Assessing Tax Reforms when Human Capital is

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Abstract. How does the long term tax incidence of capital and labor income taxes change within a computable life cycle model with allowance for endogenous ...

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Assessing Tax Reforms when Human Capital is Endogenous Morten I. Lau Working Paper No. 2000-3

ISSN 1397-3843

Assessing Tax Reforms when Human Capital is Endogenous

by

Morten I. Lau*

Abstract. How does the long term tax incidence of capital and labor income taxes change within a computable life cycle model with allowance for endogenous formation of human capital? The presence of life cycle time allocation provides one explanation for the existence and persistence of wage differentials over the life cycle. The goal is to evaluate how capital and labor income taxes may affect human capital accumulation, employment, retirement, and welfare within this framework. The present study departs from previous analyses with respect to the allocation of time over the life cycle. In particular, the point in time at which the individual retires is endogenous in the present analysis. The results indicate that it may be more efficient to have a lower tax rate on capital income compared to the tax rate on labor income instead of one comprehensive tax rate on all types of income.

January 1999

* The MobiDK Project, Danish Ministry of Business and Industry, Copenhagen, Denmark, and Institute of Economics, University of Copenhagen, Denmark. I am grateful to Thomas F. Rutherford, Glenn W. Harrison, Tobias N. Rasmussen, and a referee for helpful suggestions and comments. None of the views presented here should be attributed to any of my employers. The paper is accepted for publication in Glenn W. Harrison, Svend Erik H. Jensen, and Lars H. Pedersen (eds.) Using Dynamic General Equilibrium Models for policy Analysis (Amsterdam: North Holland, 1999). 1

1. Introduction Although human capital formation is an important factor in the labor supply choice, it has received little attention in the literature on dynamic tax incidence. How does the long term incidence of capital and labor income taxes change within a computable life cycle model with allowance for endogenous formation of human capital? The presence of life cycle time allocation decisions and endogenous human capital formation provides one explanation for the existence and persistence of wage differentials over the life cycle. The goal is to evaluate how capital and labor income taxes may affect human capital accumulation, employment, retirement and welfare within this framework. The present study departs in two respects from previous analyses with respect to the allocation of time over the life cycle. First, analytical studies of lifetime allocation decisions typically assume that the retirement age is fixed. This assumption makes these models easier to solve analytically. On the other hand, existing computable life cycle models with endogenous human capital accumulation do not determine the point in time at which the individual retires. In contrast, the point in time at which the individual retires is determined endogenously in the present analysis. Hence, time is divided between work and training during the first part of the individual’s life, and between work and leisure during the second part. Second, the labor supply curve in the present framework seems realistic. McGratten and Rogerson [1998] estimate the average individual labor supply curve over the life cycle using American data, and the model in the present study provides a reasonable approximation of the estimated individual labor supply curve in their empirical study. The public finance literature has identified several channels for tax incidence to be influenced by human capital formation. Heckman [1976] demonstrates that income taxes encourage human capital accumulation. Using a life cycle model of labor supply, earnings, consumption, and non-market benefits of education, Heckman shows that a comprehensive income tax tends to discriminate against financial investment and favor investment in human capital. The logic is that an increase in the income tax reduces the return to both human and physical capital, but lowers only the opportunity cost to investment in human capital. This analysis is extended by Kotlikoff and Summers [1979], who consider the effects of human capital 2

taxation in a general equilibrium framework. They find that conventional conclusions with respect to long run tax incidence change when endogenous retirement and human capital formation is allowed for. In particular, the capital income tax may be shifted to labor through human capital variation, even if consumption is unaffected by changes in the interest rate, because a decrease in the net interest rate increases the present value of the return to education and leads to an increase in human capital accumulation. Building on this literature, Nielsen and Sørensen [1997] show that a tax system with progressive tax rates on labor income combined with a proportional tax rate on capital income may be defended on efficiency grounds, given a fixed level of public expenditures and the absence of lump sum taxes. Relatively few quantitative analyses include human capital formation. Driffill and Rosen [1983] use a life cycle model to obtain partial equilibrium estimates of the distortionary effects of moving from a comprehensive income tax system to taxation of consumption, and they find that the welfare effects of adopting a consumption based tax system instead of income taxation may be large. However, partial equilibrium analyses ignore the general equilibrium response of factor markets. Addressing the same question, Perroni [1995] finds that the welfare effects are reduced when a general equilibrium framework is applied, since changes in the capital-labor ratio affect the returns to physical and human capital. The general equilibrium effect partially counteracts the inter-asset substitution effect in the partial equilibrium framework and reduces the changes in net factor prices. Heckman et alii [1998] formulate and estimate a dynamic general equilibrium model with exogenous retirement and heterogenous human capital accumulation to study the impacts on skill formation of proposals to switch from progressive taxes to flat income and consumption taxes. They show that a revenue-neutral flattening of the labor income tax, holding the flat tax on capital income constant, has a positive impact on both high school and college attendance, which corresponds to the analytical findings by Nielsen and Sørensen [1997]. In this analysis attention will be given to the distortionary effects of capital and labor income taxes in the long run. A single tax experiment is carried out. The capital income tax is reduced, and the labor income tax is determined endogenously to balance the public budget. The introduction of the tax reform has a positive effect on investment in physical capital while reducing investment in human capital. Hence, employment falls, and people retire earlier. The 3

results also indicate that it may be more efficient to have differentiat tax rates on capital and labor income, instead of one comprehensive tax rate on all types of income. In particular, it may be more efficient to have a lower tax rate on capital income compared to the tax rate on labor income, which corresponds to the theoretical findings above. This result is based on a secondbest argument, relying on the assumption that the government cannot levy lump sum taxes in order to finance public expenditures. The time allocation decision for each individual is described in Section 2, and the general equilibrium model determining the steady state behavior is outlined in Section 3. Section 4 describes the calibration of the model, and the effects of replacing a capital income tax with a labor income tax are then discussed in Section 5. Finally, Section 6 concludes.

2. Allocation of Time over the Life Cycle Including human capital accumulation in a life cycle model provides a way of explaining the existence of wage differentials over the life cycle. New human capital can be produced through schooling, formal or informal on-the-job training, or as a by-product of experience (learning by doing) together refered to henceforth as training. Investment in human capital is viewed here as a private investment decision by households. Hence, I do not distinguish between various forms of human capital production. The life cycle of the representative household is divided into 60 periods, each period representing one year. I assume that each individual begins his active life at age 20 and dies at age 80. The consumer spends his time working, learning or playing. When human capital is endogenous, private households can choose between two types of investment: they can either invest in financial assets, or invest in human capital. Since investment in human capital is costly and specific to each individual, all investment in human capital is concentrated in the beginning of the life cycle and ends when the representative individual begins to retire from the labor market. Individuals are identical across generations, and I assume that households are endowed with perfect foresight. Perfect competition prevails on each market, so output and factor prices are given to all agents in the model. The intertemporal maximization problem by the representative household is formulated as a primal, non-linear programming problem (NLP). The 4

corresponding complementarity problem is derived in the Appendix. The complementarity problem can be interpreted as a first-order necessary condition for the NLP model, and it provides some insight about the corresponding shadow prices. The primal NLP formulation is based on an explicit representation of the utility function for the representative household. The representative household maximizes the present value of lifetime utility, U, subject to a time endowment constraint, the law of motion with respect to human capital, and an intertemporal budget constraint. Hence, each household maximizes:

1   c t( 1 −γ )   U =∑  + α ⋅ ls t  t ⋅   1 − γ   t =0 (1 + ρ ) 59

where ct is consumption of goods in period t, lst is leisure in period t, ' is the rate of time preference,  is a weight parameter for leisure, and is the inverse of the intertemporal elasticity of substitution, ). Since the instantaneous utility function is specified as an additively separable function in consumption goods and leisure, the demand for leisure does not have a direct effect on the demand for goods. However, the supply of labor is not perfectly elastic because the instantaneous utility function exhibits decreasing marginal utility with respect to consumption goods. Contrary to popular constant elasticity of substitution formulations of the utility function, the additively separable utility function allows the point in time at which the individual retires to be endogenous. Retirement begins when the individual starts to demand a positive amount of leisure. In each period of his active life the individual allocates his endowment of time across leisure, work effort, and education. Total use of time cannot exceed the total endowment of time, which is equal to 3 in each time period, i.e.

3 ≥ ls t + q t + s t where qt is work effort in period t, and st is training in period t. Wage earnings in period t, which is denoted by et , is a linear function of the wage rate. 5

The supply of labor services to the labor market by each individual is a function of work effort and the accumulated individual human capital stock, ht. The total wage earnings at time t for an individual is:

e t = w t ⋅ ( h tβ ⋅ q t( 1 − β ) )

where wt is the wage rate in period t, and  denotes the value share of human capital in the production of labor services. Using a constant-returns-to-scale technology for labor services, human capital is treated in a similar way as physical capital. Gross investment in human capital equals training, and the stock of human capital in period t+1 is equal to the stock of human capital at the start of period t less depreciation plus training in period t. Hence, the stock of human capital evolves according to the following law of motion:

h t +1 = ( 1 − δ H ) ⋅ h t + s t

where H is the rate of depreciation with respect to human capital. The initial stock of human capital is positive, so each individual enters his active life with a minimum of skills. I assume that each individual is born without financial assets, so the lifetime budget constraint states that the present value of lifetime expenditures on consumption goods cannot exceed the present value of wage income over the life cycle: t =5 9

t =5 9 1 1 ∑ t ⋅ ct t ⋅ et ≥ ∑ + + r r ( ) ( ) 1 1 t =0 t =0

where r is the interest rate. I have implicitly assumed that the price of goods is equal to 1.

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3. General Equilibrium The overlapping generations model represents a closed economy and is used to simulate the steady state general equilibrium effects of equal yield tax reforms. Since population growth is not important to the analysis, the population is kept constant and normalized at unity. Prices and aggregate quantities are constant in steady state, so the variables do not carry time indices unless otherwise indicated. The consumption and investment decisions by the representative household reflect the behavior of current generations in the steady state. It is therefore a simple task to derive the aggregate supply of labor services to the labor market and the aggregate private demand for leisure, education and consumption in the steady state. For example, the aggregate supply of labor services to the labor market is equal to the sum of the supply of labor services over the life cycle for the representative household:

59

L=

∑ (h t =0

β t

⋅ q t( 1 − β ) )

where L is aggregate supply of labor services to the labor market in steady state. Aggregate output is produced by a Cobb-Douglas technology, so the elasticity of substitution between physical capital and labor services is equal to 1. Total output in steady state is therefore determined by:

Y = K ϕ ⋅ L ( 1 −ϕ ) where Y is aggregate output, and K is the aggregate stock of physical capital. The capital stock in period t equals the capital stock at the start of the previous period less depreciation plus investment in the previous period. Since the capital stock is constant in the steady state, the level of gross investment in the steady state is given by:

I = δK ⋅K 7

where I is gross investment in physical capital, and K is the rate of depreciation with respect to physical capital. The price of output is determined by the market clearing condition for goods. Aggregate output is either invested or consumed by private households and the public sector, i.e. the market clearing condition for goods is:

Y ≥C +I +G where C is total private demand for goods, and G is public demand for goods. The public sector levies taxes on labor and capital income and spends the tax revenue on consumption goods. Moreover, the public budget is balanced in each period, so there is no public debt financing in the model. In the absence of public debt, aggregate net savings by private households is equal to the value of the physical capital stock in the economy:

5 9 ( 5 9 − tl )

∑∑ tl = 0

t =0

1 ⋅ [c ( t + tl ) − e ( t + tl ) ] ≥ P K ⋅ K (1 + r ) t

where PK is the price of capital in the steady state.1 The left hand side is equal to the net value of accumulated assets by current generations in period 0, and the right hand side is the value of capital. The domestic interest rate is determined by this market clearing condition. The model is now closed and the description of the calibration follows next.

1

The price of capital in the steady state is equal to (1+r), where r is the interest rate in the steady state. The opportunity cost of one unit of physical capital is equal to the cost of one unit of investment goods plus foregone interest on savings. I implicitly assume that a bond market is present in the economy. Since each individual must end up with zero net debt and the capital stock must be owned by someone in the economy, aggregate net savings are equal to the value of the physical capital stock. 8

4. Calibration The model is calibrated to the data set presented in Table 1. The first step in the calibration procedure is to specify the allocation of time over the life cycle for the representative household. When calibrating the time allocation decision by the representative household, special attention is given to the labor supply curve. In a recent study, McGratten and Rogerson [1998] estimate individual labor supply curves across different cohorts using U.S. decennial censuses from 1950 to 1990. Following this study, the parameters determining the allocation of time over the life cycle are chosen such that the labor supply profile over the life cycle for the representative household resembles the estimated labor supply profile for recent generations. Figure 1 illustrates the allocation of time over the life cycle for each individual in the initial steady state. Time is allocated between training and work during the first 32 years of the individual’s active life. During the last 28 years of the life cycle, time is divided between work and leisure, i.e. each individual begins to retire from the labor market when he is 52 years old. The retirement age is endogenous in this model. Time spent on work increases sharply during the first 10 years of the individual’s active life. It then stays constant for almost 20 years and begins to fall when he retires from the labor market. The model is calibrated such that the capital value share in the production of goods is equal to one third, so the labor-capital ratio is equal to 2. This ratio implies that the level of investment in percent of GDP is equal to 16 percent, given a net interest rate of 5 percent and a depreciation rate with respect to physical capital of 10 percent. To get a sufficiently high saving rate, I have used a value of 4.6 percent for the rate of time preference. Compared to Auerbach and Kotlikoff [1987] and Perroni [1995], a rate of time preference of 4.6 percent may seem excessive. However, a recent experimental study using Danish data by Harrison et alii [1998] suggests that the rate of time preference may be considerably higher. Most applied studies apply an intertemporal elasticity of substitution between 0.5 and 1. I have used a value of 0.67 for the intertemporal elasticity of substitution. Two tax rates are included in the model: a labor income tax and a capital income tax. To illustrate the distortionary effects of the two types of income taxes, I assume that the two tax rates initially have the same value: the government levies a 33 percent gross tax rate on both labor and 9

capital income. This corresponds to a net tax rate of approximately 25 percent on both types of income. Accordingly, government spending in percent of GDP is equal to 25 percent in the initial steady state.

5. Reducing the Tax Rate on Capital Income The long run behavior of the overlapping generations model is illustrated in this section using a specific policy application.2 In particular, the tax rate on capital income is reduced successively by 2 percent per experiment from the initial level of 33 percent to 25 percent. In order to abstract from welfare effects due to changes in the supply of public services, the level of government expenditure is fixed, and the tax rate on labor income is determined endogenously to balance the public budget, see Table 2. To assess the long run impacts of the policy reform, the simulated policy changes are compared with a baseline simulation reflecting the initial steady state. Figures 2-6 illustrate the relative change for a given variable compared to the Business as Usual (BaU) scenario. Replacing the capital income tax with a labor income tax has positive, but small, effects in the long run on income and consumption. Figure 2 illustrates that GDP increases by almost 1.5 percent and private consumption increases by approximately 0.5 percent when the tax rate on capital income is reduced to 25 percent. The tax reform leads to a significant increase in the capital intensity – the physical capital stock increases while employment falls – and drives down the gross return on physical capital. On the other hand, the increase in the capital intensity raises the gross wage rate. Figure 3 illustrates that the increase in the gross wage rate is large enough to offset the higher tax on labor income, so the net returns on both capital and labor increase when the capital income tax is replaced by a labor income tax. The increase in the net returns to capital and labor have opposite effects on retirement and training. An increase in the net interest rate reduces the present value of the return to education and leads to a reduction in human capital accumulation. This reduction in training makes leisure

2

I have used GAMS/MPSGE to solve the model numerically. Rutherford [1995, 1998] documents this modelling system and software. 10

more attractive in the last part of the life cycle. On the other hand, an increase in the net wage rate increases the price of leisure, which leads to a decrease in the demand for leisure in the second half of the individual’s life. Postponing retirement increases the amortization period for human capital investment and leads to more training. Figure 4 illustrates that the tax reform leads to earlier retirement when the capital income tax is reduced by 8 percent – retirement begins at age 51 after the tax reform is introduced. Accordingly, the level of training falls. Training is reduced by one year and is not affected during the intermediate years. Figure 5 depicts the effects on training. Figure 6 illustrates the effect on individual labor supply over the life cycle of replacing the capital income tax with a labor income tax. The individual labor supply increases just before retirement begins due to the decrease in human capital accumulation. Since the tax reform encourages earlier retirement, the individual labor supply falls during the last part of the life cycle. As mentioned above, the tax reform has a negative effect on aggregate employment.

Welfare Effects The equivalent variation measure is used to assess the welfare effects of the tax reform. This is implemented by calculating the percentage change in lifetime earnings (in base year prices) necessary to yield the welfare level reached in the new steady state, i.e.

U ( E 0 ⋅ ( 1 + τ ), r0 , w 0 ) = U ( E 1 , r1 , w 1 ) where E is the present value of lifetime earnings, and the subscripts 0 and 1 denote the initial and the new steady state, respectively. The parameter - is a measure of the change in welfare between the two steady states. This welfare measure can be compared across different steady state paths and is applicable for changes of any size and not only differential approximations. The first row of Table 3 presents the welfare effects across different combinations of tax rates on labor and capital income. The results indicate that it is more efficient to have different tax rates on labor and capital income instead of a uniform tax rate on all types of income, provided that the government can not use a lump sum tax to finance public services. In particular, 11

it is more efficient to reduce the capital income tax and increase the tax rate on labor income when the initial tax system is based on a comprehensive income tax. However, the welfare effects are rather small – in the long term welfare increases by only 0.5 percent when the (gross) tax rate on capital income is reduced from 33 percent to 25 percent. Although the steady state reflects the long-run position of the economy, the positive welfare effect may come at the expense of earlier generations. The only way to consider this inter-generational redistribution is to examine the dynamic transition from the initial steady state to the final steady state. This result suggests that a comprehensive income tax may favor investment in human capital over financial investment, which corresponds to the observations by Heckman [1976] and Driffill and Rosen [1983]. Increasing the proportional tax rate on labor income does not affect the private return to human capital investment because the opportunity cost of education is reduced by the same rate. Hence, a proportional labor income tax implies that the private and social rates of return on human capital investment are identical. However, the proportional tax rate on labor income distorts the intra-temporal allocation between private consumption of goods and leisure. On the other hand, a capital income tax reduces the private return to financial saving compared to the social return. The results indicate that the inter-temporal distortion due to the capital income tax is more significant than the intra-temporal distortion due to the labor income tax, and a shift in the tax burden from capital to labor could lead to a more efficient tax system. To gain insight about the behavior of the model, it is useful to consider two special cases. In the first case, human capital accumulation is held constant. The same tax experiments as before are carried out, and the results in Table 3 show that welfare increases marginally compared to the previous scenario. Holding human capital accumulation constant implies that the effect on retirement is relatively small, and the negative effect on employment is reduced compared to the previous scenario. The increase in the tax rate on labor income is therefore smaller. However, the tax burden on labor increases when investment in human capital is constant because the supply of labor is less responsive to changes in the net wage rate. Moreover, as Kotlikoff and Summers [1979] point out, labor bears a smaller part of the tax on capital when human capital accumulation is less responsive to changes in the capital income tax. So the welfare gain to labor of reducing the capital income tax is reduced. Hence labor bears a larger part of the labor income tax, and a 12

smaller part of the tax on capital is shifted to labor when human capital accumulation is constant. These effects are more pronounced when leisure instead of human capital formation is held constant. The results in Table 3 show that welfare increases when leisure is held constant compared to the situation where training in held constant. In this case, the labor income tax does not distort the intra-temporal allocation between consumption goods and leisure. Hence, the inter-temporal distortion due to the capital income tax is reduced and replaced by a neutral labor income tax, and welfare increases compared to the reference scenario. This result suggests that alternative model specifications with a small labor supply elasticity will support the findings in the present analysis. Finally, I consider the welfare effects of the same tax reform when the intertemporal elasticity of substitution for consumption, ), is increased from 0.67 to 1.1. When calibrating the model to a higher intertemporal elasticity of substitution, it is only necessary to change the weight parameter with respect to leisure in the utility function in order to arrive at roughly the same initial steady state equilibrium – the corresponding value is  = 0.34. From the previous discussion, it is clear that the welfare effect of replacing the capital income tax with a labor income tax increases when the intertemporal elasticity of substitution is higher. A higher intertemporal elasticity of substitution raises the interest rate elasticity of savings, thereby increasing the distortionary effect of the capital income tax. So, the qualitative results in the reference scenario still hold when the intertemporal elasticity of substitution is increased.

6. Conclusions A single tax experiment is carried out using a computable multi-period life cycle model with endogenous human capital formation. The capital income tax is reduced, and the labor income tax is then determined endogenously to balance the public budget. The introduction of the tax reform has a positive effect on investment in physical capital while reducing investment in human capital. Hence, employment falls and people retire earlier. The results also indicate that it may be more efficient in the long run to have different tax rates on capital and labor instead of a uniform tax rate on all types of income. In particular, it may be more efficient to have a lower tax rate on capital income compared to the labor income tax, and thereby shift some of the tax burden 13

from capital to labor. These qualitative findings do not change when the interest elasticity of savings is increased. Since the analysis is based on a very stylized model, the specific quantitative results should not be taken too literally. Moreover, the model does not examine the dynamic transition. Although the steady state reflects the long-run position of the economy, future generations may benefit because earlier generations suffered. The only way to consider this inter-generational redistribution is to examine the dynamic transition from the initial steady state to the final steady state. When perfect competition prevails on all markets, theoretical analyses suggest that a comprehensive income tax tends to favor investment in human capital relative to investment in financial assets. The present analysis confirms these findings and also indicates that a more efficient tax system reduces the level of investment in human capital and leads to lower employment. However, many countries experience high rates of unemployment and rigid wage structures. Since flat wage structures tend to reduce the incentive for human capital investment, it could be desirable to shift the tax burden from labor to capital and encourage investment in human capital.

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References Auerbach, Alan J. and Laurence J. Kotlikoff, Dynamic Fiscal Policy (Cambridge, UK: Cambridge University Press, 1987).

Driffill, E. John and Harvey S. Rosen, “Taxation and Excess Burden: A Life-Cycle Perspective,” International Economic Review, 24(3), 1983, 671-683.

Harrison, Glenn W., Morten I. Lau and Melonie B. Williams, “Estimating Individual Discount Rates in Denmark,” Unpublished Manuscript, MobiDK Project, Ministry of Business and Industry, Copenhagen, Denmark, 1998.

Heckman, James J., “A Life Cycle Model of Earnings, Learning and Consumption,” Journal of Political Economy, 84, 1976, S11-S44.

Heckman, James J., Lance Lochner and Christopher Taber, “Tax Policy and Human Capital Formation,” Working Paper 6462, National Bureau of Economic Research, 1998.

Kotlikoff, Laurence J. and Lawrence H. Summers, “Tax Incidence in a Life Cycle Model with Variable Labour Supply,” Quarterly Journal of Economics, 93(4), 1979, 705-718.

McGratten Ellen R. and Richard Rogerson, “Changes in Hours Worked Since 1950,” Quarterly Review, Winter 1998, Federal Reserve Bank of Minneapolis.

Nielsen, Søren B. and Peter B. Sørensen, “On the Optimality of the Nordic System of Dual Income Taxation,” Journal of Public Economics, 63, 1997, 311-329.

Perroni, Carlo, “Assessing the Dynamic Efficiency Gains of Tax Reform when Human Capital is Endogenous,” International Economic Review, 36(4), 1995, 907-925.

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Rutherford, Thomas F., “Extensions of GAMS for Complementarity Problems Arising in Applied Economics,” Journal of Economic Dynamics and Control, 19, 1995, 1299-1324.

Rutherford, Thomas F., “Applied General Equilibrium Modeling using MPSGE as a GAMS Subsystem,” Computational Economics, 1998 forthcoming.

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Tables and Figures Table 1. Parameter Values and Initial Tax Rates r

Annual Interest Rate after Tax

0.050

w

Wage Rate after Tax

1.000

'

Time Preference Rate

0.046

)

Intertemporal Elasticity of Substitution

0.667

H

K

Depreciation Rate for Human Capital

0.050

Depreciation Rate for Physical Capital

0.100



Value Share of Human Capital in the Production of Labor Services

0.300

Q

Value Share of Physical Capital in Production of Goods

0.320



Weight Parameter wrt. Leisure in Utility Function

0.175

h0

Initial Human Capital Stock

3.000

TaxL

Gross Tax Rate on Labor Income

0.330

TaxK

Gross Tax Rate on Capital Income

0.330

Table 2. Tax Rates Across Experiments Tax Rate on Capital Income

33.0%

31.0%

29.0%

27.0%

25.0%

Tax Rate on Labor Income

33.0%

33.7%

34.5%

35.3%

36.2%

Table 3. Welfare Gains (Percentage of Lifetime Income) Tax Rate on Capital Income

31%

29%

27%

25%

Reference Case

0.15

0.29

0.41

0.49

Exogenous Human Capital

0.16

0.30

0.43

0.54

Exogenous Leisure

0.20

0.39

0.56

0.72

Intertemporal Elast. of Substitution () = 1.1)

0.18

0.36

0.53

0.67

17

100 90 80 70 60 50

Work

40

Training

30

Retirement

20 10 0 20

30

40

50 Age

60

70

Figure 1. Allocation of Time Over the Life Cycle. Time is allocated between training and work during the first 32 years of the individual’s active life. The individual retires at age 52, and during the last 28 years of his life time is divided between work and leisure. The retirement age is endogeous in this model.

8 7 6 5 4

Physical Capital

3

Employment GDP

2

Private Consumption

1 0 -1 -2 TaxK=31%

TaxK=29%

TaxK=27%

TaxK=25%

Figure 2. Factor Demands, Output and Consumption (Percentage Change from BaU). Replacing the capital income tax with a labor income tax has positive, but small, effects in the long run on income and private consumption. The tax reform leads to a significant increase in the capital intensity. Demand for capital increases while employment falls.

0.8 0.7 0.6 0.5

Net Wage Rate Net Interest Rate

0.4 0.3 0.2 0.1 TaxK=31%

TaxK=29%

TaxK=27%

TaxK=25%

Figure 3. Net Factor Prices (Percentage Change from BaU). The net returns on both capital and labor increase when the capital income tax is replaced with a labor income tax. The increase in the gross wage rate is large enough to offset the higher tax on labor income.

90 80 70 60 50 Capital Income Tax: 33% 40

Capital Income Tax: 25%

30 20 10 0 60

70 Age

Figure 4. Individual Retirement (Percent of Time Endowment). The tax reform leads to earlier retirement when the capital income tax is reduced by 8% and the labor income tax is adjusted to balance the public budget. Retirement begins at age 51 after the tax reform is introduced.

80 70 60 50 Capital Income Tax: 33%

40

Capital Income Tax: 25% 30 20 10 0 20

30

40

50

Figure 5. Individual Training (Percent of Time Endowment). Earlier retirement reduces the amortisation period for human capital investment. Hence, the individual level of training falls when the capital income tax is replaced by a labor income tax.

100 90 80 70 60

Capital Income Tax: 33%

50

Capital Income Tax: 25%

40 30 20 10 20

30

40

50

60

70

Figure 6. Individual Labor Supply (Percent of Time Endowment). The tax reform has a negative effect on investment in human capital which leads to an increase in the individual supply of labor just before retirement begins. Since the tax reform encourages earlier retirement, the individual labor supply falls during the last part of the life cycle.

Appendix A. MCP Formulation of Household Behavior When analyzing the model, it is useful to derive the complementarity problem corresponding to the NLP formulation. The complementarity problem can be interpreted as a first-order necessary condition for the NLP model, and it provides some insight about shadow prices. The representative household maximizes the present value of lifetime utility subject to an intertemporal budget constraint, the law of motion with respect to human capital, and a time endowment constraint. Using the same notation as before, each household maximizes overall utility, which is given by:

1   c t( 1 −γ )   U =∑  + α ⋅ ls t  t ⋅   1 − γ   t =0 (1 + ρ ) 59

This formulation of the utility function assumes that the representative household’s utility at time 0 is a weighted sum of future flows of utility over the life cycle. The complementarity problem is derived by forming the Lagrangian and differentiating with respect to the instrumental variables and the state variable. Given the optimization problem for the representative household, the Lagrangian to the NLP formulation is:

L At =

59

∑ t =0

  c t( 1 −γ )    59  1 1 ( 1 −β ) β   α λ ⋅ + ⋅ + ⋅ ⋅ − ls h q c ( )  ∑   t t t t   t =0 (1 + r ) t  (1 + ρ ) t   1 − γ   + ηt +1 ⋅ ((1 − δ H ) ⋅ h t + s t − h t +1 ) + ν t ⋅ ( 3 − q t − s t − ls t )

where the Lagrange multipliers can be referred to as shadow prices or shadow values of a given constraint. The first-order necessary conditions for the NLP problem can be interpreted as zero-profit conditions for corresponding production activities. Differentiating the Lagrangian with respect to consumption yields:

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1 1 ( −γ ) P ⋅ = t t ⋅ P U ⋅ ct (1 + r ) (1 + ρ )

where PU is the price of one unit of intertemporal utility, and P is the price of one unit of intertemporal consumption. The price of one unit of consumption at time t (left hand side) is equal to the marginal benefit of one unit of consumption at time t (right hand side). The demand for leisure, lst, by the representative household over the life cycle is determined by the following first-order condition:

P Lt ≥

1 ⋅ P U ⋅α (1 + ρ ) t

where PLt is the price of one unit of time at time t. If the price of one unit of time is larger than the marginal benefit of one unit of leisure, then the demand for leisure is equal to 0. In contrast, the demand for leisure is positive if this constraint is binding. Differentiating the Lagrangian with respect to work effort, qt, gives the following zeroprofit condition:

P Lt ≥

1 β t ⋅ P ⋅ (1 − β ) ⋅ ( h t / q t ) (1 + r )

This equation determines the individual labor supply over the life cycle. Hence, the individual labor supply is positive if the price of one unit of time is equal to the marginal product of one unit of labor. Note that the supply of labor services to the labor market is a function of work effort and the accumulated individual human capital stock, ht. The first-order condition with respect to education implies:

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P L t ≥ P H t +1 where PHt is the price of one unit of human capita in period t. The cost of spending one unit of time on education in period t is greater or equal to the price of human capital in period t+1, because one unit of education today provides one unit of human capital tomorrow. Since investment in human capital is costly and specific to each individual, all investment in human capital is concentrated in the beginning of the life cycle. Finally, the individual stock of human capital over the life cycle is determined by differentiating the Lagrangian with respect to the state variable:

PH t ≥

1 ( 1−β ) + P H t +1 ⋅ ( 1 − δ H ) t ⋅ P ⋅ β ⋅ ( q t / ht ) (1 + r )

The first term on the right hand side is the return to human capital in period t, and the last term on the right hand side is the net present value of human capital in the next period.

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