The American System of economic growth

Abstract

The early history of industrialization in the United States—famously known as “The American System of Manufactures”—exhibited four key features: the substitution of specialized intermediate inputs for skilled work in assembling final goods, the freedom with which knowledge has long been shared in the United States, a learning technology that leverages existing mechanical know-how in human capital accumulation, and increasing returns to intermediate inputs in processing final goods. Our endogenous growth model embodies these components and utilizes historical time series data on labor force “operatives” and the Census of Manufactures to calibrate the model’s parameters. Our simulation closely matches the 1.88% average per capita product growth in the United States from 1860 to date. The simulation predicts that growth will peak in 1980 and ultimately converge to 1.31%—a growth slowdown rooted from the beginning in the economization of skilled labor inherent in the American System. By 2000, simulated per capita product is 2.21 times larger than a counterfactual in which the American System of manufactures never existed.

Appendices

Appendix 1: Zero-profit equilibrium in the conventional sector

Here we provide more details of the production equilibrium for the conventional sector alone. The first step is to find the value ${\tilde{x}}_{c}$ that each intermediate firm produces for sale to final-good firms, when $M_{c}$ is given (along with $e_{c}$, h, and N). To do so, we first observe that $e_{ic}{\bar{h}}N=M_{c}V_{c}\left( x_{c}\right)$: the supply of effective labor to the intermediate firms $e_{ic}{\bar{h}}N$ must equal the economy-wide demand for those inputs by all intermediate firms $M_{c}V_{c}\left( x_{c}\right)$. Solve for $e_{ic}$:

$$\begin{aligned} e_{ic}=\frac{M_{c}\left( v_{0}+v_{1}x_{c}\right) }{{\bar{h}}N} \end{aligned}$$

(59)

To find ${\tilde{x}}_{c}$, first use (11) and (10) to substitute for $p_{c}$ and $w_{c}$ in the markup condition (12); then substitute $e_{sc}=e_{c}-e_{ic}$ in the result. Finally, use (59) to substitute for $e_{ic}$, and solve for $x_{c}$:

$$\begin{aligned} {\tilde{x}}_{c}=\frac{\alpha ^{2}\left( e_{c}{\bar{h}}N-v_{0}M_{c}\right) }{\left( 1-\alpha +\alpha ^{2}\right) v_{1}M_{c}}\equiv X\left( e_{c}{\bar{h}}N,M_{c}\right) \end{aligned}$$

(60)

We call the function that represents intermediate output ${\tilde{x}}_{c}=X\left( e_{c}{\bar{h}}N,M_{c}\right)$. The $X\left( \ldots \right)$ function allows us to find $e_{ic}/e_{c}$ [using (59)] and $e_{sc}/e_{c}=1-e_{ic}/e_{c}$ as functions of $e_{c}{\bar{h}}N$ and $M_{c}$. Because $e_{c}$ is given at this point, we can then find $e_{sc}$ and $e_{ic}$ separately, which allows us to find $w_{c}$, $p_{c}$, and $Y_{c}$ using the structural equations. The value of ${\tilde{x}}_{c}$ as a function of $M_{c}$ is represented by the EQ locus in Fig. 10.

Fig. 10

Monopolistically competitive equilibrium

We are not going to spend time analyzing the equilibrium for a given $M_{c}$, because when profit is positive new firms will enter the intermediate-good market and increase the number of specialized inputs. That is, $M_{c}$ is endogenous.

Given $e_{c}$, h, and N, the ZP locus in Fig. 10 shows the momentary zero-profit combinations $x_{c}$ and $M_{c}$ in the conventional intermediate-good sector. To derive the ZP locus set profit $p_{c}x_{c}-w_{c}\left( v_{0}+v_{1}x_{c}\right)$ to zero, and substitute for $p_{c}$, $w_{c}$, $e_{sc}$, and $e_{ic}$ as above to yield:

$$\begin{aligned} x_{c}=\frac{\alpha e_{c}hN-v_{0}M_{c}}{v_{1}M_{c}} \end{aligned}$$

(61)

The ZP locus is also downward sloping.

The ZP locus crosses the EQ locus once from above. Conventional intermediate firm’s profits are positive below the ZP locus, and negative above. Hence, the free-entry, zero-profit monopolistically competitive equilibrium in the conventional sector is unique and stable at point A in Fig. 10. Equate (60) and (61) to see that the equilibrium number of firms (range of specialization) is given by $M_{c}^{*}=\frac{\alpha \left( 1-\alpha \right) }{v_{0}}e_{c}{\bar{h}}N$, which appears as Eq. (16) in the text. Substitute $M_{c}^{*}$ back into either (60) or (61) to get $x_{c}^{*}=x^{*}=\frac{\alpha v_{0}}{\left( 1-\alpha \right) v_{1}}$, which is (13) in the text. Another way to find $x^{*}$ is to substitute (2) and (12) into the profit expression $p_{c}x_{c}-w_{c}\left( v_{0}+v_{1}x_{c}\right)$ and set the result to zero.

To find the equilibrium allocation of effort to intermediate production, we use (59) after substituting the expressions for $M_{c}^{*}$ and $x^{*}$. This yields $e_{ic}=\alpha e_{c}$ from which it follows that $e_{sc}=\left( 1-\alpha \right) e_{c}$. These appear as (15) and (14) in the text. Finally, to obtain the equilibrium wage $w_{c}$ in (17) in the text, we use (10) and then substitute in the equilibrium values for $e_{sc}$, $x_{c}$, and $M_{c}$ that we have found in this appendix.

Using the corresponding equations from Sects. 3 and 5, we can show that the momentary equilibrium in the advanced intermediate sector is also unique and stable. The equilibrium values are derived in the same way as above and appear in the text.

Appendix 2: Intertemporal optimization

The representative household maximizes (40) in the text subject to two constraints, a time constraint:

$$\begin{aligned} 1=e_{sc}+e_{ic}+e_{ua}+e_{ia}+e_{l} \end{aligned}$$

(62)

and a resource budget constraint:

$$\begin{aligned} c=w_{c}h\left( e_{sc}+e_{ic}\right) +w_{u}e_{ua}+w_{a}he_{ia} \end{aligned}$$

(63)

Individuals consider base wages—$w_{c}$ and $w_{a}$—to be given along with the wage for unskilled tasks $w_{u}$, even though these wages depend on aggregate effort. Accumulating h allows them to increase their actual wages $w_{c}h$ and $w_{a}h$ in a manner that they perceive to be proportional.

The Hamiltonian for the problem is:

$$\begin{aligned} {\mathcal {H}}= \,& {} u\left( c\right) +\lambda \left( L^{\gamma }\,h^{1-\gamma }e_{l}-\eta h\right) \\&\quad + \theta _{1}\left( w_{c}h\left( e_{sc}+e_{ic}\right) +w_{u}e_{ua}+w_{a}he_{ia}-c\right) \\&\quad + \theta _{2}\left( 1-e_{sc}-e_{ic}-e_{ua}-e_{ia}-e_{l}\right) \end{aligned}$$

where $\lambda$ is the co-state, shadow price of h, we attach the constraints (62) and (63) with Lagrangian multipliers $\theta _{1}$ and $\theta _{2}$, and utility is assumed to be logarithmic: $u\left( c\right) =\ln c$.

The static FOC’s are as follows.

$$\begin{aligned} \frac{\partial {\mathcal {H}}}{\partial c}=0&\Rightarrow \,\,\frac{1}{c}=\theta _{1} \end{aligned}$$

(64)

$$\begin{aligned} \frac{\partial {\mathcal {H}}}{\partial e_{sc}}=0&\Rightarrow \,\,\theta _{1}w_{c}h=\theta _{2} \end{aligned}$$

(65)

$$\begin{aligned} \frac{\partial {\mathcal {H}}}{\partial e_{ic}}=0&\Rightarrow \,\,\theta _{1}w_{c}h=\theta _{2} \end{aligned}$$

(66)

$$\begin{aligned} \frac{\partial {\mathcal {H}}}{\partial e_{ua}}=0&\Rightarrow \,\,\theta _{1}w_{u}=\theta _{2} \end{aligned}$$

(67)

$$\begin{aligned} \frac{\partial {\mathcal {H}}}{\partial e_{ia}}=0&\Rightarrow \,\,\theta _{1}w_{a}h=\theta _{2} \end{aligned}$$

(68)

$$\begin{aligned} \frac{\partial {\mathcal {H}}}{\partial e_{l}}=0&\Rightarrow \,\,\lambda L^{\gamma }h^{1-\gamma }=\theta _{2} \end{aligned}$$

(69)

In addition, the shadow utility price $\lambda$ of human capital must change constantly to equate the cost and benefit of accumulating human capital:

$$\begin{aligned} {\dot{\lambda }}&=\left( \rho -\eta \right) \lambda -\frac{\partial {\mathcal {H}}}{\partial h}\nonumber \\ =&\left( \rho -\eta \right) \lambda -\lambda \left( \left( 1-\gamma \right) L^{\gamma }h^{-\gamma }e_{l}-\eta \right) -\theta _{1}\left( w_{c}\left( e_{sc}+e_{ic}\right) +w_{a}e_{ia}\right) \end{aligned}$$

(70)

Finally, we require the transversality condition (41) in the text.

The representative household takes L and the economy-wide averages ${\bar{h}}$ and ${\bar{e}}_{w}$ as given when undertaking the maximization. After that, we impose aggregate consistency so that the representative household’s choices match economy-wide averages: ${\bar{h}}=h$ and ${\bar{e}}_{w}=e_{w}$.

As noted in the text, the solution involves two differential equations, one for ${\dot{q}}$ and one for ${\dot{z}}$. We derived (49), the equation for ${\dot{q}}$, in the text. Here we show how to find (50), the expression for ${\dot{z}}$.

To derive (50), use (70), along with (64), (43), and wage equalization, to get:

$$\begin{aligned} \frac{{\dot{\lambda }}}{\lambda }=\rho -\eta -\left( \left( 1-\gamma \right) A\Omega \left( q\right) e_{l}-\eta \right) -\frac{w_{c}}{c\lambda }e_{sw} \end{aligned}$$

(71)

where $e_{sw}$ is effort in skilled tasks: $e_{sw}=e_{sc}+e_{ic}+e_{ia}$. Multiply and divide the last term by h and use (63) for c to get:

$$\begin{aligned} \frac{{\dot{\lambda }}}{\lambda }=\rho -\eta -\left( \left( 1-\gamma \right) A\Omega \left( q\right) e_{l}-\eta \right) -\left( \frac{e_{sw}}{e_{w}}\right) \frac{1}{z} \end{aligned}$$

(72)

The last term can be expressed as $\Phi \left( q\right) \frac{1}{z}$ , which can be seen from (37) in the text. Since $z=\lambda h$, we know that $\frac{{\dot{z}}}{z}=\frac{{\dot{\lambda }}}{\lambda }+\frac{{\dot{h}}}{h}=\frac{{\dot{\lambda }}}{\lambda }+\frac{{\dot{q}}}{q}$ . So, we add (72) to (49) to obtain:

$$\begin{aligned} \frac{{\dot{z}}}{z}=\rho -\eta -\left( \left( 1-\gamma \right) A\Omega \left( q\right) e_{l}-\eta \right) -\Phi \left( q\right) \frac{1}{z}+A\Omega \left( q\right) -\frac{1}{z}-\eta \end{aligned}$$

(73)

From here, we use (48) and simplify to get (50) in the text.

Appendix 3: Restricting $\left( A,k\right)$ to support perpetual growth

Two conditions on the A and k parameters are necessary to ensure perpetual growth. PG1 guarantees that for the initial value of human capital $q\left( 0\right)$ the ZZ locus lies above the HH locus in Fig. 3. PG2 makes sure that the endpoint of ZZ, as $q\rightarrow \infty$, lies above the endpoint of HH. If these two conditions are satisfied, then the ZZ lies above HH for all $q>q\left( 0\right)$. We prove this informally after we discuss the region defined by these two conditions.

To find PG1, set $q=q\left( 0\right)$, equate (51) to (52) and solve for k to get:

$$\begin{aligned} k_{PG1}=\frac{1}{q\left( 0\right) }\left[ \left( 1+q\left( 0\right) \right) \left( \frac{\rho +\eta \left[ \gamma +\Phi \left( q\left( 0\right) \right) -1\right] }{A\Phi \left( q\left( 0\right) \right) }\right) ^{\left( 1/\gamma \right) }-1\right] \end{aligned}$$

(74)

where $\Phi \left( q\right)$ is defined in (37). PG1 requires that $k>k_{PG1}$ in (74).

To find PG2, note that as q tends to infinity, the HH locus (51) becomes horizontal at:

$$\begin{aligned} {\hat{z}}_{H}=\frac{1}{Ak^{\gamma }-\eta } \end{aligned}$$

(75)

Equate (75) and (53) in the text and solve for k to get:

$$\begin{aligned} k_{PG2}\equiv \left( \frac{\rho +\eta \left( \alpha +\gamma -1\right) }{\alpha A}\right) ^{\frac{1}{\gamma }} \end{aligned}$$

(76)

PG2 requires that $k>k_{PG2}$ in (76).

Fig. 11

Region and grid (Color figure online)

Figure 11 shows the two boundary loci, PG1 and PG2, defined by, respectively, (74) and (76). These loci bound the perpetual growth region on the left and bottom. The figure also shows the grid within the region over which the baseline search is conducted (4233 points), as well as the calibrations for each base/target pair (small dots), and the aggregate calibrations, labeled “G_med” and “G_ave” (large red dots). Each base/target pair is well within the region—with two exceptions, those for 1860/1920 and 1870/1920. Those two dots (labeled) are right on the PG2-border of the region and the grid. Even so, the two calibrations corresponding to those points fit the model well in the sense that the loss used by the algorithm is very low. That is, define the loss as:

$$\begin{aligned} l\left( b,T\right) =\left( ops\left( b,T\right) -opa\left( T\right) \right) ^{2}*10^{9} \end{aligned}$$

(77)

where $ops\left( b,T\right)$ is the simulated operative share using base year b and target year T, and $opa\left( T\right)$ is the actual operative share in target year T. The losses for these two points are 0.1 and 0.0, which are as small, or smaller, than the losses for the other points in Fig. 11. We did, however, exclude two other calibrations on the border based on losses that exceeded the norm for interior calibrations. These were for 1880/1900 (loss = 525) and 1910/1920 (loss = 11,744). All other losses were less than 1.5. Figure 11 only shows the 26 calibration points that were used to calculate the median and average values for A and k.

The remainder of this appendix explains why all calibrations that satisfy PG1 and PG2 are such that for all q the ZZ locus lies above the HH locus, ensuring perpetual growth.

The PG1 and PG2 loci are also shown in Fig. 12, where we write $PG1\left( q\right)$ as an explicit function of q. Only for calibrations above $PG1\left( q_{0}\right)$ and PG2—which we say reside in “Region P” in Fig. 12—does ZZ lie above HH for both $q_{0}$ and as $q\rightarrow \infty$.

The above statements are sufficient to establish the proposition—that perpetual growth only occurs for values of A and k in Region P—since we can prove that ZZ and HH never cross for $q>q_{0}$ for $\left( A,k\right)$ calibrations in Region P.

To show this, begin with an $\left( A,k\right)$ calibration like Point Q or Point S and an initial $q=q_{0}$. We know that ZZ is greater than HH for either calibration at $q_{0}$ so ${\dot{q}}>0$. As q rises, use (74), and the limiting values of $\Omega \left( q\right)$ and $\Phi \left( q\right)$, to show that $PG1\left( q\right)$ rotates counterclockwise, as its intercept with PG2 moves left, in such a way that as $q\rightarrow \infty$, $PG1\left( q\right)$ becomes indistinguishable from PG2. The arrows show the movement in $PG1\left( q\right)$ as q rises. Therefore, for all $q>q_{0}$, there is no $PG1\left( q\right)$ locus that passes through either Q or S, so there is no $q>q_{0}$ for which ZZ and HH intersect. This is true for all calibrations in Region P when $q=q_{0}$.

We can make this argument for any initial $q\in {{\left( 0,\infty \right) }}$. The reason is that for $q=0$, the $PG1\left( q\right)$ locus coincides with the vertical line labeled CA in Fig. 12. The equation of CA is $A=\rho +\gamma \eta$, which is derived by solving (74) for A and noting that $\Omega \left( 0\right) =\Phi \left( 0\right) =1$. It follows that for any initial $q_{1}>0$, no matter how small, there is a Region P for which the argument above works. If $q_{1}=0$, Region P becomes bounded by CA on the left and PG2 below. If $q_{1}=1{,}000{,}000$, Region P is bounded below, effectively, by PG2 alone.

The above observations imply that for calibrations in Region P per capita human capital growth will converge from above to a positive perpetual rate of growth. Hence, we search for $\left( A,k\right)$ pairs in Region P in Fig. 12 because model simulations using these calibrations have the potential to track historical per capita product growth.

Fig. 12

Growth regimes in $\left( A,k\right)$ space

There is one other region in $\left( A,k\right)$ space in Fig. 12 with the potential for the simulation to hit the historical per capita product growth path. This region is to the right of CA and $PG1\left( q_{0}\right)$ and to the left of PG2. Points like R support saddle-point growth of q. That is, Point R is in the region where (see above) ZZ is above HH at $q_{0}$ but ends up below HH as $q\rightarrow \infty$. That is, ZZ cuts HH from above for the calibration given by R, establishing a saddle-point equilibrium for a value $q_{R}>q_{0}$. It follows that at $q_{0}$, ${\dot{q}}>0$ as q converges to $q_{R}$; but ${\dot{q}}\rightarrow 0$ as $t\rightarrow \infty$: human capital growth converges asymptotically from above to zero.

Recall that the $PG1\left( q\right)$ locus moves leftward and rotates counterclockwise as q grows. But q growth slows as q approaches the saddle-point equilibrium $q_{R}$, so $PG1\left( q\right)$ asymptotically approaches but never reaches Point R. In other words, per capita human capital growth is falling toward zero in this saddle-point regime. And if R is very close to PG2, falling per capita human capital growth might be imperceptible, with the potential for the model simulation to track the nearly constant historic per capita product growth rate reasonably well.

To check this possibility, we searched for $\left( A,k\right)$ pairs that mimic perpetual growth in the region around points like R. Using the methodology from Sect. 8, every calibration we tried between $PG1\left( q_{0}\right)$ and PG2 (26 primary and 3 aggregated calibrations) resulted in growth in per capita product that fell unmistakably over the twentieth century. In addition to this declining trend—which is not in the historical record—simulations based on these calibrations also show average growth that is between 1.6% and 1.7%, far below the actual historical average of 1.88%. We discard the saddle-point calibration because the simulated growth rates decline monotonically from 1860 to date and average only 1.65%, failing to track the historical record.

Finally, consider Point T. Since T is below $PG1\left( q_{0}\right)$ and PG2, it must be true that ${\dot{q}}<0$. It is also true that q falls forever since $PG1\left( q\right)$ can never go through Point T. This follows from our observation above that, as $q\rightarrow 0$, the $PG1\left( q\right)$ locus moves rightward and clockwise until it coincides with the CA vertical line.

There is a region of Fig. 12—centered southwest of Point E but close to PG2 and CA—where a single calibration of $\left( A,k\right)$ supports two equilibria of the dynamic system. One of these is unstable and one is saddle-point stable. This must be true because, as we vary q, we can trace out a region in which the $PG1\left( q\right)$ loci so generated cross one another. That is, there is a region where $PG1\left( q_{1}\right) =PG1\left( q_{2}\right)$ for $q_{1}\ne q_{2}$. That region is confined to a relatively small area beneath PG2 and to the left of CA. In the phase plane, it corresponds to calibrations for which ZZ intersects HH twice, which is conceivable given their downward slope and convex curvature.

For these reasons, we confine our $\left( A,k\right)$ calibrations to the region supporting perpetual growth of per capita human capital in Fig. 12.

Appendix 4: Simulation method

Here we show how, for an arbitrary $\left( A,k\right)$ pair, we eliminate time, and find the policy function $z=F\left( q\right)$ that corresponds to the PP path in the phase plane of Fig. 3. We pointed out in Sect. 7 that the optimal path PP gets arbitrarily close to ZZ where it is horizontal at ${\hat{z}}_{Z}$ in (53) as q gets large. We construct the policy function numerically by working backwards from the point $\left( {\hat{z}}_{Z},{\hat{q}}\right)$ where ${\hat{q}}$ is an arbitrarily large value. To eliminate time, we take the ratio of (50) and (49), to get $\frac{dz}{dt}/\frac{dq}{dt}=dz/dq$. The differential equation dz/dq allows us to work backwards from the point $\left( {\hat{z}}_{Z},{\hat{q}}\right)$ to trace out the optimal PP path. We solve for this path numerically to obtain the policy function $z=F\left( q\right)$.

Then, we substitute the policy function $F\left( q\right)$ for z in (49), which yields a differential equation in ${\dot{q}}$ that depends only on q, the six model parameters, and the initial value $q\left( 0\right)$. We get $q\left( 0\right)$ from (58) using operative share data from Table 1. This allows us to find the time path $q=W\left( t\right)$ using numerical methods. The time path of z is obtained by substitution as $z=F\left( W\left( t\right) \right)$. With the state and co-state as functions of time, we can find all effort allocations, output levels, and growth rates—contingent on our arbitrary choice of $\left( A,k\right)$.

We use the following steps to simulate the growth rate over time. Take the growth rate of y in (39):

$$\begin{aligned} g_{y}=\left( 2-\alpha \right) \left( g_{h}+\frac{{\dot{e}}_{w}}{e_{w}}\right) +\left( 1-\alpha \right) \left( \eta -\frac{{\dot{Q}}}{Q}\right) \end{aligned}$$

(78)

where $Q\equiv 1+q$ and $g_{h}$ is the growth rate of h.

To find a workable expression for (78), begin with $g_{h}=g_{q}$. For this, we use (49) and the policy function $z=F\left( q\right)$. This yields:

$$\begin{aligned} g_{q}=A\Omega \left( q\right) -\frac{1}{F\left( q\right) }-\eta \end{aligned}$$

(79)

where $\Omega$ is given by (45). The policy function, recall, depends on the six calibrated parameters. From the definition of Q above, we have:

$$\begin{aligned} \frac{{\dot{Q}}}{Q}=\frac{g_{q}*q}{1+q} \end{aligned}$$

(80)

where we substitute $g_{q}$ from (79).

To find $\frac{{\dot{e}}_{w}}{e_{w}}$ in (78), we begin with (47). This yields:

$$\begin{aligned} \frac{{\dot{e}}_{w}}{e_{w}}=-\frac{{\dot{z}}}{z}-\frac{{\dot{\Omega }}}{\Omega } \end{aligned}$$

(81)

Use (50) to see that:

$$\begin{aligned} \frac{{\dot{z}}}{z}=\rho -\eta +\gamma A\Omega \left( q\right) -\frac{\gamma +\Phi \left( q\right) }{F\left( q\right) } \end{aligned}$$

(82)

where $\Phi \left( q\right)$ is given by (37). The last term in (81) can be found by differentiating the expression for $\Omega \left( q\right)$ in (45). This yields:

$$\begin{aligned} \frac{{\dot{\Omega }}}{\Omega }=\gamma g_{q}\left( \frac{k*q}{1+k*q}-\frac{q}{1+q}\right) \end{aligned}$$

(83)

Putting all these pieces into (78) gives us an expression for $g_{y}$ as a function of q—and of all the parameters. Call this function:

$$\begin{aligned} g_{y}=\Gamma \left( P,q\right) \end{aligned}$$

(84)

where P stands for the vector of parameters. Above, we showed how to find q as a function of time: $q=W\left( t\right)$. Substituting that into the $\Gamma$ function gives us:

$$\begin{aligned} g_{y}=G\left( P,t\right) \end{aligned}$$

(85)

We use (85) to simulate the paths of $g_{y}$.

Appendix 5: Counterfactual conventional-only economy

If the advanced sector did not exist, the time constraint would be $1=e_{c}+e_{l}=e_{sp}+e_{ic}+e_{l}$ and the budget constraint would be:

$$\begin{aligned} c=w_{b}he_{c} \end{aligned}$$

(86)

The learning technology (9) becomes:

$$\begin{aligned} {\dot{h}}=L_{c}^{\gamma }h^{1-\gamma }e_{l}-\eta h \end{aligned}$$

(87)

where

$$\begin{aligned} L_{c}\equiv \frac{M_{c}}{\overline{e_{c}}\,N} \end{aligned}$$

(88)

since $M_{a}=0$. Learning productivity is still enhanced by the specialization spillover $L_{c}^{\gamma }$. Substitute (16) into (88) to see that $\frac{L_{c}}{h}$ is a constant. Then define $A\equiv \left( \frac{L_{c}}{h}\right) ^{\gamma }=\left( \frac{\alpha \left( 1-\alpha \right) }{v_{0}}\right) ^{\gamma }$. First order condition (69) is replaced by $\lambda L_{c}^{\gamma }h^{1-\gamma }=\theta _{2}$, but we retain (64) and (65). The first-order conditions and the constraint (86) yield:

$$\begin{aligned} e_{c}=\frac{1}{zA} \end{aligned}$$

(89)

It follows that: $e_{l}=1-\frac{1}{zA}$ so that the accumulation equation (87) is:

$$\begin{aligned} {\dot{h}}=h\left( A-\frac{1}{z}-\eta \right) \end{aligned}$$

(90)

assuming that $e_{l}>0$. The arbitrage condition (70) is replaced with:

$$\begin{aligned} {\dot{\lambda }}&=\left( \rho -\eta \right) \lambda -\frac{\partial {\mathcal {H}}_{c}}{\partial h}\nonumber \\&=\left( \rho -\eta \right) \lambda -\lambda \left( \left( 1-\gamma \right) L_{c}^{\gamma }h^{-\gamma }e_{l}-\eta \right) -\theta _{1}w_{c}e_{w} \end{aligned}$$

(91)

where ${\mathcal {H}}$ $_{c}$ is the Hamiltonian in this case of no advanced technology.

To get the motion equation for z, recall that $\frac{{\dot{z}}}{z}=\frac{{\dot{\lambda }}}{\lambda }+\frac{{\dot{h}}}{h}$. Using the first-order conditions, along with (90) and (91), allows us to express the motion of z as follows:

$$\begin{aligned} {\dot{z}}=z[\rho -\eta +\gamma A]-\left( 1+\gamma \right) \end{aligned}$$

where we assume that $e_{l}>0$.

The optimal policy for the representative household that satisfies the first-order conditions and transversality condition (41) is to set $z\left( t\right)$ to the following constant:

$$\begin{aligned} z\left( t\right) =z_{c}\equiv \frac{1+\gamma }{\rho -\eta +\gamma A} \end{aligned}$$

(92)

There are no transitional dynamics in the conventional-only economy. Substituting $z_{c}$ from (92) into (90) shows that the growth of per capita human capital h is constant in this dynamic equilibrium:

$$\begin{aligned} g_{hc}=\frac{A-\rho -\gamma \eta }{1+\gamma } \end{aligned}$$

(93)

Output per capita is given by $y_{c}=w_{s}e_{c}$. Substitute (92) into (89) to see that $e_{c}$ is constant, so equilibrium growth in $w_{s}$ and $y_{c}$ is identical and may be inferred from (18) as:

$$\begin{aligned} g_{yc}=\left( 1-\alpha \right) \eta +\left( 2-\alpha \right) g_{hc} \end{aligned}$$

(94)

where $g_{hc}$ is given in (93).