Tax Smoothing in a Business Cycle Model with Capital-Skill Complementarity

This paper undertakes a normative investigation of the quantitative properties of optimal tax smoothing in a business cycle model with state contingent debt, capital-skill complementarity, endogenous skill formation and stochastic shocks to public consumption as well as total factor and capital equipment productivity. Our main finding is that an empirically relevant restriction which does not allow the relative supply of skilled labour to adjust in response to aggregate shocks, significantly changes the cyclical properties of optimal labour taxes. Under a restricted relative skill supply, the government finds it optimal to adjust labour income tax rates so that the average net returns to skilled and unskilled labour hours exhibit the same dynamic behaviour as under flexible skill supply.


Introduction
The celebrated labour tax smoothing result of Barro (1979) in a partial equilibrium setting has lead to a number of important studies on optimal …scal policy over the business cycle in representative agent general equilibrium models. For example, Lucas and Stokey (1983) formalised labour tax smoothing within a complete markets neoclassical setup without capital when the government has access to state-contingent debt. Chari et al. (1994) generalised this result in a model with capital taxation and showed that Ramsey policy dictates that the labour income tax ‡uctuates very little in response to aggregate shocks and the ex ante capital income tax is approximately zero in each period.
The literature has also examined the implications of policy frictions and incomplete asset markets for optimal tax and debt policy, through a variety of restrictions to the policy instrument set, government debt and capital income taxation (see e.g. Stockman (2001), Aiyagari et al. (2002), Angeletos (2002), Buera and Nicolini (2004) and Farhi (2010)). In contrast, assuming complete asset markets and a complete instrument set, Arseneau and Chugh (2012) consider labour market frictions associated with a division of the labour force into employed and unemployed workers. Their model, with state-contingent debt but no capital, suggests that optimal labour tax volatility depends on whether wages are set e¢ ciently.
Another important division of the labour force is with respect to the type of labour services workers provide and, in particular, how these complement capital in the production process. This is especially pertinent given the empirical relevance of the wage premium accruing to skilled labour and the roles attributed to capital-skill complementarity, the relative supply of skilled labour and capital augmenting technical progress (see e.g. Katz and Murphy (1992), Krusell et al. (2000) and Hornstein et al. (2005)). In an important contribution, which also considers non-homogenous labour, Werning (2007) establishes the conditions under which optimal labour tax smoothing holds in a model with redistribution under complete asset markets when workers di¤er with respect to their productivity. However, since this research treats distinct types of labour as perfect substitutes in production, it does not capture how labour may exhibit di¤erent degrees of complementarity with capital as in e.g. Katz and Murphy (1992) and Krusell et al. (2000). Moreover, since the distribution of productivity di¤erentials is taken as given, this approach also does not account for the endogenous determination of employment type (see e.g. Matsuyama (2006), who also reviews the literature on job mobility).
In this paper we aim to contribute to the tax smoothing literature by focusing on the above two features of an economy where the labour force is divided into skilled and unskilled workers. In particular, we examine the importance of di¤erences in the complementarity between capital and skill and unskilled labour as well as the endogenous determination of the relative skill supply for Ramsey tax policy over the business cycle. Compared to Werning (2007), we focus on aggregate outcomes and abstract from redistribution incentives, by following the literature that examines a division of the labour force into two types of workers. To this end, we work with a representative household which guarantees its members'the same level of consumption (see e.g. Arseneau and Chugh (2012)). We thus stay as close as possible to the representative agent Ramsey analysis of Chari et al. (1994) and extend their model to allow for capital-skill complementarity and endogenous skill formation. 1 Our goal is thus to undertake a normative investigation of the quantitative properties of optimal taxation of capital and labour income, as well as skill-acquisition expenditure, in the presence of aggregate shocks to total factor productivity (TFP), capital equipment productivity and government spending. We assume complete asset markets, however, to capture the importance of endogenous versus …xed relative skill supply, we also consider a labour market distortion that restricts the ratio of skilled to total workers to remain constant. This extension is motivated by empirical evidence suggesting that the share of college educated or skilled workers in the data has low relative volatility and is e¤ectively uncorrelated with output over the business cycle. For example, the standard deviation of the cyclical component of this share relative to the standard deviation of output is 0.27 and its correlation with output is -0.18. 2 In our setup, the government can borrow, tax skill acquisition expenditure, capital, skilled and unskilled labour income separately, to …nance exogenous public spending. All policy instruments are allowed to be statecontingent. In this environment, the optimal taxes on labour income and skill acquisition expenditure are uniquely determined. However, as is well known, when the government has access to both state contingent debt and state contingent capital taxation, the second-best Ramsey allocations do not uniquely pin down optimal debt and capital taxes (see Chari et al. (1994)). Hence, following the literature, in this instance we discuss the properties of the ex ante capital tax rate. Moreover, we also examine the case where debt is restricted to be state uncontingent, which allows us to calculate the ex post capital tax or, if we also allow for state-contingent taxation of income from bonds, the private assets tax. 3 Our main …nding is that under capital-skill complementarity, a friction that does not allow the relative supply of skill to adjust in response to aggregate shocks, signi…cantly changes the cyclical properties of optimal labour taxes. In particular, we …rst show that under endogenous relative skill supply, the optimal labour taxes for both skilled and unskilled labour income are smooth, with the volatility of the skilled income tax being marginally lower. We also …nd that the skilled tax moves pro-cyclically with output and the unskilled tax is mildly counter-cyclical. These results are largely consistent with the literature and extend previous …ndings to a setup with capital-skill complementarity and endogenous skill supply.
However, when the relative skill supply is constrained to remain constant over the business cycle, the prescriptions for optimal policy markedly change.
In particular, we …nd that the volatility of taxes increases signi…cantly, so that the standard deviation of the e¤ective average labour income tax is about seven times higher than the perfect labour markets case, while the volatility of the skilled labour income tax is about two-and-a-half times higher than that of the unskilled labour income tax. Moreover, both taxes become strongly counter-cyclical. We show that the key to understanding these changes is that the government …nds it optimal to minimise the e¤ects of the relative skill supply distortion by keeping the marginal rates of substitution between leisure and consumption for the two types of labour at roughly the same levels as under a fully ‡exible labour market. In other words, the government adjusts labour income tax rates and thus alters the average net returns to skilled and unskilled labour hours to minimise the wedge introduced by the labour market friction.
Compared with the extension of Chari et al. (1994) undertaken by Werning (2007), our extension does not allow for redistribution. However, our results add to the …ndings in Werning (2007) in the following way. Werning (2007) shows that exogenous skill heterogeneity does not alter the basic optimal tax smoothing result for a large class of utility functions, when the assumption regarding the neoclassical production function is maintained and the di¤erent skill-adjusted labour inputs are perfect substitutes in the production function. In contrast, we analyse a case where skill-adjusted labour inputs have di¤erent degrees of complementarity with capital and …nd that whether this skill heterogeneity is endogenous or exogenous does indeed matter for the cyclical properties of optimal labour taxes.
Our results further show that the skill heterogeneity considered, irrespective of the presence of the labour market friction, does not a¤ect the results obtained in the literature regarding the cyclical behaviour of asset taxes. In particular, the ex ante tax rate on capital is around zero for every period, the state contingent private assets and ex post capital taxes are near zero and are the most volatile of the tax instruments. We also …nd that the skill-acquisition tax is the least smooth of the tax instruments when debt is state-contingent and ‡uctuates nearly as much as output. Finally, irrespective of the model variant examined, all of the policy instruments, except for the ex post capital tax and the private assets tax inherit the persistence properties of the shocks.
The remainder of the paper is organised as follow. Sections 2 and 3 present the theoretical model and the Ramsey problem respectively. Section 4 contains the quantitative results and Section 5 draws the conclusions.

Model
We develop a model that extends the complete markets neoclassical setup in Zhu (1992) and Chari et al. (1994) by allowing for a division of the labour force into skilled and unskilled workers, an endogenous skill supply on the household side and capital-skill complementarity on the production side. This setup implies a wage premium for skilled labour, the relative supply of which can be increased by a cost to the household in the form of earmarked training expenditure. 4 As in Chari et al. (1994) households save in the form of physical capital and state-contingent government bonds.
The household is modelled as an in…nitely-lived representative dynasty. The head of the household makes all choices on behalf of its members by maximising the aggregate welfare of the family, ensuring that each household member experiences the same level of consumption irrespective of individual labour market status. This is a commonly employed assumption since Merz (1995), given that it allows for tractability when studying aggregate ‡uctuations under heterogeneities in the labour market (see e.g. Arseneau and Chugh (2012) for an example with optimal tax policy).
Firms use capital, skilled and unskilled labour to produce a homogeneous product. Following Katz and Murphy (1992), Krusell et al. (2000) and Horn-stein et al. (2005), skilled labour is assumed to be more complementary to capital than unskilled labour. Hence, capital accumulation as well as technological developments and government policies that are capital augmenting, increase the skilled wage premium. In contrast, increases in the relative supply of skilled labour reduce the skill premium. Finally, the government can borrow, tax skill acquisition expenditure, capital, skilled and unskilled labour income separately, to …nance exogenous public spending.

Notation
The notation employed throughout follows Ljungqvist and Sargent (2012). In particular, we assume that in every period t 0, there is a realization of shocks (stochastic events) s t 2 S. Therefore, at each period t there is a history of events s t = [s 0 ; s 1 ; s 2 ; :::; s t ] which is known. The unconditional probability of observing a speci…c history of events s t is de…ned as t (s t ).
For t > , the conditional probability of having s t sequence of events given the realization of s is de…ned as: t (s t j s ).

Households
A representative household is comprised of two types of members who provide skilled and unskilled labour services. 5 The household can invest in capital and in state-contingent sequentially traded government bonds that mature fully within a period. The objective function of the representative household is given by: where, h s t (s t ) and h u t (s t ) denote, respectively, skilled and unskilled labour hours per member. The household can determine its relative skill supply by incurring an average (over all its members) skill-acquisition expenditure, e t (s t ), according to the following relation: where e g(:) is increasing, strictly concave and three times continuously di¤erentiable with respect to e t (s t ).
The household also faces a sequence of budget constraints given by: with one element of the vector for each possible realisation of s t+1 . This yields six …rst-order conditions which are reported in Appendix A.
Combining the …rst-order conditions for consumption, skilled and unskilled labour supply as well as the relative skill supply gives the following atemporal equilibrium conditions: Conditions (6)-(7) equate the marginal rates of substitution between consumption and each type of labour with the average returns to skilled and unskilled labour net of taxes. The …nal relation given by (8) states that the marginal rate of substitution between consumption and the relative skill supply is equal to the net marginal bene…t of increasing the household's share of skilled workers. The latter includes the post-tax labour income from an additional skilled member, = t+1 (s t+1 j s t ) and E t is the expectation conditional on in- x t+1 (s t+1 ), and the summation over s t+1 denotes the sum over all possible histories e s t+1 such that e s t = s t . By combining the intertemporal conditions we obtain: which ensures no-arbitrage between the investment opportunities in bonds and capital.

Firms
Firms rent capital as well as skilled and unskilled labour from households to maximize their pro…ts using a production technology, F ( ), that exhibits constant returns to scale in its three inputs: This yields the standard …rst-order conditions:

Government budget and market clearing
Given a history s t , the government …nances an exogenous stream of expenses g e t (s t ) and its debt obligation b t (s t j s t 1 ), by taxing capital and labour income and skill acquisition expenditure, and by issuing state-contingent debt. Hence, the within-period government budget constraint is given by: Finally, the aggregate consistency condition and market clearing conditions for skilled labour, unskilled labour and capital are given respectively by:

The Ramsey problem
To solve the Ramsey problem we follow the primal approach and …rst derive the present discounted value (PDV) of the household's lifetime budget constraint using the Arrow-Debreu price of the bond and the transversality conditions for bonds and capital. Second, we derive the implementability constraint by substituting out prices and tax rates from the household's present value budget constraint using the …rst-order conditions for the household and …rm. Finally, we derive the optimal Ramsey allocations by maximising the planner's objective function subject to the implementability constraint and the aggregate resource constraint.

Present value of budget constraint
Starting from period 0 and by repeatedly substituting forward one-period budget constraints for the household, we obtain the PDV of the household's lifetime budget constraint: where we have imposed the series of no-arbitrage conditions (11) 8t and the following transversality conditions for any s 1 : which specify that for any possible future history the household does not hold positive or negative valued wealth at in…nity. De…ning is the Arrow-Debreu price, we can re-write (21) as: Notice that the Arrow-Debreu price satis…es the recursion: Using the …rst-order condition from the sequential equilibrium for pricing contingent claims (9) and noting that 0 (s 0 ) = 1, since, at period 0 the state s 0 is known, the above recursion can be written as:

Implementability constraint
First, notice that (26) implies: Substituting (27) for q 0 t (s t ); the …rst-order conditions of the …rm, (13), (14) and (15) for w s t (s t ), w u t (s t ) and r 0 , respectively; and the …rst-order conditions of the household, (6), (7), and (8) for s t (s t ), u t (s t ) and a t (s t ), respectively into the present value budget constraint (24), we obtain the implementability constraint: 8 is obtained by substituting the market clearing condition (20) into F k f (s 0 ).

Pseudo value function
Substituting the constraints (2)-(3) into the utility function u(:), the government maximises the resulting objective function subject to the implementability constraint (28) and the sequence of aggregate resource constraints in (17) To achieve this, we follow Ljungqvist and Sargent (2012) and 8 Note that the intertemporal …rst-order condition (11) has been used already in deriving (24), while the government budget constraint is redundant, since it is a linear combination of the household's budget constraint and the aggregate resource constraint. Therefore, (28) and (17) summarise all the constraints that the government needs to respect. 9 Note that following the literature we do not examine the problem of initial capital taxation and thus do not allow the government to choose k 0 .
10 …rst specify the following within-period pseudo value function: where is the Lagrange multiplier with respect to the implementability constraint. 10 The Lagrangian of the Ramsey planner is de…ned as: where e F ( ) is obtained by substituting market clearing conditions (18) t=0 is a sequence of Lagrange multipliers attached to the aggregate resource constraint. For a given level yielding the following …rst-order conditions respectively: t=0 respectively. The …rst-order conditions derived in (31)-(39) imply that the system of equations to be solved will be di¤erent for t = 0 and for t > 0. These conditions in a non-stochastic environment are presented in Appendix B.

Quantitative implementation
In this section we quantitatively solve both the non-stochastic and stochastic optimal policy models. Our solution approach follows Arseneau and Chugh (2012). In particular, we …rst calibrate the non-stochastic model with exogenous policy. Next, we solve the deterministic Ramsey problem, starting from the exogenous policy steady-state, using non-linear methods. Since we are interested in tax smoothing over the business cycle, we then approximate around the steady-state of the deterministic Ramsey problem to solve the stochastic problem and obtain near steady-state dynamics.

Functional forms
Following Chari et al. (1994) and Stockman (2001), we use a CRRA utility function: where, 1 and 2 are the weights to leisure in the utility function and 3 is the relative risk aversion parameter. The production side is given by a CES production function that allows for capital-skill complementarity, since the latter has been shown to match the dynamics of the skill premium in the data (see e.g. Krusell et al. (2000), Lindquist (2004), and Pourpourides (2011)): where, A t is total factor productivity; A k t is the e¢ ciency level of capital equipment; < 1, and < 1 are the parameters determining the factor elasticities, i.e. 1=(1 ) is the elasticity of substitution between capital and unskilled labour and between skilled and unskilled labour, whereas 1=(1 ) is the elasticity of substitution between equipment capital and skilled labour; and 0 < ; < 1 are the factor share parameters. In this speci…cation, capital-skill complementarity is obtained if 1=(1 ) > 1=(1 ). The above functional form implies that the skill premium, de…ned as w s (s t ) w u (s t ) , can be obtained as: . The restrictions placed above on the parameters of the production function imply that the skill premium is decreasing in (s t ) and increasing in k t (s t 1 ), see Appendix C.
The functional form for the relative skill supply is: where > 0 is the productivity of skill-acquisition; and 0 < 1 is the elasticity of the relative skill supply with respect to skill-acquisition expenditure.
Finally, we calculate the e¤ective labour tax rate as the ratio of total tax revenues from both skilled and unskilled sources as a share of total labour income: n t (s t ) = s Utility Table 1 below reports the model's quantitative parameters along with an indication of their source. Starting with the share of leisure for each skill type in utility, 1 and 2 , we calibrate these to 0:35 each so that, in the steady-state, the household devotes about one third of its time to working. The relative risk aversion parameter, 3 = 2 is commonly employed in business cycle models.  Census data and the share of labour income in GDP is from the BEA data on personal income. 11 We also normalize the steady-state values of TFP and capital equipment to unity (i.e. A = A k = 1).
Depreciation and time preference The depreciation rate of capital = 0:07 is calibrated to obtain an annual capital to output ratio of about 1:94, which is consistent with the annual data reported by the BEA on capital stocks. 12 The time discount factor, = 0:96, is set to obtain a post-tax post-depreciation annual real rate of return on capital of roughly 4:17%, which coheres with the 4:19% obtained in the data from the World Bank. 13 Relative skill supply To match the share of skilled workers in total population, , of roughly 44% in the data, we set the elasticity of relative skill supply with respect to skill-acquisition, , equal to 0:2334. This share is consistent with the data from the 2010 U.S. Census which indicates that 43% of the population has a college degree. 14 It also adheres with a related data set by Acemoglu and Autor (2011) which implies that the average share of the labour force with a college degree is approximately 45%. We normalise skill-acquisition productivity, to unity.
Tax rates and government spending Finally, we use the ECFIN e¤ective capital and labour tax rates from Martinez-Mongay (2000) to obtain an average tax rate for capital and labour. 15 Therefore, we set the tax rate for capital income k = 0:31 and the two labour income tax rates u = 0:20 and s = 0:25. 16 Given that it is di¢ cult to obtain data which match well with the skill-acquisition expenditure tax rate, a , we set it to zero for the exogenous policy model. We …nally set the steady-state value g e = 0:0469, 11 The data source is the Current Population Survey, 2011 Annual Social and Economic Supplement from the U.S. Census Bureau. 12 Speci…cally, the BEA Table 1.1 on …xed-assets has been used to obtain the time series for capital stock for 1970-2011. 13 The data refers to the annual real interest rate from World Bank Indicators database for the period 1970-2011 (i.e. FR.INR.RINR). 14 This information is obtained from Table 4 of the Census Bureau, Survey of Income and Program Participation. 15 In particular, we use the LITR and KITN rates for e¤ective average labour and capital taxes respectively for 1970-2011, as they treat self-employed income as capital income in the calculations. 16 Note that the calculation of the e¤ective labour income tax rate is equal to 0.22. But since we assume that the skilled and unskilled labour income is taxed di¤erently we decompose the labour income tax into skilled and unskilled tax so as the weighted average of the two tax rates equals 0. 22. to obtain a steady-state debt to output ratio, b=Y = 53%, which is equal to the average debt to GDP ratio obtained in the data. 17 Steady-state The steady-state of the DCE de…ned and calibrated above is presented in Table 2. The results indicate that the model's predictions for the great ratios match those implied by the data quite well. For example, in the data for 1970-2011: k y = 1:895, c y = 0:640, i y = 0:146, g e y = 0:203 and b y = 0:530. 18 Moreover, the share of skill acquisition expenditure in GDP, e y , roughly coheres with US total expenditures for colleges and universities as a share of output equal to 6% for 1970-2010. This data is obtained from the U.S. National Center for Education Statistics, Digest of Education Statistics. As pointed out above, the remaining steady-state variables in the exogenous model, have been calibrated to match their values in the data.

Deterministic Ramsey
The deterministic version of the Ramsey problem in (31)-(39) is summarised in Appendix B, (B1-B16) and is solved iteratively, conditional on the calibration described in the previous section. In particular, we …rst guess a value for and solve equations (B1-B15) for an allocation fc t ; h s t ; h u t ; t ; k t+1 g T t=0 . Then we test whether equation (B16) is binding and increase or decrease the value of if the budget is in de…cit or surplus respectively.
The initial conditions for the model's state variables are given by the non-stochastic exogenous steady-state (see Table 2). For the terminal values of the forward looking variables, we assume that after T years the dynamic system has converged to its Ramsey steady-state. This implies that the appropriate terminal conditions are obtained by setting the values for these variables equal to those of the preceding period.
The …nal system is given by [(15 T ) + 1] equations, which is solved non-linearly using standard numeric methods (see, e.g. Garcia-Milà et al.  )). This gives the dynamic transition path from the exogenous to the optimal steady-state. We set T = 250 to ensure that convergence is achieved. Our results show 17 The source of that time series is: FRED Economic Data on Gross Federal Debt as a percentage of GDP, 1970-2011. 18 Note that if model prediction for the cost of becoming skilled, e y = 0:0659, is added to the c y ratio from the model, the sum is very close to the c y ratio in the data. that this occurs for all endogenous variables within 150 years. 19 After we …nd the optimal allocation for fc t ; h s t ; h u t ; t ; k t+1 g T t=0 we obtain w s t = e F h s (t), w u t = e F h u (t) and r t = e F k (t). Additionally, we solve for s t , u t , a t , k t and n t using the non-stochastic form of (6), (7), (8), (10) and (44) respectively. The Ramsey steady-state is reported in Table 3. The results are consistent with the messages from the literature initiated by Chamley (1986) on dynamic Ramsey taxation in a deterministic environment (see e.g. Ljungqvist and Sargent (2012), ch. 16 for a review of this literature). As expected, allowing the government a complete instrument set results in a zero capital tax rate in the long-run. Compared with the steady-state of exogenous policy, a Ramsey government would increase capital accumulation in the steady-state, by eliminating the intertemporal wedge. Moreover, since skilled labour is complementing capital more than unskilled, the Ramsey government would …nd it optimal to encourage an increase in the relative skill supply, since a higher relative quantity of skilled labour increases the returns to, and thus the accumulation of, physical capital. This is achieved by a small subsidy to skill acquisition expenditure. The fall in the skill premium under Ramsey policy suggests that the increase in the relative skill supply has a relatively stronger quantitative impact than the increase in the capital stock. The Ramsey equilibrium also implies a mild regressivity regarding the long-run labour income taxes, revealing an incentive to encourage the labour supply of skilled hours, consistent with the discussion above. Finally, the government is able to reduce the overall burden of taxation, since it can …nance part of the required public spending from accumulated assets. Note that all taxes are reduced compared with the exogenous policy regime.  We next study the transition dynamics associated with Ramsey policy. Figure 1 illustrates the dynamic paths implied by optimal policy for the capital tax, the two labour taxes, the skill-acquisition expenditure tax and debt to output as the economy evolves from the exogenous steady-state to the Ramsey steady-state. The …rst panel of Figure 1 shows that in period 1 skilled and unskilled labour are subsidised at rates of 16% and 14.57% respectively; and skill-acquisition expenditure is taxed at a rate of 26.76%. In 19 See Figure 1 below for an illustration of convergence using the policy instruments. period 2, skilled and unskilled labour taxes are 15.24% and 14.36% respectively and eventually converge to their steady-state values reported in Table  3. Also in period 2, skill-acquisition is subsidised at a rate of 2.11% and converges to 3.53% in the steady-state. The second panel of Figure 1 shows that in period 1, since capital already in place, capital income is taxed at a con…scatory rate (approximately 210%). In period 2, the capital income tax is 0.27% and then converges slowly to zero. The high capital taxation in the …rst period allows the government to create a …rst period stock of assets of approximately the size of GDP, by lending to the household. Government assets increase in future periods and their income is used to subsidise skill-acquisition expenditure and to compensate for the losses from foregone capital income taxation, without the need to resort to high labour income taxes. These transition paths are consistent with previous research.

Stochastic processes
To move to the analysis of the stochastic Ramsey problem, we need to de…ne the stochastic processes that drive economic ‡uctuations. In what follows we designate a stochastic state s t at time t that determines exogenous shocks to both the …rm's production technologies, (A t , A k t ), and to government expenditures (g e t ). Therefore, the optimal allocation of households will depend on the history of events s t at time t. Following the literature, A t , A k t and g e t are assumed to follow stochastic AR(1) processes: where " A t , " A k t and " g e t are independently and identically distributed Gaussian random variables with zero means and standard deviations given respectively by A , A k and g e .
The values for the AR(1) coe¢ cients and the standard deviations for the government expenditures and capital productivity exogenous processes are data based and are estimated to be: A k = 0:90, g e = 0:70, A k = 0:007 and g = 0:012. 20 The autocorrelation parameter of TFP is set equal to 0:95, following Lindquist (2004) and Pourpourides (2011), while A is calibrated to match the volatility of output observed in the BEA data. 21 More speci…cally, the standard deviation for TFP is set A = 0:8% to obtain a volatility for output from 1970-2011 equal to 1:2%.

Stochastic Ramsey
We next approximate the dynamic equilibrium paths due to three exogenous shocks using …rst-order accurate decision rules of the equilibrium conditions under optimal policy in (31)-(35), around the optimal deterministic steadystate of these conditions described above. 22 As is common in the literature when characterizing policy dynamics, we also make the auxiliary assumption that the initial state of the economy at t = 0 is the steady-state under optimal policy. As is well known (see e.g. Zhu (1992), Chari et al. (1994) and Ljungqvist and Sargent (2012)), the Ramsey problem with state-contingent debt cannot uniquely pin down the capital tax rate. Hence, we follow the literature and calculate the optimal ex-ante capital income tax rate (see Appendix D for details): Alternatively, by assuming that government debt is not state-contingent, we can calculate the ex post state contingent capital tax (see Appendix E for that there is no data available prior to 1988 for the productivity of capital. To calculate the statistical properties of the cyclical component of the series, we take logs and apply the HP-…lter with smoothing parameter equal to 6.25. 21 The time series for GDP from 1970-2011 is obtained from NIPA Table 1.1.5. Cyclical output is again calculated using the HP-…lter as above. 22 We use the perturbation methods in Schmitt-Grohé and Uribe (2003) to solve the dynamic model. the derivation): where R t (s t ) is the state uncontingent or the risk free return to holding government debt. Alternatively, assuming the government employs a statecontingent tax on income from government bonds, we can calculate the private assets tax, (s t+1 js t ) that applies to taxing jointly the income from assets as (see Appendix E for the derivation): To calculate the business cycle statistics of the relevant quantities of the model under optimal policy, we conduct simulations by shocking all of the exogenous processes, obtain the required moments for each simulation and then calculate their mean value across the simulations. We undertake 1000 simulations, each 242 periods long and drop the …rst 200 periods to ensure that the initial conditions do not a¤ect the results. We retain 42 periods in our analysis to match the number of years between 1970 and 2011 used in the calibration.

Cyclical properties
We next present the results regarding the key second moments of the stochastic optimal policy problem. We conduct this analysis for both the model developed above and the model where the relative skill supply is exogenously determined over the business cycle. This is followed by an impulse response analysis, which allows to investigate the channels through which tax policy works over the business cycle.

Endogenous relative skill supply
We start with the cyclical properties of Ramsey taxation under endogenous relative skill supply. The results on standard deviations and correlations with output, for the endogenous variables of the model as well as the various tax rates that were explained above are summarised in the …rst three columns of Table 5. The results regarding optimal taxation are largely consistent with 20 the literature and thus extend previous …ndings to a setup with capital-skill complementarity and endogenous skill supply.  In particular, the ex ante tax rate on capital is e¤ectively zero and is around zero for every period. Moreover, when debt is not allowed to be state-contingent, the state contingent private assets and ex post capital taxes are near zero, have low correlations with output and are the most volatile of the tax instruments. These results are similar to …ndings in the literature to date. Also consistent with the labour tax-smoothing results in the literature, both labour taxes have very low standard deviations relative to output, as the government …nds it optimal to minimise the distortions introduced by labour taxes over the business cycle by keeping them relatively smooth and letting the remaining state-contingent policy instruments respond to exogenous shocks. However, they exhibit di¤erent correlations with output. The tax rate on skilled labour income is pro-cyclical, whereas the tax rate on unskilled labour income is mildly counter-cyclical. The skill-acquisition tax is the least smooth of the tax instruments when debt is state-contingent and ‡uctuates nearly as much as output. Moreover, it is mildly pro-cyclical. Finally, the labour income taxes and the ex ante capital income tax in this model inherit the properties of the exogenous processes. As can be seen in Table 6, the autocorrelations of these instruments follow the autocorrelations of the exogenous processes, so that when shocks are autocorrelated as in Table 4, so are the tax rates. However, if we assume that the shocks follow iid processes, the autocorrelation of the tax rates generally becomes very small. On the contrary, the autocorrelations of the ex post capital tax and of the private assets tax do not follow the autocorrelations of the exogenous processes. This is again similar to previous …ndings.

Exogenous relative skill supply
We next examine how the prescriptions for optimal policy are a¤ected by a friction in the labour market that does not permit changes in the relative skill supply over the business cycle. As discussed in the introduction, this restriction is empirically relevant. 23 To analyse the e¤ects of a …xed relative skill supply over the business cycle, we obtain the …rst-order conditions for optimal policy incorporating this rigidity and then approximate these conditions around the Ramsey deterministic steady-state with endogenous the returns to skilled and unskilled labour, w s t and w u t , respectively, now follow di¤erent paths, summarised by the increase in the skill premium. Ceteris paribus, this drives a wedge between the average net returns to skilled and unskilled labour hours under the restricted model, relative to those from the ‡exible labour markets model. To minimise the e¤ects of the relative labour supply distortion, the government adjusts the optimal response of the labour income taxes, as can be seen in the plots for these returns. It achieves this by by keeping the marginal rates of substitution between leisure and consumption for the two types of labour at roughly the same levels as under a fully ‡exible labour market. Indeed, the response becomes more counter-cyclical, to smooth the response of average net returns to skilled and unskilled labour, so that these last two quantities exhibit, post shock, e¤ectively identical responses with their corresponding quantities in the ‡exible labour market. Note also that the change in s t is larger than u t , since, given capital-skill complementarity, w s t is a¤ected more by the increase in the capital stock than w u t . Thus a larger adjustment in policy is required. A temporary reduction in government spending in Figure 4, does not have direct productivity e¤ects in these models. However, it allows the government to brie ‡y reduce the tax burden on labour income and thus encourage labour supply. In the model with endogenous relative skill supply, a small reduction in s t increases the average net return to skilled labour both directly and indirectly, via the induced increase in t . The latter happens because the increase in skilled labour raises the return to capital as well and thus the returns to investing into skill-acquisition. On the contrary, under the restricted relative skill supply assumption, the indirect e¤ect is missing and thus s t needs to be increased by more, to maintain the same average net return to skilled labour hours. The unskilled labour supply does not a¤ect capital accumulation as much (given capital-skill complementarity). Hence it does not need to be changed by as much under endogenous relative skill supply. In turn, this implies that no big changes are required in the optimal response to u t when relative skill supply is …xed, to maintain the same average net return to unskilled labour hours.

Conclusions
Motivated by the empirical relevance of the wage-skill premium and the roles played by capital-skill complementarity, the relative supply of skilled labour and capital augmenting technical change, this paper contributed to the tax smoothing literature by undertaking a normative investigation of the quantitative properties of optimal taxation of capital and labour income, as well as skill-acquisition expenditure, in the presence of aggregate shocks to total factor productivity (TFP), capital equipment productivity and government spending.
Our main …nding was that under capital-skill complementarity, a friction that did not allow the relative supply of skill to adjust in response to aggregate shocks, signi…cantly changed the cyclical properties of optimal labour taxes. In particular, we …rst showed that under endogenous relative skill supply, the optimal labour taxes for both skilled and unskilled labour income were smooth, with the volatility of the skilled income tax being marginally lower. We also found that the skilled tax moves pro-cyclically with output and the unskilled tax was mildly counter-cyclical. These results were largely consistent with the literature and extended previous …ndings to a setup with capital-skill complementarity and endogenous skill supply.
We also found that, when the relative skill supply was constrained to remain constant over the business cycle, the prescriptions for optimal policy markedly changed. In particular, we found that the volatility of taxes increased signi…cantly, so that the standard deviation of the e¤ective average labour income tax was about seven times higher than the perfect labour markets case, while the volatility of the skilled labour income tax was about two-and-a-half times higher than that of the unskilled labour income tax. Moreover, both taxes became strongly counter-cyclical. We further demonstrated that the key to explaining these changes was that the government found it optimal to adjust labour income tax rates to alter the average net returns to skilled and unskilled labour hours, so that their dynamic behaviour under restricted skill supply is the same as under ‡exible skill supply.
Our results additionally revealed that the skill heterogeneity considered, irrespective of the presence of the labour market friction, did not a¤ect the results obtained in the literature regarding the cyclical behaviour of asset taxes. In particular, the ex ante tax rate on capital was around zero for every period, the state contingent private assets and ex post capital taxes were near zero and are the most volatile of the tax instruments. We also found that the skill-acquisition tax was the least smooth of the tax instruments when debt was state-contingent and ‡uctuated nearly as much as output. Finally, irrespective of the model variant examined, all of the policy instruments, except the ex post capital tax and the private assets tax inherited the persistence properties of the shocks.
In a non-stochastic environment, the …rst-order conditions derived in (31)-(39) of the main text become: for t = 1; 2; 3:::T 1: lifetime implementability constraint: , the Lagrange multiplier t has been replaced with V c (t) using (31) and (36) in the main text and the notation X (t) denotes the time period t quantity of X.
Appendix C: The e¤ects of k t and t on the skill premium Di¤erentiating the skill premium, given by (42) in the main text,with respect to k t we have (note that we do not use the s t notation to keep the presentation more parsimonious): This is positive if a > ; a; < 1; 0 < ; < 1. Di¤erentiating (42) with respect to t gives: This expression is negative if [(1 ) + t ( a)] > 0 or 1 > t , which is true because 1 > 1, since 1 > ) 1 > and 0 < t < 1.

Appendix D: Ex ante capital tax
Assume that the government uses a capital tax that is not state-contingent, so that its value for period t + 1 is decided using the history s t . De…ne this uncontingent tax as k t+1 (s t ) and note that it needs to satisfy the Eulerequation from (10)  By comparing (C2) with (C1), we see that k t+1 (s t ) needs to satisfy: This gives k t+1 (s t ) the ex ante capital tax interpretation, since, by multiplying both numerator and denominator in (C4) by k t+1 (s t ), this expression provides the expected tax revenue from capital income as share of the expected capital income, where the expectation is calculated using information at period t.
To obtain the ex ante rate stated in equation (48) of the main text, we …rst expand the Euler-equation (C1): (1 ) and note that E t u c (s t+1 ) k t+1 (s t+1 ) e F k (s t+1 ) in (C5) equals k t+1 (s t )E t u c (s t+1 ) e F k (s t+1 ), using (C4). Substituting this expression back into (C5) we obtain: Finally solving (C6) for k t+1 (s t ) gives the ex ante capital tax rate reported in equation (48) of the main text.   Figure 4: Impulse responses to 1% temporary shock to goverment spending