Technology, skill, and growth in a global economy

Abstract

This paper develops an endogenous growth model based on a Roy-like assignment model in which heterogeneous workers endogenously sort into different technologies/tasks according to their comparative advantage. By modeling explicit distinction between worker skills and tasks as well as incorporating task-specific technologies, we analyze in depth the technology-skill-growth and offshoring-growth links that are absent in traditional models of endogenous growth. We show that offshoring increases domestic aggregate productivity, welfare and growth due to technology-upgrading mechanism.

Appendices

Appendix A: Numerical illustration

1.1 A.1 The initial equilibrium

In this section we illustrate our theoretical discussions with numerical simulations. To do this, we first set the stage by characterizing the initial equilibrium roughly calibrated to US data. We assume a log-normal skill distribution²⁹ and linear technologies consistent with Eq. (10):

$$\begin{aligned} \begin{array}{l} g(z)=\frac{1}{z\sqrt{2\pi }\varepsilon }\mathrm{e}^{-\frac{(\ln z-\mu )^{2}}{ 2\varepsilon ^{2}}},\ \ \ \ z\in (0,\infty ) \\ \varphi _{j}(z)=1+a_{j}z,\ \ \ \ \ \ \ \ \ \ j\in \left\{ M,L,H\right\} . \end{array} \end{aligned}$$

Multinational firms do not, of course, offshore outsource all the intermediate components. To reflect more the reality, here we assume that high-tech multinationals use $\beta$ share of foreign intermediate components and ($1-\beta$) share of domestic intermediate components. ILO provides us with unit labor costs (relative to the USA) in manufacturing for a number of cheap labor countries. From which we choose $C_{M}^{*}=0.63$, approximately a value of Mexico at the end of 1980’ before NAFTA was signed. We set $\rho =0.05$ for the discount rate and $\sigma =4$ for the elasticity of substitution between varieties. Empirical evidence on the level of fixed costs is scarce, but it seems reasonable that the ratio of fixed costs for a vertically fragmented firm to those for a domestic firm lies between 1 and 2 (Markusen 2002); we choose the following relative fixed costs: $f_{L}=1.0,$$f_{H}=1.2$ and $f_{O}=0.1$.³⁰

Given these parameter values and functional forms, we calibrate other key parameter values—$\mu$, $\varepsilon$, $a_{M}$, $a_{L}$, $a_{H}$, $\lambda$ and $\beta$– by characterizing the initial equilibrium as follows.

(i):

M-workers represent 70% of the population. US Economic Census reports that the ratio of production workers to total employees in manufacturing is about 70%. From the cumulative distribution function (CDF) of log-normal distribution, we set: $\frac{1}{2}+\frac{1}{2}$erf${\left( \frac{\ln z_{1}-\mu }{\sqrt{2}\varepsilon }\right) } =0.7;$

(ii):

From the same source, we pick the non-production workers’ wage share in total value added from labor as: $\frac{ C_{L}\int _{z_{1}}^{z_{2}}\varphi _{L}(z)g(z)\mathrm{d}z+C_{H}\int _{z_{2}}^{\infty }\varphi _{H}(z)g(z)\mathrm{d}z}{\widetilde{W}}=0.42;$

(iii):

US Economic Census also reports industry statistics by employment size. We approximate MNs’ total output value share as that of establishments with 2,500 or more employees: $\frac{p_{H}x_{H}N_{H}}{ p_{L}x_{L}N_{L}+p_{H}x_{H}N_{H}}=0.14;$

(iv):

For productivity difference between MNs and non-MNs, we set: $\frac{a_{H}}{a_{L}}=1.15;$³¹

(v):

For initial income inequality level, we choose: Gini index $=0.32;$³²

(vi):

For the initial growth rate we set $g=0.03$, the average US real GDP growth rate between 1980 and 2006.

(vii):

Finally, Antràs et al. (2017) reports 14% for the average foreign-input share of US offshoring firms; we set: $\frac{\beta C_{M}^{*}}{(1-\beta )C_{M}+\beta C_{M}^{*}+C_{H}} =0.14$.

The calibrated parameter values are: $\mu =-0.80$, $\varepsilon =0.83$, $a_{M}=0.35$, $a_{L}=1.98$, $a_{H}=2.27$, $\lambda =0.09$ and $\beta =0.20$.³³

1.2 A.2 The effects of offshoring

Table 1 reports the effects of globalization: falls in $f_{O}$ and $C_{M}^{*}$, respectively, which confirms our theoretical analyses. Results are percentage change from the initial equilibrium.

Table 1 The effects of falls in $f_{O}$ and $C_{M}^{*}$

Though both shocks provide the same qualitative results, it can be seen that the results are relatively more sensitive to a fall in $C_{M}^{*}$. As can be seen from Eq. (27), a fall in $C_{M}^{*}$ induces a direct price advantage of MNs vis-à-vis non-MNs, while a fall in $f_{O}$ is mitigated from the power term of $\frac{1}{1-\sigma }$ as is common in this type of monopolistic competition settings. Note also that offshoring increases not only $L^{*}$ but also $L^{eff}$ ($=\int _{0}^{z_{1}}\varphi _{M}(z)g(z)\mathrm{d}z+\int _{z_{1}}^{z_{2}}\varphi _{L}(z)g(z)\mathrm{d}z+\int _{z_{2}}^{\infty }\varphi _{H}(z)g(z)\mathrm{d}z$), total efficiency units of labor in the North, due to the technology-upgrading mechanisms.

1.3 A.3 Welfare effects

Table 2 reports then the induced percentage changes of welfare for each skill group.

Table 2 Welfare effects of falls in $f_{O}$ and $C_{M}^{*}$

$Welf_{Agg}$, $Welf_{M}$, $Welf_{L}$, and $Welf_{H}$ measure real incomes $\frac{E}{P_{C}}$, $\frac{C_{M}}{P_{C}}$, $\frac{C_{L}}{P_{C}}$, and $\frac{ C_{H}}{P_{C}}$, respectively. As discussed in Sect. 4, offshoring leads to different welfare implications for each skill group. M-workers lose in terms of their real income due to an increase of $P_{C}$, while H-workers would be the main beneficiary.

This welfare conclusion is of course incomplete since it does not take into account the dynamic gains to be reaped from growth. From Eqs. (12) and ( 28), a rise in growth rate implies an increase of total number of firms (varieties) at the same rate. This, in turn, implies from Eq. (5) that $P_{C}$ falls over time at the rate of $\frac{g}{\sigma -1}$. To evaluate the intertemporal welfare consequences, we compute the equivalent variation index $\phi$ from the following utility indifference condition:

$$\begin{aligned} \left( 1+\phi \right) \int _{t=0}^{\infty }\mathrm{e}^{-\rho t}\ln \left( \frac{E\mathrm{e}^{\frac{g_{0}}{\sigma -1}}}{P_{C0}}\right) \mathrm{d}t=\int _{t=0}^{\infty }\mathrm{e}^{-\rho t}\ln \, \left( \frac{E\mathrm{e}^{\frac{g_{1} }{\sigma -1}}}{P_{C1}}\right) \mathrm{d}t, \end{aligned}$$

(44)

where subscripts 0 and 1 indicate before and after shocks, respectively.³⁴

$\int _{t=0}^{\infty }\mathrm{e}^{-\rho t}\ln \left( \frac{E\mathrm{e}^{\frac{g_{0}}{ \sigma -1}}}{P_{C0}}\right) \mathrm{d}t$ represents the present value of total reference real consumption streams before the shock, while $\int _{t=0}^{\infty }\mathrm{e}^{-\rho t}\ln \left( \frac{E\mathrm{e}^{\frac{g_{1}}{ \sigma -1}}}{P_{C1}}\right) \mathrm{d}t$ represents the corresponding value computed after the shock at $t=0$. The welfare gain resulting from the shock is equivalent from the perspective of the households to increasing the reference real consumption profile by $\phi$ percent. This is equivalent to the most frequently used welfare measure $\left( 1+\phi \right) C_{0}=C_{1}$ in static applied general equilibrium analysis. In Table 2, ItWelfs report these computed values of $\phi$.³⁵ As can be seen, the dynamic welfare gains are large enough to dominate any static losses.

Finally, Table 3 reports the effects of a rise in $\mu$—at a given $\varepsilon$ and at a given mean by adjusting $\varepsilon$, respectively.

Table 3 The effects of a rise in $\mu$

A rise in $\mu$ at a given $\varepsilon$ (skill upgrading) increases both $Welf_{Agg}$ and $ItWelf_{Agg}$ by increased high-tech activities on the right tail of the distribution. An increase in $\mu$ raises the dispersion of worker skill heterogeneity and the average skill level rises in a monotonous way: i.e., less least skilled and more highest skilled.³⁶ Skill upgrading at a given population should be isomorphic to labor-augmenting technological progress, as in the case of offshoring with access to cheap labor. Both contribute to an aggregate productivity increase, and thus enhance growth.

On the other hand, keeping the mean following a rise in $\mu$ implies a decrease in $\varepsilon$, which results in a concentration of density relatively in the middle of the distribution. This favors non-MNs over MNs leading to decrease in aggregate productivity and growth rate; both $Welf_{Agg}$ and $ItWelf_{Agg}$ fall by decreased high-tech activities.

Appendix B: Proofs

1.1 B.1 Proof of Proposition 1

Totally differentiating Eqs. (16) and (27), and using Eq. (24), we get:

$$\begin{aligned} \left[ \begin{array}{cc} j_{11} &{} j_{12} \\ j_{21} &{} j_{22} \end{array} \right] \left[ \begin{array}{c} dz_{1} \\ dz_{2} \end{array} \right] =\left[ \begin{array}{c} 0 \\ 1 \end{array} \right] df_{O}, \end{aligned}$$

where

$$\begin{aligned} \begin{array}{l} j_{11}=\varphi _{M}(z_{1})g(z_{1})+\varphi _{L}(z_{1})g(z_{1}),\\ j_{12}=-\varphi _{L}(z_{2})g(z_{2}), \\ j_{21}=\left( \sigma -1\right) f_{L}\left( \frac{\alpha _{1}\left( z_{1}\right) +C_{M}}{\alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}}\right) ^{\sigma -2}\frac{\alpha _{1}^{\prime }\left( z_{1}\right) \left[ C_{M}^{*}-\alpha _{2}\left( z_{2}\right) C_{M}\right] }{\left[ \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}\right] ^{2}}, \\ j_{22}=\left( \sigma -1\right) f_{L}\left( \frac{\alpha _{1}\left( z_{1}\right) +C_{M}}{\alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}}\right) ^{\sigma -2}\frac{-\alpha _{1}\left( z_{1}\right) \alpha _{2}^{\prime }\left( z_{2}\right) \left[ \alpha _{1}\left( z_{1}\right) +C_{M}\right] }{\left[ \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}\right] ^{2}}. \end{array} \end{aligned}$$

The Jacobian determinant is:

$$\begin{aligned} \left| J\right|= & {} \left[ \varphi _{M}(z_{1})g(z_{1})+\varphi _{L}(z_{1})g(z_{1})\right] \\&\times \left[ \left( \sigma -1\right) f_{L}\left( \frac{ \alpha _{1}\left( z_{1}\right) +C_{M}}{\alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}}\right) ^{\sigma -2}\frac{-\alpha _{1}\left( z_{1}\right) \alpha _{2}^{\prime }\left( z_{2}\right) \left[ \alpha _{1}\left( z_{1}\right) +C_{M}\right] }{\left[ \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}\right] ^{2}} \right] \\&+\,\varphi _{L}(z_{2})g(z_{2})\left[ \left( \sigma -1\right) f_{L}\left( \frac{\alpha _{1}\left( z_{1}\right) +C_{M}}{\alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}} \right) ^{\sigma -2}\frac{\alpha _{1}^{\prime }\left( z_{1}\right) \left[ C_{M}^{*}-\alpha _{2}\left( z_{2}\right) C_{M}\right] }{\left[ \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}\right] ^{2}}\right] . \end{aligned}$$

Note from Eqs. (9), (23), and (24),

$$\begin{aligned} C_{M}^{*}-\alpha _{2}\left( z_{2}\right) C_{M}<0. \end{aligned}$$

(45)

From Eqs. (24) and (45), it follows then that: $\left| J\right| >0.$

Using Cramer’s rule, we now obtain:

$$\begin{aligned} \begin{array}{l} \dfrac{dz_{1}}{df_{O}}=\frac{1}{\left| J\right| }\left[ \varphi _{L}(z_{2})g(z_{2})\right]>0,\\ \dfrac{dz_{2}}{df_{O}}=\frac{1}{\left| J\right| }\left[ \varphi _{M}(z_{1})g(z_{1})+\varphi _{L}(z_{1})g(z_{1})\right] >0. \end{array} \end{aligned}$$

1.2 B.2 Proof of Proposition 3

From (16), (17), (20), and (30) we have:³⁷

$$\begin{aligned} \frac{\sigma -1}{\sigma }E= & {} \left[ \left( C_{M}+C_{L}\right) \int _{0}^{z_{1}}\varphi _{M}(z)g(z)\mathrm{d}z+C_{H}\int _{z_{2}}^{\overline{z} }\varphi _{H}(z)g(z)\mathrm{d}z\right] \nonumber \\&+\frac{C_{M}^{*}}{\sigma }\int _{z_{2}}^{ \overline{z}}\varphi _{H}(z)g(z)\mathrm{d}z. \end{aligned}$$

(46)

From (39), (41), and (46) we then get:

$$\begin{aligned} \begin{array}{l} \left( \frac{E}{P_{C}}\right) ^{\sigma -1} =\left[ \left( C_{M}+C_{L}\right) \int _{0}^{z_{1}}\varphi _{M}(z)g(z)\mathrm{d}z+\left( C_{H}+\frac{C_{M}^{*}}{\sigma }\right) \int _{z_{2}}^{\overline{z}}\varphi _{H}(z)g(z)\mathrm{d}z\right] ^{\sigma -1} \\ \quad \cdot \left[ \frac{C_{L}+C_{M}}{\left( \sigma -1\right) \pi f_{L}} \int _{0}^{z_{1}}\varphi _{M}(z)g(z)\mathrm{d}z\left( C_{L}+C_{M}\right) ^{1-\sigma }+ \frac{C_{H}+C_{M}^{*}}{\left( \sigma -1\right) \pi \left( f_{H}+f_{O}\right) }\int _{z_{2}}^{\overline{z}}\varphi _{H}(z)g(z)\mathrm{d}z\left( C_{H}+C_{M}^{*}\right) ^{1-\sigma }\right] . \end{array}\nonumber \\ \end{aligned}$$

(47)

Differentiating the RHS of this expression with respect to $z_{2}$ (at a given growth rate), and making use of Eqs. (11), (27), and (37) yields:

$$\begin{aligned} \begin{array}{l} \frac{\mathrm{d}RHS(47)}{\mathrm{d}z_{2}}=\frac{1}{\pi f_{L}}\left( \frac{ \widetilde{E}}{C_{L}+C_{M}}\right) ^{\sigma -2} \\ \cdot \left[ \begin{array}{l} \left[ C_{L}^{^{\prime }}\int _{0}^{z_{1}}\varphi _{M}(z)g(z)\mathrm{d}z\frac{dz_{1}}{ dz_{2}}+C_{H}^{^{\prime }}\int _{z_{2}}^{\overline{z}}\varphi _{H}(z)g(z)\mathrm{d}z- \frac{C_{M}^{*}}{\sigma }\varphi _{H}(z_{2})g(z_{2})\right] \cdot \\ \left[ \int _{0}^{z_{1}}\varphi _{M}(z)g(z)\mathrm{d}z+\left( \frac{f_{L}}{f_{H}+f_{O}} \right) ^{\frac{1}{\sigma -1}}\int _{z_{2}}^{\overline{z}}\varphi _{H}(z)g(z)\mathrm{d}z\right] \\ +\left( \frac{\widetilde{E}}{C_{L}+C_{M}}\right) \left[ C_{L}^{^{\prime }}\int _{0}^{z_{1}}\varphi _{M}(z)g(z)\mathrm{d}z\frac{dz_{1}}{dz_{2}}+C_{H}^{^{\prime }}\int _{z_{2}}^{\overline{z}}\varphi _{H}(z)g(z)\mathrm{d}z-\left( \sigma -1\right) C_{M}^{*}\varphi _{H}(z_{2})g(z_{2})\right] \end{array} \right] \end{array} \end{aligned}$$

where $\widetilde{E}=\frac{\sigma -1}{\sigma }E$, and $C_{L}^{^{\prime }}$ and $C_{H}^{^{\prime }}$ represent derivatives with respect to $z_{1}$ and $z_{2}$. From (24) and (37), this expression is unambiguously negative.³⁸

1.3 B.3 Proof of Proposition 5

Totally differentiating Eq. (43), we get:

$$\begin{aligned} \begin{array}{l} \dfrac{\mathrm{d}L_{I}}{\mathrm{d}z_{2}}=\left[ \begin{array}{l} \dfrac{\alpha _{1}^{\prime }\left( z_{1}\right) \left[ C_{M}^{*}-\sigma \alpha _{2}\left( z_{2}\right) C_{M}\right] }{\left[ \sigma \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}\right] ^{2}}\int _{0}^{z_{1}}\varphi _{M}(z)g(z)\mathrm{d}z+\dfrac{\left( 1-\sigma \right) \alpha _{1}^{\prime }\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) C_{M}^{*}}{\left[ \sigma \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}\right] ^{2}}\int _{z_{2}}^{\infty }\varphi _{H}(z)g(z)\mathrm{d}z \\ -\dfrac{\rho \left( \sigma -1\right) \alpha _{1}^{\prime }\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) C_{M}^{*}}{\lambda ^{2}\left[ \sigma \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*} \right] ^{2}}+\dfrac{\alpha _{1}\left( z_{1}\right) +C_{M}}{\sigma \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}}\left[ \varphi _{M}(z_{1})g(z_{1})\right] \end{array} \right] \frac{dz_{1}}{dz_{2}}\\ \qquad \qquad +\left[ \begin{array}{l} \dfrac{-\sigma \alpha _{1}\left( z_{1}\right) \alpha _{2}^{\prime }\left( z_{2}\right) \left[ \alpha _{1}\left( z_{1}\right) +C_{M}\right] }{\left[ \sigma \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}\right] ^{2}}\int _{0}^{z_{1}}\varphi _{M}(z)g(z)\mathrm{d}z+\dfrac{ \left( 1-\sigma \right) \alpha _{1}\left( z_{1}\right) \alpha _{2}^{\prime }\left( z_{2}\right) C_{M}^{*}}{\left[ \sigma \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}\right] ^{2}} \int _{z_{2}}^{\infty }\varphi _{H}(z)g(z)\mathrm{d}z \\ -\dfrac{\rho \left( \sigma -1\right) \alpha _{1}\left( z_{1}\right) \alpha _{2}^{\prime }\left( z_{2}\right) C_{M}^{*}}{\lambda ^{2}\left[ \sigma \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*} \right] ^{2}}-\dfrac{\alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}}{\sigma \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}}\left[ \varphi _{H}(z_{2})g(z_{2}) \right] \end{array} \right] . \end{array}\nonumber \\ \end{aligned}$$

(48)

Given that all the other terms are based on second-order price adjustments $\alpha _{1}^{\prime }\left( z_{1}\right)$ and $\alpha _{2}^{\prime }\left( z_{2}\right)$, we compare the only two sizable terms: i.e., the last terms in each bracket. From Eqs. (23), (24) and (37 ), we obtain:

$$\begin{aligned}&\frac{\alpha _{1}\left( z_{1}\right) +C_{M}}{\sigma \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}}\left[ \varphi _{M}(z_{1})g(z_{1})\right] \frac{\varphi _{L}(z_{2})g(z_{2})}{\varphi _{M}(z_{1})g(z_{1})+\varphi _{L}(z_{1})g(z_{1})} \\&\quad -\frac{\alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}}{\sigma \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}}\left[ \varphi _{H}(z_{2})g(z_{2})\right] =-\frac{C_{M}^{*}\varphi _{H}(z_{2})g(z_{2})}{\sigma \alpha _{1}\left( z_{1}\right) \alpha _{2}\left( z_{2}\right) +C_{M}^{*}}<0. \end{aligned}$$

It can be thus believed that $\frac{dL_{I}}{dz_{2}}<0$, implying $\frac{ dL_{I}}{df_{O}}<0$ from Proposition 1.³⁹