Demographic Change, Wage Inequality, and Technology

Abstract

This chapter investigates the consequences of population ageing and demographic change for the long-run performance of the economies from the perspective of wage inequality and technology intensity. We devise two alternative models of endogenous growth with demographic variables. In the baseline model, a lower birth rate or a higher mortality rate implies a lower share of R&D workers in total high-skilled labour and a lower skill premium. However, the alternative directed technical change model may imply a rising skill premium depending on the elasticity of substitution. Quantitatively, it is shown that the decline in the birth rate and, consequently, ageing may help explain the increase in the skill premium and the decline in the R&D intensity observed in the end of the XXth century.

Appendix

In this appendix we derive the aggregate Euler equation for consumption and the aggregate law of motion for assets (Eqs. (4.21) and (4.23)). We consider that the population grows at rate n = ψ − μ > 0 and we normalize the initial population size to $\mathcal {L}(0)$ such that the size of a generation born at t₀ < t at a certain point in time t is:

$$\displaystyle \begin{aligned} N\left(t_{0},t\right)=\psi\cdot\mathcal{L}(t_{0})\cdot e^{\mu\left(t_{0}-t\right)}=\psi\cdot\mathcal{L}(0)\cdot e^{nt_{0}}\cdot e^{\mu\left(t_{0}-t\right)}=\psi\cdot\mathcal{L}(0)\cdot e^{\psi t_{0}}\cdot e^{-\mu t}.{} \end{aligned} $$

(4.80)

Integrating over all generations yields the population size as:

$$\displaystyle \begin{aligned} \mathcal{L}\left(t\right)=\int_{-\infty}^{t}\psi\cdot\mathcal{L}(0)\cdot e^{\psi t_{0}}\cdot e^{-\mu t}\cdot dt_{0}=\psi\cdot\mathcal{L}(0)\cdot e^{-\mu t}\int_{-\infty}^{t}e^{\psi t_{0}}dt_{0}.{} \end{aligned} $$

(4.81)

Hence, we can define the aggregate consumption as:

$$\displaystyle \begin{aligned} C\left(t\right)=\psi\cdot\mathcal{L}(0)\cdot e^{-\mu t}\int_{-\infty}^{t}c(t_{0},t)\cdot e^{\psi t_{0}}dt_{0}.{} \end{aligned} $$

(4.82)

Differentiating this equation with respect to time yields:

$$\displaystyle \begin{aligned} \dot{C}(t)=\psi\cdot\mathcal{L}(0)\cdot e^{-\mu t}\int_{-\infty}^{t}\dot{c}\left(t_{0},t\right)\cdot e^{\psi t_{0}}dt_{0}-\psi C(t)+\psi\cdot\mathcal{L}(0)\cdot e^{-\mu t}c\left(t,t\right)e^{\psi t}.{} \end{aligned} $$

(4.83)

In order to find g_C(t), we can reformulate the household’s optimization problem in Sect. 4.2.2.1, by considering:

$$\displaystyle \begin{aligned} \underset{c(t_{0},\tau)}{\mathrm{Max}}U=\int_{t}^{\infty}\mathrm{log}c(t_{0},\tau)\cdot e^{\left(\mu+\rho\right)\left(t-\tau\right)}d\tau,{} \end{aligned} $$

(4.84)

subject to its lifetime budget restriction, stating that the present value of lifetime consumption expenditures has to be equal to the present value of lifetime wage income plus initial assets:

$$\displaystyle \begin{aligned} & \int_{t}^{\infty}c(t_{0},\tau)\cdot e^{-\int_{t}^{\tau}\left(r(s)+\mu\right)ds}d\tau=k(t_{0},t)\\ & \quad +\int_{t}^{\infty}\left(1-s\right)\cdot w_{L}(\tau)\cdot e^{-\int_{t}^{\tau}\left(r(s)+\mu\right)ds}d\tau\\ & \quad +\int_{t}^{\infty}s\cdot w_{H}(\tau)\cdot e^{-\int_{t}^{\tau}\left(r(s)+\mu\right)ds}d\tau{} \end{aligned} $$

(4.85)

From the first order condition to this problem results:

$$\displaystyle \begin{aligned} c(t_{0},\tau)^{-1}\cdot e^{\left(\mu+\rho\right)\left(t-\tau\right)}=\lambda(t)\cdot e^{-\int_{t}^{\tau}\left(r(s)+\mu\right)ds}{} \end{aligned} $$

(4.86)

Hence, in time τ = t, we have $c(t_{0},\tau )=\frac {1}{\lambda (t)}$ and we can write $c(t_{0},\tau )^{-1}\cdot e^{\left (\mu +\rho \right )\left (t-\tau \right )}=c(t_{0},\tau )^{-1}\cdot e^{-\int _{t}^{\tau }\left (r(s)+\mu \right )ds}$, which, in turn, implies that:

$$\displaystyle \begin{aligned} \int_{t}^{\infty}c(t_{0},\tau)\cdot e^{\left(\mu+\rho\right)\left(t-\tau\right)}d\tau=\int_{t}^{\infty}c(t_{0},\tau)\cdot e^{-\int_{t}^{\tau}\left(r(s)+\mu\right)ds}d\tau.{} \end{aligned} $$

(4.87)

So, from (4.85):

$$\displaystyle \begin{aligned} c(t_{0},t)=\left(\mu+\rho\right)\cdot\left[k(t_{0},t)+\int_{t}^{\infty}\left(1-s\right)\cdot w_{L}(\tau)\cdot e^{-\int_{t}^{\tau}\left(r(s)+\mu\right)ds}d\tau\right.\\ \left.+\int_{t}^{\infty}s\cdot w_{H}(\tau)\cdot e^{-\int_{t}^{\tau}\left(r(s)+\mu\right)ds}d\tau\right], {} \end{aligned} $$

(4.88)

where the first term in brackets, k(t₀, t), is the financial wealth, which depends on the date of birth, and the second and third terms represent the wealth resulting from work, which is independent of the date of birth because the productivity is age independent. Thus, optimal consumption in the planning period is proportional to total wealth with a marginal propensity to consume of $\left (\mu +\rho \right )$.

By using (4.88), we can write the aggregate consumption in (4.82) as:

$$\displaystyle \begin{aligned} \begin{array}{c} C(t)=\left(\mu+\rho\right)\psi\cdot\mathcal{L}(0)\cdot e^{-\mu t}\int_{-\infty}^{t}k(t_{0},t)\cdot e^{\psi t_{0}}dt_{0}+\left(\mu+\rho\right)\cdot\mathcal{L}(0)\cdot e^{(\psi-\mu)t}\cdot\\ {} \cdot\left[\int_{t}^{\infty}\left(1-s\right)\cdot w_{L}(t)\cdot e^{-\int_{t}^{\tau}\left(r(s)+\mu\right)ds}dt+\int_{t}^{\infty}s\cdot w_{H}(t)\cdot e^{-\int_{t}^{\tau}\left(r(s)+\mu\right)ds}dt\right]. \end{array}{} \end{aligned} $$

(4.89)

Newborn households do not have financial wealth because there are no inheritances; thus, $c(t,t){=}\left (\mu \,{+}\,\rho \right )\cdot \left [\hspace{-2pt}\int _{t}^{\infty }\hspace{-2pt}\left (1{-}s\right )\cdot w_{L}(t)\cdot e^{-\int _{t}^{\tau }\hspace{-2pt}\left (r(s)+\mu \right )ds}dt\right .$ $+\left .\int _{t}^{\infty }s\cdot w_{H}(t)\cdot e^{-\int _{t}^{\tau }\left (r(s)+\mu \right )ds}dt\right ]$ and $C(t,t)=\mathcal {L}(0)\cdot c(t,t)\cdot e^{(\psi -\mu )t}$ holds for, respectively, each newborn household and each newborn generation. Bearing in mind these last expressions for c(t, t) and C(t, t) and putting (4.83), (4.89) and (4.19), together yields Eqs. (4.21) and (4.22) in the text.

Now, in order to get an expression for the dynamic behaviour of aggregate assets, K, we need to apply the following to integrate over all generations alive at time t. Differentiating (4.22) with respect to time yields:

$$\displaystyle \begin{aligned} \dot{K}(t)=\psi\cdot\mathcal{L}(0)\cdot e^{-\mu t}\cdot\int_{-\infty}^{t}\dot{k}\left(t_{0},t\right)\cdot e^{\psi t_{0}}dt_{0}-\psi\cdot K(t)+\psi\cdot\mathcal{L}(0)\cdot e^{-\mu t}\cdot k\left(t,t\right)\cdot e^{\psi t}.{} \end{aligned} $$

(4.90)

Since $k\left (t,t\right )=0$ and bearing in mind (4.18), it results that:

$$\displaystyle \begin{aligned} \begin{array}{c} \dot{K}\left(t\right)=-\mu\cdot K(t)+\left(r(t)+\mu\right)\cdot\psi\cdot\mathcal{L}(0)\cdot e^{-\mu t}\cdot\int_{-\infty}^{t}k(t_{0},t)\cdot e^{\psi t_{0}}dt_{0}\\ {} -\mu\cdot\mathcal{L}(0)\cdot e^{-\mu t}\int_{-\infty}^{t}c(t_{0},t)\cdot e^{\psi t_{0}}dt_{0}{+}\mathcal{L}(0)\cdot e^{-\mu t}\cdot e^{\psi t}\hspace{-1pt}\cdot\left[\left(1{-}s\right)\cdot w_{L}(t)\,{+}\,s\cdot w_{H}(t)\right]_{-\infty}^{t}, \end{array} \end{aligned}$$

which implies that the aggregate law of motion for assets (equivalent to the aggregate flow budget constraint for households) is given by Eq. (4.23) in the text.