Catching up and falling behind

Abstract

This paper studies the interaction between technology, a public input that flows in from abroad, and human capital, a private input that is accumulated domestically, as twin engines of growth in a developing economy. The model displays two types of long run behavior, depending on policies and initial conditions. One is sustained growth, where the economy keeps pace with the technology frontier. The other is stagnation, where the economy converges to a minimal technology level that is independent of the world frontier. In a calibrated version of the model, transition paths after a policy change can display rapid growth, as in modern growth ‘miracles.’ In these economies policies that promote technology inflows are much more effective than subsidies to human capital accumulation in accelerating growth. A policy reversal produces a ‘lost decade,’ a period of slow growth that permanently reduces the level of income and consumption.

Appendices

Appendix 1: Equilibrium conditions

The Hamiltonian for the household’s problem in (9) is

$$\begin{aligned} {\mathfrak {H}}&= \frac{C^{1-\theta }}{1-\theta }+\Lambda _{H}\left[ \phi _{0}\left( vH\right) ^{\eta }A^{\zeta }\overline{H}^{1-\eta -\zeta }-\delta _{H}H\right] \\&\quad +\Lambda _{K}\left[ \left( 1-v\right) \hat{w}A^{\beta }H^{\omega } \overline{H}^{1-\beta -\omega }+v\sigma \hat{w}A^{\beta }\overline{H} ^{1-\beta }+\left( R-\delta _{K}\right) K-C-\tau \right] , \end{aligned}$$

where to simplify the notation the subscript $i$’s have been dropped. Taking the first order conditions for a maximum, using the equilibrium conditions $\overline{H}=H$ and $\tau =v\sigma \hat{w}A^{\beta }H^{1-\beta },$ and simplifying gives

$$\begin{aligned} \Lambda _{H}\phi _{0}\eta v^{\eta -1}&= \Lambda _{K}\left( 1-\sigma \right) \hat{w}\left( \frac{A}{H}\right) ^{\beta -\zeta }, \nonumber \\ C^{-\theta }&= \Lambda _{K}, \nonumber \\ \frac{\dot{\Lambda }_{H}}{\Lambda _{H}}&= \rho +\delta _{H}-\phi _{0}\eta v^{\eta }\left( \frac{A}{H}\right) ^{\zeta }\left( 1+\frac{\omega }{1-\sigma }\frac{1-v}{v}\right) , \nonumber \\ \frac{\dot{\Lambda }_{K}}{\Lambda _{K}}&= \rho +\delta _{K}-R, \nonumber \\ \frac{\dot{H}}{H}&= \phi _{0}v^{\eta }\left( \frac{A}{H}\right) ^{\zeta }-\delta _{H}, \nonumber \\ \frac{\dot{K}}{K}&= \left( \frac{K}{L}\right) ^{\alpha -1}-\frac{C}{K} -\delta _{K}, \end{aligned}$$

(25)

where $L=\left( 1-v\right) A^{\beta }H^{1-\beta },$ and $\dot{A}/A$ is in ( 2). The transversality conditions are

$$\begin{aligned} \lim _{t\rightarrow \infty }e^{-\rho t}\Lambda _{K}(t)K(t)=0\qquad \text {and} \qquad \lim _{t\rightarrow \infty }e^{-\rho t}\Lambda _{H}(t)H(t)=0. \end{aligned}$$

(26)

Equations (1), (2), (25) and (26) characterize the competitive equilibrium, given the policy parameters $\left( B,\sigma \right)$ and initial values for the state variables $\left( W,A,H,K\right)$.

The law of motion for $A$ requires $H/W$ and $A/W$ to be constant along a BGP, so $A$ and $H$ must also grow at the rate $g.$ Since the production functions for effective labor in (4) and for output have constant returns to scale, the factor inputs $L$ and $K$ also grow at the rate $g,$ as do output $Y$ and consumption $C.$ Hence the factor returns $R$ and $\hat{w}$ are constant on a BGP, and the costate variable $\Lambda _{K}$ grows at the rate $-\theta g.$ The costate $\Lambda _{H}$ grows at the same rate as $\Lambda _{K}.$

The normalized conditions in (12) follow directly from (2) and (25).

Appendix 2: Linear approximations and stability

Define the constants

$$\begin{aligned} \chi \equiv \eta \frac{\omega }{1-\sigma },\qquad \qquad \qquad \Pi _{H}\equiv \frac{r^{bg}+\delta _{H}}{g+\delta _{H}},\qquad \qquad \Delta \equiv \beta \left( 1-\theta \right) -\zeta , \end{aligned}$$

$$\begin{aligned} \Gamma _{2}&\equiv \eta -\frac{\chi }{\eta v^{bg}+\chi \left( 1-v^{bg}\right) }<\eta , \\&= \eta -\frac{\chi /v^{g}}{\eta +\chi \left( 1/v^{bg}-1\right) } \\&= \eta -\frac{\chi +\Pi _{H}-\eta }{\Pi _{H}}, \\ \frac{1}{\Gamma _{3}}&\equiv \eta -1-\frac{\theta }{1/v^{bg}-1} \\&= \eta -1-\frac{\theta \chi }{\Pi _{H}-\eta }<0, \end{aligned}$$

and recall that

$$\begin{aligned} \frac{1}{v^{bg}}-1=\frac{1}{\chi }\left( \Pi _{H}-\eta \right) >0. \end{aligned}$$

In addition define the log deviations

$$\begin{aligned} x_{1}\equiv \ln \left( a/a^{bg}\right) ,\qquad x_{2}\equiv \ln \left( h/h^{bg}\right) ,\qquad x_{3}\equiv \ln \left( \lambda _{h}/\lambda _{h}^{bg}\right) , \end{aligned}$$

Take a first-order approximation to (20) to get

$$\begin{aligned} \frac{v-v^{bg}}{v^{bg}}=\Gamma _{3}\left[ \Delta x_{1}-\left( \Delta +\theta \right) x_{2}-x_{3}\right] . \end{aligned}$$

Then linearize the laws of motion for $a,h,$ and $\lambda _{h}$ in (12 ) to find that

$$\begin{aligned} \dot{x}_{1}&\approx \left( g+\delta _{A}\right) \left[ -\frac{a_{J}^{bg}}{ 1-a_{J}^{bg}}x_{1}+x_{2}\right] , \\ \dot{x}_{2}&\approx \left( g+\delta _{H}\right) \left[ \zeta \left( x_{1}-x_{2}\right) +\eta \frac{v-v^{bg}}{v^{bg}}\right] \\&= \left( g+\delta _{H}\right) \left\{ \zeta \left( x_{1}-x_{2}\right) +\eta \Gamma _{3}\left[ \Delta \left( x_{1}-x_{2}\right) -\theta x_{2}-x_{3}\right] \right\} , \\ \dot{x}_{3}&\approx -\left( r^{bg}+\delta _{H}\right) \left[ \zeta \left( x_{1}-x_{2}\right) +\Gamma _{2}\frac{v-v^{bg}}{v^{bg}}\right] \\&= -\left( r^{bg}+\delta _{H}\right) \left\{ \zeta \left( x_{1}-x_{2}\right) +\Gamma _{2}\Gamma _{3}\left[ \Delta \left( x_{1}-x_{2}\right) -\theta x_{2}-x_{3}\right] \right\} . \end{aligned}$$

Hence

$$\begin{aligned} \left( \begin{array}{c} \dot{x}_{1} \\ \dot{x}_{2} \\ \dot{x}_{3} \end{array} \right) \approx \left( \begin{array}{rrr} -q_{1J} &{} q_{2} &{} 0 \\ q_{3} &{} -q_{3}-\theta q_{4} &{} -q_{4} \\ -q_{6} &{} q_{6}+\theta q_{5} &{} q_{5} \end{array} \right) \left( \begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned} q_{1J}&= \left( g+\delta _{A}\right) a_{J}^{bg}/\left( 1-a_{J}^{bg}\right) , \\ q_{2}&= g+\delta _{A}, \\ q_{3}&= \left( g+\delta _{H}\right) \left( \zeta +\eta \Gamma _{3}\Delta \right) , \\ q_{4}&= \left( g+\delta _{H}\right) \eta \Gamma _{3}, \\ q_{5}&= \left( r^{bg}+\delta _{H}\right) \Gamma _{2}\Gamma _{3}, \\ q_{6}&= \left( r^{bg}+\delta _{H}\right) \left( \zeta +\Gamma _{2}\Gamma _{3}\Delta \right) . \end{aligned}$$

The local stability of each steady state depends on the roots of the associated characteristic equation,

$$\begin{aligned} 0&= \det \left( \begin{array}{rrr} -R-q_{1J} &{} q_{2} &{} 0 \\ q_{3} &{} -R-\left( q_{3}+\theta q_{4}\right) &{} -q_{4} \\ -q_{6} &{} q_{6}+\theta q_{5} &{} -R+q_{5} \end{array} \right) \\&= \det \left( \begin{array}{rrr} -\left( R+q_{1J}\right) &{} q_{2} &{} 0 \\ q_{3} &{} -\left( R+q_{3}\right) &{} -q_{4} \\ -q_{6} &{} q_{6}+\theta R &{} -R+q_{5} \end{array} \right) \\&= -\left( R+q_{1J}\right) \left[ \left( R+q_{3}\right) \left( R-q_{5}\right) +q_{4}\left( q_{6}+\theta R\right) \right] \\&\quad -q_{2}\left( -Rq_{3}+q_{3}q_{5}-q_{4}q_{6}\right) \\&= -\left( R+q_{1J}\right) \left[ R^{2}-Rm_{1}-m_{2}\right] +q_{2}\left( Rq_{3}-m_{2}\right) \\&= -R^{3}+\left( m_{1}-q_{1J}\right) R^{2}+\left( m_{2}+m_{1}q_{1J}+q_{2}q_{3}\right) R+m_{2}\left( q_{1J}-q_{2}\right) , \end{aligned}$$

where

$$\begin{aligned} m_{1}&\equiv q_{5}-q_{3}-\theta q_{4}, \\ m_{2}&\equiv q_{3}q_{5}-q_{4}q_{6} \\&= \left( g+\delta _{H}\right) \left( r^{bg}+\delta _{H}\right) \Gamma _{3}\zeta \left( \Gamma _{2}-\eta \right) >0, \end{aligned}$$

and the last line uses the fact that $\Gamma _{3}<0$ and $\Gamma _{2}-\eta <0.$

Write the cubic as

$$\begin{aligned} 0=\Psi _{J}(R)\equiv -R^{3}+A_{2J}R^{2}+A_{1J}R+A_{0J},\qquad J=H,L, \end{aligned}$$

where

$$\begin{aligned} A_{2J}&\equiv m_{1}-q_{1J}, \\ A_{1J}&\equiv m_{2}+m_{1}q_{1J}+q_{2}q_{3}, \\ A_{0J}&\equiv m_{2}\left( q_{1J}-q_{2}\right) ,\qquad \qquad \qquad J=H,L. \end{aligned}$$

Since $a_{H}^{bg}>1/2>a_{L}^{bg},$ it follows that $q_{1H}>q_{2}>q_{1L}.$ Hence

$$\begin{aligned} \Psi _{H}(0)=A_{0H}>0,\qquad \Psi _{L}(0)=A_{0L}<0, \end{aligned}$$

so $\Psi _{H}$ has at least one positive real root, and $\Psi _{L}$ has at least one negative real root.

The other roots of $\Psi _{H}$ are real and both are negative if and only if $\Psi _{H}$ has real inflection points, the smaller one is negative—call it $I_{H}<0$, and $\Psi _{H}(I_{H})<0$. Similarly, the other roots of $\Psi _{L}$ are real and both are positive, if and only if $\Psi _{L}$ has real inflection points, the larger one is positive—call it $I_{L}>0$, and $\Psi _{L}(I_{L})>0.$ The inflection points are solutions of the quadratic $\Psi _{J}^{\prime }(I)=0,$ so they are real if and only if $\Psi _{J}^{\prime }(0)=A_{1J}>0.$ Write this condition as

$$\begin{aligned} 0<m_{2}+\left( m_{1}+q_{3}\right) q_{2}+m_{1}\left( q_{1J}-q_{2}\right) . \end{aligned}$$

(27)

To construct examples where $\Psi _{J}$ has complex roots, consider cases where the inequality in (27) fails. As noted above, $m_{2}>0.$ The term $\left( q_{1J}-q_{2}\right)$ is small in absolute value if $a_{j}^{bg}$ is close to 1/2. As shown in Fig. 4b, this occurs if $1/B$ is close to the lowest value for which a BGP exists. For the middle term note that $q_{2}>0$ and

$$\begin{aligned} m_{1}+q_{3}&= q_{5}-\theta q_{4} \\&= \left( g+\delta _{H}\right) \Gamma _{3}\left( \Pi _{H}\Gamma _{2}-\theta \eta \right) \\&= -\left( g+\delta _{H}\right) \Gamma _{3}\left[ \theta \eta -\Pi _{H}\eta +\chi +\Pi _{H}-\eta \right] \\&= -\left( g+\delta _{H}\right) \Gamma _{3}\left[ \eta \left( \theta -1+ \frac{\omega }{1-\sigma }\right) +\Pi _{H}\left( 1-\eta \right) \right] . \end{aligned}$$

Since $\Gamma _{3}<0,$ this term has the sign of the expression in square brackets. The term $\Pi _{H}\left( 1-\eta \right)$ is positive, but for $\eta$ close to one it is small. The term $\omega /\left( 1-\sigma \right)$ is also positive, but for $\omega$ close to zero it is also small. The term $\theta -1$ is negative if $\theta$ is close to zero. Thus, complex roots can occur if $B$ is large, $\eta$ is close to unity, and $\omega ,\theta$ are close to zero.

For example, with the preference and technology parameters

$$\begin{aligned} \theta =0.10,\eta =0.95,\zeta =0.05,\omega =0.05,\beta =0.45,a_{F}=0.9179, \end{aligned}$$

and the others at the baseline values, and the policy parameters

$$\begin{aligned} \sigma =0.9026,B=1.0, \end{aligned}$$

the steady states are $a_{H}^{bg}=0.6684$ and $a_{L}^{bg}=0.3316,$ and the system has complex roots—with negative real parts—at both steady states. For nearby policy parameters, the complex roots can have positive real parts at the low steady state, and the roots at the high steady state can be real while those at the low steady state remain complex. Notice that $\omega$ close to zero implies that an individual’s wage depends very little on his own human capital, while $1-\omega -\beta =0.50$ implies that average human capital is quite important for productivity. Hence the optimal steady state subsidy is quite high—here it is $\sigma _{F}=0.9163,$ and the example uses a subsidy almost that high. In addition, $\theta$ close to zero implies a very high elasticity of intertemporal substitution. These parameter values are implausible.

Appendix 3: Computing transition paths to the SS

The following two-step procedure is used to compute transition to the stagnation SS. For the first step, let $A_{0}=A^{st}$ and conjecture that $\dot{A}=0.$ Then the transition is described by the equations for $\dot{H},$ $\dot{\Lambda }_{H},$ which do not involve $W.$ Linearize (3) and (22) around $\left( H^{st},\Lambda ^{st}\right)$, using (21) for $v$. The resulting pair of equations has roots that are real and of opposite sign. Hence for any $H_{0}$ sufficiently close to $H^{st},$ there exists a unique $\Lambda _{H0}$ near $\Lambda _{H}^{st}$ with the property that for the initial conditions $\left( H_{0},\Lambda _{H0}\right)$ and $A_{0}=A^{st},$ the linearized system converges to the steady state. This solution constitutes a competitive equilibrium provided that $\dot{A}=0$ when (2) is used. That holds provided that $H_{0}/W_{0}$ is small enough. For the second step, choose any pair $\left( H,\Lambda _{H}\right)$ generated by the first step, and a (large) value $W.$ Run the system of ODEs in (1)–(3) and (22) backward from the ‘initial’ condition $\left( W,A^{st},H,\Lambda _{H}\right)$.