Dynamic analysis of wage inequality and creative destruction

Abstract

This paper investigates the transitional dynamics of a basic Schumpeterian growth model under constant relative risk aversion. In this model, there are three patterns governing the evolution of wage inequality, but only if the intertemporal elasticity of substitution in consumption is sufficiently low: (a) skill-biased technological change, i.e., technological progress leads to a widening of wage inequality; (b) unskill-biased technological change, i.e., technological progress leads to a contraction of wage inequality; and (c) unbiased technological change, i.e., technological progress is independent of wage inequality. By conducting comparative dynamics of an unexpected permanent increase in research productivity in any sector, which we interpret as the arrival of new general purpose technologies, we show that the property of technological change shifts entirely from unskill-biased to skill-biased. The evolution of wage inequality in the model is then consistent with the shift in the trend in wage inequality beginning in the 1970s in the US economy.

Appendices

Appendix A

This appendix derives Eq. (22). According to Eqs. (10) and (12), we obtain

$$\begin{aligned} \frac{{\dot{\omega }_{{s,t}} }}{{\omega _{{s,t}} }} = r_{t} - \frac{{\tilde{\pi }_{t} }}{{V_{t} }} + \phi _{t} + \left( {1 - \beta } \right) g_{t}. \end{aligned}$$

(31)

Substituting Eq. (16) into (31), yields

$$\begin{aligned} \frac{{\dot{\omega }_{{s,t}} }}{{\omega _{{s,t}} }} = r_{t} - \frac{{\tilde{\pi }_{t} }}{{V_{t} }} + \left[ {1 + \left( {1 - \beta } \right) \sigma } \right] \phi _{t}. \end{aligned}$$

(32)

Next, we derive the required return on assets on an equilibrium path. According to Eqs. (16), (19), and (20), the growth rate of consumption satisfies

$$\begin{aligned} \frac{{\dot{C}_{t} }}{{C_{t} }} = \beta \sigma \phi _{t} - \left( {\frac{{1 - \beta }}{\beta }} \right) \frac{{\dot{\omega }_{{s,t}} }}{{\omega _{{s,t}} }} + \frac{{\dot{a}_{t} }}{{a_{t} }} = \frac{1}{\theta }\left( {r_{t} - \rho } \right) . \end{aligned}$$

(33)

Solving Eq. (33) with respect to $r_t$, yields

$$\begin{aligned} r_{t} = \rho + \theta \beta \sigma \phi _{t} - \theta \left( {\frac{{1 - \beta }}{\beta }} \right) \frac{{\dot{\omega }_{{s,t}} }}{{\omega _{{s,t}} }} + \theta \frac{{\dot{a}_{t} }}{{a_{t} }}. \end{aligned}$$

(34)

Substituting Eq. (34) into (32), we obtain

$$\begin{aligned} \left[ {1 + \theta \left( {\frac{{1 - \beta }}{\beta }} \right) } \right] \frac{{\dot{\omega }_{{s,t}} }}{{\omega _{{s,t}} }} = - \frac{{\tilde{\pi }_{t} }}{{V_{t} }} + \rho + \left[ {1 + \left( {1 - \beta } \right) \sigma + \theta \beta \sigma } \right] \phi _{t} + \theta \frac{{\dot{a}_{t} }}{{a_{t} }}. \end{aligned}$$

(35)

Finally, substituting Eq. (17) into (35), we obtain Eq. (22).

Appendix B

This appendix shows that the steady state $( {\omega _s^*,a^*})$ satisfies the transversality condition (21). Following Aghion and Howitt (1998, Ch. 3) and Howitt (1999), we can derive the long-run distribution function of relative productivity parameters $a_{it} \equiv A_{it} /A_t^{max}$

$$\begin{aligned} \text{ Pr }\left\{ {a_{it} \le a} \right\} \equiv F( a)=a^{1/\sigma }, \quad 0<a\le 1. \end{aligned}$$

(36)

In the steady state $( {\omega _s^*,a^*})$, the expected present value of future profits to be earned by an incumbent whose technology is of vintage $\tau \le t$ can be expressed as

$$\begin{aligned} \tilde{V}_{\tau ,t}^*=\int \limits _t^\infty e^{-( {r^*+\phi ^*})( {\upsilon -t})}\tilde{\pi }_{\tau ,\upsilon }^*d\upsilon , \end{aligned}$$

(37)

where superscript “${*}$” denotes the steady-state value. According to Eq. (8), we can rewrite Eq. (37) as

$$\begin{aligned} \tilde{V}_{\tau ,t}^*=\frac{\tilde{a}_{\tau ,t} \tilde{\pi }_t^*}{r^*+\phi ^*+( {1-\beta })g^*}. \end{aligned}$$

(38)

Note that $\tilde{\pi }_t^*$ represents the steady-state value of Eq. (7). According to Eqs. (36) and (38), total assets can be written as

$$\begin{aligned} (\textit{Total Assets})_t^*=\int \nolimits _0^1 f( a) \tilde{V}_{t}^{*} ( a)da, \end{aligned}$$

(39)

where $f( a)\equiv F^{'}( a)=( {1/\sigma })a^{( {1-\sigma })/\sigma }$ represents the density function, and each asset $\tilde{V}_t^*( a)$ is given by

$$\begin{aligned} \tilde{V}_t^*( a)=\frac{a\tilde{\pi }_t^*}{r^*+\phi ^*+( {1-\beta })g^*}. \end{aligned}$$

Solving Eq. (39), we obtain

$$\begin{aligned} (\textit{Total Assets})_t^*=\frac{a^*\tilde{\pi }_t^*}{r^*+\phi ^*+( {1-\beta })g^*}. \end{aligned}$$

This equation also represents the average asset value as intermediate firms are distributed as $i\in \left[ {0,1} \right]$. The growth rate of term $e^{-\rho t}C_t^{-\theta } ({Total Assets})_t$ in the steady state $( {\omega _s^*,a^*})$ obtains

$$\begin{aligned} -\rho -( {\theta -1})\beta g^*<0, \hbox { since } \theta >1. \end{aligned}$$

Thus, the steady state $( {\omega _s^*,a^*})$ satisfies the transversality condition.

Appendix C

This appendix considers the log-linearization of differential Eqs. (17) and (22). Log-linearizing around the steady state $( {\omega _s^*,a^*})$ obtains

$$\begin{aligned} \left[ \begin{array}{l@{\quad }l} {d\left[ {\ln ( {\omega _{s,t} })} \right] /dt} \\ {d\left[ {\ln ( {a_t })} \right] /dt} \\ \end{array} \right] =\mathcal{J}\left[ \begin{array}{l} {\ln ( {\omega _{s,t} /\omega _s^*})}\\ {\ln ( {a_t /a^*})}\\ \end{array} \right] \end{aligned}$$

(40)

with a Jacobian matrix

$$\begin{aligned} {\mathcal {J}}=\left[ \begin{array}{l@{\quad }l} {\text{ J }_1 } &{} {\text{ J }_2 } \\ {\text{ J }_3 } &{} {\text{ J }_4 } \\ \end{array} \right] , \end{aligned}$$

where each $\text{ J }_k$ for $k=1,2,3, \text{ and } 4$ is given by

$$\begin{aligned}&\text{ J }_1 =\frac{\rho +\lambda S\left[ {1+\sigma ( {1-\beta +\theta \beta })} \right] }{\beta +\theta ( {1-\beta })}>0,\\&\text{ J }_2 =-\frac{\theta ( {1+\sigma })\phi ^*+\lambda \left[ {1+\sigma ( {1-\beta +\theta \beta })} \right] s_m^*}{1+\theta ( {\frac{1-\beta }{\beta }})}<0,\\&\text{ J }_3 =0, \quad \hbox {and}\\&\text{ J }_4 =-( {1+\sigma })\phi ^*<0. \end{aligned}$$

Note that superscript “${*}$” denotes the steady-state value. Computing the eigenvalues, denoted by $\varepsilon _1$ and $\varepsilon _2$, we obtain

$$\begin{aligned} \varepsilon _1&= \text{ J }_1 >0\quad \hbox {and}\\ \varepsilon _2&= \text{ J }_4 <0. \end{aligned}$$

Hence, the steady state $( {\omega _s^*,a^*})$ is a saddle point. Using the stable eigenvalue, we obtain the differential equation

$$\begin{aligned} d\left[ {\ln ( {\omega _{s,t} })} \right] /dt=\varepsilon _2 \ln ( {\omega _{s,t} /\omega _s^*}). \end{aligned}$$

(41)

According to Eqs. (40) and (41), we obtain the relationship $\omega _{s,t}$ and $a_t$ on the saddle path

$$\begin{aligned} \ln ( {\omega _{s,t} /\omega _s^*})=\Psi \ln ( {a_t /a^*}), \end{aligned}$$

(42)

where $\Psi =\left[ {\text{ J }_2 /( {\varepsilon _2 -\text{ J }_1 })} \right] >0$. Thus, the saddle path is an upward-sloping curve. Additionally, differentiating Eq. (42) with respect to time, we obtain

$$\begin{aligned} \frac{{\dot{\omega }_{{s,t}} }}{{\omega _{{s,t}} }} = \Psi \frac{{\dot{a}_{t} }}{{a_{t} }}. \end{aligned}$$

Appendix D

In this Appendix, we derive the inequality (27) via a proof by contradiction. Suppose that

$$\begin{aligned} \Psi \ge \frac{\beta }{1-\beta }. \end{aligned}$$

Hence, we have

$$\begin{aligned} \frac{\left[ {1+\theta \left( {\frac{1-\beta }{\beta }}\right) } \right] ^{-1}\left[ {\theta ( {1+\sigma })\phi ^*+\lambda \left[ {1+\sigma ( {1-\beta +\theta \beta })} \right] s_m^*} \right] }{( {1+\sigma })\phi ^*+\frac{\rho +\lambda S\left[ {1+\sigma ( {1-\beta +\theta \beta })} \right] }{\beta +\theta ( {1-\beta })}}\ge \frac{\beta }{1-\beta }.\quad \quad \end{aligned}$$

(43)

Rewriting Eq. (43), yields

$$\begin{aligned}&( {1-\beta })\left[ {1+\theta \left( {\frac{1-\beta }{\beta }}\right) } \right] ^{-1}\left[ {\theta ( {1+\sigma })\phi ^*+\lambda \left[ {1+\sigma ( {1-\beta +\theta \beta })} \right] s_m^*} \right] \nonumber \\&\quad \ge \beta ( {1+\sigma })\phi ^*+\frac{\rho +\lambda S\left[ {1+\sigma ( {1-\beta +\theta \beta })} \right] }{1+\theta \left( {\frac{1-\beta }{\beta }}\right) }. \end{aligned}$$

(44)

Moving the first term on the right-hand side of Eq. (44) to the left-hand side, and subtracting the term $( {1-\beta })\left[ {1+\theta \left( {\frac{1-\beta }{\beta }}\right) } \right] ^{-1}\left[ {1+\sigma ( {1-\beta +\theta \beta })} \right] \lambda s_m^*$ from both sides, yields

$$\begin{aligned} -\left( {\frac{\beta ( {1+\sigma })}{1+\theta \left( {\frac{1-\beta }{\beta }}\right) }}\right) \phi ^*&\ge \frac{\rho }{1+\theta \left( {\frac{1-\beta }{\beta }}\right) }+\left( {\frac{1+\sigma ( {1-\beta +\theta \beta })}{1+\theta \left( {\frac{1-\beta }{\beta }}\right) }}\right) \lambda S\\&-( {1-\beta })\left( {\frac{1+\sigma ( {1-\beta +\theta \beta })}{1+\theta \left( {\frac{1-\beta }{\beta }}\right) }}\right) \lambda s_m^*. \end{aligned}$$

Using the fact that $\phi ^*=\lambda ( {S-s_m^*})$, and multiplying $\left[ {1+\theta ( {\frac{1-\beta }{\beta }})} \right]$ from both sides, yields

$$\begin{aligned} -\beta ( {1+\sigma })\phi ^*\ge \rho +\left[ {1+\sigma ( {1-\beta +\theta \beta })} \right] \phi ^*+\beta \left[ {1+\sigma ( {1-\beta +\theta \beta })} \right] \lambda s_m^*\nonumber \\ \end{aligned}$$

(45)

This is a contradiction, because the left-hand side of Eq. () has a negative value, while the right-hand side has a positive value. Hence, we obtain the result $\Psi <\beta /( {1-\beta })$.