadaqadaWdaeaa peGaaGymaiabgkHiTiabes8a09aadaWgaaWcbaWdbiaadohaa8aabe aaaOWdbiaawIcacaGLPaaacqGHRaWkcaWGKbGaamOuamaaBaaaleaa caaMc8Uaam4CaaqabaaakiaawIcacaGLPaaacqGHsislcaqGyoGaam izaiaadQhapaWaa0baaSqaa8qacaWGZbaapaqaa8qacaGGQaaaaaaa @549C@

where Ω>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyQdiabg6da+iaaicdaaaa@38FD@ , Γ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Kdiabg6da+iaaicdaaaa@38E7@ , and Θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiMdiabg6da+iaaicdaaaa@38EC@  are constants that depend entirely on the model’s parameters and second-order partial derivatives of utility (see the Appendix for derivations). Equation (2) shows that the individual’s response to a change in the own tax rate depends on: 1) a substitution (price) effect, evaluated relative to a weighted average of the marginal utilities of consumption for both spouses, λ u c i +( 1λ ) u c s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4UdWMaamyDa8aadaqhaaWcbaWdbiaadogaa8aabaWdbiaadMga aaGccqGHRaWkdaqadaWdaeaapeGaaGymaiabgkHiTiabeU7aSbGaay jkaiaawMcaaiaadwhapaWaa0baaSqaa8qacaWGJbaapaqaa8qacaWG Zbaaaaaa@4430@ ; 2) an income effect, expressed as the sum of the infra-marginal effect of the reform relative to z i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaqhaaWcbaWdbiaadMgaa8aabaWdbiaacQcaaaaaaa@3912@  and the change in virtual income, d R i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadkfapaWaaSbaaSqaa8qacaWGPbaapaqabaaaaa@3914@ ; and 3) a cross-spouse labour income response, d z s * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadQhapaWaa0baaSqaa8qacaWGZbaapaqaa8qacaGGQaaa aaaa@3A05@ . This expression forms the basis for the estimating equation of individuals’ responsiveness to changes in taxes carried out in Section 4.

Further, Equation (3) shows that a change in the spouse’s tax liability only has income and cross-spouse labour effects on the income adjustment of the individual. In this case, the income effect is expressed as the sum of an infra-marginal effect relative to z s * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaqhaaWcbaWdbiaadohaa8aabaWdbiaacQcaaaaaaa@391C@  and the change in virtual income, d R s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadkfapaWaaSbaaSqaa8qacaWGZbaapaqabaaaaa@391E@ . The lack of direct price effect arises because, despite the joint optimization, the spouse’s tax rate is only levied on spousal labour income; a small deviation in ( 1 τ s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaSbaaSqaa8qa caWGZbaapaqabaaak8qacaGLOaGaayzkaaaaaa@3C8D@  does not affect the marginal value of an extra $1 earned by the individual.Note  Hence, combining Equations (2) and (3), exogenous variation in the net-of-tax share of the spouse, d( 1 τ s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamizamaabmaapaqaa8qacaaIXaGaeyOeI0IaeqiXdq3damaaBaaa leaapeGaam4CaaWdaeqaaaGcpeGaayjkaiaawMcaaaaa@3D76@ , is a valid excluded instrument for d z s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadQhapaWaaSbaaSqaa8qacaWGZbaapaqabaaaaa@3946@  to estimate how a change in the taxable labour income of the spouse directly affects the taxable labour income of the individual, d z i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadQhapaWaaSbaaSqaa8qacaWGPbaapaqabaaaaa@393C@ . The tests of cross-spouse income and labour effects are performed in Section 5, using these model implications to guide the analysis.

2.3 The extensive margin

The labour market participation decision depends on the utility gain from working relative to the cost. Suppose the utility cost of supplying labour is k j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4Aa8aadaWgaaWcbaWdbiaadQgaa8aabeaaaaa@3845@  for each j{ i,s } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiabgIGiopaacmaapaqaa8qacaWGPbGaaiilaiaadohaaiaa wUhacaGL9baaaaa@3D65@ . As in Alpert and Powell (2014), the agent solves the following maximization problem:

max z i ' { 0, z i * },  z s ' { 0, z s * }  U( z i ' ( 1 τ i )+ z s ' ( 1 τ s )+ R i + R s , z i ' , z s ' ; ψ i , ψ s )    (4) 1( z i ' >0 ) k i 1( z s ' >0 ) k s   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaWaaCbeae aaqaaaaaaaaaWdbiGac2gacaGGHbGaaiiEaaWcpaqaa8qacaWG6bWd amaaDaaameaapeGaamyAaaWdaeaapeGaai4jaaaaliabgIGiopaacm aapaqaa8qacaaIWaGaaiilaiaadQhapaWaa0baaWqaa8qacaWGPbaa paqaa8qacaGGQaaaaaWccaGL7bGaayzFaaGaaiilaiaacckacaWG6b WdamaaDaaameaapeGaam4CaaWdaeaapeGaai4jaaaaliabgIGiopaa cmaapaqaa8qacaaIWaGaaiilaiaadQhapaWaa0baaWqaa8qacaWGZb aapaqaa8qacaGGQaaaaaWccaGL7bGaayzFaaaapaqabaGcpeGaaiiO aiaadwfadaqadaWdaeaapeGaamOEa8aadaqhaaWcbaWdbiaadMgaa8 aabaWdbiaacEcaaaGcdaqadaWdaeaapeGaaGymaiabgkHiTiabes8a 09aadaWgaaWcbaWdbiaadMgaa8aabeaaaOWdbiaawIcacaGLPaaacq GHRaWkcaWG6bWdamaaDaaaleaapeGaam4CaaWdaeaapeGaai4jaaaa kmaabmaapaqaa8qacaaIXaGaeyOeI0IaeqiXdq3damaaBaaaleaape Gaam4CaaWdaeqaaaGcpeGaayjkaiaawMcaaiabgUcaRiaadkfapaWa aSbaaSqaa8qacaWGPbaapaqabaGcpeGaey4kaSIaamOua8aadaWgaa WcbaWdbiaadohaa8aabeaak8qacaGGSaGaamOEa8aadaqhaaWcbaWd biaadMgaa8aabaWdbiaacEcaaaGccaGGSaGaamOEa8aadaqhaaWcba Wdbiaadohaa8aabaWdbiaacEcaaaGccaGG7aGaeqiYdK3damaaBaaa leaapeGaamyAaaWdaeqaaOWdbiaacYcacqaHipqEpaWaaSbaaSqaa8 qacaWGZbaapaqabaaak8qacaGLOaGaayzkaaaabaGaeyOeI0IaaGym amaabmaapaqaa8qacaWG6bWdamaaDaaaleaapeGaamyAaaWdaeaape Gaai4jaaaakiabg6da+iaaicdaaiaawIcacaGLPaaacaWGRbWdamaa BaaaleaapeGaamyAaaWdaeqaaOWdbiabgkHiTiaaigdadaqadaWdae aapeGaamOEa8aadaqhaaWcbaWdbiaadohaa8aabaWdbiaacEcaaaGc cqGH+aGpcaaIWaaacaGLOaGaayzkaaGaam4Aa8aadaWgaaWcbaWdbi aadohaa8aabeaak8qacaGGGcaaaaa@9568@

where 1( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymamaabm aabaGaeyyXICnacaGLOaGaayzkaaaaaa@3A85@  is an indicator function, and z j * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaqhaaWcbaWdbiaadQgaa8aabaWdbiaacQcaaaaaaa@3913@  for each j{ i,s } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiabgIGiopaacmaapaqaa8qacaWGPbGaaiilaiaadohaaiaa wUhacaGL9baaaaa@3D65@  is the solution to the intensive-margin optimization problem outlined above. The participation decision only depends on the effect of total after-tax income on consumption—an income effect. To see this, notice that a tax reform that is revenue-neutral for the agent (dc/d( 1 τ i )=0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaiikaiaadsgacaWGJbGaai4laiaadsgadaqadaWdaeaapeGaaGym aiabgkHiTiabes8a09aadaWgaaWcbaWdbiaadMgaa8aabeaaaOWdbi aawIcacaGLPaaacqGH9aqpcaaIWaGaaiykaaaa@4309@  has no effect on participation. For a decrease in the total tax liability of either the individual or spouse, the benefit of working increases, in turn raising the likelihood that the individual is employed.

3 Data and empirical methodology

This section first describes the dataset and sample selection used. Then, a brief overview of Canada’s income tax system is provided, including details of the tax reform exploited in the empirical analysis. This section concludes by presenting the estimating equations derived from the theoretical framework and outlining how the predicted tax measures were constructed.

3.1 Data and sample selection

The Longitudinal Administrative Databank (LAD) is used to carry out this study. The LAD is a panel dataset comprising a 20% nationally representative subset of the T1 Family File (T1FF). Importantly, the T1FF is a yearly cross-sectional dataset of taxfilers and their families based on records from Canada’s central tax authorities. While individuals file tax returns independently in Canada, census families (both legal and common-law) were created in the T1FF based on the spousal social insurance number listed on each individual’s tax form or by matching based on name, address, age, sex, and marital status. Therefore, the LAD provides data on both individuals and their spouses, which is needed to carry out this intra-household analysis.Note  Since the LAD does not contain information on individuals’ tax rates, these measures were constructed using Milligan’s (2012) Canadian Tax and Credit Simulator (CTaCS) along with the wide scope of information available in the data to accurately predict tax liabilities.

The following sample restrictions are imposed. First, individuals were included in the sample only if they were observed filing taxes in every year from 2005 to 2008, a time period that encompasses the relevant tax reform. This restriction is needed for the analysis to control for fixed effects in the empirical analysis; approximately 90% of taxfilers meet this requirement. Second, both individuals and their spouses must have been 55 years of age or older in 2008 (the last year of data used), so that the analysis centers on older workers. Third, individuals and their spouses must have been 54 years of age or younger in at least one year from 1991 to 2006, for methodological reasons discussed below.

Table 1 presents descriptive statistics for this sample. Individuals are 60 years old on average, of whom approximately half are male and 6.1% are immigrants. Importantly, more than half of individuals and their spouses were employed in 2006 (62.9% and 62.6%, respectively), and many couples consisted of at least one pensioner (44.2%), which means there is a large share of households whose labour decisions were potentially affected by the tax reform. Total after-tax labour income in 2006 averaged $38,650 and $38,100 for individuals and their spouses, respectively, and the corresponding marginal tax rates with respect to labour income were 24.1% and 20.5%.

Table 1
Descriptive statistics
Table summary
This table displays the results of Descriptive statistics. The information is grouped by Statistic (appearing as row headers), Median and Average, calculated using years, percent and nominal dollars units of measure (appearing as column headers).
Statistic Median Average
years
Demographics
Individual's age 60.0 60.3
Spouse's age 60.0 60.3
percent
Female Note ...: not applicable 49.8
Male Note ...: not applicable 50.2
Married Note ...: not applicable 100.0
Immigrant Note ...: not applicable 6.1
Non-pension income
Individual has labour income Note ...: not applicable 62.9
Spouse has labour income Note ...: not applicable 62.6
Family has capital gains Note ...: not applicable 26.8
Family has investment income Note ...: not applicable 62.5
Family has Employment Insurance income Note ...: not applicable 12.5
Family has social assistance income Note ...: not applicable 1.8
Pension income
Individual has private pension income Note ...: not applicable 28.4
Spouse has private pension income Note ...: not applicable 28.2
Family has private pension income Note ...: not applicable 44.2
nominal dollars
Earnings
Individual's labour income 10,000 27,500
Spouse's labour income 9,750 26,950
Individual's total after-tax income 28,950 38,650
Spouse's total after-tax income 28,700 38,100
percent
Allowances
Family has disability allowances Note ...: not applicable 3.9
Family has medical expense allowances Note ...: not applicable 42.5
Marginal tax rates
Individual's marginal tax rate 26.7 24.1
Spouse's marginal tax rate 19.2 20.5
Average tax rates
Individual's average tax rate 18.2 14.8
Spouse's average tax rate 14.7 14.3

3.2 Pension income splitting reform

Personal income tax in Canada is calculated based on a measure of taxable income (net of permitted deductions), then tax credits are applied to determine the net amount of income tax payable. While the unit of taxation is the individual, the personal income tax system recognizes taxpayers’ reduced ability to pay taxes when individuals have dependent spouses and in certain cases when taxpayers support other dependent relatives (e.g., parents or grandparents). This recognition is generally provided in the form of additional tax credits and through permitting the transfer of certain dependents’ unused portion of personal tax credits to the taxpayers.

Taxes are determined at the federal and provincial levels, with the federal and provincial governments each applying separate tax rates to a uniform measure of taxable income and each applying separate tax credits to determine the net amount of federal and provincial tax owing.Note  At the federal level, taxable income in 2006 was divided into four brackets: 1) the first $36,378 of income; 2) from $36,379 to $72,756; 3) from $72,757 to $118,285; and 4) over $118,285. The income tax rates for these brackets were 15.25%, 22%, 26%, and 29%, respectively, and the federal basic exemption was $9,039. At the provincial level, there is significant heterogeneity in income tax structures and rates. For example, Alberta levied provincial taxes using the same bracket structure as the federal level, whereas Nova Scotia, Ontario, and British Columbia had eight brackets; basic exemptions ranged from $7,231 in Nova Scotia to $14,799 in Alberta. Given the plethora of factors that affect taxable income, some researchers regard the Income Tax Act as one of Canada’s most complex pieces of legislation (Wolfson et al. 2016).

On January 1, 2007, the federal government implemented pension income splitting, which permits individuals to notionally “split” private pension income with their spouses. In particular, pension recipients with high income (the “pensioners”) may allocate up to 50% of their eligible pension income to their spouses (the “transferees”) to reduce household tax liabilities. To qualify, the pension income must satisfy certain criteria. If a pensioner is 65 years of age or older, eligible pension income includes lifetime annuity payments under employer-sponsored pension plans (EPPs), registered retirement savings plans, and payments out of Registered Retirement Income Funds. However, if a pensioner is less than 65 years old, eligible pension income only includes lifetime annuity payments from EPPs and certain payments received as a result of the death of a spouse.

3.3 Empirical model

The objective of this study is to estimate both individual and cross-spouse responses to changes in tax rates. The empirical model extends the work of Gruber and Saez (2002), Gelber (2014), and the related literature by expressing the change in the individual’s log of labour income to the changes in the individual’s log of the net-of-tax share and log of total after-tax income, as well as the direct adjustment to the spouse’s log of labour income. The baseline estimating equation for the intensive-margin analysis, derived from Equation (2), is as follows:

Δln( z it )= α 0 + β 0 Δln( 1 τ it )+ γ 0 Δln( Y it T it )+ ζ 0 Δln( z st )+ X it ' θ 0 +f[ ln( z i,t1 ) ]+ ϵ it    (5) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaciiBaiaac6gadaqadaWdaeaapeGaamOEa8aadaWgaaWc baWdbiaadMgacaWG0baapaqabaaak8qacaGLOaGaayzkaaGaeyypa0 JaeqySde2damaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabgUcaRiab ek7aI9aadaWgaaWcbaWdbiaaicdaa8aabeaakiabfs5ae9qaciGGSb GaaiOBamaabmaapaqaa8qacaaIXaGaeyOeI0IaeqiXdq3damaaBaaa leaapeGaamyAaiaadshaa8aabeaaaOWdbiaawIcacaGLPaaacqGHRa WkcqaHZoWzpaWaaSbaaSqaa8qacaaIWaaapaqabaGccqqHuoarpeGa ciiBaiaac6gadaqadaWdaeaapeGaamywa8aadaWgaaWcbaWdbiaadM gacaWG0baapaqabaGcpeGaeyOeI0Iaamiva8aadaWgaaWcbaWdbiaa dMgacaWG0baapaqabaaak8qacaGLOaGaayzkaaGaey4kaSIaeqOTdO 3damaaBaaaleaapeGaaGimaaWdaeqaaOGaeuiLdq0dbiGacYgacaGG UbWaaeWaa8aabaWdbiaadQhapaWaaSbaaSqaa8qacaWGZbGaamiDaa WdaeqaaaGcpeGaayjkaiaawMcaaiabgUcaRiaadIfapaWaa0baaSqa a8qacaWGPbGaamiDaaWdaeaapeGaai4jaaaakiabeI7aX9aadaWgaa WcbaWdbiaaicdaa8aabeaak8qacqGHRaWkcaWGMbWaamWaa8aabaWd biGacYgacaGGUbWaaeWaa8aabaWdbiaadQhapaWaaSbaaSqaa8qaca WGPbGaaiilaiaadshacqGHsislcaaIXaaapaqabaaak8qacaGLOaGa ayzkaaaacaGLBbGaayzxaaGaey4kaSYefv3ySLgznfgDOfdaryqr1n gBPrginfgDObYtUvgaiuGacqWF1pG8paWaaSbaaSqaa8qacaWGPbGa amiDaaWdaeqaaaaa@90A0@

where Y jt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamywa8aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaaaaa@392C@  is total pre-tax income and T jt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiva8aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaaaaa@3927@  is tax liability for each j{ i,s } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiabgIGiopaacmaapaqaa8qacaWGPbGaaiilaiaadohaaiaa wUhacaGL9baaaaa@3D65@  at time t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@367D@ ; τ jt = T jt / z jt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdq3damaaBaaaleaapeGaamOAaiaadshaa8aabeaak8qacqGH 9aqpcqGHciITcaWGubWdamaaBaaaleaapeGaamOAaiaadshaa8aabe aak8qacaGGVaGaamOEa8aadaWgaaWcbaWdbiaadQgacaWG0baapaqa baaaaa@43C2@  is the marginal tax rate with respect to labour income; X jt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiwa8aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaaaaa@392B@  is a set of observed covariates; ϵ it MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfiaeaaaaaaaaa8qacqWF1pG8 paWaaSbaaSqaa8qacaWGPbGaamiDaaWdaeqaaaaa@4448@  is the statistical residual; and Δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfs5aebaa@3752@  is the difference operator, Δ A t = A t+1 A t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaamyqa8aadaWgaaWcbaWdbiaadshaa8aabeaak8qacqGH 9aqpcaWGbbWdamaaBaaaleaapeGaamiDaiabgUcaRiaaigdaa8aabe aak8qacqGHsislcaWGbbWdamaaBaaaleaapeGaamiDaaWdaeqaaaaa @4181@ . Thus, Δln( 1 τ jt ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaciiBaiaac6gadaqadaWdaeaapeGaaGymaiabgkHiTiab es8a09aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaaak8qacaGLOa Gaayzkaaaaaa@40C7@  measures the change in the log of the net-of-tax share, and Δln( Y jt T jt ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaciiBaiaac6gadaqadaWdaeaapeGaamywa8aadaWgaaWc baWdbiaadQgacaWG0baapaqabaGcpeGaeyOeI0Iaamiva8aadaWgaa WcbaWdbiaadQgacaWG0baapaqabaaak8qacaGLOaGaayzkaaaaaa@425A@  measures the change in the log of total after-tax income. The regression parameters of interest are: 1)  β 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOSdi2damaaBaaaleaapeGaaGimaaWdaeqaaaaa@38C1@ , the substitution response to a marginal deviation in the price of labour; and 2)  γ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4SdC2damaaBaaaleaapeGaaGimaaWdaeqaaaaa@38C7@ , the income effect of the change in total after-tax income. The model predicts that β ^ 0 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqOSdi2dayaajaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaeyOp a4JaaGimaaaa@3AAD@  and γ ^ 0 <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gafq4SdC2dayaajaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaeyip aWJaaGimaaaa@3AAF@ .

Implicit in Equation (5) is the fact that tax liability is a function of the income and personal characteristics of both the individual and spouse (which affect taxable income through various allowances), as well as a vector of tax parameters. Formally, T jt =T( Ψ jt ; Π t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiva8aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaGcpeGaeyyp a0Jaamivamaabmaapaqaa8qacaqGOoWdamaaBaaaleaapeGaamOAai aadshaa8aabeaak8qacaGG7aGaaeiOd8aadaWgaaWcbaWdbiaadsha a8aabeaaaOWdbiaawIcacaGLPaaaaaa@43A4@  subject to Ψ jt ={ Y it , Y st , X it , X st } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiQd8aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaGcpeGaeyyp a0ZaaiWaa8aabaWdbiaadMfapaWaaSbaaSqaa8qacaWGPbGaamiDaa WdaeqaaOWdbiaacYcacaWGzbWdamaaBaaaleaapeGaam4Caiaadsha a8aabeaak8qacaGGSaGaamiwa8aadaWgaaWcbaWdbiaadMgacaWG0b aapaqabaGcpeGaaiilaiaadIfapaWaaSbaaSqaa8qacaWGZbGaamiD aaWdaeqaaaGcpeGaay5Eaiaaw2haaaaa@4BF2@  and Π t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiOd8aadaWgaaWcbaWdbiaadshaa8aabeaaaaa@3885@  is the set of relevant tax parameters at time t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@367D@ . As is common in the elasticity of taxable income literature, lagged labour income, z i,t1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaWgaaWcbaWdbiaadMgacaGGSaGaamiDaiabgkHiTiaa igdaa8aabeaaaaa@3BA3@ , is controlled for flexibly in order to account for mean-reversion bias and potential changes in income inequality around the time of the tax reform (Auten and Carroll 1999; Gruber and Saez 2002; Saez, Slemrod and Giertz 2012; Gelber 2014; Kleven and Schultz 2014). Specifically, f[ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOzamaadmaapaqaa8qacqGHflY1aiaawUfacaGLDbaaaaa@3B52@  is set to be a 10-piece spline based on percentiles. Most of the literature cited above estimates tax elasticities using policy-induced variation in marginal income tax schedules over time. That approach assumes individuals at one point in the income distribution are a reasonable control group for individuals at the other points in the distribution, which makes controlling for mean-reversion and a changing income distribution especially important. Yet including controls that are too flexible invariably absorbs much of the tax rate variation created by the reform that is useful in the identification. This issue is less of a concern in this analysis because variation in effective tax rates brought on by the introduction of a new tax allowance is exploited. The pension income splitting reform affected individuals differently across multiple dimensions including, but not limited to, the income distribution. This analysis will show how the results change as lagged income becomes controlled for.

Based on Equation (4), the extensive-margin analysis relates the change in labour market participation to the change in the log of total after-tax income, as well as the change in spousal labour market participation. Specifically, the estimating equation is:

Δ1( z it >0 )= ι 0 + κ 0 Δln( Y it T it )+ μ 0 Δ1( z st >0 )+ X it ' ϕ 0 +f[ ln( z i,t1 ) ]+ ν it    (6) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaaGymamaabmaapaqaa8qacaWG6bWdamaaBaaaleaapeGa amyAaiaadshaa8aabeaak8qacqGH+aGpcaaIWaaacaGLOaGaayzkaa Gaeyypa0JaeqyUdK2damaaBaaaleaapeGaaGimaaWdaeqaaOWdbiab gUcaRiabeQ7aR9aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaqGuo GaciiBaiaac6gadaqadaWdaeaapeGaamywa8aadaWgaaWcbaWdbiaa dMgacaWG0baapaqabaGcpeGaeyOeI0Iaamiva8aadaWgaaWcbaWdbi aadMgacaWG0baapaqabaaak8qacaGLOaGaayzkaaGaey4kaSIaeqiV d02damaaBaaaleaapeGaaGimaaWdaeqaaOWdbiaabs5acaaIXaWaae Waa8aabaWdbiaadQhapaWaaSbaaSqaa8qacaWGZbGaamiDaaWdaeqa aOWdbiabg6da+iaaicdaaiaawIcacaGLPaaacqGHRaWkcaWGybWdam aaDaaaleaapeGaamyAaiaadshaa8aabaWdbiaacEcaaaGccqaHvpGz paWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaey4kaSIaamOzamaadm aapaqaa8qaciGGSbGaaiOBamaabmaapaqaa8qacaWG6bWdamaaBaaa leaapeGaamyAaiaacYcacaWG0bGaeyOeI0IaaGymaaWdaeqaaaGcpe GaayjkaiaawMcaaaGaay5waiaaw2faaiabgUcaRiabe27aU9aadaWg aaWcbaWdbiaadMgacaWG0baapaqabaaaaa@78B2@

where these variables are defined above. The expectation is that an increase in the after-tax income of the individual makes leisure more affordable and reduces the incidence of being employed, κ ^ 0 <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqOUdS2dayaajaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaeyip aWJaaGimaaaa@3ABA@ . While mean-reversion and distributional changes in income are less of an issue in the extensive-margin analysis, a 10-piece spline for lagged labour income is still included, as described in the Table 2 notes.

An implicit assumption of the empirical model is that hourly wages are uncorrelated with the change in taxes, so that the full burden of an income tax reform operates through its effect on time spent working. As in Gelber (2014), this analysis partly addresses concerns regarding this assumption by controlling for an array of job-specific characteristics in the intensive-margin analysis, including indicators for union status, EPP coverage, and sector of employment, although excluding these variables would have no qualitative effect on the results.

3.4 Instruments

The instruments for the changes in the net-of-tax shares and total after-tax income are predicted tax measures, calculated by estimating how tax liabilities would have changed due to the reform holding everything else constant. Denote π t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiWda3damaaBaaaleaapeGaamiDaaWdaeqaaaaa@391C@  as the parameter governing pension income splitting and Π ˜ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafuiOda1dayaaiaWaaSbaaSqaa8qacaWG0baapaqabaaaaa@38EC@  as the vector of all other tax parameters. In addition, for each j{ i,s } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiabgIGiopaacmaapaqaa8qacaWGPbGaaiilaiaadohaaiaa wUhacaGL9baaaaa@3D65@ , denote Y ^ jt = Y jt + s ^ jt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gabmywa8aagaqcamaaBaaaleaapeGaamOAaiaadshaa8aabeaak8qa cqGH9aqpcaWGzbWdamaaBaaaleaapeGaamOAaiaadshaa8aabeaak8 qacqGHRaWkceWGZbWdayaajaWaaSbaaSqaa8qacaWGQbGaamiDaaWd aeqaaaaa@41C2@  as the pre-reform total income ( Y jt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamywa8aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaaaaa@392C@  ) net of a predicted amount that would have been split ( s ^ jt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gabm4Ca8aagaqcamaaBaaaleaapeGaamOAaiaadshaa8aabeaaaaa@3956@  ) in this period had this practice been permissible, where s ^ jt >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gabm4Ca8aagaqcamaaBaaaleaapeGaamOAaiaadshaa8aabeaak8qa cqGH+aGpcaaIWaaaaa@3B32@  if individual j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaaaa@36FB@  is the transferee and s ^ jt <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gabm4Ca8aagaqcamaaBaaaleaapeGaamOAaiaadshaa8aabeaak8qa cqGH8aapcaaIWaaaaa@3B2E@  if j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaaaa@36FB@  is the pensioner. In other words, ( Y ^ jt Y jt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gabmywa8aagaqcamaaBaaaleaapeGaamOAaiaadshaa8aabeaak8qa cqGHsislcaWGzbWdamaaBaaaleaapeGaamOAaiaadshaa8aabeaaaa a@3D63@  ) is the predicted change in total income around the introduction of pension income splitting resulting exclusively from this tax reform; the process for calculating s ^ jt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gabm4Ca8aagaqcamaaBaaaleaapeGaamOAaiaadshaa8aabeaaaaa@3956@  is described below. Taken together, the instruments for the changes in the marginal net-of-tax share and total after-tax income are:

Δln ( 1 τ jt ) IV =ln( 1 T( Ψ ^ jt ; π t+1 , Π ˜ t ) z jt )ln( 1 T( Ψ jt ; π t , Π ˜ t ) z jt )   (7) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaciiBaiaac6gadaqadaWdaeaapeGaaGymaiabgkHiTiab es8a09aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaaak8qacaGLOa GaayzkaaWdamaaCaaaleqabaWdbiaadMeacaWGwbaaaOGaeyypa0Ja ciiBaiaac6gadaqadaWdaeaapeGaaGymaiabgkHiTmaalaaapaqaa8 qacqGHciITcaWGubWaaeWaa8aabaWdbiqbfI6az9aagaqcamaaBaaa leaapeGaamOAaiaadshaa8aabeaak8qacaGG7aGaeqiWda3damaaBa aaleaapeGaamiDaiabgUcaRiaaigdaa8aabeaak8qacaGGSaGafuiO da1dayaaiaWaaSbaaSqaa8qacaWG0baapaqabaaak8qacaGLOaGaay zkaaaapaqaa8qacqGHciITcaWG6bWdamaaBaaaleaapeGaamOAaiaa dshaa8aabeaaaaaak8qacaGLOaGaayzkaaGaeyOeI0IaciiBaiaac6 gadaqadaWdaeaapeGaaGymaiabgkHiTmaalaaapaqaa8qacqGHciIT caWGubWaaeWaa8aabaWdbiabfI6az9aadaWgaaWcbaWdbiaadQgaca WG0baapaqabaGcpeGaai4oaiabec8aW9aadaWgaaWcbaWdbiaadsha a8aabeaak8qacaGGSaGafuiOda1dayaaiaWaaSbaaSqaa8qacaWG0b aapaqabaaak8qacaGLOaGaayzkaaaapaqaa8qacqGHciITcaWG6bWd amaaBaaaleaapeGaamOAaiaadshaa8aabeaaaaaak8qacaGLOaGaay zkaaaaaa@79B5@

Δln ( Y jt T jt ) IV =ln( Y ^ jt T( Ψ ^ jt ; π t+1 , Π ˜ t ) )ln( Y jt T( Ψ jt ; π t , Π ˜ t ) )   (8) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaciiBaiaac6gadaqadaWdaeaapeGaamywa8aadaWgaaWc baWdbiaadQgacaWG0baapaqabaGcpeGaeyOeI0Iaamiva8aadaWgaa WcbaWdbiaadQgacaWG0baapaqabaaak8qacaGLOaGaayzkaaWdamaa CaaaleqabaWdbiaadMeacaWGwbaaaOGaeyypa0JaciiBaiaac6gada qadaWdaeaapeGabmywa8aagaqcamaaBaaaleaapeGaamOAaiaadsha a8aabeaak8qacqGHsislcaWGubWaaeWaa8aabaWdbiqbfI6az9aaga qcamaaBaaaleaapeGaamOAaiaadshaa8aabeaak8qacaGG7aGaeqiW da3damaaBaaaleaapeGaamiDaiabgUcaRiaaigdaa8aabeaak8qaca GGSaGafuiOda1dayaaiaWaaSbaaSqaa8qacaWG0baapaqabaaak8qa caGLOaGaayzkaaaacaGLOaGaayzkaaGaeyOeI0IaciiBaiaac6gada qadaWdaeaapeGaamywa8aadaWgaaWcbaWdbiaadQgacaWG0baapaqa baGcpeGaeyOeI0Iaamivamaabmaapaqaa8qacqqHOoqwpaWaaSbaaS qaa8qacaWGQbGaamiDaaWdaeqaaOWdbiaacUdacqaHapaCpaWaaSba aSqaa8qacaWG0baapaqabaGcpeGaaiilaiqbfc6aq9aagaacamaaBa aaleaapeGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaaaGaayjkaiaa wMcaaaaa@736D@

where Ψ ^ jt ={ Y ^ it , Y ^ st , X it , X st } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GabeiQd8aagaqcamaaBaaaleaapeGaamOAaiaadshaa8aabeaak8qa cqGH9aqpdaGadaWdaeaapeGabmywa8aagaqcamaaBaaaleaapeGaam yAaiaadshaa8aabeaak8qacaGGSaGabmywa8aagaqcamaaBaaaleaa peGaam4Caiaadshaa8aabeaak8qacaGGSaGaamiwa8aadaWgaaWcba WdbiaadMgacaWG0baapaqabaGcpeGaaiilaiaadIfapaWaaSbaaSqa a8qacaWGZbGaamiDaaWdaeqaaaGcpeGaay5Eaiaaw2haaaaa@4C22@ . Equations (7) and (8) show that the only factor for each instrument that changes within individuals over time is due to pension income splitting, namely as a result of the reform’s introduction ( π t π t+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiWda3damaaBaaaleaapeGaamiDaaWdaeqaaOWdbiabgkziUkab ec8aW9aadaWgaaWcbaWdbiaadshacqGHRaWkcaaIXaaapaqabaaaaa@3FD0@  ) and the amount of pension income predicted to be sent or received resulting from the reform ( Y jt Y ^ jt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamywa8aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaGcpeGaeyOK H4Qabmywa8aagaqcamaaBaaaleaapeGaamOAaiaadshaa8aabeaaaa a@3E63@  ). The variation across individuals comes from the differential effects of the reform on transferees versus pensioners, age groups, and whether the pension income is derived from EPPs. Exploiting policy-induced variation in net-of-tax shares is necessary to estimate the elasticity of taxable labour income consistently because of tax progressivity. Since the marginal tax rate increases with income, the ordinary least squares (OLS) estimator is biased downwards. Figure 1 shows this downward bias, as well as a reduced-form look at how the instrument overcomes this problem.

The process for predicting the amount of pension income split ( s ^ jt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gabm4Ca8aagaqcamaaBaaaleaapeGaamOAaiaadshaa8aabeaaaaa@3956@  ) is based on two assumptions. First, although eligibility to split pension income depends on whether this income is derived from EPPs, the tax data do not reveal such information. To address this issue, sources of pension income are inferred using the longitudinal component of the data: workers who were observed with a positive EPP contribution from 1991 to 2006 at least once during normal working years (defined here as 54 years of age or younger) are assumed to draw all their pension income from an EPP.Note  All other pension recipients are assumed to be drawing from other private savings accounts and are, therefore, subject to the age restriction for splitting.Note  Second, it is assumed that the person in each family with the highest pension income always transfers 50% of this income in order to predict changes in taxes resulting from the reform. Splitting a lesser amount may be optimal in practice, especially when spouses’ incomes are similar. However, predicting splitting using an optimization approach is difficult to implement in practice, and would require a strong assumption that all taxfilers are sufficiently knowledgeable about the tax system to optimize correctly. Imposing that 50% is always transferred avoids any behavioural assumptions and is correlated with the reform’s true effect since this method is strictly based on a maximum eligibility rule.Note 

Figure 1 Graphical inspection of the net-of-tax shares, actual and instruments

Description for Figure 1

The title of Figure 1 is “Graphical inspection of the net-of-tax shares, actual and instruments.”

This figure contains two charts (Panels A and B), which are described below.

Chart title: Panel A: Marginal net-of-tax share

This chart is a combination of a scatter plot and a line chart.
The vertical axis (y-axis) title is “percentage change in labour income.”
The axis graduation begins at -100 and ends at 100, with tick marks at every 50 units.
The horizontal axis (x-axis) title is “Percentage change in the marginal net-of-tax share.”
The axis graduation begins at -50 and ends at 50, with tick marks at every 25 units.
There are two series in this chart: one is called “Actual” and the other, “Predicted.”

Chart title: Panel B: Average net-of-tax share

This chart is a combination of a scatter plot and a line chart.
The vertical axis (y-axis) title is “percentage change in labour income.”
The axis graduation begins at -100 and ends at 100, with tick marks at every 50 units.
The horizontal axis (x-axis) title is “Percentage change in the average net-of-tax share.”
The axis graduation begins at -20 and ends at 20, with tick marks at every 10 units.
There are two series in this chart: one is called “Actual” and the other, “Predicted.”

Notes: The relationship between labour income and the marginal (Panel A) and average (Panel B) net-of-tax shares are shown. Each dot corresponds to a unique observation, reflecting the individual’s percentage changes in labour income and actual net-of-tax share. The kernel-weighted local polynomial smoothing of these relationships are also shown, for both the actual and predicted net-of-tax shares.

Source: Statistics Canada, Longitudinal Administrative Databank.

To illustrate the actual and predicted variations in the net-of-tax shares and total after-tax income due to splitting, Charts 1 and 2 show the distributions of individuals who experienced different changes in each variable among couples with at least one pensioner, based on whether the individual was predicted to be the transferee or pensioner. These charts illustrate that there are large shares of individuals who experienced both decreases and increases in the tax measures owing to the tax reform. On balance, the instruments reasonably approximate the actual changes in taxes around the time that pension income splitting was introduced. However, the mass of individuals who experienced no change in each instrument is disproportionately large relative to the corresponding actual variable given that no splitting is predicted to occur if: 1) individuals and spouses have equal pension incomes; or 2) the age and EPP requirements for splitting eligibility are not met. The predicted after-tax income is shown to be non-decreasing for transferees and non-increasing for pensioners, which occurs by design. As a result, the marginal and average net-of-tax shares tend to decrease by a lesser amount than the actual declines in these variables for transferees, while the opposite is true for pensioners.

Description of Chart 1 Mechanical variation in the tax variables, among predicted transferees

Data table for Chart 1
Data table for Chart 1
Table summary
This table displays the results of Data table for Chart 1 Less than -15.00, -15.00 to -5.01, -5.00 to -0.01, 0.00, 0.01 to 5.00, 5.01 to 15.00 and Greater than 15.00 (appearing as column headers).
Less than -15.00 -15.00 to -5.01 -5.00 to -0.01 0.00 0.01 to 5.00 5.01 to 15.00 Greater than 15.00
Marginal net-of-tax share
Actual 26.42 21.00 11.63 3.07 25.81 6.57 5.50
Predicted 11.86 25.54 16.27 26.52 15.78 3.43 0.60
Average net-of-tax share
Actual 21.56 19.30 21.62 3.14 24.19 8.00 2.19
Predicted 6.63 20.71 40.90 7.08 22.93 0.92 0.83
Total after-tax income
Actual 10.81 5.32 5.15 0.10 11.42 14.11 53.09
Predicted 0.00 0.00 0.00 3.73 7.03 13.40 75.85

Description of Chart 2 Mechanical variation in the tax variables, among predicted pensioners

Data table for Chart 2
Data table for Chart 2
Table summary
This table displays the results of Data table for Chart 2 Less than -15.00, -15.00 to -5.01, -5.00 to -0.01, 0.00, 0.01 to 5.00, 5.01 to 15.00 and Greater than 15.00 (appearing as column headers).
Less than -15.00 -15.00 to -5.01 -5.00 to -0.01 0.00 0.01 to 5.00 5.01 to 15.00 Greater than 15.00
Marginal net-of-tax share
Actual 3.83 7.00 15.89 1.12 26.57 20.01 25.57
Predicted 0.70 4.18 17.50 22.01 22.36 25.65 7.60
Average net-of-tax share
Actual 0.90 5.16 17.05 0.92 34.52 26.48 14.96
Predicted 0.04 0.08 17.08 4.50 50.74 27.19 0.38
Total after-tax income
Actual 36.60 13.67 9.22 0.06 15.00 12.30 13.15
Predicted 72.85 15.63 8.25 3.26 0.00 0.00 0.00

4 Labour income responses to tax rates

The regression estimates of individuals’ labour income responses to changes in their net-of-tax shares and total after-tax income are presented in this section, based on empirical model specifications derived from Equations (2) and (4) of the theoretical framework. To estimate these responses consistently, the predicted tax measures defined in Equations (7) and (8) are used as instruments in a two-stage least squares (2SLS) approach.

4.1 Primary results

In Table 2, the first-stage effects of the instruments for the marginal net-of-tax share and total after-tax income on the endogenous regressors in Equations (5) and (6) are shown, for both the intensive-margin analysis in Panel A and the extensive-margin analysis in Panel B. As expected, these results indicate that the instruments are strong predictors of the true variation in taxes around the time of the pension income splitting reform, with large F-statistics for the tests of excluded instruments. Each regression controls for many observed characteristics across both individuals and their spouses, including age and spousal age fixed effects, sex, marital status, immigrant status, and province of residence; log values of family income from capital gains, investments, Employment Insurance, and social assistance; and log values of family tax allowances for disability and medical expenses. In addition, several job characteristics of the individual are controlled for in the intensive-margin analysis: indicators of union status, EPP coverage status, and sector of employment based on the 2-digit North American Industrial Classification System (NAICS) code. Throughout the analysis, standard errors are always clustered by individual.

Table 2
First-stage inspection of the instruments
Table summary
This table displays the results of First-stage inspection of the instruments Uncompensated effect, Compensated effect, Income effect, Marginal net-of-tax share and Total after-tax income, calculated using estimates units of measure (appearing as column headers).
Uncompensated effect Compensated effect Income effect
Marginal net-of-tax share Marginal net-of-tax share Total after-tax income Total after-tax income
estimates
Panel A: Intensive margin
Coefficient estimates
Predicted marginal net-of-tax share 0.669Note *** 0.531Note *** -0.136Note *** Note ...: not applicable
Predicted total after-tax income Note ...: not applicable -0.127Note *** 0.570Note *** Note ...: not applicable
F-statistic 3,782.96Note *** 3,730.15Note *** 1,801.85Note *** Note ...: not applicable
Panel B: Extensive margin
Coefficient estimates
Predicted marginal net-of-tax share 0.734Note *** 0.649Note *** -0.389Note *** Note ...: not applicable
Predicted total after-tax income Note ...: not applicable -0.070Note *** 0.706Note *** 0.731Note ***
F-statistic 27,607.80Note *** 27,362.69Note *** 11,077.97Note *** 16,810.34Note ***

The second-stage regression results for the intensive-margin analysis are shown in Table 3 using both OLS and IV (reduced-form and 2SLS) estimators. The downward bias of the OLS estimator for the marginal net-of-tax share caused by tax progressivity is apparent. In addition, there is significant upward bias in the OLS estimates of the effects of total after-tax income on labour earnings, which arises because individuals with higher labour earnings are more likely to also have higher total income. The predicted measure of total after-tax income effectively and consistently corrects for this bias. Along the intensive margin, in Panels A and B of Table 3, the uncompensated regression estimates—not instrumenting for Δln( Y it T it ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiaabYgacaqGUbWaaeWaa8aabaWdbiaadMfapaWaaSbaaSqa a8qacaWGPbGaamiDaaWdaeqaaOWdbiabgkHiTiaadsfapaWaaSbaaS qaa8qacaWGPbGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaaaaa@4208@  using the corresponding predicted variable—suggest that individuals respond meaningfully to changes in their marginal net-of-tax shares. Specifically, each 1% increase in the net-of-tax share appears to induce a 1.004% to 1.036% increase in labour income, on average, depending on whether the lagged labour income controls are included. However, controlling for income effects of the tax reform by instrumenting for the change in total after-tax income using Δln ( Y it T it ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiGacYgacaGGUbWaaeWaa8aabaWdbiaadMfapaWaaSbaaSqa a8qacaWGPbGaamiDaaWdaeqaaOWdbiabgkHiTiaadsfapaWaaSbaaS qaa8qacaWGPbGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaa8aadaah aaWcbeqaa8qacaWGjbGaamOvaaaaaaa@4401@  absorbs this effect—a finding that is consistent with previous studies. Hence, these results ultimately indicate that individuals’ labour income decisions are unresponsive to exogenous changes in their marginal tax rates.

Table 3
Intensive margin—Labour income responses to changes in the marginal net-of-tax share and total after-tax income
Table summary
This table displays the results of Intensive margin—Labour income responses to changes in the marginal net-of-tax share and total after-tax income Ordinary least squares, Instrumental variables, Reduced form and Two-stage least squares, calculated using coefficient estimates units of measure (appearing as column headers).
Ordinary least squares Instrumental variables
Reduced form Two-stage least squares
coefficient estimates
Panel A: No lagged income controls
Uncompensated effect
Marginal net-of-tax share -0.884Note *** 0.693Note *** 1.036Note ***
Compensated effect
Marginal net-of-tax share -0.884Note *** 0.010 0.015
Total after-tax income 0.515Note *** -0.014 -0.021
Panel B: 10-piece spline for the one-period lagged labour income
Uncompensated effect
Marginal net-of-tax share -0.890Note *** 0.671Note *** 1.004Note ***
Compensated effect
Marginal net-of-tax share -0.890Note *** -0.023 -0.054
Total after-tax income 0.509Note *** -0.016 -0.041

In contrast, Table 4 shows that each 1% increase in total after-tax income reduces the likelihood that individuals are employed by 1.6 percentage points, on average. This finding indicates that older individuals do respond to tax incentives along the extensive margin. Since the probability that individuals are employed in the pre-reform period is 62.9%, as shown in Table 1, the implied elasticity of employment to total after-tax income is ΔPr( z it >0 )/Pr( z it >0)=0.016/0.692=0.025 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiGaccfacaGGYbWaaeWaa8aabaWdbiaadQhapaWaaSbaaSqa a8qacaWGPbGaamiDaaWdaeqaaOWdbiabg6da+iaaicdaaiaawIcaca GLPaaacaGGVaGaaeiuaiaabkhacaGGOaGaamOEa8aadaWgaaWcbaWd biaadMgacaWG0baapaqabaGcpeGaeyOpa4JaaGimaiaacMcacqGH9a qpcqGHsislcaaIWaGaaiOlaiaaicdacaaIXaGaaGOnaiaac+cacaaI WaGaaiOlaiaaiAdacaaI5aGaaGOmaiabg2da9iabgkHiTiaaicdaca GGUaGaaGimaiaaikdacaaI1aaaaa@582C@ . As the model predicts, the change in after-tax income is the only relevant factor in explaining labour market participation, whereas the marginal net-of-tax share is statistically insignificant in the preferred model specification that controls for lagged labour income.

Table 4
Extensive margin—Employment responses to changes in the marginal net-of-tax share and total after-tax income
Table summary
This table displays the results of Extensive margin—Employment responses to changes in the marginal net-of-tax share and total after-tax income Ordinary least squares, Instrumental variables, Reduced form and Two-stage least squares, calculated using coefficient estimates units of measure (appearing as column headers).
Ordinary least squares Instrumental variables
Reduced form Two-stage least squares
coefficient estimates
Panel A: No lagged income controls
Uncompensated effect
Marginal net-of-tax share -0.691Note *** 0.177Note *** 0.245Note ***
Compensated effect
Marginal net-of-tax share -0.691Note *** 0.057Note *** 0.094Note ***
Total after-tax income 0.011Note *** -0.001 0.009Note **
Income effect
Total after-tax income 0.050Note *** -0.004Note *** -0.006Note ***
Panel B: 10-piece spline for the one-period lagged labour income
Uncompensated effect
Marginal net-of-tax share -0.675Note *** 0.149Note *** 0.202Note ***
Compensated effect
Marginal net-of-tax share -0.675Note *** 0.011 0.009
Total after-tax income 0.007Note *** -0.011Note *** -0.014Note ***
Income effect
Total after-tax income -0.045Note *** -0.011Note *** -0.016Note ***

4.2 Average versus marginal net-of-tax shares

There are several reasons why workers’ labour income may be unresponsive to a change in the marginal price of labour especially in the short run, such as inflexible employment contracts or work preferences. Since the demand for work diminishes as workers age, and retirement or partial retirement become viable substitutes for labour market participation, the ex ante expectation is for older workers to be at least as responsive as young and middle-aged workers to changes in tax rates. However, under-responsiveness may stem from behavioural factors including an imperfect understanding of the tax system. As Milligan (2009) shows, marginal income tax schedules in Canada are complex functions of taxfilers’ province of residence, demographics, sources of income, and a plethora of other factors determined at the federal and provincial levels. For this reason, individuals may struggle to know their true marginal tax rates at the time of making labour decisions, especially when some sources of income are uncertain.

Liebman and Zeckhauser (2004) have posited that, when nonlinear price schedules are complex and knowing what marginal price is actually faced is difficult, individuals may use the average price as a proxy to help guide their decision-making. Figure 2 plots the typical marginal and average income tax schedules for the sample of individuals included in this study over a range of labour income from $0 to $60,000. The marginal tax rate exceeds the average tax rate due to progressivity, and the marginal tax rate has clear discontinuities at various points where specific features of the tax code and social programs start or stop taking effect, whereas the average tax rate is a smooth function of labour income.

 

Figure 2 Marginal and average tax rate schedules, illustrated

Description for Figure 2

The title of Figure 2 is “Marginal and average tax rate schedules, illustrated.”

This figure is a line chart.
The vertical axis (y-axis) title is “percent.”
The axis graduation begins at 0 and ends at 40, with tick marks at every 10 units.
The horizontal axis (x-axis) title is “Labour income (dollars).”
The axis graduation begins at 0 and ends at 60,000, with tick marks at every 20,000.
There are two series in this chart: one is called “Marginal tax rate” and the other, “Average tax rate.”

There are two vertical broken lines in the chart, extending from the x-axis to the last graduation in the y-axis.
The first one, located at $3,500 on the x-axis, is called “CPP/QPP basic exemption.
The second one, starting at $9,039 on the x-axis, is called “Federal basic exemption.”
The third one, starting at $36,378 on the x-axis, is called “Second federal tax bracket.”
The fourth one, starting at $42,100 on the x-axis, is called “CPP/QPP YMPE” (Canada Pension Plan and Quebec Pension Plan Year’s Maximum Pensionable Earnings).

Notes: The marginal and average tax rate schedules are shown for the relevant sample of taxfilers, among those earning from $1 to $60,000 in 2006. Individuals were binned according to income in $250 increments and the average values of the tax rates within each bin are shown. Several income thresholds are illustrated: the point where the Canada Pension Plan and Quebec Pension Plan (CPP/QPP) social program contributions begin ($3,500 in 2006); the federal basic income tax exemption ($9,039 in 2006); the start of the second federal income-tax bracket ($36,378 in 2006); and the CPP/QPP Year’s Maximum Pensionable Earnings threshold ($42,100 in 2006) at which point the CPP/QPP marginal contribution rate falls to zero. These program features result in clear discontinuities in effective marginal tax rates, whereas average effective tax rates remain smooth through the income thresholds.

Source: Statistics Canada, Longitudinal Administrative Databank.

To determine whether individuals are more responsive to the average or marginal net-of-tax share along the intensive margin, the approach from Ito (2014) is used of performing an encompassing test of these alternative price measures. For each j{ i,s } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOAaiabgIGiopaacmaapaqaa8qacaWGPbGaaiilaiaadohaaiaa wUhacaGL9baaaaa@3D65@ , denote τ jt a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiXdq3damaaDaaaleaapeGaamOAaiaadshaa8aabaWdbiaadgga aaaaaa@3B0A@  as the average net-of-tax share for individual j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGQbaaaa@3673@  at time t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@367D@ . The change in the log of the average net-of-tax share is Δln( 1 τ jt a )=ln( 1 T jt / Y jt )ln( 1 T j,t1 / Y j,t1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiGacYgacaGGUbWaaeWaa8aabaWdbiaaigdacqGHsislcqaH epaDpaWaa0baaSqaa8qacaWGQbGaamiDaaWdaeaapeGaamyyaaaaaO GaayjkaiaawMcaaiabg2da9iGacYgacaGGUbWaaeWaa8aabaWdbiaa igdacqGHsislcaWGubWdamaaBaaaleaapeGaamOAaiaadshaa8aabe aak8qacaGGVaGaamywa8aadaWgaaWcbaWdbiaadQgacaWG0baapaqa baaak8qacaGLOaGaayzkaaGaeyOeI0IaciiBaiaac6gadaqadaWdae aapeGaaGymaiabgkHiTiaadsfapaWaaSbaaSqaa8qacaWGQbGaaiil aiaadshacqGHsislcaaIXaaapaqabaGcpeGaai4laiaadMfapaWaaS baaSqaa8qacaWGQbGaaiilaiaadshacqGHsislcaaIXaaapaqabaaa k8qacaGLOaGaayzkaaaaaa@60B1@ , and the instrument for this endogenous regressor is given by:

Δln ( 1 τ jt a ) IV =ln( 1 T( Ψ ^ jt ; π t+1 , Π ˜ t ) Y ^ jt )ln( 1 T( Ψ jt ; π t , Π ˜ t ) Y jt )   (9) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiGacYgacaGGUbWaaeWaa8aabaWdbiaaigdacqGHsislcqaH epaDpaWaa0baaSqaa8qacaWGQbGaamiDaaWdaeaapeGaamyyaaaaaO GaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaWGjbGaamOvaaaakiab g2da9iGacYgacaGGUbWaaeWaa8aabaWdbiaaigdacqGHsisldaWcaa WdaeaapeGaamivamaabmaapaqaa8qaceqGOoWdayaajaWaaSbaaSqa a8qacaWGQbGaamiDaaWdaeqaaOWdbiaacUdacqaHapaCpaWaaSbaaS qaa8qacaWG0bGaey4kaSIaaGymaaWdaeqaaOWdbiaacYcaceqGGoWd ayaaiaWaaSbaaSqaa8qacaWG0baapaqabaaak8qacaGLOaGaayzkaa aapaqaa8qaceWGzbWdayaajaWaaSbaaSqaa8qacaWGQbGaamiDaaWd aeqaaaaaaOWdbiaawIcacaGLPaaacqGHsislciGGSbGaaiOBamaabm aapaqaa8qacaaIXaGaeyOeI0YaaSaaa8aabaWdbiaadsfadaqadaWd aeaapeGaaeiQd8aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaGcpe Gaai4oaiabec8aW9aadaWgaaWcbaWdbiaadshaa8aabeaak8qacaGG SaGabeiOd8aagaacamaaBaaaleaapeGaamiDaaWdaeqaaaGcpeGaay jkaiaawMcaaaWdaeaapeGaamywa8aadaWgaaWcbaWdbiaadQgacaWG 0baapaqabaaaaaGcpeGaayjkaiaawMcaaaaa@7315@

The estimating equation assesses the extent to which average and marginal net-of-tax shares independently affect labour income. Specifically, the statistical model for the intensive-margin analysis is given by:

Δln( z it )= α 1 + β 1 Δln( 1 τ it )+ β 1 a Δln( 1 τ it a )+ γ 1 Δln( Y it T it )+ ζ 1 Δln( z st )    (10) + X it ' θ 1 +f[ ln( z i,t1 ) ]+ η it MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqqHuoarciGGSbGaaiOBamaabmaapaqaa8qacaWG6bWdamaa BaaaleaapeGaamyAaiaadshaa8aabeaaaOWdbiaawIcacaGLPaaacq GH9aqpcqaHXoqypaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4k aSIaeqOSdi2damaaBaaaleaapeGaaGymaaWdaeqaaOGaeuiLdq0dbi GacYgacaGGUbWaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWa aSbaaSqaa8qacaWGPbGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaai abgUcaRiabek7aI9aadaqhaaWcbaWdbiaaigdaa8aabaWdbiaadgga aaGcpaGaeuiLdq0dbiGacYgacaGGUbWaaeWaa8aabaWdbiaaigdacq GHsislcqaHepaDpaWaa0baaSqaa8qacaWGPbGaamiDaaWdaeaapeGa amyyaaaaaOGaayjkaiaawMcaaiabgUcaRiabeo7aN9aadaWgaaWcba Wdbiaaigdaa8aabeaakiabfs5ae9qaciGGSbGaaiOBamaabmaapaqa a8qacaWGzbWdamaaBaaaleaapeGaamyAaiaadshaa8aabeaak8qacq GHsislcaWGubWdamaaBaaaleaapeGaamyAaiaadshaa8aabeaaaOWd biaawIcacaGLPaaacqGHRaWkcqaH2oGEpaWaaSbaaSqaa8qacaaIXa aapaqabaGccqqHuoarpeGaciiBaiaac6gadaqadaWdaeaapeGaamOE a8aadaWgaaWcbaWdbiaadohacaWG0baapaqabaaak8qacaGLOaGaay zkaaaabaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRaWkcaWGyb WdamaaDaaaleaapeGaamyAaiaadshaa8aabaWdbiaacEcaaaGccqaH 4oqCpaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaey4kaSIaamOzam aadmaapaqaa8qaciGGSbGaaiOBamaabmaapaqaa8qacaWG6bWdamaa BaaaleaapeGaamyAaiaacYcacaWG0bGaeyOeI0IaaGymaaWdaeqaaa GcpeGaayjkaiaawMcaaaGaay5waiaaw2faaiabgUcaRiabeE7aO9aa daWgaaWcbaWdbiaadMgacaWG0baapaqabaaaaaa@C375@

Note that Δln( 1 τ it ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiGacYgacaGGUbWaaeWaa8aabaWdbiaaigdacqGHsislcqaH epaDpaWaaSbaaSqaa8qacaWGPbGaamiDaaWdaeqaaaGcpeGaayjkai aawMcaaaaa@407A@ , Δln( 1 τ it a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiGacYgacaGGUbWaaeWaa8aabaWdbiaaigdacqGHsislcqaH epaDpaWaa0baaSqaa8qacaWGPbGaamiDaaWdaeaapeGaamyyaaaaaO GaayjkaiaawMcaaaaa@4161@  and Δln( Y it T it ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaciiBaiaac6gadaqadaWdaeaapeGaamywa8aadaWgaaWc baWdbiaadMgacaWG0baapaqabaGcpeGaeyOeI0Iaamiva8aadaWgaa WcbaWdbiaadMgacaWG0baapaqabaaak8qacaGLOaGaayzkaaaaaa@4258@  are separately identified in this environment because of tax progressivity and the fact that both Y jt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamywa8aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaaaaa@392C@  and T jt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiva8aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaaaaa@3927@  are varying exogenously in the instruments due to the tax reform, as shown graphically in Figure 3 and derived formally in the Appendix. To the extent that individuals understand the income tax schedule and the lack of response to the marginal net-of-tax share correctly indicates that labour income is unresponsive to changes in tax rates, the expectation is that β ^ 1 = β ^ 1 a =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqOSdi2dayaajaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyyp a0JafqOSdi2dayaajaWaa0baaSqaa8qacaaIXaaapaqaa8qacaWGHb aaaOGaeyypa0JaaGimaaaa@3F79@ . In contrast, the schmeduling hypothesis predicts that β ^ 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqOSdi2dayaajaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyyp a0JaaGimaaaa@3AAC@  and β ^ 1 a >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqOSdi2dayaajaWaa0baaSqaa8qacaaIXaaapaqaa8qacaWGHbaa aOGaeyOpa4JaaGimaaaa@3B95@ .

For expositional purposes, an encompassing test is also performed for the extensive-margin analysis. The corresponding statistical model in this case is the following:

Δ1( z it >0 )= ι 1 + ϑ 1 Δln( 1 τ it )+ ϑ 1 a Δln( 1 τ it a )+ κ 1 Δln( Y it T it )+ μ 1 Δ1( z st >0 )    (11) + X it ' ϕ 1 +f[ ln( z i,t1 ) ]+ ϱ it MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqqHuoarcaaIXaWaaeWaa8aabaWdbiaadQhapaWaaSbaaSqa a8qacaWGPbGaamiDaaWdaeqaaOWdbiabg6da+iaaicdaaiaawIcaca GLPaaacqGH9aqpcqaH5oqApaWaaSbaaSqaa8qacaaIXaaapaqabaGc peGaey4kaSIaeqy0dO0damaaBaaaleaapeGaaGymaaWdaeqaaOGaeu iLdq0dbiGacYgacaGGUbWaaeWaa8aabaWdbiaaigdacqGHsislcqaH epaDpaWaaSbaaSqaa8qacaWGPbGaamiDaaWdaeqaaaGcpeGaayjkai aawMcaaiabgUcaRiabeg9ak9aadaqhaaWcbaWdbiaaigdaa8aabaWd biaadggaaaGcpaGaeuiLdq0dbiGacYgacaGGUbWaaeWaa8aabaWdbi aaigdacqGHsislcqaHepaDpaWaa0baaSqaa8qacaWGPbGaamiDaaWd aeaapeGaamyyaaaaaOGaayjkaiaawMcaaiabgUcaRiabeQ7aR9aada WgaaWcbaWdbiaaigdaa8aabeaak8qacaqGuoGaciiBaiaac6gadaqa daWdaeaapeGaamywa8aadaWgaaWcbaWdbiaadMgacaWG0baapaqaba GcpeGaeyOeI0Iaamiva8aadaWgaaWcbaWdbiaadMgacaWG0baapaqa baaak8qacaGLOaGaayzkaaGaey4kaSIaeqiVd02damaaBaaaleaape GaaGymaaWdaeqaaOWdbiaabs5acaaIXaWaaeWaa8aabaWdbiaadQha paWaaSbaaSqaa8qacaWGZbGaamiDaaWdaeqaaOWdbiabg6da+iaaic daaiaawIcacaGLPaaaaeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlabgUcaRiaadIfapaWaa0baaSqaa8qacaWG PbGaamiDaaWdaeaapeGaai4jaaaakiabew9aM9aadaWgaaWcbaWdbi aaigdaa8aabeaak8qacqGHRaWkcaWGMbWaamWaa8aabaWdbiGacYga caGGUbWaaeWaa8aabaWdbiaadQhapaWaaSbaaSqaa8qacaWGPbGaai ilaiaadshacqGHsislcaaIXaaapaqabaaak8qacaGLOaGaayzkaaaa caGLBbGaayzxaaGaey4kaSYefv3ySLgznfgDOfdaryqr1ngBPrginf gDObYtUvgaiuGacqWFXpq8paWaaSbaaSqaa8qacaWGPbGaamiDaaWd aeqaaaaaaa@D32E@

The encompassing test of an extensive-margin response serves as a useful placebo test of whether individuals may respond to changes in their average tax rates as a proxy for their marginal rates, or whether this is simply due to an income effect of the tax reform given that Δln ( 1 τ it a ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaciiBaiaac6gadaqadaWdaeaapeGaaGymaiabgkHiTiab es8a09aadaqhaaWcbaWdbiaadMgacaWG0baapaqaa8qacaWGHbaaaa GccaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaadMeacaWGwbaaaaaa @43A2@  and Δln ( Y it T it ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiGacYgacaGGUbWaaeWaa8aabaWdbiaadMfapaWaaSbaaSqa a8qacaWGPbGaamiDaaWdaeqaaOWdbiabgkHiTiaadsfapaWaaSbaaS qaa8qacaWGPbGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaa8aadaah aaWcbeqaa8qacaWGjbGaamOvaaaaaaa@4401@  are correlated. Since the theoretical framework predicts that individuals only respond meaningfully to changes in their total after-tax income along the extensive margin, the expectation is that ϑ ^ 1 = ϑ ^ 1 a =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gafqy0dO0dayaajaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyyp a0Jafqy0dO0dayaajaWaa0baaSqaa8qacaaIXaaapaqaa8qacaWGHb aaaOGaeyypa0JaaGimaaaa@3F87@  and κ ^ 1 <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqOUdS2dayaajaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyip aWJaaGimaaaa@3ABB@ . In contrast, if Δln ( 1 τ it a ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaciiBaiaac6gadaqadaWdaeaapeGaaGymaiabgkHiTiab es8a09aadaqhaaWcbaWdbiaadMgacaWG0baapaqaa8qacaWGHbaaaa GccaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaadMeacaWGwbaaaaaa @43A2@  captures an income effect, the likely outcome in this case is ϑ ^ 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gafqy0dO0dayaajaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyyp a0JaaGimaaaa@3AB3@  and ϑ ^ 1 a , κ ^ 1 <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gafqy0dO0dayaajaWaa0baaSqaa8qacaaIXaaapaqaa8qacaWGHbaa aOGaaiilaiqbeQ7aR9aagaqcamaaBaaaleaapeGaaGymaaWdaeqaaO WdbiabgYda8iaaicdaaaa@3F39@ .

The results from Equation (10) are shown in Panel A of Table 5. These findings indicate that individuals are very responsive to contemporaneous changes in their average—not their marginal—tax rates. For each 1% increase in the average net-of-tax share, labour income is estimated to increase by approximately 0.421% in the preferred model specification. In contrast, the marginal net-of-tax share and total after-tax income continue to be insignificant determinants of labour earnings, consistent with the previous section’s findings. Ito (2014) posits for the case of domestic electricity consumption that individuals may initially respond to the average price, but then start to use the lagged marginal price as this information becomes available over time. In this setting, Table 5 finds no evidence of a delayed responsiveness in Δln( z i,t+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiaabYgacaqGUbWaaeWaa8aabaWdbiaadQhapaWaaSbaaSqa a8qacaWGPbGaaiilaiaadshacqGHRaWkcaaIXaaapaqabaaak8qaca GLOaGaayzkaaaaaa@4055@  to either the marginal or average net-of-tax share, although an increase in total after-tax income does (weakly) induce a reduction in labour earnings in the next period, which is consistent with the initial expectation that γ ^ 0 <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gafq4SdC2dayaajaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaeyip aWJaaGimaaaa@3AAF@  from the previous section.

Figure 3 The relationship between the changes in the actual and predicted tax variables

Description for Figure 3

The title of Figure 3 is “The relationship between the changes in the actual and predicted tax variables.”

This figure contains four charts: two charts in Panel A two charts in Panel B, which are described below.

Panel title: Panel A: Marginal net-of-tax share versus average net-of-tax share

Chart title: Actual

This chart is a combination of a scatter plot and a line chart.
The vertical axis (y-axis) title is “percentage change in the average net-of-tax share.”
The axis graduation begins at -50 and ends at 50, with tick marks at every 25 units.
The horizontal axis (x-axis) title is “Percentage change in the marginal net-of-tax share.”
The axis graduation begins at -50 and ends at 50, with tick marks at every 25 units.
The chart contains a diagonal line extending from -50 to 50 on the x-axis.

Chart title: Predicted

This chart is a combination of a scatter plot and a line chart.
The vertical axis (y-axis) title is “percentage change in the predicted average net-of-tax share.”
The axis graduation begins at -20 and ends at 20, with tick marks at every 10 units.
The horizontal axis (x-axis) title is “Percentage change in the predicted marginal net-of-tax share.”
The axis graduation begins at -20 and ends at 20, with tick marks at every 10 units.
The chart contains a diagonal line extending from -20 to 20 on the x-axis.

Panel title: Panel B: Average net-of-tax share versus total after-tax income

Chart title: Actual

This chart is a combination of a scatter plot and a line chart.
The vertical axis (y-axis) title is “percentage change in the total after-tax income.”
The axis graduation begins at -75 and ends at 75, with tick marks at every 25 units.
The horizontal axis (x-axis) title is “Percentage change in the average net-of-tax share.”
The axis graduation begins at -15 and ends at 15, with tick marks at every 5 units.
The chart contains a curved line extending from -15 to 15 on the x-axis.

Chart title: Predicted

This chart is a combination of a scatter plot and a line chart.
The vertical axis (y-axis) title is “percentage change in the predicted total after-tax income.”
The axis graduation begins at -20 and ends at 20, with tick marks at every 10 units.
The horizontal axis (x-axis) title is “Percentage change in the predicted average net-of-tax share.”
The axis graduation begins at -5.0 and ends at 5.0, with tick marks at every 2.5 units.
The chart contains a curved line extending from -5.0 to 5.0 on the x-axis.

Notes: The relationship between the change in the marginal net-of-tax share relative to the change in the average net-of-tax share (Panel A) and between the change in the average net-of-tax share relative to the change in total after-tax income (Panel B) are shown, for both the actual and predicted tax variables. Each dot corresponds to a unique observation. The kernel-weighted local polynomial smoothing of these relationships are also shown. The predicted measures maintain the same basic relationships as the actual measures, but are less noisy.

Source: Statistics Canada, Longitudinal Administrative Databank.

 

Panel B of Table 5 shows the results from the extensive-margin analysis of Equation (11). Notably, in the preferred model specification that controls for income effects of the tax reform, changes in the marginal and average net-of-tax shares are both insignificant predictors of contemporaneous changes in labour market participation, as expected. This provides additional support for the notion that individuals respond to changes in their average tax rates along the intensive margin as a proxy for their marginal rates and not simply because of income effects resulting from the reform. The results for an initial response also continue to indicate that each 1% increase in total after-tax income reduces the likelihood of being employed, in this case by an average of 1.5 percentage points, the implied elasticity being -0.024%. However, the assessment of a delayed response shows that increases in the average net-of-tax share and total after-tax income both reduce the likelihood that individuals are employed in the next period, suggesting that individuals may use the average tax rate as a measure of the reform’s income effect in a lagged context. This finding partly supports modelling employment as a function of the average net-of-tax share, which is an approach that has been used in related studies (Eissa and Hoynes 2004; Gelber and Mitchell 2012).

Table 5
Encompassing tests—Labour income and employment responses to changes in the marginal versus average net-of-tax shares
Table summary
This table displays the results of Encompassing tests—Labour income and employment responses to changes in the marginal versus average
net-of-tax shares Initial timing of response, net-of-tax shares, Delayed timing of response, net-of-tax shares, Marginal, Average and Marginal and average, calculated using coefficient estimates units of measure (appearing as column headers).
Initial timing of response, net-of-tax shares Delayed timing of response, net-of-tax shares
Marginal Average Marginal and average Marginal and average Marginal Average Marginal and average Marginal and average
coefficient estimates
Panel A: Intensive margin
Marginal net-of-tax share 1.004Note *** Note ...: not applicable 0.573Note *** -0.112 -0.276Note *** Note ...: not applicable -0.178 -0.122
Average net-of-tax share Note ...: not applicable 1.838Note *** 1.293Note *** 0.421Note ** Note ...: not applicable -0.457Note *** -0.288Table 5 Note  -0.208
Total after-tax income Note ...: not applicable Note ...: not applicable Note ...: not applicable 0.028 Note ...: not applicable Note ...: not applicable Note ...: not applicable -0.134Table 5 Note 
Panel B: Extensive margin
Marginal net-of-tax share 0.202Note *** Note ...: not applicable 0.145Note *** 0.011 -0.033Note ** Note ...: not applicable -0.018 -0.019
Average net-of-tax share Note ...: not applicable 0.148Note *** 0.104Note *** -0.005 Note ...: not applicable -0.033Note *** -0.028Note *** -0.028Note **
Total after-tax income Note ...: not applicable Note ...: not applicable Note ...: not applicable -0.015Note *** Note ...: not applicable Note ...: not applicable Note ...: not applicable -0.006Note *

5 Intra-household effects

The extent to which changes in the tax liabilities of spouses have spillover effects on the employment and labour income decisions of individuals is investigated in this section. More precisely, based on the comparative statics of Equations (2) and (3) derived in the theoretical framework, this section estimates whether individuals respond directly to: 1) changes in their spouses’ total after-tax income, which affect the household budget set (a cross-spouse income effect); and 2) changes in their spouses’ employment and labour income decisions (a cross-spouse labour effect). This analysis is important because the labour supply decisions of older workers may be codependent, and the pension income splitting reform is known to have had different effects on the tax liabilities of individuals and spouses.

5.1 Test of cross-spouse income effects

This analysis estimates whether individuals respond equally to changes in their own total after-tax income and to the after-tax income of their spouses, as predicted by the unitary model. To this end, variants of Equations (5) and (6) are estimated, given by:

Δln( z it )= α 2 + β 2 a Δln( 1 τ it a )+ γ 2 Δln( Y it T it )+ δ 2 Δln( Y st T st )+ ζ 2 Δln( z st )    (12) + X it ' θ 2 +f[ ln( z i,t1 ) ]+ ε it MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqqHuoarciGGSbGaaiOBamaabmaapaqaa8qacaWG6bWdamaa BaaaleaapeGaamyAaiaadshaa8aabeaaaOWdbiaawIcacaGLPaaacq GH9aqpcqaHXoqypaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaey4k aSIaeqOSdi2damaaDaaaleaapeGaaGOmaaWdaeaapeGaamyyaaaak8 aacqqHuoarpeGaciiBaiaac6gadaqadaWdaeaapeGaaGymaiabgkHi Tiabes8a09aadaqhaaWcbaWdbiaadMgacaWG0baapaqaa8qacaWGHb aaaaGccaGLOaGaayzkaaGaey4kaSIaeq4SdC2damaaBaaaleaapeGa aGOmaaWdaeqaaOGaeuiLdq0dbiGacYgacaGGUbWaaeWaa8aabaWdbi aadMfapaWaaSbaaSqaa8qacaWGPbGaamiDaaWdaeqaaOWdbiabgkHi TiaadsfapaWaaSbaaSqaa8qacaWGPbGaamiDaaWdaeqaaaGcpeGaay jkaiaawMcaaiabgUcaRiabes7aK9aadaWgaaWcbaWdbiaaikdaa8aa beaak8qacaqGuoGaciiBaiaac6gadaqadaWdaeaapeGaamywa8aada WgaaWcbaWdbiaadohacaWG0baapaqabaGcpeGaeyOeI0Iaamiva8aa daWgaaWcbaWdbiaadohacaWG0baapaqabaaak8qacaGLOaGaayzkaa Gaey4kaSIaeqOTdO3damaaBaaaleaapeGaaGOmaaWdaeqaaOGaeuiL dq0dbiGacYgacaGGUbWaaeWaa8aabaWdbiaadQhapaWaaSbaaSqaa8 qacaWGZbGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaaaqaaiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabgUcaRiaadIfapaWaa0ba aSqaa8qacaWGPbGaamiDaaWdaeaapeGaai4jaaaakiabeI7aX9aada WgaaWcbaWdbiaaikdaa8aabeaak8qacqGHRaWkcaWGMbWaamWaa8aa baWdbiGacYgacaGGUbWaaeWaa8aabaWdbiaadQhapaWaaSbaaSqaa8 qacaWGPbGaaiilaiaadshacqGHsislcaaIXaaapaqabaaak8qacaGL OaGaayzkaaaacaGLBbGaayzxaaGaey4kaSIaeqyTdu2damaaBaaale aapeGaamyAaiaadshaa8aabeaaaaaa@C65F@

Δ1( z it >0 )= ι 2 + κ 2 Δln( Y it T it )+ λ 2 Δln( Y st T st )+ μ 2 Δ1( z st >0 )    (13) + X it ' ϕ 2 +f[ ln( z i,t1 ) ]+ φ it MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqqHuoarcaaIXaWaaeWaa8aabaWdbiaadQhapaWaaSbaaSqa a8qacaWGPbGaamiDaaWdaeqaaOWdbiabg6da+iaaicdaaiaawIcaca GLPaaacqGH9aqpcqaH5oqApaWaaSbaaSqaa8qacaaIYaaapaqabaGc peGaey4kaSIaeqOUdS2damaaBaaaleaapeGaaGOmaaWdaeqaaOWdbi aabs5aciGGSbGaaiOBamaabmaapaqaa8qacaWGzbWdamaaBaaaleaa peGaamyAaiaadshaa8aabeaak8qacqGHsislcaWGubWdamaaBaaale aapeGaamyAaiaadshaa8aabeaaaOWdbiaawIcacaGLPaaacqGHRaWk cqaH7oaBpaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaaeiLdiGacY gacaGGUbWaaeWaa8aabaWdbiaadMfapaWaaSbaaSqaa8qacaWGZbGa amiDaaWdaeqaaOWdbiabgkHiTiaadsfapaWaaSbaaSqaa8qacaWGZb GaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaaiabgUcaRiabeY7aT9aa daWgaaWcbaWdbiaaikdaa8aabeaak8qacaqGuoGaaGymamaabmaapa qaa8qacaWG6bWdamaaBaaaleaapeGaam4Caiaadshaa8aabeaak8qa cqGH+aGpcaaIWaaacaGLOaGaayzkaaaabaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRaWkcaWGybWdamaaDa aaleaapeGaamyAaiaadshaa8aabaWdbiaacEcaaaGccqaHvpGzpaWa aSbaaSqaa8qacaaIYaaapaqabaGcpeGaey4kaSIaamOzamaadmaapa qaa8qaciGGSbGaaiOBamaabmaapaqaa8qacaWG6bWdamaaBaaaleaa peGaamyAaiaacYcacaWG0bGaeyOeI0IaaGymaaWdaeqaaaGcpeGaay jkaiaawMcaaaGaay5waiaaw2faaiabgUcaRiabeA8aQ9aadaWgaaWc baWdbiaadMgacaWG0baapaqabaaaaaa@BA00@

Intuitively, the test for a cross-spouse income effect is carried out by directly inserting the variable for the change in the spouse's log of total after-tax income into the estimating equation to determine its significance. As before, to credibly identify the effects of the endogenous regressors, the instruments Δln ( 1 τ it a ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiGacYgacaGGUbWaaeWaa8aabaWdbiaaigdacqGHsislcqaH epaDpaWaa0baaSqaa8qacaWGPbGaamiDaaWdaeaapeGaamyyaaaaaO GaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaWGjbGaamOvaaaaaaa@4356@ , Δln ( Y it T it ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiaabYgacaqGUbWaaeWaa8aabaWdbiaadMfapaWaaSbaaSqa a8qacaWGPbGaamiDaaWdaeqaaOWdbiabgkHiTiaadsfapaWaaSbaaS qaa8qacaWGPbGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaa8aadaah aaWcbeqaa8qacaWGjbGaamOvaaaaaaa@43FD@  and Δln ( Y st T st ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiaabYgacaqGUbWaaeWaa8aabaWdbiaadMfapaWaaSbaaSqa a8qacaWGZbGaamiDaaWdaeqaaOWdbiabgkHiTiaadsfapaWaaSbaaS qaa8qacaWGZbGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaa8aadaah aaWcbeqaa8qacaWGjbGaamOvaaaaaaa@4411@  are used in a 2SLS approach. While pension income splitting mechanically shifts the total after-tax incomes of individuals and spouses in opposite directions, these effects are separately identified within couples since the percentage changes in total after-tax income depend on the initial (pre-reform) levels of pension and total income. Note that the average rather than the marginal net-of-tax share is used in Equation (12) based on the results in the previous section. The unitary model predicts that individuals respond equally to changes in their own as well as their spouses’ tax liabilities, hence the expectation is that γ ^ 2 = δ ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gafq4SdC2dayaajaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyyp a0JafqiTdq2dayaajaWaaSbaaSqaa8qacaaIYaaapaqabaaaaa@3CC4@  from Equation (12), and κ ^ 2 = λ ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqOUdS2dayaajaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyyp a0Jafq4UdW2dayaajaWaaSbaaSqaa8qacaaIYaaapaqabaaaaa@3CDE@  from Equation (13).

The results of this analysis are shown in Table 6 based on the OLS and IV (reduced-form and 2SLS) estimators. Notice that this analysis is carried out separately across individuals (Panel A) and their spouses (Panel B), where “individuals” are the taxfilers observed in the data and “spouses” are the taxfilers matched to those individuals with whom they are married or in a common-law relationship. Since the unit of taxation in Canada is the individual, and taxfilers are randomly selected into the LAD dataset, individuals and spouses should exhibit roughly symmetric cross-spouse responses to the tax reform despite the fact that the reform is known to have had opposite effects on each spouse's total after-tax income, as shown in Charts 1 and 2.

Table 6
Tests of cross-spouse income effects
Table summary
This table displays the results of Tests of cross-spouse income effects Ordinary least squares, Instrumental variables, Reduced form and Two-stage least squares, calculated using coefficient estimates units of measure (appearing as column headers).
Ordinary least squares Instrumental variables
Reduced form Two-stage least squares
coefficient estimates
Panel A: Responses of individuals to changes in the couples' total after-tax incomes
Intensive margin
Individual's average net-of-tax share -1.537Note *** 0.336Note * 0.409Note *
Individual's total after-tax income 0.497Note *** -0.007 0.031
Spouse's total after-tax income -0.052Note *** -0.041Table 6 Note  -0.076Note *
Extensive margin
Individual's total after-tax income 0.044Note *** -0.018Note *** -0.023Note ***
Spouse's total after-tax income -0.002Note *** -0.014Note *** -0.018Note ***
Panel B: Responses of spouses to changes in the couples' total after-tax incomes
Intensive margin
Spouse's average net-of-tax share -3.215Note *** 0.468Table 6 Note  0.913Note *
Spouse's total after-tax income 0.408Note *** 0.013 0.121
Individual's total after-tax income -0.032Note *** -0.052Note * -0.117Note **
Extensive margin
Spouse's total after-tax income 0.044Note *** -0.016Note *** -0.021Note ***
Individual's total after-tax income -0.001Table 6 Note  -0.013Note *** -0.017Note ***

The findings indicate, first, that individuals’ and spouses’ labour incomes respond meaningfully to exogenous changes in their own average net-of-tax shares, but not to changes in their own total after-tax incomes, consistent with the results presented in Section 4. In addition, large cross-spouse responses are observed: each 1% increase in total after-tax income leads to a decrease in the other spouse’s labour earnings of approximately 0.076% to 0.117% on average. Hence, older workers’ intensive-margin decisions are influenced by changes in their own average tax rates, and by changes in disposable income at the household level brought on by their spouses. These results are quite robust given that they are qualitatively similar across panels, as expected.

Table 6 also shows that there are large cross-spouse effects of a tax reform along the extensive margin. For individuals, a 1% increase in their own and their spouses’ total after-tax incomes reduces the likelihood of being employed by 2.3 and 1.8 percentage points; implied elasticities are -0.037 and -0.029, respectively. These findings are again similar for spouses. Importantly, while the own elasticity is only approximately 20% larger than the cross-spouse elasticity for both individuals and spouses, a test of equality rejects that they are equal (p=0.014 in Panel A, and p=0.022 in Panel B). As a result, the prediction of income pooling based on the unitary model is rejected (this finding is consistent with that of Gelber [2014]), although it is interesting to note that the own and cross-spouse elasticities are at least qualitatively similar.

5.2 Test of cross-spouse labour effects

The comparative statics in Subsection 2.2 show that a spouse’s marginal net-of-tax share is a valid excluded instrument for estimating how a change in the labour income of the spouse affects an individual’s employment and labour income decisions, provided that the spouse’s tax rate varies exogenously. This is true because the marginal income tax rate of each spouse is only levied on that spouse’s income, since the unit of taxation in Canada is the individual. With this in mind, the following variants of Equations (5) and (6) are estimated:

Δln( z it )= α 3 + β 3 a Δln( 1 τ it a )+ γ 3 Δln( Y it T it )+ δ 3 Δln( Y st T st )+ ζ 3 Δln( z st )    (14) + X it ' θ 3 +f[ ln( z i,t1 ) ]+ ω it MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqqHuoarciGGSbGaaiOBamaabmaapaqaa8qacaWG6bWdamaa BaaaleaapeGaamyAaiaadshaa8aabeaaaOWdbiaawIcacaGLPaaacq GH9aqpcqaHXoqypaWaaSbaaSqaa8qacaaIZaaapaqabaGcpeGaey4k aSIaeqOSdi2damaaDaaaleaapeGaaG4maaWdaeaapeGaamyyaaaak8 aacqqHuoarpeGaciiBaiaac6gadaqadaWdaeaapeGaaGymaiabgkHi Tiabes8a09aadaqhaaWcbaWdbiaadMgacaWG0baapaqaa8qacaWGHb aaaaGccaGLOaGaayzkaaGaey4kaSIaeq4SdC2damaaBaaaleaapeGa aG4maaWdaeqaaOGaeuiLdq0dbiGacYgacaGGUbWaaeWaa8aabaWdbi aadMfapaWaaSbaaSqaa8qacaWGPbGaamiDaaWdaeqaaOWdbiabgkHi TiaadsfapaWaaSbaaSqaa8qacaWGPbGaamiDaaWdaeqaaaGcpeGaay jkaiaawMcaaiabgUcaRiabes7aK9aadaWgaaWcbaWdbiaaiodaa8aa beaak8qacaqGuoGaciiBaiaac6gadaqadaWdaeaapeGaamywa8aada WgaaWcbaWdbiaadohacaWG0baapaqabaGcpeGaeyOeI0Iaamiva8aa daWgaaWcbaWdbiaadohacaWG0baapaqabaaak8qacaGLOaGaayzkaa Gaey4kaSIaeqOTdO3damaaBaaaleaapeGaaG4maaWdaeqaaOGaeuiL dq0dbiGacYgacaGGUbWaaeWaa8aabaWdbiaadQhapaWaaSbaaSqaa8 qacaWGZbGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaaaqaaiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8Uaey4kaSIaamiwa8aadaqhaaWcbaWd biaadMgacaWG0baapaqaa8qacaGGNaaaaOGaeqiUde3damaaBaaale aapeGaaG4maaWdaeqaaOWdbiabgUcaRiaadAgadaWadaWdaeaapeGa ciiBaiaac6gadaqadaWdaeaapeGaamOEa8aadaWgaaWcbaWdbiaadM gacaGGSaGaamiDaiabgkHiTiaaigdaa8aabeaaaOWdbiaawIcacaGL PaaaaiaawUfacaGLDbaacqGHRaWkcqaHjpWDpaWaaSbaaSqaa8qaca WGPbGaamiDaaWdaeqaaaaaaa@C4FF@

Δ1( z it >0 )= ι 3 + κ 3 Δln( Y it T it )+ λ 3 Δln( Y st T st )+μ ' 3 Δln( z st )    (15) + X it ' ϕ 3 +f[ ln( z i,t1 ) ]+ χ it MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacqqHuoarcaaIXaWaaeWaa8aabaWdbiaadQhapaWaaSbaaSqa a8qacaWGPbGaamiDaaWdaeqaaOWdbiabg6da+iaaicdaaiaawIcaca GLPaaacqGH9aqpcqaH5oqApaWaaSbaaSqaa8qacaaIZaaapaqabaGc peGaey4kaSIaeqOUdS2damaaBaaaleaapeGaaG4maaWdaeqaaOWdbi aabs5aciGGSbGaaiOBamaabmaapaqaa8qacaWGzbWdamaaBaaaleaa peGaamyAaiaadshaa8aabeaak8qacqGHsislcaWGubWdamaaBaaale aapeGaamyAaiaadshaa8aabeaaaOWdbiaawIcacaGLPaaacqGHRaWk cqaH7oaBpaWaaSbaaSqaa8qacaaIZaaapaqabaGcpeGaaeiLdiGacY gacaGGUbWaaeWaa8aabaWdbiaadMfapaWaaSbaaSqaa8qacaWGZbGa amiDaaWdaeqaaOWdbiabgkHiTiaadsfapaWaaSbaaSqaa8qacaWGZb GaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaaiabgUcaRiabeY7aTjaa cEcapaWaaSbaaSqaa8qacaaIZaaapaqabaGcpeGaaeiLdiGacYgaca GGUbWaaeWaa8aabaWdbiaadQhapaWaaSbaaSqaa8qacaWGZbGaamiD aaWdaeqaaaGcpeGaayjkaiaawMcaaaqaaiaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8Uaey4kaSIaamiwa8aadaqhaa WcbaWdbiaadMgacaWG0baapaqaa8qacaGGNaaaaOGaeqy1dy2damaa BaaaleaapeGaaG4maaWdaeqaaOWdbiabgUcaRiaadAgadaWadaWdae aapeGaciiBaiaac6gadaqadaWdaeaapeGaamOEa8aadaWgaaWcbaWd biaadMgacaGGSaGaamiDaiabgkHiTiaaigdaa8aabeaaaOWdbiaawI cacaGLPaaaaiaawUfacaGLDbaacqGHRaWkcqaHhpWypaWaaSbaaSqa a8qacaWGPbGaamiDaaWdaeqaaaaaaa@BA11@

To obtain causal estimates of the effects of interest, namely ζ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOTdO3damaaBaaaleaapeGaaG4maaWdaeqaaaaa@38E0@  and μ ' 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiVd0Maai4ja8aadaWgaaWcbaWdbiaaiodaa8aabeaaaaa@3984@ , the change in the spouse’s predicted net-of-tax share is used as the instrumental variable for Δln( z st ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiGacYgacaGGUbWaaeWaa8aabaWdbiaadQhapaWaaSbaaSqa a8qacaWGZbGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaaaaa@3E16@  in a 2SLS approach. While the theory indicates that the marginal tax rate should be used, the encompassing test of Subsection 4.2 suggests the average tax rate should be used in its place. Thus, the results will be shown using both Δln ( 1 τ st ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaciiBaiaac6gadaqadaWdaeaapeGaaGymaiabgkHiTiab es8a09aadaWgaaWcbaWdbiaadohacaWG0baapaqabaaak8qacaGLOa GaayzkaaWdamaaCaaaleqabaWdbiaadMeacaWGwbaaaaaa@42C5@  and Δln ( 1 τ st a ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaciiBaiaac6gadaqadaWdaeaapeGaaGymaiabgkHiTiab es8a09aadaqhaaWcbaWdbiaadohacaWG0baapaqaa8qacaWGHbaaaa GccaGLOaGaayzkaaWdamaaCaaaleqabaWdbiaadMeacaWGwbaaaaaa @43AC@  as the excluded instrument, although the results do not differ meaningfully in either case. Based on the model, the expectation is that individuals compensate for an exogenous decline in their spouses’ labour income by working more, and vice versa, such that ξ ^ 3 , μ ^ 3 ' <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GafqOVdG3dayaajaWaaSbaaSqaa8qacaaIZaaapaqabaGcpeGaaiil aiqbeY7aT9aagaqcamaaDaaaleaapeGaaG4maaWdaeaapeGaai4jaa aakiabgYda8iaaicdaaaa@3F21@ .

Several features of these model specifications are important to mention. In Equation (14), the preferred specification uses the average net-of-tax share, although the marginal tax rate is used when Δln ( 1 τ st ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaciiBaiaac6gadaqadaWdaeaapeGaaGymaiabgkHiTiab es8a09aadaWgaaWcbaWdbiaadohacaWG0baapaqabaaak8qacaGLOa GaayzkaaWdamaaCaaaleqabaWdbiaadMeacaWGwbaaaaaa@42C5@  is used as the excluded instrument. As before, this analysis conditions on observations where both individuals and their spouses have strictly positive labour income. In Equation (15), the parameter μ ' 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiVd0Maai4ja8aadaWgaaWcbaWdbiaaiodaa8aabeaaaaa@3984@  captures the effect of a change in the labour income of the spouse on the individual’s labour market participation, and is conditional on the spouse being employed. This restriction is necessary to ensure the spouse’s net-of-tax share is a valid excluded instrument based on the prediction from the theoretical framework. Lastly, as in Subsection 5.1, the analysis is repeated to assess whether there is symmetry of the findings across individuals and spouses, in this case using Δln ( 1 τ it ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeuiLdqKaciiBaiaac6gadaqadaWdaeaapeGaaGymaiabgkHiTiab es8a09aadaWgaaWcbaWdbiaadMgacaWG0baapaqabaaak8qacaGLOa GaayzkaaWdamaaCaaaleqabaWdbiaadMeacaWGwbaaaaaa@42BB@  as the excluded instrument.

The results of this analysis are shown in Table 7, for both individuals (Panel A) and spouses (Panel B). In both cases, the OLS regressions indicate that there is a significant positive correlation between the labour incomes of individuals and their spouses, which likely occurs as a result of positive assortative matching. In contrast, the intensive-margin 2SLS regressions indicate that each 1% change in one spouse’s labour income induces the other spouse to reduce their labour income by approximately 1.008% to 1.567%, a finding that is quite robust both across spouses and the choice of excluded instrument. The sign and magnitude of this result are both consistent with expectations from the model. Moreover, the extensive-margin analysis indicates that there are large cross-spouse income effects of a tax reform on the labour market participation decisions of individuals—the implied elasticity of employment to the spouse’s labour income is approximately -0.636% to -0.196%.

Table 7
Tests of cross-spouse labour effects
Table summary
This table displays the results of Tests of cross-spouse labour effects. The information is grouped by Excluded instrument (appearing as row headers), Ordinary least squares, Two-stage least squares, Marginal net-of-tax share and Average net-of-tax share, calculated using coefficient estimates units of measure (appearing as column headers).
Excluded instrument Ordinary least squares Two-stage least squares
Marginal net-of-tax share Average net-of-tax share
coefficient estimates
Panel A: Responses of individuals to changes in spouses' labour income
Intensive margin
Spouse's labour income 0.114Note *** -1.152Note *** -1.567Note ***
Extensive margin
Spouse's labour income 0.007Note *** -0.388Note *** -0.123Note ***
Panel B: Responses of spouses to changes in individuals' labour income
Intensive margin
Individual's labour income 0.107Note *** -1.008Note *** -1.506Note ***
Extensive margin
Individual's labour income 0.007Note *** -0.398Note *** -0.206Note ***

Lastly, Table 8 carries out the tests of cross-spouse income and labour effects for various subsamples to test for heterogeneity by sex, income, and family composition. Several findings are noteworthy. In the intensive-margin analysis of Panel A, men appear more responsive than women to changes in their tax rates and the tax liabilities of their spouses, whereas women respond more to changes in the labour income of their spouses. Tax responses are also robust across income groups and appear to be primarily driven by the middle of the income distribution, whereas cross-spouse labour responses are the largest among high-income households. Couples without children residing in the household are the most responsive to changes in their average tax rates, whereas those with children exhibit the largest cross-spouse income and labour effects. This finding may arise because households without children are less financially constrained and are, therefore, able to adjust their labour supply in response to price incentives, whereas those with children make labour decisions in a manner that is most consistent with the model.

Along the extensive margin shown in Panel B, the results continue to indicate that women are more responsive than men to cross-spouse labour effects. In this case, however, income effects of a tax reform have the largest effect on the labour market participation of older workers from high-income households, which may arise because these workers are the most likely to be able to afford to retire early. The results are also similar across couples with and without children. Overall, the results of this heterogeneity analysis are generally consistent with the primary findings, and are quite robust and symmetric across both individuals and spouses.

Table 8
Heterogeneous responses
Table summary
This table displays the results of Heterogeneous responses. The information is grouped by Endogenous regressor (appearing as row headers), Individuals' cross-spouse effect on, Spouses' cross-spouse effect on, Income, Labour, Individual's average net-of-tax share, Individual's total after-tax income, Spouse's total after-tax income, Spouse's labour income, Spouse's average net-of-tax share and Individual's labour income, calculated using coefficient estimates units of measure (appearing as column headers).
Endogenous regressor Individuals' cross-spouse effect on Spouses' cross-spouse effect on
Income Labour Income Labour
Individual's average net-of-tax share Individual's total after-tax income Spouse's total after-tax income Spouse's labour income Spouse's average net-of-tax share Spouse's total after-tax income Individual's total after-tax income Individual's labour income
coefficient estimates
Panel A: Intensive margin
Sex
Female 0.186 0.121Note * 0.184Table 8 Note  -1.900Note *** -0.343 0.079 0.234Note * -2.011Note ***
Male 2.068 0.260 -0.221Note *** -1.496Note *** 3.696Note * 0.441 -0.328Note *** -1.269Note ***
Household income
Top 50 percent -0.526 -0.136 -0.053 -2.208Note *** -1.522 -0.342 -0.089 -1.670Note ***
Top 75 percent 0.546Note * 0.062 -0.101Note * -1.710Note *** 0.412 0.039 -0.113Note * -1.539Note ***
Top 90 percent 0.529Note * 0.044 -0.100Note ** -1.565Note *** 0.944Table 8 Note  0.135 -0.105Note * -1.518Note ***
Exclude top 5 percent 0.425Note * 0.042 -0.069Table 8 Note  -1.501Note *** 0.981Note * 0.131 -0.119Note ** -1.539Note ***
Exclude top 10 percent 0.426Note ** 0.011 -0.086Note ** -1.482Note *** 1.315Note ** 0.173Note * -0.150Note *** -1.538Note ***
Family composition
Has children 0.104 -0.167Table 8 Note  -0.168Note ** -1.840Note *** -0.337 -0.111 -0.202Note ** -1.264Note ***
No children 0.666Note ** 0.136Table 8 Note  -0.048 -1.476Note *** 1.293Note ** 0.179Table 8 Note  -0.083 -1.629Note ***
Panel B: Extensive margin
Sex
Females Note ...: not applicable -0.014Note *** 0.023Note ** -0.267Note *** Note ...: not applicable -0.016Note *** 0.010 -0.378Note ***
Males Note ...: not applicable -0.013 -0.019Note *** -0.038Table 8 Note  Note ...: not applicable -0.026Note ** -0.021Note *** -0.129Note ***
Household income
Top 50 percent Note ...: not applicable -0.023Note *** -0.037Note *** -0.035 Note ...: not applicable -0.026Note *** -0.040Note *** -0.125Note ***
Top 75 percent Note ...: not applicable -0.025Note *** -0.022Note *** -0.108Note *** Note ...: not applicable -0.025Note *** -0.022Note *** -0.193Note ***
Top 90 percent Note ...: not applicable -0.024Note *** -0.018Note *** -0.125Note *** Note ...: not applicable -0.022Note *** -0.017Note *** -0.210Note ***
Exclude top 5 percent Note ...: not applicable -0.023Note *** -0.017Note *** -0.126Note *** Note ...: not applicable -0.021Note *** -0.016Note *** -0.216Note ***
Exclude top 10 percent Note ...: not applicable -0.022Note *** -0.016Note *** -0.139Note *** Note ...: not applicable -0.022Note *** -0.015Note *** -0.232Note ***
Family composition
Has children Note ...: not applicable -0.024Note *** -0.024Note *** -0.047Note *** Note ...: not applicable -0.027Note *** -0.022Note *** -0.057Note ***
No children Note ...: not applicable -0.022Note *** -0.016Note *** -0.073Note *** Note ...: not applicable -0.019Note *** -0.015Note *** -0.066Note ***

6 Conclusion

This paper assesses the extent to which the labour decisions of older workers respond to policy-induced variation in effective income tax rates, in Canada. To credibly identify these effects, the analysis exploits exogenous variation in tax rates following a recent reform that reduced the income tax liabilities of older couples by permitting pensioners to split this income with their spouses, using an instrumental variables design. The results show that older workers are very responsive to changes in taxes in terms of both their employment and labour income adjustments. However, in contrast with standard predictions, workers respond to changes in their average—not their marginal—tax rates. The compensated elasticities of taxable labour income with respect to the average and marginal net-of-tax shares are estimated to be 0.421 and -0.112, respectively, of which only the former is statistically significant.

An explanation for this result is that the individuals use the average price of labour as a proxy for the marginal price given that the income tax schedule in Canada is complex and may be difficult to understand (Liebman and Zeckhauser 2004). Given that the marginal tax rate generally exceeds the average tax rate due to progressivity, this finding suggests that workers oversupply labour at older ages because of tax illiteracy as they are not taking their true marginal costs of working into account properly. This finding also raises questions about the extent to which the weak responsiveness to income taxes found in related studies of young and middle-aged workers can also be explained by low tax salience, this being an important issue for future research.

In addition, individuals appear responsive to variation in household income resulting from changes in the tax liabilities and earnings of their spouses following a tax reform. This result is sensible in the context of this study given that income splitting necessarily involves some collaboration and tax planning between spouses. While the test of cross-spouse income effects rejects the income pooling prediction of the unitary model of labour supply, individuals and spouses nonetheless still take changes in each other’s incomes into account when making their own employment decisions, which is qualitatively consistent with expectations. These findings provide new insight into intra-household labour supply.

Appendix

Derivations of d z i * d( 1 τ i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaadsgacaWG6bWdamaaDaaaleaapeGaamyAaaWd aeaapeGaaiOkaaaaaOWdaeaapeGaamizamaabmaapaqaa8qacaaIXa GaeyOeI0IaeqiXdq3damaaBaaaleaapeGaamyAaaWdaeqaaaGcpeGa ayjkaiaawMcaaaaaaaa@41B3@  and d z i * d( 1 τ s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaSaaa8aabaWdbiaadsgacaWG6bWdamaaDaaaleaapeGaamyAaaWd aeaapeGaaiOkaaaaaOWdaeaapeGaamizamaabmaapaqaa8qacaaIXa GaeyOeI0IaeqiXdq3damaaBaaaleaapeGaam4CaaWdaeqaaaGcpeGa ayjkaiaawMcaaaaaaaa@41BD@

The optimal levels of income { z i * , z s * } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaiWaa8aabaWdbiaadQhapaWaa0baaSqaa8qacaWGPbaapaqaa8qa caGGQaaaaOGaaiilaiaadQhapaWaa0baaSqaa8qacaWGZbaapaqaa8 qacaGGQaaaaaGccaGL7bGaayzFaaaaaa@3F36@  are given by:

z i * , z s * arg max  z i , z s  λ u i ( c, z i , z s ; ψ i )+( 1λ ) u s ( c, z i , z s ; ψ s )   (16) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaqhaaWcbaWdbiaadMgaa8aabaWdbiaacQcaaaGccaGG SaGaamOEa8aadaqhaaWcbaWdbiaadohaa8aabaWdbiaacQcaaaGccq GHiiIZciGGHbGaaiOCaiaacEgapaWaaCbeaeaapeGaaeyBaiaabgga caqG4bGaaeiOaaWcpaqaa8qacaWG6bWdamaaBaaameaapeGaamyAaa WdaeqaaSWdbiaacYcacaWG6bWdamaaBaaameaapeGaam4CaaWdaeqa aaWcbeaak8qacaGGGcGaeq4UdWMaamyDa8aadaahaaWcbeqaa8qaca WGPbaaaOWaaeWaa8aabaWdbiaadogacaGGSaGaamOEa8aadaWgaaWc baWdbiaadMgaa8aabeaak8qacaGGSaGaamOEa8aadaWgaaWcbaWdbi aadohaa8aabeaak8qacaGG7aGaeqiYdK3damaaBaaaleaapeGaamyA aaWdaeqaaaGcpeGaayjkaiaawMcaaiabgUcaRmaabmaapaqaa8qaca aIXaGaeyOeI0Iaeq4UdWgacaGLOaGaayzkaaGaamyDa8aadaahaaWc beqaa8qacaWGZbaaaOWaaeWaa8aabaWdbiaadogacaGGSaGaamOEa8 aadaWgaaWcbaWdbiaadMgaa8aabeaak8qacaGGSaGaamOEa8aadaWg aaWcbaWdbiaadohaa8aabeaak8qacaGG7aGaeqiYdK3damaaBaaale aapeGaam4CaaWdaeqaaaGcpeGaayjkaiaawMcaaaaa@71C9@

where c= z i ( 1 τ i )+ z s ( 1 τ s )+R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaam4yaiabg2da9iaadQhapaWaaSbaaSqaa8qacaWGPbaapaqabaGc peWaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaSbaaSqaa8 qacaWGPbaapaqabaaak8qacaGLOaGaayzkaaGaey4kaSIaamOEa8aa daWgaaWcbaWdbiaadohaa8aabeaak8qadaqadaWdaeaapeGaaGymai abgkHiTiabes8a09aadaWgaaWcbaWdbiaadohaa8aabeaaaOWdbiaa wIcacaGLPaaacqGHRaWkcaWGsbaaaa@4C59@ . Given z s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaWgaaWcbaWdbiaadohaa8aabeaaaaa@385D@ , the first-order condition to this optimization problem with respect to z i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOEa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaaa@3853@  is:

λ( u c i ( 1 τ i )+ u z i i )+( 1λ )( u c s ( 1 τ i )+ u z i s )=0   (17) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4UdW2aaeWaa8aabaWdbiaadwhapaWaa0baaSqaa8qacaWGJbaa paqaa8qacaWGPbaaaOWaaeWaa8aabaWdbiaaigdacqGHsislcqaHep aDpaWaaSbaaSqaa8qacaWGPbaapaqabaaak8qacaGLOaGaayzkaaGa ey4kaSIaamyDa8aadaqhaaWcbaWdbiaadQhapaWaaSbaaWqaa8qaca WGPbaapaqabaaaleaapeGaamyAaaaaaOGaayjkaiaawMcaaiabgUca Rmaabmaapaqaa8qacaaIXaGaeyOeI0Iaeq4UdWgacaGLOaGaayzkaa WaaeWaa8aabaWdbiaadwhapaWaa0baaSqaa8qacaWGJbaapaqaa8qa caWGZbaaaOWaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaS baaSqaa8qacaWGPbaapaqabaaak8qacaGLOaGaayzkaaGaey4kaSIa amyDa8aadaqhaaWcbaWdbiaadQhapaWaaSbaaWqaa8qacaWGPbaapa qabaaaleaapeGaam4CaaaaaOGaayjkaiaawMcaaiabg2da9iaaicda aaa@6148@

To derive the effect of a change in the marginal net-of-tax share on the taxable labour income of the individual, totally differentiate Equation (17) with respect to ( 1 τ i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaSbaaSqaa8qa caWGPbaapaqabaaak8qacaGLOaGaayzkaaaaaa@3C83@  and evaluate at { z i * , z s * } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaiWaa8aabaWdbiaadQhapaWaa0baaSqaa8qacaWGPbaapaqaa8qa caGGQaaaaOGaaiilaiaadQhapaWaa0baaSqaa8qacaWGZbaapaqaa8 qacaGGQaaaaaGccaGL7bGaayzFaaaaaa@3F36@  per the envelope condition, then solve for d z i * /d( 1 τ i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadQhapaWaa0baaSqaa8qacaWGPbaapaqaa8qacaGGQaaa aOGaai4laiaadsgadaqadaWdaeaapeGaaGymaiabgkHiTiabes8a09 aadaWgaaWcbaWdbiaadMgaa8aabeaaaOWdbiaawIcacaGLPaaaaaa@4218@  to obtain:

d z i * d( 1 τ i ) = 1 υ( 1 τ i )+ψ  ( λ u c i +( 1λ ) u c s ) υ υ( 1 τ i )+ψ ( z i * + dR d( 1 τ i ) )    (18) υ( 1 τ s )+ω υ( 1 τ i )+ψ ( d z s * d( 1 τ i ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qadaWcaaWdaeaapeGaamizaiaadQhapaWaa0baaSqaa8qacaWG Pbaapaqaa8qacaGGQaaaaaGcpaqaa8qacaWGKbWaaeWaa8aabaWdbi aaigdacqGHsislcqaHepaDpaWaaSbaaSqaa8qacaWGPbaapaqabaaa k8qacaGLOaGaayzkaaaaaiabg2da9iabgkHiTmaalaaapaqaa8qaca aIXaaapaqaa8qacqaHfpqDdaqadaWdaeaapeGaaGymaiabgkHiTiab es8a09aadaWgaaWcbaWdbiaadMgaa8aabeaaaOWdbiaawIcacaGLPa aacqGHRaWkcqaHipqEaaGaaiiOamaabmaapaqaa8qacqaH7oaBcaWG 1bWdamaaDaaaleaapeGaam4yaaWdaeaapeGaamyAaaaakiabgUcaRm aabmaapaqaa8qacaaIXaGaeyOeI0Iaeq4UdWgacaGLOaGaayzkaaGa amyDa8aadaqhaaWcbaWdbiaadogaa8aabaWdbiaadohaaaaakiaawI cacaGLPaaacqGHsisldaWcaaWdaeaapeGaeqyXduhapaqaa8qacqaH fpqDdaqadaWdaeaapeGaaGymaiabgkHiTiabes8a09aadaWgaaWcba WdbiaadMgaa8aabeaaaOWdbiaawIcacaGLPaaacqGHRaWkcqaHipqE aaWaaeWaa8aabaWdbiaadQhapaWaa0baaSqaa8qacaWGPbaapaqaa8 qacaGGQaaaaOGaey4kaSYaaSaaa8aabaWdbiaadsgacaWGsbaapaqa a8qacaWGKbWaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaS baaSqaa8qacaWGPbaapaqabaaak8qacaGLOaGaayzkaaaaaaGaayjk aiaawMcaaaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7cqGHsisldaWcaaWdaeaapeGaeqyXdu3aaeWa a8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaSbaaSqaa8qacaWGZb aapaqabaaak8qacaGLOaGaayzkaaGaey4kaSIaeqyYdChapaqaa8qa cqaHfpqDdaqadaWdaeaapeGaaGymaiabgkHiTiabes8a09aadaWgaa WcbaWdbiaadMgaa8aabeaaaOWdbiaawIcacaGLPaaacqGHRaWkcqaH ipqEaaWaaeWaa8aabaWdbmaalaaapaqaa8qacaWGKbGaamOEa8aada qhaaWcbaWdbiaadohaa8aabaWdbiaacQcaaaaak8aabaWdbiaadsga daqadaWdaeaapeGaaGymaiabgkHiTiabes8a09aadaWgaaWcbaWdbi aadMgaa8aabeaaaOWdbiaawIcacaGLPaaaaaaacaGLOaGaayzkaaaa aaa@C8A2@

and,

υ=λ( u cc i ( 1 τ i )+ u z i c i )+( 1λ )( u cc s ( 1 τ i )+ u z i c s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyXduNaeyypa0Jaeq4UdW2aaeWaa8aabaWdbiaadwhapaWaa0ba aSqaa8qacaWGJbGaam4yaaWdaeaapeGaamyAaaaakmaabmaapaqaa8 qacaaIXaGaeyOeI0IaeqiXdq3damaaBaaaleaapeGaamyAaaWdaeqa aaGcpeGaayjkaiaawMcaaiabgUcaRiaadwhapaWaa0baaSqaa8qaca WG6bWdamaaBaaameaapeGaamyAaaWdaeqaaSWdbiaadogaa8aabaWd biaadMgaaaaakiaawIcacaGLPaaacqGHRaWkdaqadaWdaeaapeGaaG ymaiabgkHiTiabeU7aSbGaayjkaiaawMcaamaabmaapaqaa8qacaWG 1bWdamaaDaaaleaapeGaam4yaiaadogaa8aabaWdbiaadohaaaGcda qadaWdaeaapeGaaGymaiabgkHiTiabes8a09aadaWgaaWcbaWdbiaa dMgaa8aabeaaaOWdbiaawIcacaGLPaaacqGHRaWkcaWG1bWdamaaDa aaleaapeGaamOEa8aadaWgaaadbaWdbiaadMgaa8aabeaal8qacaWG Jbaapaqaa8qacaWGZbaaaaGccaGLOaGaayzkaaaaaa@6633@

ψ=λ( u c z i i ( 1 τ i )+ u z i z i i )+( 1λ )( u c z i s ( 1 τ i )+ u z i z i s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiYdKNaeyypa0Jaeq4UdW2aaeWaa8aabaWdbiaadwhapaWaa0ba aSqaa8qacaWGJbGaamOEa8aadaWgaaadbaWdbiaadMgaa8aabeaaaS qaa8qacaWGPbaaaOWaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaD paWaaSbaaSqaa8qacaWGPbaapaqabaaak8qacaGLOaGaayzkaaGaey 4kaSIaamyDa8aadaqhaaWcbaWdbiaadQhapaWaaSbaaWqaa8qacaWG PbaapaqabaWcpeGaamOEa8aadaWgaaadbaWdbiaadMgaa8aabeaaaS qaa8qacaWGPbaaaaGccaGLOaGaayzkaaGaey4kaSYaaeWaa8aabaWd biaaigdacqGHsislcqaH7oaBaiaawIcacaGLPaaadaqadaWdaeaape GaamyDa8aadaqhaaWcbaWdbiaadogacaWG6bWdamaaBaaameaapeGa amyAaaWdaeqaaaWcbaWdbiaadohaaaGcdaqadaWdaeaapeGaaGymai abgkHiTiabes8a09aadaWgaaWcbaWdbiaadMgaa8aabeaaaOWdbiaa wIcacaGLPaaacqGHRaWkcaWG1bWdamaaDaaaleaapeGaamOEa8aada WgaaadbaWdbiaadMgaa8aabeaal8qacaWG6bWdamaaBaaameaapeGa amyAaaWdaeqaaaWcbaWdbiaadohaaaaakiaawIcacaGLPaaaaaa@6BAA@

ω=λ( u c z s i ( 1 τ i )+ u z i z s i )+( 1λ )( u c z s s ( 1 τ i )+ u z i z s s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyYdCNaeyypa0Jaeq4UdW2aaeWaa8aabaWdbiaadwhapaWaa0ba aSqaa8qacaWGJbGaamOEa8aadaWgaaadbaWdbiaadohaa8aabeaaaS qaa8qacaWGPbaaaOWaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaD paWaaSbaaSqaa8qacaWGPbaapaqabaaak8qacaGLOaGaayzkaaGaey 4kaSIaamyDa8aadaqhaaWcbaWdbiaadQhapaWaaSbaaWqaa8qacaWG PbaapaqabaWcpeGaamOEa8aadaWgaaadbaWdbiaadohaa8aabeaaaS qaa8qacaWGPbaaaaGccaGLOaGaayzkaaGaey4kaSYaaeWaa8aabaWd biaaigdacqGHsislcqaH7oaBaiaawIcacaGLPaaadaqadaWdaeaape GaamyDa8aadaqhaaWcbaWdbiaadogacaWG6bWdamaaBaaameaapeGa am4CaaWdaeqaaaWcbaWdbiaadohaaaGcdaqadaWdaeaapeGaaGymai abgkHiTiabes8a09aadaWgaaWcbaWdbiaadMgaa8aabeaaaOWdbiaa wIcacaGLPaaacqGHRaWkcaWG1bWdamaaDaaaleaapeGaamOEa8aada WgaaadbaWdbiaadMgaa8aabeaal8qacaWG6bWdamaaBaaameaapeGa am4CaaWdaeqaaaWcbaWdbiaadohaaaaakiaawIcacaGLPaaaaaa@6BD1@

denote Γ=υ/( υ( 1 τ i )+ψ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Kdiabg2da9iabew8a1jaac+cadaqadaWdaeaapeGaeqyXdu3a aeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaSbaaSqaa8qaca WGPbaapaqabaaak8qacaGLOaGaayzkaaGaey4kaSIaeqiYdKhacaGL OaGaayzkaaaaaa@473B@ , Θ=( υ( 1 τ s )+ω )/( υ( 1 τ i )+ψ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiMdiabg2da9maabmaapaqaa8qacqaHfpqDdaqadaWdaeaapeGa aGymaiabgkHiTiabes8a09aadaWgaaWcbaWdbiaadohaa8aabeaaaO WdbiaawIcacaGLPaaacqGHRaWkcqaHjpWDaiaawIcacaGLPaaacaGG VaWaaeWaa8aabaWdbiabew8a1naabmaapaqaa8qacaaIXaGaeyOeI0 IaeqiXdq3damaaBaaaleaapeGaamyAaaWdaeqaaaGcpeGaayjkaiaa wMcaaiabgUcaRiabeI8a5bGaayjkaiaawMcaaaaa@5218@ , and Ω=1/( υ( 1 τ i )+ψ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyQdiabg2da9iabgkHiTiaaigdacaGGVaWaaeWaa8aabaWdbiab ew8a1naabmaapaqaa8qacaaIXaGaeyOeI0IaeqiXdq3damaaBaaale aapeGaamyAaaWdaeqaaaGcpeGaayjkaiaawMcaaiabgUcaRiabeI8a 5bGaayjkaiaawMcaaaaa@4732@ . Based on the assumptions imposed on the utility function, Γ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4Kdiabg6da+iaaicdaaaa@38E7@ , Θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiMdiabg6da+iaaicdaaaa@38EC@ , and Ω>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeyQdiabg6da+iaaicdaaaa@38FD@ . Substituting these terms into Equation (18) gives Equation (2), as desired.

To solve for Equation (3), totally differentiate Equation (17) with respect to ( 1 τ s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaSbaaSqaa8qa caWGZbaapaqabaaak8qacaGLOaGaayzkaaaaaa@3C8D@  and solve for d z i * /d( 1 τ s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamizaiaadQhapaWaa0baaSqaa8qacaWGPbaapaqaa8qacaGGQaaa aOGaai4laiaadsgadaqadaWdaeaapeGaaGymaiabgkHiTiabes8a09 aadaWgaaWcbaWdbiaadohaa8aabeaaaOWdbiaawIcacaGLPaaaaaa@4222@ . The derivation is the same as above, except that the first term on the right-hand side of Equation (18) is not present.

Separately identifying Δln( 1 τ jt ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiaabYgacaqGUbWaaeWaa8aabaWdbiaaigdacqGHsislcqaH epaDpaWaaSbaaSqaa8qacaWGQbGaamiDaaWdaeqaaaGcpeGaayjkai aawMcaaaaa@4077@ , Δln( 1 τ jt a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiaabYgacaqGUbWaaeWaa8aabaWdbiaaigdacqGHsislcqaH epaDpaWaa0baaSqaa8qacaWGQbGaamiDaaWdaeaapeGaamyyaaaaaO GaayjkaiaawMcaaaaa@415E@  and Δln( Y jt T jt ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeiLdiaabYgacaqGUbWaaeWaa8aabaWdbiaadMfapaWaaSbaaSqa a8qacaWGQbGaamiDaaWdaeqaaOWdbiabgkHiTiaadsfapaWaaSbaaS qaa8qacaWGQbGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaaaaa@420A@

The predicted marginal tax rate, ( 1 τ jt ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaaSbaaSqaa8qa caWGQbGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaa8aadaahaaWcbe qaa8qacaWGjbGaamOvaaaaaaa@3F72@ , is separately identified from the other two predicted tax variables given the nonlinearity (convexity) of the tax schedule, due to progressivity. For the remaining two variables, note that the percentage change in the predicted total after-tax income, ( Y jt T jt ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaeWaa8aabaWdbiaadMfapaWaaSbaaSqaa8qacaWGQbGaamiDaaWd aeqaaOWdbiabgkHiTiaadsfapaWaaSbaaSqaa8qacaWGQbGaamiDaa WdaeqaaaGcpeGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaWGjbGa amOvaaaaaaa@4105@ , can be expressed algebraically as follows:

%Δ ( Y jt T jt ) IV = ( Y ^ jt T( Ψ ^ jt ; π t+1 , Π ˜ t ) )( Y jt T( Ψ jt ; π t , Π ˜ t ) ) ( Y jt T( Ψ jt ; π t , Π ˜ t ) ) = ( Y ^ jt Y jt T( Ψ ^ jt ; π t+1 , Π ˜ t ) Y jt )( Y jt Y jt T( Ψ jt ; π t , Π ˜ t ) Y jt ) ( Y jt Y jt T( Ψ jt ; π t , Π ˜ t ) Y jt ) = ( Y ^ jt Y jt T( Ψ ^ jt ; π t+1 , Π ˜ t ) Y jt )( 1 T( Ψ jt ; π t , Π ˜ t ) Y jt ) ( 1 T( Ψ jt ; π t , Π ˜ t ) Y jt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaaeaaaaaa aaa8qacaGGLaGaaeiLdmaabmaapaqaa8qacaWGzbWdamaaBaaaleaa peGaamOAaiaadshaa8aabeaak8qacqGHsislcaWGubWdamaaBaaale aapeGaamOAaiaadshaa8aabeaaaOWdbiaawIcacaGLPaaapaWaaWba aSqabeaapeGaamysaiaadAfaaaGccqGH9aqpdaWcaaWdaeaapeWaae Waa8aabaWdbiqadMfapaGbaKaadaWgaaWcbaWdbiaadQgacaWG0baa paqabaGcpeGaeyOeI0Iaamivamaabmaapaqaa8qaceqGOoWdayaaja WaaSbaaSqaa8qacaWGQbGaamiDaaWdaeqaaOWdbiaacUdacqaHapaC paWaaSbaaSqaa8qacaWG0bGaey4kaSIaaGymaaWdaeqaaOWdbiaacY caceqGGoWdayaaiaWaaSbaaSqaa8qacaWG0baapaqabaaak8qacaGL OaGaayzkaaaacaGLOaGaayzkaaGaeyOeI0YaaeWaa8aabaWdbiaadM fapaWaaSbaaSqaa8qacaWGQbGaamiDaaWdaeqaaOWdbiabgkHiTiaa dsfadaqadaWdaeaapeGaaeiQd8aadaWgaaWcbaWdbiaadQgacaWG0b aapaqabaGcpeGaai4oaiabec8aW9aadaWgaaWcbaWdbiaadshaa8aa beaak8qacaGGSaGabeiOd8aagaacamaaBaaaleaapeGaamiDaaWdae qaaaGcpeGaayjkaiaawMcaaaGaayjkaiaawMcaaaWdaeaapeWaaeWa a8aabaWdbiaadMfapaWaaSbaaSqaa8qacaWGQbGaamiDaaWdaeqaaO WdbiabgkHiTiaadsfadaqadaWdaeaapeGaaeiQd8aadaWgaaWcbaWd biaadQgacaWG0baapaqabaGcpeGaai4oaiabec8aW9aadaWgaaWcba Wdbiaadshaa8aabeaak8qacaGGSaGabeiOd8aagaacamaaBaaaleaa peGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaaaGaayjkaiaawMcaaa aaaeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7cqGH9aqpdaWcaaWdaeaapeWaaeWaa8aabaWdbmaalaaa paqaa8qaceWGzbWdayaajaWaaSbaaSqaa8qacaWGQbGaamiDaaWdae qaaaGcbaWdbiaadMfapaWaaSbaaSqaa8qacaWGQbGaamiDaaWdaeqa aaaak8qacqGHsisldaWcaaWdaeaapeGaamivamaabmaapaqaa8qace qGOoWdayaajaWaaSbaaSqaa8qacaWGQbGaamiDaaWdaeqaaOWdbiaa cUdacqaHapaCpaWaaSbaaSqaa8qacaWG0bGaey4kaSIaaGymaaWdae qaaOWdbiaacYcaceqGGoWdayaaiaWaaSbaaSqaa8qacaWG0baapaqa baaak8qacaGLOaGaayzkaaaapaqaa8qacaWGzbWdamaaBaaaleaape GaamOAaiaadshaa8aabeaaaaaak8qacaGLOaGaayzkaaGaeyOeI0Ya aeWaa8aabaWdbmaalaaapaqaa8qacaWGzbWdamaaBaaaleaapeGaam OAaiaadshaa8aabeaaaOqaa8qacaWGzbWdamaaBaaaleaapeGaamOA aiaadshaa8aabeaaaaGcpeGaeyOeI0YaaSaaa8aabaWdbiaadsfada qadaWdaeaapeGaaeiQd8aadaWgaaWcbaWdbiaadQgacaWG0baapaqa baGcpeGaai4oaiabec8aW9aadaWgaaWcbaWdbiaadshaa8aabeaak8 qacaGGSaGabeiOd8aagaacamaaBaaaleaapeGaamiDaaWdaeqaaaGc peGaayjkaiaawMcaaaWdaeaapeGaamywa8aadaWgaaWcbaWdbiaadQ gacaWG0baapaqabaaaaaGcpeGaayjkaiaawMcaaaWdaeaapeWaaeWa a8aabaWdbmaalaaapaqaa8qacaWGzbWdamaaBaaaleaapeGaamOAai aadshaa8aabeaaaOqaa8qacaWGzbWdamaaBaaaleaapeGaamOAaiaa dshaa8aabeaaaaGcpeGaeyOeI0YaaSaaa8aabaWdbiaadsfadaqada WdaeaapeGaaeiQd8aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaGc peGaai4oaiabec8aW9aadaWgaaWcbaWdbiaadshaa8aabeaak8qaca GGSaGabeiOd8aagaacamaaBaaaleaapeGaamiDaaWdaeqaaaGcpeGa ayjkaiaawMcaaaWdaeaapeGaamywa8aadaWgaaWcbaWdbiaadQgaca WG0baapaqabaaaaaGcpeGaayjkaiaawMcaaaaaaeaacaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua eyypa0ZaaSaaa8aabaWdbmaabmaapaqaa8qadaWcaaWdaeaapeGabm ywa8aagaqcamaaBaaaleaapeGaamOAaiaadshaa8aabeaaaOqaa8qa caWGzbWdamaaBaaaleaapeGaamOAaiaadshaa8aabeaaaaGcpeGaey OeI0YaaSaaa8aabaWdbiaadsfadaqadaWdaeaapeGabeiQd8aagaqc amaaBaaaleaapeGaamOAaiaadshaa8aabeaak8qacaGG7aGaeqiWda 3damaaBaaaleaapeGaamiDaiabgUcaRiaaigdaa8aabeaak8qacaGG SaGabeiOd8aagaacamaaBaaaleaapeGaamiDaaWdaeqaaaGcpeGaay jkaiaawMcaaaWdaeaapeGaamywa8aadaWgaaWcbaWdbiaadQgacaWG 0baapaqabaaaaaGcpeGaayjkaiaawMcaaiabgkHiTmaabmaapaqaa8 qacaaIXaGaeyOeI0YaaSaaa8aabaWdbiaadsfadaqadaWdaeaapeGa aeiQd8aadaWgaaWcbaWdbiaadQgacaWG0baapaqabaGcpeGaai4oai abec8aW9aadaWgaaWcbaWdbiaadshaa8aabeaak8qacaGGSaGabeiO d8aagaacamaaBaaaleaapeGaamiDaaWdaeqaaaGcpeGaayjkaiaawM caaaWdaeaapeGaamywa8aadaWgaaWcbaWdbiaadQgacaWG0baapaqa baaaaaGcpeGaayjkaiaawMcaaaWdaeaapeWaaeWaa8aabaWdbiaaig dacqGHsisldaWcaaWdaeaapeGaamivamaabmaapaqaa8qacaqGOoWd amaaBaaaleaapeGaamOAaiaadshaa8aabeaak8qacaGG7aGaeqiWda 3damaaBaaaleaapeGaamiDaaWdaeqaaOWdbiaacYcaceqGGoWdayaa iaWaaSbaaSqaa8qacaWG0baapaqabaaak8qacaGLOaGaayzkaaaapa qaa8qacaWGzbWdamaaBaaaleaapeGaamOAaiaadshaa8aabeaaaaaa k8qacaGLOaGaayzkaaaaaaaaaa@7FC7@

The percentage change in the predicted average net-of-tax share, ( 1 τ jt a ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape WaaeWaa8aabaWdbiaaigdacqGHsislcqaHepaDpaWaa0baaSqaa8qa caWGQbGaamiDaaWdaeaapeGaamyyaaaaaOGaayjkaiaawMcaa8aada ahaaWcbeqaa8qacaWGjbGaamOvaaaaaaa@4059@ , is expressed algebraically as follows:

%Δ ( 1 τ it a ) IV = ( 1 T( Ψ ^ jt ; π t+1 , Π ˜ t ) Y ^ jt )( 1 T( Ψ jt ; π t , Π ˜ t ) Y jt ) ( 1 T( Ψ jt ; π t , Π ˜ t ) Y jt ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaiyjaiaabs5adaqadaWdaeaapeGaaGymaiabgkHiTiabes8a09aa daqhaaWcbaWdbiaadMgacaWG0baapaqaa8qacaWGHbaaaaGccaGLOa GaayzkaaWdamaaCaaaleqabaWdbiaadMeacaWGwbaaaOGaeyypa0Za aSaaa8aabaWdbmaabmaapaqaa8qacaaIXaGaeyOeI0YaaSaaa8aaba WdbiaadsfadaqadaWdaeaapeGabeiQd8aagaqcamaaBaaaleaapeGa amOAaiaadshaa8aabeaak8qacaGG7aGaeqiWda3damaaBaaaleaape GaamiDaiabgUcaRiaaigdaa8aabeaak8qacaGGSaGabeiOd8aagaac amaaBaaaleaapeGaamiDaaWdaeqaaaGcpeGaayjkaiaawMcaaaWdae aapeGabmywa8aagaqcamaaBaaaleaapeGaamOAaiaadshaa8aabeaa aaaak8qacaGLOaGaayzkaaGaeyOeI0YaaeWaa8aabaWdbiaaigdacq GHsisldaWcaaWdaeaapeGaamivamaabmaapaqaa8qacaqGOoWdamaa BaaaleaapeGaamOAaiaadshaa8aabeaak8qacaGG7aGaeqiWda3dam aaBaaaleaapeGaamiDaaWdaeqaaOWdbiaacYcaceqGGoWdayaaiaWa aSbaaSqaa8qacaWG0baapaqabaaak8qacaGLOaGaayzkaaaapaqaa8 qacaWGzbWdamaaBaaaleaapeGaamOAaiaadshaa8aabeaaaaaak8qa caGLOaGaayzkaaaapaqaa8qadaqadaWdaeaapeGaaGymaiabgkHiTm aalaaapaqaa8qacaWGubWaaeWaa8aabaWdbiaabI6apaWaaSbaaSqa a8qacaWGQbGaamiDaaWdaeqaaOWdbiaacUdacqaHapaCpaWaaSbaaS qaa8qacaWG0baapaqabaGcpeGaaiilaiqabc6apaGbaGaadaWgaaWc baWdbiaadshaa8aabeaaaOWdbiaawIcacaGLPaaaa8aabaWdbiaadM fapaWaaSbaaSqaa8qacaWGQbGaamiDaaWdaeqaaaaaaOWdbiaawIca caGLPaaaaaaaaa@827D@

Hence, %Δ ( 1 τ it a ) IV %Δ ( Y jt T jt ) IV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaiyjaiaabs5adaqadaWdaeaapeGaaGymaiabgkHiTiabes8a09aa daqhaaWcbaWdbiaadMgacaWG0baapaqaa8qacaWGHbaaaaGccaGLOa GaayzkaaWdamaaCaaaleqabaWdbiaadMeacaWGwbaaaOGaeyiyIKRa aiyjaiaabs5adaqadaWdaeaapeGaamywa8aadaWgaaWcbaWdbiaadQ gacaWG0baapaqabaGcpeGaeyOeI0Iaamiva8aadaWgaaWcbaWdbiaa dQgacaWG0baapaqabaaak8qacaGLOaGaayzkaaWdamaaCaaaleqaba WdbiaadMeacaWGwbaaaaaa@50A8@  on average. These two predicted variables are separately identified in this environment, as desired.

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