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Optimal taxation in the Uzawa–Lucas model with externality in human capital

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Abstract

In this paper we study the optimal policy in the Uzawa–Lucas model with externality in human capital when agents value both consumption and leisure. We find that the government pursuing the first best can achieve its goal by a subsidy which depends on foregone earnings while studying and which is financed through a lump sum tax. Anyway, the optimal policy, that should be designed to provide incentives for agents to devote more time to schooling and cut both on leisure and working, is not unique. There exists an infinite number of combinations of consumption, capital income, labor income and lump sum taxes that can decentralize the first best.

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Notes

  1. Just to mention a few, see Caballé and Santos (1993), Chamley (1993), Mulligan and Sala-i-Martin (1993), Benhabib and Perli (1994), García-Castrillo and Sanso (2000), Ben-Gad (2003), Gómez (2003), Gómez (2004), Bethmann (2007), Boucekkine and Ruiz-Tamarit (2008), among many others.

  2. Given the mathematical complexities, most of the cited papers deal with continuous time models. Bethmann (2007) is an exception doing so in a discrete time setup.

  3. Caballé and Santos (1993) show that the balanced growth path exists for production functions that fullfil the properties imposed on \(F(.,.)\) here.

  4. As shown in Benhabib and Perli (1994), if the utility is separable in consumption and leisure, logarithmic utility in consumption is needed for the balanced growth path with positive growth and constant supply of labor hours to exist.

  5. Let us define \(\lambda _{i}, i=1,2,3\) as the real part of the eigenvalues of the Jacobian matrix of our dynamic system. If the Jacobian matrix \(J\) has two stable roots, say \(\left|\lambda _{1}\right| <1, \left| \lambda _{2}\right| <1, \left|\lambda _{3}\right|>1,\) we have indeterminacy around a given steady state. If the Jacobian matrix \(J\) has just one stable root, say \(\left| \lambda _{1}\right| <1, \left| \lambda _{2}\right| >1, \left| \lambda _{3}\right|>1,\) there will exist a saddle path towards a given steady state.

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Acknowledgments

We would like to thank the editor and two anonymous referees for helpful comments on an earlier version of the paper. Financial support from the Spanish Ministry of Education and Science through Grants ECO 2009-009732 (Gorostiaga), ECO2009-10003 (López-García), ECO2010-16353 (Hromcová), Basque Government Grant IT-214-07 (Gorostiaga), Generalitat de Catalunya through SGR 2009-0600 and XREPP (Hromcová and López-García), and Instituto Valenciano de Investigaciones Económicas are gratefully acknowledged.

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A Appendix

A Appendix

Following Benhabib and Perli (1994) we rewrite the necessary conditions in the planner’s problem (18)–(22), (4) and (17) in terms of three new variables, \(m_{t}, q_{t}\) and \(p_{t}\), defined as:

$$\begin{aligned} p_{t}=\varepsilon _{t}^{SP}h_{t}^{SP}, \quad m_{t}=A( k_{t}^{SP}) ^{\alpha -1}( h_{t}^{SP}) ^{1-\alpha +\gamma }, \qquad q_{t}=\frac{c_{t}^{SP}}{k_{t}^{SP}}. \end{aligned}$$

To simplify notation, we drop superscripts \(SP\) in the planner’s problem allocations. Under Cobb–Douglas production function and utility function (39), the set of equations to solve for the first best allocation gets

$$\begin{aligned} l_{t}&= \left( \frac{b}{\phi p_{t}}\right) ^{\frac{1}{\sigma }}, \end{aligned}$$
(42)
$$\begin{aligned} u_{t}&= \left[ \frac{(1-\alpha )m_{t}}{\phi p_{t}q_{t}}\right] ^{\frac{1}{\alpha }}, \end{aligned}$$
(43)
$$\begin{aligned} \frac{m_{t+1}}{m_{t}}&= \left[ m_{t}u_{t}^{1-\alpha }+1-\delta _{k}-q_{t} \right] ^{\alpha -1}\left[ \phi \left( 1-u_{t}-l_{t}\right) +1-\delta _{h} \right] ^{1-\alpha +\gamma }, \end{aligned}$$
(44)
$$\begin{aligned} \frac{q_{t+1}}{q_{t}}&= \beta \frac{\alpha m_{t+1}u_{t+1}^{1-\alpha }+1-\delta _{k}}{m_{t}u_{t}^{1-\alpha }+1-\delta _{k}-q_{t}}, \end{aligned}$$
(45)
$$\begin{aligned} \frac{p_{t+1}}{p_{t}}&= \frac{\phi \left( 1-u_{t}-l_{t}\right) +1-\delta _{h}}{\beta \left[ \phi \frac{\gamma }{\left( 1-\alpha \right) }u_{t+1}+\phi \left( 1-l_{t+1}\right) +1-\delta _{h}\right] }. \end{aligned}$$
(46)

Since \(p_{t},m_{t}\) and \(q_{t}\) are stationary in the balanced growth path, we can write the steady state of the system above as

$$\begin{aligned} l^{*}&= \left( \frac{b}{\phi p^{*}}\right) ^{\frac{1}{\sigma }}, \\ u^{*}&= \left[ \frac{(1-\alpha )m^{*}}{\phi p^{*}q^{*}} \right] ^{\frac{1}{\alpha }}, \\ ( m^{*}u^{*1-\alpha }+1-\delta _{k}-q^{*}) ^{1-\alpha }&= \left[ \phi ( 1-u^{*}-l^{*}) +1-\delta _{h}\right] ^{1-\alpha +\gamma }, \\ m^{*}u^{*1-\alpha }+1-\delta _{k}-q^{*}&= \beta ( \alpha m^{*}u^{*1-\alpha }+1-\delta _{k}) , \\ \beta \left[ \frac{\phi \gamma }{( 1-\alpha ) }u^{*}+\phi ( 1-l^{*}) +1-\delta _{h}\right]&= \phi ( 1-u^{*}-l^{*}) +1-\delta _{h}. \end{aligned}$$

We follow Novales et al. (2010) to study the local stability properties of the balanced growth path. We first make the system tractable by loglinearizing it around its steady state. We define a log-deviation of a variable \(x_{t}\) from its steady state \(x^{*}\) (balanced growth path) as \(\tilde{x}_{t}=\ln x_{t}-\ln x^{*}.\) Then we can write the log-linearized form of (44)–(46), using (42) and (43), as

$$\begin{aligned} \Gamma _{0}\left( \begin{array}{l} \tilde{m}_{t+1} \\ \tilde{q}_{t+1} \\ \tilde{p}_{t+1} \end{array} \right) =\Gamma _{1}\left( \begin{array}{l} \tilde{m}_{t} \\ \tilde{q}_{t} \\ \tilde{p}_{t} \end{array} \right) \end{aligned}$$

where

$$\begin{aligned} \Gamma _{0}=\left( \begin{array}{c@{\quad }c@{\quad }c} \chi _{1}&0&0 \\ -\beta \chi _{4}&\chi _{1}+\beta (1-\alpha )\chi _{4}&\beta (1-\alpha )\chi _{4} \\ 1&-1&\beta \left[ -\frac{(1-\alpha )}{\beta }+\alpha \chi _{7}+\chi _{8}\right] \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} \Gamma _{1}=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} -\chi _{3}-(1-\alpha )\chi _{4}&&\chi _{3}+(1-\alpha )q^{*}&&\chi _{3}+\frac{\phi l^{*}\chi _{1}}{\sigma \chi _{2}} \\ -\frac{\chi _{4}}{\alpha }&&\chi _{5}&&\frac{(1-\alpha )\chi _{4}}{ \alpha } \\ -\chi _{7}&&\chi _{7}&&\chi _{7}+\chi _{8} \end{array} \right) \end{aligned}$$

with

$$\begin{aligned} \chi _{1}&= m^{*}( u^{*}) ^{1-\alpha }+1-\delta _{k}-q^{*}, \\ \chi _{2}&= \phi ( 1-u^{*}-l^{*}) +1-\delta _{h}, \\ \chi _{3}&= \frac{\chi _{1}\phi u^{*}+\chi _{2}(1-\alpha )^{2}m^{*}( u^{*}) ^{1-\alpha }}{\alpha \chi _{2}}, \\ \chi _{4}&= m^{*}( u^{*}) ^{1-\alpha }, \\ \chi _{5}&= q^{*}+\beta ( \alpha \chi _{4}+1-\delta _{k}) + \frac{(1-\alpha )\chi _{4}}{\alpha }, \\ \chi _{6}&= \frac{\phi \gamma u^{*}}{( 1-\alpha ) }, \\ \chi _{7}&= \frac{( 1-\alpha ) }{\beta \gamma }, \\ \chi _{8}&= \frac{\alpha }{\beta \chi _{6}}\left( \frac{\phi l^{*}}{\sigma }+\chi _{2}\right). \end{aligned}$$

We define the Jacobian matrix at the steady state as

$$\begin{aligned} J=\left( \Gamma _{0}\right) ^{-1}\Gamma _{1} \end{aligned}$$

and check how many of the three eigenvalues have real parts with modulus above/below one.Footnote 5

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Gorostiaga, A., Hromcová, J. & López-García, MÁ. Optimal taxation in the Uzawa–Lucas model with externality in human capital. J Econ 108, 111–129 (2013). https://doi.org/10.1007/s00712-012-0285-5

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