Lecture on Taxable Income Elasticities PhD Course in Uppsala

Transcription

1 Lecture on Taxable Income Elasticities PhD Course in Uppsala Håkan Selin Institute for Evaluation of Labour Market and Education Policy Uppsala, May 15,

2 TAXABLE INCOME ELASTICITIES Modern public finance literature focuses on taxable income elasticities instead of hours of work elasticities Two main reasons: 1) What matters for policy is the total behavioral response to tax rates (not only hours of work but also occupational choices, avoidance, etc.) 2) Data availability: taxable income is precisely measured in tax return data 2

3 OUTLINE FOR TODAY S LECTURE 1) Conceptual framework a) Basic model b) When is the taxable income elasticity a sufficient statistic? 2) Issues in estimation a) Standard panel data approach b) Repeated cross section vs. panel data 3) Examples on U.S. and Nordic taxable income papers 4) Bunching 3

4 LITERATURE Some of the lecture material builds on Emmanuel Saez graduate course in Berkeley, see Saez webpage. Key reading: Recent overview of the ETI literature: Saez- Slemrod-Giertz JEL other papers discussed in the lecture. 4

5 STANDARD STATIC LABOR SUPPLY MODEL In the standard model the individual maximizes u(c, l), where c is consumption and l is work hours, subject to the linearized budget constraint c = wl (1 τ) + R, where w is the hourly wage rate, τ is the marginal tax rate and R is virtual income. The budget constraint is linearized around the individual s optimum. Virtual income is defined as R = τwl T (wl) + m, where m is non-labor income and T is the non-linear income tax function. The dual optimization problem generates a Marshallian hours supply function of the form l((1 τ)w, R) and a Hicksian supply function of the form l((1 τ)w, u), where u is a fixed utility level. 5

6 LABOR SUPPLY MODEL WITH AVOIDANCE Now suppose that the individual, in addition to the labor supply choice, is able to reduce her taxable income by an amount A to a utility cost; u(c, l, A) depends negatively both on l and A. The invidual maximizes u(c, l, A), subject to the linearized budget constraint c = wl τ(z) + R, where z = wl A is the individual s taxable income. The dual optimization problem generates Marshallian hours supply and avoidance functions of the form l((1 τ)w, τ, R) and A((1 τ)w, τ, R) and the corresponding Hicksian supply functions. 6

7 THE TAXABLE INCOME ELASTICITY In this model, the taxable income elasticity can now be defined as e = dz 1 τ d(1 τ) z = dwl 1 τ d(1 τ) z da 1 τ d(1 τ) z Note that both Marshallian and Hicksian functions can be used to obtain the elasticity (depends on purpose). 7

8 FELDSTEIN RESTAT 99 Deadweight loss of taxation = money metric of the individual s utility loss of the tax - tax revenues collected Feldstein: The compensated taxable income elasticity with respect to 1 τ is a sufficient statistic for calculating the deadweight loss (excess burden) of taxation. Key point: It is irrelevant if the response to a tax increase comes from hours adjustments or tax avoidance. All that matters is the taxable income elasticity. Intuition for a small tax change: Due to the envelope theorem, the individual s utility loss (measured in dollar terms) is equal to the mechanical tax revenue effect (given constant behavior). However, the actual tax revenue collected depends on e. 8

9 TOP TAX RATES The taxable income elasticity for high income taxpayers can be used to obtain the revenue maximizing marginal tax rate in the top bracket: τ = 1 1+a e a is the Pareto paramater if the right tail of the income distribution is Pareto distributed. a is a measure of the thinness of the top tail: the thicker is the tail the smaller is a. More generally, as shown by Saez (2001), taxable income elasticities across the income distribution can be used to determine the shape of the optimal tax schedule. 9

10 Source: Pirttilä and Selin (2011b)

11 FISCAL EXTERNALITIES Sometimes, a reduction in reported income is due in part to a shift away from taxable individual income toward other forms of taxable income such as corporate income, or deferred compensation that will be taxable at a later date. If so, the taxable income elasticity is not a sufficient statistic for welfare evaluation; the effects on other tax bases must also be accounted for. Income shifting behavior have been analyzed e.g. by Gordon and Slemrod (2000) in the U.S., Pirttilä and Selin (2011a) for Finland and Alstadsæter and Jacob (2012) for Sweden. 11

12 TAX BASE ISSUES In contrast to the hours of work elasticity, the taxable income elasticity can be directly affected by the government. In the model of Slemrod and Kopczuk (2002) the more tax deductions that are allowed, the higher will be the taxable income elasticity. Suppose that we estimate a large elasticity in a tax system with many loopholes. Conclusion in the Feldstein model: a lower tax rate reduces deadweight loss. Alternative conclusion: closing down loopholes reduces deadweight loss. 12

13 ESTIMATION ISSUES To be able to recover both Hicksian and Marshallian taxable income elasticities we need to estimate both the response to the net-of-tax rate and virtual income. Suppose the following taxable income function: log z it = β 0 + β 1 log (1 τ) it + β 2 log R it + ɛ it (1), where i is an individual index and t is a time index. In this model, β 1 represents the Marshallian (uncompensated) taxable income elasticity. The Hicksian (compensated ) elasticity can be obtained by β 2 and the Slutsky relationship (see labor supply lecture). 13

14 FUNDAMENTAL PROBLEM Suppose we would estimate equation (1) by OLS. Problem (1): τ it is a direct function of the individual s choice of z it. Problem (2): The same holds for R it. In addition, some sources of non-labor income might be correlated with the choice of z it. Instruments are needed! The main approach in the taxable income literature is to exploit tax reform variation to recover β 1 while assuming away income effects. 14

15 ESTIMATION ISSUES Henceforth, we assume that β 2 = 0, i.e. no income effects. Note that credible estimates of the income effect also can be obtained from other quasi-experimental settings (e.g. Imbens et al. (2001) on lottery winners). 15

16 COMMON SOLUTION A majority of papers in the literature exploits individual level panel data and estimates (1) in first differences by 2SLS. log z it z i,t k = β 1 log 1 τ it 1 τ i,t k + ɛ it ɛ i,t k (2) One or several tax reforms that treat individuals differentially in the period t k to t. Typically, different income groups are treated differently. Well cited paper by Gruber and Saez (2002) construct instruments as a function of base period taxable income, z i,t k. 16

17 COMMON SOLUTION A common approach is to compute synthetic tax rates The marginal tax rate in period t is calculated based on the tax law of period t and the taxable income of period t k. 1 τ(z it ) it Accordingly, the instrument for log[ ] can then be 1 τ(z i,t k ) i,t k written as log[ 1 τ(z i,t k) it 1 τ(z i,t k ) i,t k ] Intuition: The instrument for the change in the net-of-tax rate between t and t k reflects tax law changes only, no behavioral changes. 17

18 TWO PROBLEMS OF VALIDITY The synthetic tax rate instrument is typically strongly correlated with the endogenous regressor. However, there are two main threats to instrument validity. (1) Mean reversion. Transitory shocks in z i,t k causes a correlation between the instrument and ɛ it ɛ i,t k. (1) Non-tax related trends in income inequality. logz might have grown differentially for individuals with different values of z i,t k even in the absence of a tax reform. This will also introduce a correlation between the instrument and the first differenced error term. To address these two problems, Gruber and Saez (2002) includes a (spline) function of z i,t k to equation (2). It is not clear, however, that this solves the problems. 18

19 SOLUTION TO MEAN REVERSION PROBLEM Blomquist and Selin (2010) propose an instrumentation procedure that is not subject to the mean reversion problem under the assumption that the series ɛ it is covariance stationary (neither its mean neither nor its autocovariances depend are allowed to depend on the time). Blomquist and Selin: construct instruments as a function of the middle year in the year difference OR take an average of all years in the year difference. In their case, t = 1991 and t k = Hence, they construct instruments based on income in Important: This is not a solution to the problem with non-tax related income growth! 19

20 PANEL DATA OR CROSS-SECTION DATA? The mean reversion issue leads Saez et al. (2012, JEL) to conclude that the advantage of panel data relative repeatedcross section has been exaggerated. Cross-section analysis typically compares taxable income growth in different income groups over time (see Section in Saez et al. (2012)). Key issue in cross-section analysis: If the tax reform changes the composition of treatment- and control groups over time (which is quite likely) there will be a bias. 20

21 SPECIFIC TAX REFORM STUDIES Recent survey by Saez-Slemrod-Giertz JEL 12. Some papers exploit very long time series. Here we will focus on studies that use tax reforms to identify the taxable income elasticity. Lindsey JpubE 87 analyzes ERTA 81 using repeated crosssection tax data and finds large elasticities Feldstein JPE 95 uses panel tax data to study TRA 86 Gruber-Saez JpubE 02 uses panel tax data Several Scandinavian studies as well, most recently Kleven and Schultz (2014, AEJ EP) 21

22 FELDSTEIN JPE 95: METHODOLOGY Feldstein (1995) estimates the effect of TRA86 on taxable income for top earners using panel tax data 1) Constructs three income groups M (Medium), H (High), HH (Highest) based on before reform income in ) Looks at how incomes and MTRs evolve from 1985 to 1988 for individuals in each group using panel: forms DD estimates ê = log(z H ) log(z M ) log(1 τ H ) log(1 τ M ) where z H, z M and τ H, τ M are income and MTRs of the H and M groups 22

23 Source: Feldstein (1995), p. 561

24 Source: Feldstein (1995), p. 565

25 FELDSTEIN JPE 95: RESULTS Results: Feldstein obtains very high elasticities (above 1) for top earners US was on the wrong side of the Laffer curve for the rich Laffer rate τ = 1/(1 + a e) = 1/( ) = 33% Cutting top tax rate from 50% to 28% raised revenue 24

26 FELDSTEIN JPE 95: ISSUES 1) Non-tax related changes in inequality : panel helps only if inequality changes due to arrival of new people 2) Short-term vs. Long-term response 3) Mean reversion: rich people in year t tend to revert to the mean in year t + 1 Panel analysis introduces downward bias in e [when τ for rich] 4) Very small sample in panel data [57 tax filers in HH group] [Auten-Carroll RESTAT 99 use larger Treasury panel data and find smaller elasticity 0.65] In net, not clear panel data adds value relative to repeatedcross-section 25

27 FELDSTEIN JPE 95: ISSUES 5) DD can give very biased results when elasticity differs across groups: Example: (a) M group has e M = 0 so that log(z M ) = 0 and that H group has e H = e > 0 so that log(z H ) = e log(1 τ H ). Suppose that log(1 τ M ) = 0.5 log(1 τ H ). Then, the estimated elasticity ê DD = e log(1 τ H )/[ log(1 τ H ) log(1 τ M )] = 2e In Feldstein JPE 95: Simple Difference log(z)/ log(1 τ) uniformly smaller than DD 26

28 GRUBER AND SAEZ JPUBE 02 Generalization of Feldstein JPE 95 using IV regression analysis Use panel data from on all tax changes available rather than a single reform Model: z it = zit 0 (1 τ it) e where zit 0 is potential income (if MTR=0), e is elasticity ( ) ( ) zit+3 1 τit+3 log = α + e log + ε it z it 1 τ it τ it+3 and ε it are correlated [because τ it+3 = T t+3 (z it+3)] Instrument: predicted change in MTR assuming income stays constant: log[(1 τ p it+3 )/(1 τ it)] where τ p it+3 = T t+3 (z it) Isolates changes in tax law (T t (.)) as the only source of variation in tax rates 27

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30 GRUBER AND SAEZ JPUBE 02 Find an elasticity of roughly BUT results are very fragile [Saez-Slemrod-Giertz JEL 12] 1) Sensitive to exclusion of low incomes (min income threshold) 2) Sensitive to controls for mean reversion 3) Subsequent studies find smaller elasticities using data from other countries [Kleven-Schultz AEJ-EP 14 for Denmark] 4) Bundles together small tax changes and large tax changes: if individuals respond only to large changes in short-medium run, then estimated elasticity is too low [Chetty et al. QJE 11] 29

31 BLOMQUIST AND SELIN (2010,JPUBE) Exploits the Swedish Level of Living Survey (LNU) + register data. Survey information on hourly wage rates Blomquist and Selin are able to separarately study the response in the hourly wage rate. The estimates of the hourly wage rate elasticity are in the range 0.14 to 0.16 for males and 0.41 to 0.57 for females. The taxable labor income elasticity estimates are in the range 0.19 to 0.21 for males and 0.96 to (Estimates for females more imprecise however.) Moreover, Blomquist and Selin estimate significant income effects for taxable labor income. 30

32 KLEVEN AND SCHULTZ AEJ-EP 14 Key Advantages: a) Use full population of tax returns in Denmark since 1980 (large sample size, panel structure, many demographic variables, stable inequality) b) A number of reforms changing tax rates differentially across three income brackets and across tax bases (capital income taxed separately from labor income) c) Show compelling visual DD-evidence of tax responses around the 1986 large reform 31

33 Figure 2. Two Decades of Danish Tax Reform Panel A. Marginal Tax Rate on Labor Income Panel B. Marginal Tax Rate on Negative Capital Income Marginal Tax Rate Marginal Tax Rate Bottom bracket Middle bracket Top bracket Bottom bracket Middle bracket Top bracket Panel C. Marginal Tax Rate on Positive Capital Income Panel D. Share of Taxpayers in the Three Tax Brackets Marginal Tax Rate Share of all taxpayers (%) Source: Kleven and Schultz '12 Bottom bracket Middle bracket Top bracket Bottom bracket Middle bracket Top bracket

34 Notes: Panel A considers the effect on labor income under two treatment group definitions using the grouping in Figure 3. Treatment 1 includes all groups in Figure 3 who experience an increase in the marginal net of tax rate on labor income as a result of the reform ( difference), while treatment 2 includes the same groups except those in the middle bracket ("stay middle" group in Figure 3) who experience a relatively small Figure 6. Graphical Evidence on the Effects of the 1987 reform on Taxable Income Source: Kleven and Schultz '12Panel A. Labor Income Labor income (ind dex 1986=100) DD 1 Elasticity = (0.011) DD 2 Elasticity = (0.013) Treatment 1 Treatment 2 Control Panel B. Positive Capital Income Positive Cap pital Income (index 1986=1 100) 130 DD Elasticity = (0.063) Treatment Control

35 KLEVEN AND SCHULTZ AEJ-EP 14 Key Findings: a) Small labor income elasticity (.05-.1) b) bigger capital income elasticities (.2) c) bigger elasticities for large reforms d) modest income shifting between labor and capital in Denmark (top rates on labor and capital are carefully aligned) Danish tax system optimized to have broad base and few avoidance opportunities 33

36 RELATED: NON-LINEAR ELECTRICITY PRICING Nonlinear pricing used in other settings (electricity, utility, cell phones consumption, etc.) Ito AER 14 is a great example using household electricity consumption to estimate consumer responses: 1) Electricity is priced nonlinearly based on monthly consumption 2) Ito compares people at the border of two electricity areas in Southern California: comparable groups facing different schedules and changes over time 3) Finds very compelling evidence that consumers respond to average price and not marginal price To date, no such compelling design has been found for income taxation 34

37 Introduction Research Design Estimation Welfare Conclusion I specifically focus on households located within 1 mile of the utility border Edison (Southern California Edison) provides electricity for the north side San Diego Source: Ito, (San 2011Diego Gas & Electric) provides electricity for the south side 19 / 69

38 Introduction Research Design Estimation Welfare Conclusion In contrast, they experience substantially different nonlinear pricing Edison and San Diego: Cents per kwh in Edison 20 San Diego Monthly Consumption Source: Ito, / 69

39 DD = (mean % change in San Diego) - (mean % change in Edison) Relative changes for SDG&E customers relative to SCE customers. Panel A: Top Decile (90% - 100%) of Consumption Distributions Difference in Differences in Price (%) Year Source: Ito, 2011 Marginal Price Average Price Consumption Difference in Differences in Consumption (%)

40 DD = (mean % change in San Diego) - (mean % change in Edison) Relative changes for SDG&E customers relative to SCE customers. Panel A: Top Decile (90% - 100%) of Consumption Distributions Difference in Differences in Price (%) Year Source: Ito, 2011 Marginal Price Average Price Consumption Difference in Differences in Consumption (%)

41 DD = (mean % change in San Diego) - (mean % change in Edison) Relative changes for SDG&E customers relative to SCE customers. Panel A: Top Decile (90% - 100%) of Consumption Distributions Difference in Differences in Price (%) Year Source: Ito, 2011 Marginal Price Average Price Consumption Difference in Differences in Consumption (%)

42 DD = (mean % change in San Diego) - (mean % change in Edison) Relative changes for SDG&E customers relative to SCE customers. Panel B. Fifth Decile (40% - 50%) of Consumption Distributions Difference in Differences in Price (%) Year Source: Ito, 2011 Marginal Price Average Price Consumption Difference in Differences in Consumption (%)

43 Introduction Research Design Estimation Welfare Conclusion Estimation results: Marginal Price v.s. Average Price 2SLS Estimates: Marginal Price vs. Average Price Distance from border 1 mile 0.5 mile (1) (2) (3) (4) (5) (6) ln(mp) (.007) (.015) (.011) (.020) ln(ap) (.006) (.013) (.011) (.017) Observations 6,513,600 3,520,320 Dependent variable: ln(electricity consumption) Standard errors are clustered at city-deciles levels Source: Ito, / 69

44 ALTERNATIVE METHOD: BUNCHING (SAEZ AEJ-EP 10) Traditionally, the piece-wise linear tax structure has been considered as a problem when estimating labor supply and taxable income elasticities. Saez (2010) shows how kink points (where marginal tax rates increase) can be exploited to recover taxable income elasticities. 36

45 BUNCHING THEORY Consider a hypothetical reform scenario. Individuals derive utility from consumption, c, and disutility from supplying taxable income, z. Preferences are smoothly distributed and quasi-linear; the optimal choice of z is independent of the level of non-labor income. Before a hypothetical reform all individuals are taxed at the proportional rate τ 1. After the reform individuals with taxable income z k will face a marginal tax rate of τ 2 > τ 1, whereas individuals with z < k still face the marginal tax rate τ 1. 37

46 CONSEQUENCES OF THE REFORM 1. The taxable income distribution to the left of k is unaffected. 2. All individuals who before the reform reported taxable incomes with z > k will reduce their taxable income in response to the tax increase. 3. We will observe a spike in the income distribution. The specific mass of taxpayers B = k+ z k h 0 (z)dz, will move to k where [k, k + z] is the interval of taxpayers who choose to locate at the kink after the reform. 38

47 THE BUNCHING FORMULA From the definition of B and the definition of the compensated elasticity it follows that: ẽ(k) = B( z) k h 0 (ξ) (1 τ) (1 τ 1 ) = b k (1 τ) (1 τ 1 ) (3), where the excess mass b = B needs to be estimated. h 0 (ξ) When the tax change is small ẽ(k) non-parametrically identifies the compensated ETI. In fact, for small tax changes it is not needed to assume away income effects. Bastani and Selin (2014, JPUBE) show that income effects, in practice, do not bias the estimate of the compensated elasticity even if the tax change is large. 39

48 184 American Economic Journal: economic policy AUgust 2010 Panel A. Indifference curves and bunching Individual L indifference curve Individual H indifference curves After-tax income c = z T(z) Slope 1 t Slope 1 t dt Individual L chooses z* before and after reform Individual H chooses z*+ dz* before and z* after reform dz*/z* = e dt/(1 t) with e compensated elasticity Source: Saez (2010), p. 184 z* z*+ dz* Before tax income z Panel B. Density distributions and bunching

49 Slope 1 t z* z*+ dz* Panel B. Density distributions and bunching Before tax income z Density distribution Pre-reform incomes between z* and z*+ dz* bunch at z* after reform After reform density Before reform density Source: Saez (2010), p. 184 z* z*+ dz* Before tax income z Figure 1. Bunching Theory Notes: Panel A displays the effect on earnings choices of introducing a (small) kink in the budget set by increasing the tax rate t by dt above income level z*. Individual L who chooses z* before the reform stays at z* after the reform.

50 BUNCHING AT KINKS (SAEZ AEJ-EP 10) 1) Uses individual tax return micro data (IRS public use files) from 1960 to ) Advantage of dataset over survey data: very little measurement error 3) Finds bunching around: a) First kink point of the Earned Income Tax Credit (EITC), especially for self-employed b) At threshold of the first tax bracket where tax liability starts, especially in the 1960s when this point was very stable 4) However, no bunching observed around all other kink points 41

51 Notes: The figure displays the kernel density of earnings for wage earners (those with no self-employment earnings) and for the self-employed (those with nonzero self employment earnings). Panel A reports the density for tax filers with one dependent child and panel B for tax filers with two or more dependent children. The charts include all years The bandwidth is $400 in all kernel density estimations. The fraction self-employed in 16.1 percent and 20.5 percent in the population depicted on panels A and B (in the data sample, the unweighted fraction self-employed is 32 percent and 40 percent). We display in dotted vertical lines around the first kink point the three bands used for the elasticity estimation with δ = $1,500. Earnin 0 0 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000 Earnings (2008 $) Panel B. Two or more children Earnings density Wage earners Self-employed EIC amount 2,000 1,000 5,000 4,000 3,000 2,000 EIC amount ($) EIC a 1,000 0 Source: Saez (2010), p ,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000 Earnings (2008 $) Figure 4. Earnings Density and the EITC: Wage Earners versus Self-Employed

52 BASTANI AND SELIN (JPUBE 2014) Bastani and Selin (2014) exploit population wide register data from Sweden Very large kink point in the Swedish tax schedule. During the period of study the change in the log net-of-tax rate reached a maximum value of 45.6%. Bastani and Selin find no bunching among wage earners. Selfemployed do bunch, but the implied compensated elasticity is low (0.02). 43

53 Frequency (all individuals) (b) Wage Earners, Total Sample, Excess mass (b) = Standard error = Implied elasticity = Taxable Income Relative to First Central Government Kink (1000s SEK)

54 Frequency (self employed) (a) All Self Employed, Excess mass (b) = Standard error = Implied elasticity = Taxable Income Relative First Central Government Kink (1000s SEK)

55 KLEVEN AND WASEEM (QJE 2013) Notch: a discontinuity in the budget set where the tax liability increases Increase in the average tax rate at the notch. Standard theory implies 1) bunching at the notch. 2) a hole in the income distribution to the right of the notch. Kleven and Waseem (QJE 2013) develop a method to estimate both the structural elasticity and optimizing frictions on population wide data from Pakistan. A lot of bunching, but structural elasticities are low. Many individuals in regions that should be empty This is due to frictions. 46

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58 WHY NOT MORE BUNCHING AT KINKS? 1) True intensive elasticity of response may be small 2) Randomness in income generation process: Saez, 2002 shows that year-to-year income variation too small to erase bunching if elasticity is large 3) Information and salience: a) Liebman and Zeckhauser: Schmeduling (behavioral model where individuals confuse MTR with average tax rate) b) Chetty and Saez AEJ:AP 13: information significantly affects bunching in EITC field experiment 4) Frictions: Adjustment costs and institutional constraints (Chetty et al. QJE 11) 49

59 SUMMARY OF TODAY S LECTURE Under some conditions the taxable income elasticity is a sufficient statistic for welfare evaluation. The first studies, e.g. Feldstein (1995), found large elasticities, often in excess of 1. Subsequent studies found smaller elasticities; estimates are often in the range 0 to 0.4. Problematic to compare income growth of different income groups; both with cross section and panel data. Bunching analysis produces elasticity estimates close to zero and casts doubts on the standard static labor supply/taxable income model. 50

60 REFERENCES Alstadsæter, A. and M. Jacob (2012) Income Shifting in Sweden. An empirical evaluation of the 3:12 rules. Report to the Expert Group for Public Economics, Ministry of Finance, Stockholm, (web) Blomquist, S. and H. Selin Hourly wage rate and taxable labor income responsiveness to changes in marginal tax rates, Journal of Public Economics, Vol. 94, 2010, Chetty, R., J. Friedman, T. Olsen and L. Pistaferri Adjustment Costs, Firms Responses, and Micro vs. Macro Labor Supply Elasticities: Evidence from Danish Tax Records, Quarterly Journal of Economics, 126(2), 2011, (web) Feldstein, M. The Effect of Marginal Tax Rates on Taxable Income: A Panel Study of the 1986 Tax Reform Act,, Journal of Political Economy, Vol. 103, 1995, (web) Feldstein, M. Tax Avoidance and the Deadweight Loss of the Income Tax, Review of Economics and Statistics, Vol. 81, 1999, (web) Gordon, R.H. and J. Slemrod Are Real Responses to Taxes Simply Income Shifting Between Corporate and Personal Tax Bases?, NBER Working Paper No. 6576, (web)

61 Gruber, J. and E. Saez The Elasticity of Taxable Income: Evidence and Implications, Journal of Public Economics, Vol. 84, 2002, (web) Imbens, G.W., D.B. Rubin and B.I. Sacerdote Estimating the Effect of Unearned Income on Labor Earnings, Savings, and Consumption: Evidence from a Survey of Lottery Players. American Economic Review, 91(4), 2001, Kleven, Henrik and Esben Schultz Estimating Taxable Income Responses using Danish Tax Reforms, American Economic Journal: Economic Policy, 2014 (web) Kleven, Henrik and Mazhar Waseem Tax Notches in Pakistan: Tax Evasion, Real Responses, and Income Shifting, forthcoming Quarterly Journal of Economics, 2013 (web) Ito, Koichiro. Do Consumers Respond to Marginal or Average Price? Evidence from Nonlinear Electricity Pricing, NBER Working Paper No , 2012, American Economic Review, 2014 (web) Lindsey, L. Individual Taxpayer Response to Tax Cuts, : With Implications for the Revenue Maximizing Tax Rate, Journal of Public Economics, 33, 1987, (web) Pirttilä, Jukka and Håkan Selin, Income shifting within a dual income tax system: evidence from the Finnish tax reform, Scandinavian Journal of Economics, 113(1), , 2011a. (web)

62 Pirttilä, Jukka and Håkan Selin, Tax Policy and Employment. How Does the Swedish System Fare UCFS working paper 2011:2, 2011b. (web) Saez, E. Using Elasticities to Derive Optimal Income Tax Rates, Review of Economics Studies, Vol. 68, 2001, (web) Saez, E., J. Slemrod, and S. Giertz The Elasticity of Taxable Income with Respect to Marginal Tax Rates: A Critical Review, Journal of Economic Literature 50(1), 2012, (web) Slemrod, J. and W. Kopczuk The Optimal Elasticity of Taxable Income, Journal of Public Economics 84(1), 2002,

63 APPENDIX: SOLUTION TO MEAN REVERSION PROBLEM Blomquist and Selin (2010): Let m be the middle year of the year difference t k and t (assuming that m = t k/2 is an integer). Suppose that ɛ it follows an AR(1) process: ɛ it = ρɛ i,t 1 + υ i,t, where ρ < 1 and υ i,t N(0, σ 2 υ). Given this assumption, we know that ɛ i,t m and (ɛ it ɛ it k ) will have a joint normal distribution. 51

64 APPENDIX The covariance between ɛ i,t m and (ɛ it ɛ it k ) can be expressed as = E E ɛ i,t m(ρ m ɛ i,t m + (ρm ɛ i,t k + E[ɛ i,t m (ɛ it ɛ i,t k )] = m 1 q=0 m 1 q=0 ρ q υ t q ) ρ q υ t m q )ɛ i,t k = ρ m σ 2 ɛ ρ m σ 2 ɛ = 0 (4) 52

65 Hence, ɛ i,t m and (ɛ it ɛ i,t k ) will be uncorrelated. Accordingly, functions of ɛ i,t m and (ɛ it ɛ i,t k ) will be stochastically independent.