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Stochastic accumulation of human capital and welfare in the Uzawa–Lucas model: an analytical characterization

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Abstract

Stochastic growth models are often solved numerically, because they are not tractable in general. However, recent several studies find the closed-form solution to the stochastic Uzawa–Lucas model in which technological progress or population dynamics follow a Brownian motion process with one or two parameter restriction(s). However, they assume that the return on the accumulation of human capital is deterministic, which is inconsistent with empirical evidence. Therefore, I develop the Uzawa–Lucas model in which the accumulation of human capital follows a mixture of a Brownian motion process and many Poisson jump processes, and obtain the closed-form solution. Moreover, I use it to examine the nexus between human capital uncertainty, technological progress, expected growth rate of human capital, and welfare.

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Notes

  1. The other approach is to abstract from the depreciation of capital. See the introduction of Bucci et al. (2011) and Wälde (2011a) for a list of studies that have tried getting the explicit solution to stochastic growth models.

  2. In virtue of a variety of mathematical techniques, Hiraguchi (2009), Naz et al. (2016) and Chaudhry and Naz (2018) also find the closed-form solution to the Uzawa–Lucas model in continuous time, though in the deterministic setup.

  3. An explicit incorporation of leisure makes it extremely difficult to obtain the analytical solution. Therefore, I abstract from it. See Ladrón-De-Guevara et al. (1999) for the deterministic Uzawa–Lucas model with leisure. No study has found the closed-form solution to the stochastic Uzawa–Lucas model with leisure.

  4. We can be sure that \(u\in (0,1)\) as long as the inequality

    $$\begin{aligned} \alpha b-\alpha \delta _h-\frac{\sigma _h^2}{2}\alpha (1-\alpha )+\mu \gamma<\rho <b-\alpha \delta _h-\frac{\sigma _h^2}{2}\alpha (1-\alpha )+\mu \gamma \end{aligned}$$

    holds. Moreover, one can establish that the TVC

    $$\begin{aligned} \lim _{t\rightarrow \infty } E[e^{-\rho t}K^{\alpha +\gamma }] = \lim _{t\rightarrow \infty } E[e^{-\rho t}A^{\gamma }H^\alpha ] = 0 \end{aligned}$$

    is satisfied. The proof requires the verification theorem. See Chang (2004, Ch. 4) for details of this theorem. Hiraguchi (2013, Appendix B) provides an excellent proof of the TVC for the stochastic Uzawa–Lucas model in which technology follows a geometric Brownian motion process. To save space, I will not mention the proof of TVC in what follows.

  5. For example, on the one hand, Lucas (2003) claims that \(\phi \) ranges from 1 (log utility) to 4, but on the other, Smith (2007) says that \(\phi \) should be smaller than 1.

  6. See also Eq. (16) in Bucci et al. (2011), Eq. (11) in Marsiglio and Torre (2012a), and Eq. (50) in Marsiglio and Torre (2012b).

  7. Following the seminal paper of Mankiw et al. (1992, p. 432), I set the human capital share \(\alpha =1/3\). For physical capital share, it has been commonplace in macroeconomics to assume that it is also 1/3. However, as Karabarbounis and Neiman (2014) document, the labor share is declining (or, put differently, physical capital share is rising) globally. Therefore, I set \(\gamma =0.27\) so that the physical capital share roughly equals 0.40, the value used by Ahn et al. (2017). \(b=0.11\) is used when Barro and Sala-i-Martin (2004) simulate the Uzawa–Lucas model. I choose \(\mu =0.02\) and \(\delta _k=\delta _h=0.03\), again following Mankiw et al. (1992). Finally, following Caballé and Santos (1993) and Moll (2014), I set \(\rho =0.05\). In Fig. 1, I use \(K=10\) to make it transparent.

  8. I thank one anonymous referee for pointing out this property.

  9. One more marginal channel through which \(\mu \) increases A can be seen by solving the differential equation (3). However, this somewhat overlaps the first channel.

  10. I thank one anonymous referee for encouraging me to pursue this.

  11. Specifically, I use the solution to (1)

    $$\begin{aligned} H(t)=H(0)e^{\left( b(1-u)-\delta _h-\frac{\sigma _h^2}{2}\right) t}e^{\sigma _h z_h} \end{aligned}$$

    for simulation. Higham (2001) provides a concise explanation of simulation technique for stochastic differential equation driven by a Brownian motion process.

  12. According to Lee and Lee (2016) data set on human capital stock, we can also see the “bell-shaped” pattern in US between 2000 and 2005. During this period, the number declined from 3.708 (in 2000) to 3.673 (in 2005). In fact, this phenomenon is not unique in US. For instance, in Switzerland between 1980 and 2000, the number consecutively declined from 3.103 (in 1980) to 2.758 (in 2000); in Spain between 1915 and 1920, the number declined from 1.403 (in 1915) to 1.389 (in 1920); in Portugal between 2000 and 2005, the number declined from 2.318 (in 2000) to 2.253 (in 2005). These evidence suggests that the above exercise can be applied not only to the period of unprecedentedly big events (such as World War II) or to the specific country, but also to other (possibly) disruptive events across time and space.

  13. I thank one anonymous referee for pointing out this possibility.

  14. Steger (2005) compares a Brownian motion process with a Poisson jump process in the AK model. He shows that, to conduct a sensible comparison between these requires some unrealistic restrictions. Furthermore, even when they are imposed, he finds that insights from the comparison is quantitatively negligible. Following his findings, I will not do empirical simulation in what follows. In principle, with Poisson jump processes, we would see occasional jumps in Fig. 3, in addition to random fluctuations driven by a Brownian motion process.

  15. We can be sure that \(u\in (0,1)\) as long as the inequality

    $$\begin{aligned}&\alpha b-\alpha \delta _h-\frac{\sigma _h^2}{2}\alpha (1-\alpha )+\mu \gamma +\sum _{i=1}^N \lambda _i\left( (1+\beta _i)^{\alpha }-1\right)<\rho <b-\alpha \delta _h \end{aligned}$$
    $$\begin{aligned}&\quad -\frac{\sigma _h^2}{2}\alpha (1-\alpha )+\mu \gamma +\sum _{i=1}^N \lambda _i\left( (1+\beta _i)^{\alpha }-1\right) \end{aligned}$$
    (19)

    holds. Moreover, one can establish that the appropriate TVC is satisfied. Sennewald (2007) provides the proof of the TVC for the Poisson jump case.

  16. The notion of stochastic physical capital accumulation is first proposed by Eaton (1981). He assumes that the depreciation rate of physical capital follows a Brownian motion process. Similarly, Rebelo and Xie (1999) assume that it follows both a Brownian motion process and one Poisson process.

  17. We can be sure that \(u\in (0,1)\) as long as the inequality (19) holds. Moreover, one can establish that the appropriate TVC is satisfied. See Sennewald (2007) for the proof.

  18. See also Proposition 6 and 7 in Rebelo and Xie (1999) and Eq. (16) in Wälde (2011a).

  19. To be precise, as \(\mathcal {B}_X\) is contained in \(\mathcal {B}_Y\), shock terms in (22) affect both A and H, and hence welfare J. However, this channel would be too obvious to explain in detail.

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Acknowledgements

I thank Prof. Noritsugu Nakanishi, Assoc. Prof. Quoc Hung Nguyen, Prof. Hiroyuki Nishiyama, Prof. Masao Oda, Prof. Yoshifumi Okawa and seminar participants at the summer 2017 JSIE Kansai Branch Meeting for their many extensive and thoughtful comments. I would particularly like to thank Assoc. Prof. Shiro Kuwahara for his encouragement, pointing out troublesome typos, and constructive comments, “from the cradle to completion” of this paper. I am especially grateful to Prof. Yoichi Gokan, my discussant, and two anonymous referees of this journal for their detailed and exceptionally helpful comments that lead to the unthinkably substantial improvement of the earlier version of the manuscript. All remaining mistakes are my own. This paper was accepted to the J Econ under the guidance of Prof. Giacomo Corneo (Editor). Figure 2 is created with Eviews 9.5 Student Version, while the others are with MATLAB R2016b (Version 9.1, MATLAB and Simulink Student Suite). This research does not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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  1. Graduate School of Economics, University of Hyogo, 8-2-1, Nishi-Ku, Kobe, 651-2197, Japan

    Mizuki Tsuboi

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Appendix: Analytical solution

Appendix: Analytical solution

This appendix briefly describes how to find the closed-form representation of the value function in Theorems 1, 2, and 3. For this purpose, postulate the tentative value function of the form

$$\begin{aligned} J(K,A,H)=x K^{\theta _1}+yA^{\theta _2}H^{\theta _3}+z \end{aligned}$$

where x, y, z, \(\theta _1\), \(\theta _2\), and \(\theta _3\) are all unknown constants to be determined. The relevant partials are \(J_K = x\theta _1 K^{\theta _1 -1}\), \(J_{KK} = x \theta _1(\theta _1 -1) K^{\theta _1-2}\), \(J_A = y \theta _2 A^{\theta _2 -1} H^{\theta _3}\), \(J_H = y \theta _3 A^{\theta _2} H^{\theta _3 -1}\), and \(J_{HH} = y \theta _3 (\theta _3-1) A^{\theta _2} H^{\theta _3 -2}\).

To obtain the explicit expression, substitute these partials into the maximized HJB equation (21). Then, set \(\theta _1=\alpha +\gamma \), \(\theta _2=\gamma \), and \(\theta _3=\alpha \). Finally, by imposing the parameter restriction (8), you can find the explicit expressions for x, y, and z, and consequently, those for control variables C and u and for the value function J(K, A, H) in Theorem 3. Expressions in Theorem 2 are available by abstracting from the shock terms associated with the stochastic depreciation of physical capital, while those in Theorem 1 are obtained by abstracting from many Poisson jump processes in Eq. (16).

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Tsuboi, M. Stochastic accumulation of human capital and welfare in the Uzawa–Lucas model: an analytical characterization. J Econ 125, 239–261 (2018). https://doi.org/10.1007/s00712-018-0604-6

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