3.1. Preferences

Infinitely lived households obtain utility from the consumption, $C$ , of the unique aggregate final good, whose price we normalize to $1$ , and collect income from investments in financial assets (equity) and labor. They supply labor to both sectors, $s=\{N,R\}$ . Preferences are identical across workers $L_{N}^{i}$ , $L_{N}^{h}$ , $L_{R}^{i}$ , and $L_{R}^{h}$ . Thus, there is a representative household with preferences at time $t=0$ given by $U_{C}=\int _{0}^{\infty }\left (\frac{C(t)^{1-\theta }-1}{1-\theta }\right )e^{-\rho t}dt$ , where $\rho \gt 0$ is the subjective discount rate and $\theta \gt 0$ is the inverse of the inter-temporal elasticity of substitution. The flow budget constraint is

(3) \begin{equation} \dot{a}(t)=r(t)\cdot a(t)+\sum _{s=N,R}\left (w_{s}^{h}\cdot L_{s}^{h}+w_{s}^{i}\cdot L_{s}^{i}\right )-C(t), \end{equation}

where $a(t)=\sum _{s=N,R}\left [a_{s}^{h}(t)+a_{s}^{i}(t)\right ]$ denotes household’s real financial assets/wealth holdings (composed of equity of machine producers, considering the profits seized by the top-quality producers), $r$ is the real interest rate, and $w_{s}^{h}$ and $w_{s}^{i}$ are the wage for labor type $h$ and $i$ employed in sector $s=\{N,R\}$ , respectively. The initial level of wealth $a(0)$ is given, and the non-Ponzi games condition $\lim _{t\rightarrow \infty }e^{-\int _{0}^{t}r(s)ds}a(t)\geq 0$ is imposed. The representative household chooses the path of aggregate consumption $\left [C(t)\right ]_{t\geq 0}$ to maximize the discounted lifetime utility, resulting in the following optimal consumption path Euler equation:

(4) \begin{equation} \frac{\dot{C}(t)}{C(t)}=g=\frac{1}{\theta }\cdot \left [r(t)-\rho \right ]. \end{equation}

Moreover, the transversality condition is also standard: $\underset{t\rightarrow \infty }{lim}e^{-\rho t}\cdot C(t)^{-\theta }\cdot a(t)=0$ .

3.2. Technologies, output, and prices

3.2.1. Aggregate economy

The aggregate output $Y$ is produced with a CES aggregate production function of nonroutine and routine competitively produced final goods:

(5) \begin{equation} \begin{array}{ccc} Y(t) & =\left [\sum _{s=N,R}\chi _{s}\cdot Y_{s}(t)^{\frac{\varepsilon -1}{\varepsilon }}\right ]^{\frac{\varepsilon }{\varepsilon -1}} &,\end{array}\;\varepsilon \in \left (0,+\infty \right ), \end{equation}

where $Y_{N}$ and $Y_{R}$ are the total outputs of the $N$ - and the $R$ -sector, respectively, $\chi _{N}$ and $\chi _{R}$ , with $\sum _{s=N,R}\chi _{s}=1$ , are the distribution parameters, measuring the importance of the sectors, and $\varepsilon \geq 0$ is the elasticity of substitution between sectors, wherein $\varepsilon \gt 1$ $\left (\varepsilon \lt 1\right )$ means that they are gross substitutes (complements) in the production of $Y$ . The assumption of competitive final-good firms implies the following maximization problem: $\max _{Y_{s}}\,\Pi _{Y}=P_{Y}\cdot Y-\sum _{s=N,R}P_{s}\cdot Y_{s}$ . From the first-order conditions emerge the inverse demand for $Y_{s}$ :Footnote ¹¹

(6) \begin{equation} \frac{P_{s}}{P_{Y}}=\chi _{s}\left (\frac{Y}{Y_{s}}\right )^{\frac{1}{\varepsilon }}\Leftrightarrow Y_{s}=\left (\frac{P_{s}}{P_{Y}\cdot \chi _{s}}\right )^{-\varepsilon }Y. \end{equation}

Thus, we obtain the following expression for relative demand for output from the $N$ -sector:

(7) \begin{equation} \frac{Y_{N}}{Y_{R}}=\left (\frac{\chi _{N}}{\chi _{R}}\right )^{\epsilon }\left (\frac{P_{N}}{P_{R}}\right )^{-\epsilon }, \end{equation}

which depends positively on the $N$ -sector relative weight parameter and negatively on the $N$ -sector relative price of output. Replacing (6) in (5), we have that $P_{Y}=\left [\sum _{s=N,R}\chi _{s}^{\varepsilon }\cdot P_{s}^{1-\varepsilon }\right ]^{\frac{1}{1-\varepsilon }}$ , where $P_{N}$ and $P_{R}$ are the prices of the outputs of, respectively, the $N$ - and the $R$ -sector, and thus the right-hand side of the expression is the unit production cost. Summing across sectors, from (6) we have that $P_{Y}\cdot Y=P_{N}\cdot Y_{N}+P_{R}\cdot Y_{R}$ .

3.2.2. Sectors of the economy

The output $Y_{s}$ of each sector $s=\{N,R\}$ is produced in perfect competition by the following production function with constant returns to scale $Y_{s}=\exp\! \left (\int _{0}^{1}\ln Y_{v_{s}}dv_{s}\right )$ , that is, $Y_{s}$ is a continuum of the output produced by tasks $Y_{v_{s}}$ indexed, respectively, by $v_{N}\in \left [0,1\right ]$ and $v_{R}\in \left [0,1\right ]$ . Tasks $v_{R}$ are routine tasks. The producer of $Y_{s}$ maximizes profits given by ${{\Pi _{s}}}=P_{s}\cdot Y_{s}-\int _{0}^{1}P_{v_{s}}\cdot Y_{v_{s}}dv_{s}$ , where $P_{v_{s}}$ is the price of output of task $v_{s}$ , subject to the restriction imposed by the functional form of the production function of $Y$ . Assuming perfect competition, the maximization problem results in the following first-order conditions: $\frac{\partial{\Pi }_{s}}{\partial Y_{v_{s}}}=0\Rightarrow Y_{v_{s}}=\frac{P_{s}\cdot Y_{s}}{P_{v_{s}}}$ . From here $P_{v_{s}}\cdot Y_{v_{s}}=P_{s}\cdot Y_{s}$ , which can be replaced in the profits function and the production function results, respectively, in ${\Pi _{s}}=P_{s}\cdot Y_{s}-\int _{0}^{1}P_{s}\cdot Y_{s}dv_{s}=0$ and also in

(8) \begin{equation} Y_{s}=\exp\! \left (\int _{0}^{1}\ln \frac{P_{s}\cdot Y_{s}}{P_{v_{s}}}dv_{s}\right )\Leftrightarrow P_{s}=\exp\! \left (\int _{0}^{1}\ln P_{v_{s}}dv_{s}\right ). \end{equation}

3.2.3. Tasks in each sector

Task producers in sector $s=N$ must choose to produce them either with domestic low-skilled labor type “ $h$ ” or with domestic high-skilled labor type “ $i$ ,” and task producers in sector $s=R$ must choose to produce them with foreign labor, relocating tasks in developing countries “ $h$ ” or with domestic medium-skilled labor employed in automated tasks “ $i$ ,” which implies choosing between the following two Cobb–Douglas production functions:

(9) \begin{eqnarray} Y_{v_{s}}^{h}(t) & = & \left [\int _{0}^{J}\left (q^{k(j,t)}{{\cdot }}x_{v_{s}}^{h}(k,j,t)\right )^{1-\alpha }dj\right ]\left [\left (1-v_{s}(t)\right )\cdot l_{s}^{h}\cdot L_{v_{s}}^{h}\right ]^{\alpha }, \end{eqnarray}

(10) \begin{eqnarray} Y_{v_{s}}^{i}(t) & = & \left [\int _{J}^{1}\left (q^{k(j,t)}{{\cdot }}x_{v_{s}}^{i}(k,j,t)\right )^{1-\alpha }dj\right ]\left [v_{s}(t)\cdot l_{s}^{i}\cdot L_{v_{s}}^{i}\right ]^{\alpha }. \end{eqnarray}

Each uses two factors: labor of type $L_{N}^{i},$ $L_{N}^{h}$ , and $L_{R}^{i}$ —domestic—or $L_{R}^{h}$ —foreign—(the second term on the right-hand side) and intermediate goods—machines/robots—(the first term on the right-hand side) with a share in the income of $\alpha$ and $1-\alpha$ , respectively. Each machine $j$ used in $v_{s}$ production is quality-adjusted: the constant quality upgrade is $q\gt 1$ , $k$ is the top-quality rung at $t$ , and $x_{v_{s}}^{h}(k,j,t)$ and $x_{v_{s}}^{i}(k,j,t)$ represent the units of machines or robots demanded for task $v_{s}$ if it is produced to be used by $L_{s}^{h}$ or by $L_{s}^{i}$ , respectively. The labor term includes the quantities employed in the production of $v_{s}$ , $L_{v_{s}}^{i}$ or $L_{v_{s}}^{h}$ , and two types of corrective factors accounting for productivity differentials such that workers are assigned to tasks according to location and the most efficient firm in production, that is, we take into account:

• • The absolute net advantage of labor. For $s=N$ , we consider $l_{N}^{i}\gt l_{N}^{h}$ since $L_{N}^{i}$ is more qualified than $L_{N}^{h}$ , implying that $L_{N}^{i}$ operates in increasingly abstract/cognitive nonroutine tasks, while $L_{N}^{h}$ operates in “purely manual” nonroutine tasks. In $s=R$ , the production can be performed by foreign workers if relocated, $L_{R}^{h}$ ,Footnote ¹² or by domestic medium-skilled workers if automated, $L_{R}^{i}$ , and the quantities used should be corrected by the term $l_{R}^{h}$ and $l_{R}^{i}$ due to factors that are specific to automation and relocations; we consider that $l_{R}^{i}\gt l_{R}^{h}$ since domestic medium-skilled labor has an absolute productivity advantage over foreign labor in developing countries.Footnote ¹³

• • The relative productivity advantage of labor. Following the point of view proposed by, for example, Acemoglu and Zilibotti (Reference Acemoglu and Zilibotti2001), through the terms $(1-v_{s})$ and $v_{s}$ in (9) and (10), $L^{i}$ is relatively more productive in tasks indexed by larger $v_{s}$ and vice versa.

In each sector $s$ , there is substitutability between tasks that use labor type $h$ and tasks that use labor type $i$ . On the other hand, it is assumed that, regardless of the labor type used by sector $s$ , there is complementarity between labor and a specific set of machines or robots. To determine the tasks that use labor type “ $h$ ” and labor type “ $i$ ” in each sector, firstly, we need to solve the respective maximization problems:

(11) \begin{eqnarray} \max _{x_{v_{s}}^{h}(k,j,t),L_{v_{s}}^{h}}{{\Pi _{v_{s}}^{h}}}(t) & = & P_{v_{s}}^{h}(t){\cdot }Y_{v_{s}}^{h}(t)-\int _{0}^{J}p(k,j,t){{\cdot }}x_{v_{s}}^{h}(k,j,t){\cdot }dj-w_{s}^{h}(t)\cdot L_{v_{s}}^{h}, \end{eqnarray}

(12) \begin{eqnarray} \max _{x_{v_{s}}^{i}(k,j,t),L_{v_{s}}^{i}}{{\Pi _{v_{s}}^{i}}}(t) & = & P_{v_{s}}^{i}(t){\cdot }Y_{v_{s}}^{i}(t)-\int _{J}^{1}p(k,j,t){{\cdot }}x_{v_{s}}^{i}(k,j,t){\cdot }dj-w_{s}^{i}(t)\cdot L_{v_{s}}^{i}, \end{eqnarray}

bearing in mind (9) and (10), where $P_{v_{s}}^{h}(t)$ and $P_{v_{s}}^{i}(t)$ are the price of task $v_{s}$ produced by labor type $h$ and $i$ , respectively, at time $t$ , $p(k,j,t)$ denotes the price paid for the machine $j$ with quality $k$ , at time $t$ , $w_{s}^{h}(t)$ and $w_{s}^{i}(t)$ are the price of each unit of labor type $h$ and $i$ , respectively, at time $t$ —these prices are given for the perfectly competitive producers of the tasks. From the first-order conditions with respect to machines/robots results:

(13) \begin{eqnarray} x_{v_{s}}^{h}(k,j,t) & = & \left [\frac{P_{v_{s}}^{h}(t)\cdot (1-\alpha )}{p(k,j,t)}\right ]^{\frac{1}{\alpha }}\cdot q^{k(j,t)\frac{1-\alpha }{\alpha }}\cdot \left (1-v_{s}(t)\right )\cdot l_{s}^{h}\cdot L_{v_{s}}^{h}, \end{eqnarray}

(14) \begin{eqnarray} x_{v_{s}}^{i}(k,j,t) & = & \left [\frac{P_{v_{s}}^{i}(t)\cdot (1-\alpha )}{p(k,j,t)}\right ]^{\frac{1}{\alpha }}\cdot q^{k(j,t)\frac{1-\alpha }{\alpha }}\cdot v_{s}(t)\cdot l_{s}^{i}\cdot L_{v_{s}}^{i}. \end{eqnarray}

Replacing (13) and (14) in the corresponding production functions (9) and (10), we have that:

(15) \begin{eqnarray} Y_{v_{s}}^{h}(t) & = & \left [\frac{P_{v_{s}}^{h}(t)\cdot (1-\alpha )}{p(k,j,t)}\right ]^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}^{h}(t)\cdot \left (1-v_{s}(t)\right )\cdot l_{s}^{h}\cdot L_{v_{s}}^{h}, \end{eqnarray}

(16) \begin{eqnarray} Y_{v_{s}}^{i}(t) & = & \left [\frac{P_{v_{s}}^{i}(t)\cdot (1-\alpha )}{p(k,j,t)}\right ]^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}^{i}(t)\cdot v_{s}(t)\cdot l_{s}^{i}\cdot L_{v_{s}}^{i} \end{eqnarray}

where $Q_{s}^{h}\equiv \int _{0}^{J}q^{k(j,t)\frac{1-\alpha }{\alpha }}dj$ and $Q_{s}^{i}\equiv \int _{J}^{1}q^{k(j,t)\frac{1-\alpha }{\alpha }}dj$ are measures of the quality level of machines/robots used in sector $s$ to be endogenously determined in Section 3, thereby originating the dynamic effects of the model.

3.2.4. Wages and threshold task in each sector

From the first-order conditions with respect to labor units results:

(17) \begin{eqnarray} w_{s}^{h}(t) & = & \left [P_{v_{s}}^{h}(t)\right ]^{\frac{1}{\alpha }}\cdot \left [\frac{1-\alpha }{p(k,j,t)}\right ]^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}^{h}(t)\cdot \left (1-v_{s}(t)\right )\cdot l_{s}^{h}, \end{eqnarray}

(18) \begin{eqnarray} w_{s}^{i}(t) & = & \left [P_{v_{s}}^{i}(t)\right ]^{\frac{1}{\alpha }}\cdot \left [\frac{1-\alpha }{p(k,j,t)}\right ]^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}^{i}(t)\cdot v_{s}(t)\cdot l_{s}^{i} \end{eqnarray}

In equilibrium, there is a threshold task $v_{s}$ , that ensures that each type of labor gets the same wage regardless of the task it is used for. To this end, we can define the following price indexes:

(19) \begin{equation} \left [P_{s}^{h}(t)\right ]^{\frac{1}{\alpha }}=\left [P_{v_{s}}^{h}(t)\right ]^{\frac{1}{\alpha }}\cdot \left (1-v_{s}(t)\right )\,{and}\,\left [P_{s}^{i}(t)\right ]^{\frac{1}{\alpha }}=\left [P_{v_{s}}^{i}(t)\right ]^{\frac{1}{\alpha }}\cdot v_{s}(t). \end{equation}

As shown in Appendix A.1, in sector $s=\{N,R\}$ (i) tasks with a very low (high) $v_{s}$ have a lower price if produced by $L_{s}^{h}$ ( $L_{s}^{i}$ ) rather than $L_{s}^{i}$ ( $L_{s}^{h}$ ), such that perfectly competitive producers use $L_{s}^{h}$ ( $L_{s}^{i}$ ) to avoid being driven out of the market, and (ii) there is a threshold task $v_{s},$ where prices are equal and is given by the following expression:

(20) \begin{equation} \overline{v}_{s}=\left [1+\left (\frac{Q_{s}^{i}}{Q_{s}^{h}}\frac{l_{s}^{i}}{l_{s}^{h}}\frac{L_{s}^{i}}{L_{s}^{h}}\right )^{\frac{1}{2}}\right ]^{-1}, \end{equation}

which assesses the “comparative advantage” of sector $s$ . Therefore, if $\overline{v}_{s}\lt v_{s}$ firms will be biased toward producing tasks using both labor type $h$ and the respective technological-knowledge level of each intermediate goods set, in sector $s$ , while if $\overline{v}_{s}\gt v_{s}$ firms will be biased toward producing tasks using both labor type $i$ and the respective technological-knowledge level of each intermediate goods set, in sector $s.$

In particular, whenever labor abroad becomes more productive,Footnote ¹⁴ that is, $l_{R}^{h}$ increases, boosting the number of tasks relocated and promoting globalization. Automation, on the other hand, makes the use of domestic medium-skilled labor less advantageous, translating into a decrease of $l_{R}^{i}$ that negatively affects the production of routine tasks at home, thus favoring globalization. Finally, an increase in the technological-knowledge bias toward high-skilled workers decreases the incentive for automation and relocations. On the contrary, an increase in the technological-knowledge bias toward low-skilled workers increases the incentive for automation and relocations. Thus, improvements in the machines or robots’ technology that are biased toward low-skilled tasks would increase automation and relocations.

3.3. Machines sector

In the machines sector, producing the top quality $k$ of each $j$ needs an initial R&D cost to achieve the new prototype/design. This initial cost can only be recovered if, with the production of the new quality of the robot, profits are made over a certain time in the future. This is assured by a system of intellectual property rights that protect the leader firm’s monopoly. At the same time, this technological knowledge is accessible, practically free of charge, from other firms. Hence, each firm that holds the patent for the top quality $k$ of $j$ at $t$ supplies all respective tasks, $v_{s}$ , in sector $s=\{N,R\}$ . If we consider that each unit of robot $j$ requires one unit of final output $Y$ , since its price is 1 to 1 and the producer of $j$ gets profits $\pi _{s}(k,j,t)=\left [p(k,j,t)-1\right ]\cdot x_{s}(k,j,t)$ , where $x_{s}(k,j,t)=\int _{0}^{\overline{v}_{s}}x_{v_{s}}^{h}(k,j,t)\cdot dv_{s}+\int _{\overline{v}_{s}}^{1}x_{v_{s}}^{i}(k,j,t)\cdot dv_{s}$ is the demand for robot $j$ from all the producers of tasks $v_{s}$ that use such input, regardless of the labor type used in tasks.

Assuming that the monopolist charges the same price, $p(k,j,t)$ , for all these firms, we can find the optimal price by replacing $x_{s}(k,j,t)$ by the demand of the producer of a single task $v_{s}$ , that is, either by $x_{v_{s}}^{h}(k,j,t)$ or by $x_{v_{s}}^{i}(k,j,t)$ and then maximizing with respect to $p(k,j,t)$ . This can be seen by $\pi _{s}(k,j,t)=\int _{0}^{1}\pi _{v_{s}}(k,j,t)\cdot dv_{s}=\underbrace{\int _{0}^{\overline{v}_{s}}\pi _{v_{s}}^{h}(k,j,t)\cdot dv_{s}}_{\pi _{v_{s}}^{h}(k,j,t)}+\underbrace{\int _{\overline{v}_{s}}^{1}\pi _{v_{s}}^{i}(k,j,t)\cdot dv_{s}}_{\pi _{v_{s}}^{i}(k,j,t)}$ , where $\pi _{v_{s}}^{h}(k,j,t)$ and $\pi _{v_{s}}^{i}(k,j,t)$ denote the profits of the producer of $j$ for selling this robot to the producer of task $v_{s}$ . Therefore, we can find $p(k,j,t)$ by solving the following maximization problems $\max _{p(k,j,t)}\left [p(k,j,t)-1\right ]\cdot x_{v_{s}}^{h}(k,j,t)$ and $\max _{p(k,j,t)}\left [p(k,j,t)-1\right ]\cdot x_{v_{s}}^{i}(k,j,t)$ , where $x_{v_{s}}^{h}(k,j,t)$ and $x_{v_{s}}^{i}(k,j,t)$ can be done by (13) or (14). From the first-order condition $\frac{\partial \pi _{s}(k,j,t)}{\partial p(k,j,t)}$ , we have that $p(k,j,t)\equiv p=\frac{1}{1-\alpha }=q$ , assuming that the limit pricing strategy is binding.Footnote ¹⁵ Taking also into account $p=q$ , (19), (13), and (14), the demand for the machine $j$ used in sector $s$ together with $L_{s}^{h}$ and $L_{s}^{i}$ is, respectively,

(21) \begin{eqnarray} x_{s}^{h}(t) & = & \int _{0}^{\overline{v}_{s}}x_{v_{s}}^{h}(k,j,t)\cdot dv_{s}=\left [\frac{P_{s}^{h}(t)\cdot (1-\alpha )}{q}\right ]^{\frac{1}{\alpha }}\cdot Q_{s}^{h}(t)\cdot l_{s}^{h}\cdot L_{s}^{h}, \end{eqnarray}

(22) \begin{eqnarray} x_{s}^{i}(t) & = & \int _{\overline{v}_{s}}^{1}x_{v_{s}}^{i}(k,j,t)\cdot dv_{s}=\left [\frac{P_{s}^{i}(t)\cdot (1-\alpha )}{q}\right ]^{\frac{1}{\alpha }}\cdot Q_{s}^{i}(t)\cdot l_{s}^{i}\cdot L_{s}^{i}. \end{eqnarray}

Total demand for robot $j$ used in sector $s$ is $X_{s}(j)=x_{s}^{h}(k,j,t)+x_{s}^{i}(k,j,t)$ , and the profits for the machines used in sector $s$ by labor type $h$ and $i$ are $\pi _{s}^{h}(t)=\left (q-1\right )\cdot x_{s}^{h}(t)$ and $\pi _{s}^{i}(k,j,t)=\left (q-1\right )\cdot x_{s}^{i}(t)$ , respectively.

3.4. Allocation of resources

Once determined the threshold task as in (20), we can start by determining absolute values for price indexes. To this end, we use the definition of the price of output underlying the producer’s output maximization problem in sector $s$ , $Y_{s}$ , which implies $P_{s}=\exp\! \left (\int _{0}^{1}\ln P_{v_{s}}dv_{s}\right )$ —see (8). We also make use of the result that the value of each task, $P_{v_{s}}Y_{v_{s}}$ , is a constant for all $v_{s}$ , and we use (19) and (20) to have $P_{s}^{i}=\left (\frac{\overline{v}_{s}}{1-\overline{v}_{s}}\right )^{\alpha }P_{s}^{h}$ . From this analysis, we obtain the following expressions—see Appendix A.2:

(23) \begin{equation} P_{s}^{h}=P_{s}\cdot \exp\! \!\left (-\alpha \right )\cdot \overline{v}_{s}^{-\alpha }\:\text{and}\:P_{s}^{i}=P_{s}\cdot \exp\! \left (-\alpha \right )\cdot \left (1-\overline{v}_{s}\right )^{-\alpha }\Rightarrow \frac{P_{s}^{i}}{P_{s}^{h}}=\left (\frac{Q_{s}^{i}}{Q_{s}^{h}}\frac{l_{s}^{i}}{l_{s}^{h}}\frac{L_{s}^{i}}{L_{s}^{h}}\right )^{-\frac{\alpha }{2}}, \end{equation}

where $P_{N}$ and $P_{R}$ are also determined in Appendix A.2. An increase in the labor level of sector $s$ has a market-size effect on the demand for machines through the term $\overline{v}_{s}$ . However, by affecting $\overline{v}_{s}$ the same effect has, in addition, a price effect since it increases the supply of output of sector $s$ that induces a decrease in the absolute price of this output and, therefore, a decrease in the price index of tasks in the sector. This decreases the output of each task, which decreases the demand for machines in the sector—see (21) and (22).

From the profit maximization problem of the producer of $Y$ and since in each sector some tasks are produced by labor $L_{s}^{h}$ and other parts are performed by labor $L_{s}^{i}$ , the aggregate output is the following: $P_{s}Y_{s}=\int _{0}^{1}P_{v_{s}}Y_{v_{s}}dv_{s}$ $=\int _{0}^{\overline{v}_{s}}P_{v_{s}}^{h}Y_{v_{s}}^{h}dv_{s}+\int _{\overline{v}_{s}}^{1}P_{v_{s}}^{i}Y_{v_{s}}^{i}dv_{s}$ $=P_{s}Y_{s}^{h}+P_{s}Y_{s}^{i}$ . On the basis of these definitions and taking into account (15), (16), (19), and (23), the outputs in sector $s$ performed by labor type $L_{s}^{h}$ , $Y_{s}^{h}$ , and labor type $L_{s}^{i}$ , $Y_{s}^{L_{s}^{i}}$ , are as follows:

(24) \begin{eqnarray} Y_{s}^{h} & = & \exp\! \left (-1\right )\cdot \left [\frac{P_{s}\cdot (1-\alpha )}{q}\right ]^{\frac{1-\alpha }{\alpha }}\cdot \frac{Q_{s}^{h}\cdot l_{s}^{h}\cdot L_{s}^{h}}{\overline{v}_{s}}, \end{eqnarray}

(25) \begin{eqnarray} Y_{s}^{i} & = & \exp\! \left (-1\right )\cdot \left [\frac{P_{s}\cdot (1-\alpha )}{q}\right ]^{\frac{1-\alpha }{\alpha }}\cdot \frac{Q_{s}^{i}\cdot l_{s}^{i}\cdot L_{s}^{i}}{1-\overline{v}_{s}}. \end{eqnarray}

We can use equations (24) and (25) to obtain the intra-country output ratio and the output of each sector as:

(26) \begin{equation} Y_{s}=\exp\! \left (-1\right )\cdot \left [\frac{P_{s}\cdot (1-\alpha )}{q}\right ]^{\frac{1-\alpha }{\alpha }}\cdot M_{s}, \end{equation}

where, bearing in mind (20), $M_{s}=\frac{Q_{s}\cdot l_{s}^{h}\cdot L_{s}^{h}}{\overline{v}_{s}}+\frac{Q_{s}\cdot l_{s}^{i}\cdot L_{s}^{i}}{1-\overline{v}_{s}}=$ $\left [\left (Q_{s}^{h}\cdot l_{s}^{h}\cdot L_{s}^{h}\right )^{\frac{1}{2}}+\left (Q_{s}^{i}\cdot l_{s}^{i}\cdot L_{s}^{i}\right )^{\frac{1}{2}}\right ]^{2}$ evaluates the market size. Similarly, we can obtain an expression for machines produced for each sector $s$ :

(27) \begin{equation} X_{s}=\exp\! (-1)\cdot \left [\frac{P_{s}\cdotp (1-\alpha )}{q}\right ]^{\frac{1}{\alpha }}\cdot M_{s}, \end{equation}

where the aggregate resources devoted to machines production in sector $s$ , $X_{s}$ , is also expressible as a function of the currently given technological knowledge in each sector $s$ .

The output performed by each labor type in each sector, $Y_{s}^{h}$ and $Y_{s}^{i}$ , and the output of sector $s$ , $Y_{s}$ , can increase in response to advancements in technological-knowledge levels. Moreover, a rise in the labor levels implies, (i) an increase in the output produced and, thereby, in the relative demand for machines, with an increase in the labor levels due to scale effects—through the terms $L_{s}^{h}$ or $L_{s}^{i}$ ; (ii) a decrease in the output produced and, thereby, in the relative demand for machines, due to price effects through the terms $\left (\overline{v}_{s}\right )^{-1}$ and $(1-\overline{v}_{s})^{-1}$ , since it increases the share of tasks produced by the respective labor type, which decreases the price index where such an increase took place.

Moreover, bearing in mind (26) the computed inter-sector output ratio is

(28) \begin{equation} \frac{Y_{N}}{Y_{R}}=\left (\frac{\chi _{N}}{\chi _{R}}\right )^{\frac{\epsilon -\alpha \epsilon }{1-\alpha +\epsilon \alpha }}\cdot \left (\frac{M_{N}}{M_{R}}\right )^{\frac{\epsilon \alpha }{1-\alpha +\epsilon \alpha }}. \end{equation}

Furthermore, from (7) and considering (26), the relative price of the output in the $N$ -sector is—see Appendix A.2:

(29) \begin{equation} \left (\frac{P_{N}}{P_{R}}\right )=\left (\frac{\chi _{R}}{\chi _{N}}\right )^{-\frac{\epsilon \alpha }{\epsilon \alpha +1-\alpha }}\cdot \left (\frac{M_{N}}{M_{R}}\right )^{-\frac{\alpha }{\epsilon \alpha +1-\alpha }}. \end{equation}

The intuition behind (29) can be grasped by taking into account that an increase in the relative relevance of the $N$ -sector in the production of the aggregate final good, $\frac{\chi _{N}}{\chi _{R}}$ , increases the relative demand for output in this sector which leads to an increase in relative prices. Hence, through $M_{N}$ and $M_{R}$ , (29) shows that if either the technological knowledge is highly $N$ -biased or if there is a large relative supply of $N$ , the output of the $N$ -sector is large—see (28), which implies a low relative price of the $N$ -sector. In this case, the demand for $N$ -machines is low, which discourages R&D activities aimed at improving their quality, as we will see below. Thus, labor structure affects the direction of R&D through the price channel and/or by the market-size channel. This latter channel may or may not be removed, eliminated, or not in conducting the economic mechanisms.

The question of wages for labor type $L_{s}^{i}$ and $L_{s}^{h}$ and the differences in wages that can be established still need to be addressed. From (19), the wages in (17) and (18) can be rewritten in the form:

(30) \begin{eqnarray} w_{s}^{h}(t) & = & \left [P_{s}^{h}(t)\right ]^{\frac{1}{\alpha }}\cdot \left (\frac{1-\alpha }{q}\right )^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}^{h}(t)\cdot l_{s}^{h}, \end{eqnarray}

(31) \begin{eqnarray} w_{s}^{i}(t) & = & \left [P_{s}^{i}(t)\right ]^{\frac{1}{\alpha }}\cdot \left (\frac{1-\alpha }{q}\right )^{\frac{1-\alpha }{\alpha }}\cdot Q_{s}^{i}(t)\cdot l_{s}^{i}. \end{eqnarray}

Moreover, we can obtain wage differentials between types of labor in each sector $s$ , $\frac{w_{s}^{i}}{w_{s}^{h}}=\left (\frac{Q_{s}^{i}\cdot l_{s}^{i}}{Q_{s}^{h}\cdot l_{s}^{h}}\frac{L_{s}^{h}}{L_{s}^{i}}\right )^{\frac{1}{2}}$ , that allow us to obtain the (domestic) steady-state skill premium and the inter-country wage inequality in favor of the domestic country which are, respectively:

(32) \begin{equation} \frac{w_{N}^{i}}{w_{N}^{h}}=\left (\frac{Q_{N}^{i}\cdot l_{N}^{i}}{Q_{N}^{h}\cdot l_{N}^{h}}\frac{L_{N}^{h}}{L_{N}^{i}}\right )^{\frac{1}{2}}\:\text{and}\:\frac{w_{R}^{i}}{w_{R}^{h}}=\left (\frac{Q_{R}^{i}\cdot l_{R}^{i}}{Q_{R}^{h}\cdot l_{R}^{h}}\frac{L_{R}^{h}}{L_{R}^{i}}\right )^{\frac{1}{2}} \end{equation}

Finally, from (17), (18) [or (30) and (31)], (19), (20), and (23), we obtain the wage polarization in favor of domestic (nonroutine) high- and low-skilled workers:

(33) \begin{eqnarray} \frac{w_{N}^{i}}{w_{R}^{i}} & = & \left (\frac{\chi _{N}}{\chi _{R}}\right )^{\frac{\epsilon }{\epsilon \alpha +1-\alpha }}\cdot \left (\frac{M_{N}}{M_{R}}\right )^{-\frac{1}{\epsilon \alpha +1-\alpha }+\frac{1}{2}}\cdot \left (\frac{Q_{N}^{i}}{Q_{R}^{i}}\cdot \frac{l_{N}^{i}}{l_{R}^{i}}\cdot \frac{L_{R}^{i}}{L_{N}^{i}}\right )^{\frac{1}{2}}, \end{eqnarray}

(34) \begin{eqnarray} \frac{w_{N}^{h}}{w_{R}^{i}} & = & \left (\frac{\chi _{N}}{\chi _{R}}\right )^{\frac{\epsilon }{\epsilon \alpha +1-\alpha }}\cdot \left (\frac{M_{N}}{M_{R}}\right )^{-\frac{1}{\epsilon \alpha +1-\alpha }+\frac{1}{2}}\cdot \left (\frac{Q_{N}^{h}}{Q_{R}^{i}}\frac{l_{N}^{h}}{l_{R}^{i}}\frac{L_{R}^{i}}{L_{N}^{h}}\right )^{\frac{1}{2}}. \end{eqnarray}

Hence, from (32), both domestic skill premium and the inter-country wage inequality in favor of the domestic country are decreasing in the relative supply factor, since the more abundant factor is substituted for the less abundant one, given, respectively, by $\frac{L_{N}^{h}}{L_{N}^{i}}$ and $\frac{L_{R}^{h}}{L_{R}^{i}}$ , and, increasing, in the absolute productivity advantage given, respectively, by $l_{N}^{i}$ and $l_{R}^{i}$ . Moreover, a rise in technological-knowledge biased toward both $N^{i}$ , in the skill premium function, and $R^{i}$ , in the inter-country wage inequality in favor of the domestic country function positively impacts wages. This rise fosters competitiveness within their respective sectors and improves the workers’ relative productivity.

Furthermore, within the same context, from (33) and (34), the wage polarization in favor of domestic (nonroutine) high- and low-skilled workers, respectively, is positively affected by a rise in the relative relevance of the $N$ -sector since it augments the relative demand for output in this sector that leads to an increase in the wage polarization, biased to high- and low-skilled employees. Moreover, if $\frac{\epsilon }{\epsilon \alpha +1-\alpha }\gt \frac{1}{2}$ and the market size dominates the price of the price effect channel, an increase either in the technological-knowledge highly $N$ -biased, an increase in the relative supply factor of $R$ -sector, or a decrease in the absolute productivity in the $R$ -sector stimulates significant wages improvements at the tails of the distribution.

These results can be interpreted as short-run results, as R&D has not been considered yet. The exposition of the R&D sector closes the model and allows for the calculation of steady-state or long-run results. In fact, R&D is also responsible for the transitional dynamics of the model. The following subsection describes the R&D sector.

3.5. R&D sector

By producing innovative designs, R&D activities drive the rate and the direction of technological knowledge, and thus wages and economic growth. Innovative designs for the manufacture of new qualities of the machines are patented, and the leader firm in each industry—the one that produces according to the latest patent—uses limit pricing to assure monopoly. The value of the leading-edge patent relies on the profit yields accruing during each time $t$ to the monopolist and on the duration of the monopoly power. The duration, in turn, depends on the probability of a new innovation, which creatively destroys the current leading-edge design [e.g., Aghion and Howitt (Reference Aghion and Howitt1992), Grossman and Helpman (Reference Grossman and Helpman1991, ch. 12), and Barro and Sala-i-Martin (Reference Barro and Sala-i-Martin2004, ch. 7)]. The probability of successful innovation is, thus, at the heart of the R&D activity. Let $\mathcal{I}_{s}^{h}(k,j,t)$ and $\mathcal{I}_{s}^{i}(k,j,t)$ denote the instantaneous probability at time $t$ in sector $s$ for, respectively, $h$ and $i$ —a Poisson arrival rate—of successful innovation in the next higher quality $\left [k(j,t)+1\right ]$ in machine $j$ given current rung quality $k$ . We define it as follows:

(35) \begin{eqnarray} \mathcal{I}_{s}^{h}(k,j,t) & = & e_{s}^{h}(k,j,t)\cdot \beta q^{k(j,t)}\cdot \zeta ^{-1}q^{-\alpha ^{-1}k(j,t)}\cdot (L_{s}^{h})^{-\xi }, \end{eqnarray}

(36) \begin{eqnarray} \mathcal{I}_{s}^{i}(k,j,t) & = & e_{s}^{i}(k,j,t)\cdot \beta q^{k(j,t)}\cdot \zeta ^{-1}q^{-\alpha ^{-1}k(j,t)}\cdot (L_{s}^{i})^{-\xi }. \end{eqnarray}

where, for example, following Afonso and Sequeira (Reference Afonso and Sequeira2023): (i) $e_{s}^{h}(k,j,t)$ and $e_{s}^{i}(k,j,t)$ are the flow of domestic final-good resources devoted to R&D in $j$ belonging to $s$ for, respectively, $h$ and $i$ , which define our framework as a lab equipment model; (ii) $\beta$ $q^{k(j,t)}$ , $\beta \gt 0$ , is the learning-by-past domestic R&D, as a positive learning effect of public knowledge accumulated from past successful R&D; (iii) $\zeta ^{-1}q^{-\alpha ^{-1}k(j,t)}$ , $\zeta \gt 0$ , is the adverse effect—the cost of complexity—caused by the increasing complexity of quality improvements;Footnote ¹⁶ (iv) $(L_{s}^{h})^{-\xi }$ and $(L_{s}^{i})^{-\xi }$ , with $\xi \geq 0$ , are the adverse effects of market size, capturing the idea that the difficulty of introducing new quality machines and replacing old ones is proportional to the respective labor units in sector $s$ . The scale benefits on profits can be partially ( $0\lt \xi \lt 1$ ) or totally ( $\xi =1$ ) removed and thus allows us to remove (explicit) scale effects on the economic growth rate [e.g., Jones (Reference Jones1995) and Sequeira et al. (Reference Sequeira, Gil and Afonso2018)]. However if $\xi \gt 1$ , we would have (negative) scale effects. That is, for reasons of simplicity, we reflect in R&D the costs of scale increasing, due to coordination among agents, processing of ideas, and informational, organizational, marketing, and transportation costs [e.g., Dinopoulos and Thompson (Reference Dinopoulos and Thompson1999)].

We consider that the probability of innovation presented above is similar for incumbents and entrants. Thus, R&D is conducted by entrants, as shown in Appendix A.3. The value of the leading-edge patent for the producer of an intermediate good $j$ belonging to $s$ and used by, respectively, $h$ and $i$ , with quality level $k$ at time $t$ is the expected present value of the flow of profits given by the following equations:Footnote ¹⁷ $V_{s}^{i}(j,k,t,T(k))=\int _{t}^{t+T(k)}\pi _{s}^{i}(j,s)\exp\! \left (-\int _{t}^{s}r(w)dw\right )ds$ and $V_{s}^{h}(j,k,t,T(k))=\int _{t}^{t+T(k)}\pi _{s}^{h}(j,s)\exp\! \left (-\int _{t}^{s}r(w)dw\right )ds$ , where $T(k)$ is the duration of the patent during which there is no innovation in the quality level of intermediate good $j$ by another entrant.Footnote ¹⁸

Given the functional forms (35) and (36) of the probabilities of success in R&D, which rely on the resources—composite final goods—allocated to it, free-entry equilibrium is defined by the equality between expected revenue, $\mathcal{I}_{s}^{h}(j,t)\cdot V_{s}^{h}(j,t)$ and $\mathcal{I}_{s}^{i}(j,t)\cdot V_{s}^{i}(j,t)$ , and resources spent, $e_{s}^{h}(j,t)$ and $e_{s}^{i}(j,t)$ . By considering free entry in R&D activities, free access to the R&D technology, and a proportional relationship between successful R&D and the share of R&D effort, the R&D spending aimed at, for example, improving $j$ should equal the expected payoff generated by the innovation, that is,

(37) \begin{equation} \mathcal{I}_{s}^{h}(k,j,t)\cdot V_{s}^{h}(j,t)=e_{s}^{h}(j,t)\:and\:\mathcal{I}_{s}^{i}(k,j,t)\cdot V_{s}^{i}(j,t)=e_{s}^{i}(j,t). \end{equation}

Assuming that all the prices and quantities are fixed during the time in which there are no quality improvements [e.g., Aghion and Howitt (Reference Aghion and Howitt1992), Barro and Sala-i-Martin (Reference Barro and Sala-i-Martin2004), and Gil et al. (Reference Gil, Brito and Afonso2013)], then we have that—see Appendix A.4:

(38) \begin{equation}{V_{s}^{h}(j,k,t)=\frac{\pi _{s}^{h}(j,k,t)}{r(t)+\mathcal{I}_{s}^{h}(j,t)}\:and\:V_{s}^{i}(j,k,t)=\frac{\pi _{s}^{i}(j,k,t)}{r(t)+\mathcal{I}_{s}^{i}(j,t)}}{,} \end{equation}

and can be seen as the no-arbitrage condition, where $V_{s}^{h}\left (k,j,t\right )\cdot r\left (t\right )$ and $V_{s}^{i}\left (k,j,t\right )\cdot r\left (t\right )$ , the expected income generated by a successful innovation at time $t$ on rung $k$ , equals the profit flow, $\pi _{s}^{h}(j,k,t)$ and $\pi _{s}^{i}(j,k,t)$ , minus the expected capital loss, $V_{s}^{h}\left (k,j,t\right )\cdot \mathcal{I}_{s}^{h}\left (j,\tau \right )$ and $V_{s}^{i}\left (k,j,t\right )\cdot \mathcal{I}_{s}^{i}\left (j,\tau \right )$ . Then plugging (38) into (37) and solving for $\mathcal{I}_{s}^{h}$ and $\mathcal{I}_{s}^{i}$ , the equilibrium probability of successful innovation in sector $s$ and for $h$ and $i$ are, respectively—given the interest rate and the price indexes of final goods:

(39) \begin{eqnarray} \mathcal{I}_{s}^{h}(t) & = & \frac{\beta }{\zeta }\cdot \left (\frac{q-1}{q}\right )\cdot \exp\! \left (-1\right )\cdot \left [\left (1-\alpha \right )\cdot P_{s}^{h}\right ]^{\frac{1}{\alpha }}\cdot l_{s}^{h}\cdot (L_{s}^{h})^{1-\xi }-r(t), \end{eqnarray}

(40) \begin{eqnarray} \mathcal{I}_{s}^{i}(t) & = & \frac{\beta }{\zeta }\cdot \left (\frac{q-1}{q}\right )\cdot \exp\! \left (-1\right )\cdot \left [\left (1-\alpha \right )\cdot P_{s}^{i}\right ]^{\frac{1}{\alpha }}\cdot l_{s}^{i}\cdot (L_{s}^{i})^{1-\xi }-r(t). \end{eqnarray}

The equilibrium $\mathcal{I}_{s}^{i}(t)$ and $\mathcal{I}_{s}^{h}(t)$ in (35) and (40) are, respectively, independent of $j$ and $k$ since the removal of the scale of technological-knowledge effects—see the exponents of $q$ in the demand of intermediate goods above, which impacts the expression of profits, and in equations (35) and (36).

Finally, from the definition of the probabilities of achieving higher quality rungs (35) and (40), and since, by definition, $\mathcal{I}_{s}^{h}(k,j,t)$ and $\mathcal{I}_{s}^{i}(k,j,t)$ do not differentiate between different machines belonging to the same sector, we have that:

(41) \begin{equation} E_{s}(t)=\underbrace{\int _{0}^{J}e_{s}^{h}(k,j,t)dj}_{E_{s}^{h}(t)}+\underbrace{\int _{J}^{1}e_{s}^{i}(k,j,t)dj}_{E_{s}^{i}(t)}=\mathcal{I}_{s}^{h}(k,j,t)\cdot \frac{\zeta }{\beta }\cdot Q_{s}^{h}\cdot (L_{s}^{h})^{\xi }+\mathcal{I}_{s}^{i}(k,j,t)\cdot \frac{\zeta }{\beta }\cdot Q_{s}^{i}\cdot (L_{s}^{i})^{\xi }, \end{equation}

and thus more resources devoted to R&D are needed as $Q_{s}^{h}$ and $Q_{s}^{i}$ rise to offset the greater difficulty of R&D when $Q_{s}^{h}$ and $Q_{s}^{i}$ increase.

The following section derives the general equilibrium of the model, as well as our main long-run results.

4.1. Technological-knowledge indexes and bias

If a new quality of machine $j$ is introduced, the rate of change in the quality index of sector $s$ used by, for example, labor type $h$ is the following: $\varDelta Q_{s}^{h}(t)=Q_{s}^{h}\left (k+1,t\right )-Q_{s}^{h}(k,t)=\int _{0}^{J}q^{\left [k(j,t)+1\right ]\left (\frac{1-\alpha }{\alpha }\right )}-\int _{0}^{J}q^{k(j,t)\left (\frac{1-\alpha }{\alpha }\right )}$ and thus $\frac{\varDelta Q_{s}^{h}}{Q_{s}^{h}}=\left [q^{\left (\frac{1-\alpha }{\alpha }\right )}-1\right ]$ . Since the probability of this occurring per unit of time is given by $\mathcal{I}_{s}^{h}(t)$ , we have that:

(42) \begin{align} \frac{\dot{Q_{s}^{h}}}{Q_{s}^{h}}&=\mathcal{I}_{s}^{h}(t)\cdot \left [q^{\left (\frac{1-\alpha }{\alpha }\right )}-1\right ]=\left [\frac{\beta }{\zeta }\cdot \left (\frac{q-1}{q}\right )\cdot \exp\! \left (-1\right )\cdot \left [\left (1-\alpha \right )\cdot P_{s}^{h}\right ]^{\frac{1}{\alpha }}\cdot l_{s}^{h}\cdot (L_{s}^{h})^{1-\xi }-r\right ]\nonumber\\&\qquad\cdot \left [q^{\left (\frac{1-\alpha }{\alpha }\right )}-1\right ] \end{align}

Thus, bearing in mind (42), from the emergence of a shock to a parameter or an exogenous variable to the steady state, the paths of the technological-knowledge bias in sector $s=N$ , which drives the skill premium and the inter-country wage inequality in (32), are as follows:

(43) \begin{eqnarray} \frac{\dot{Q_{N}^{i}}}{Q_{N}^{i}}-\frac{\dot{Q_{N}^{h}}}{Q_{N}^{h}} & = & \left [q^{\left (\frac{1-\alpha }{\alpha }\right )}-1\right ]\cdot \frac{\beta }{\zeta }\cdot \frac{q-1}{q}\cdot \exp\! \left (-1\right )\cdot \left (1-\alpha \right )^{\frac{1}{\alpha }}\nonumber\\&&\quad\cdot \left [\left (P_{N}^{i}\right )^{\frac{1}{\alpha }}\cdot l_{s}^{i}\cdot (L_{N}^{i})^{1-\xi }-\left (P_{N}^{h}\right )^{\frac{1}{\alpha }}\cdot l_{s}^{h}\cdot (L_{N}^{h})^{1-\xi }\right ], \end{eqnarray}

(44) \begin{eqnarray} \frac{\dot{Q_{R}^{i}}}{Q_{R}^{i}}-\frac{\dot{Q_{R}^{h}}}{Q_{R}^{h}} & = & \left [q^{\left (\frac{1-\alpha }{\alpha }\right )}-1\right ]\cdot \frac{\beta }{\zeta }\cdot \frac{q-1}{q}\cdot \exp\! \left (-1\right )\cdot \left (1-\alpha \right )^{\frac{1}{\alpha }}\nonumber\\ &&\qquad\cdot \left [\left (P_{R}^{i}\right )^{\frac{1}{\alpha }}\cdot l_{R}^{i}\cdot (L_{R}^{i})^{1-\xi }-\left (P_{R}^{h}\right )^{\frac{1}{\alpha }}\cdot l_{R}^{h}\cdot (L_{R}^{h})^{1-\xi }\right ]. \end{eqnarray}

In turn, the inter-sector technological knowledge, which drives the wage polarization in (33) and (34), are

(45) \begin{eqnarray} \frac{\dot{Q_{N}^{h}}}{Q_{N}^{h}}-\frac{\dot{Q_{R}^{h}}}{Q_{R}^{h}} & = & \left [q^{\left (\frac{1-\alpha }{\alpha }\right )}-1\right ]\frac{\beta }{\zeta }\cdot \left (\frac{q-1}{q}\right )\cdot \exp\! \left (-1\right )\nonumber\\ &&\qquad\cdot \left [\left (P_{N}^{h}\right )^{\frac{1}{\alpha }}\cdot l_{N}^{h}\cdot (L_{N}^{h})^{1-\xi }-\left (P_{R}^{h}\right )^{\frac{1}{\alpha }}\cdot l_{R}^{h}\cdot (L_{R}^{h})^{1-\xi }\right ], \end{eqnarray}

(46) \begin{eqnarray} \frac{\dot{Q_{N}^{i}}}{Q_{N}^{i}}-\frac{\dot{Q_{R}^{i}}}{Q_{R}^{i}} & = & \left [q^{\left (\frac{1-\alpha }{\alpha }\right )}-1\right ]\cdot \frac{\beta }{\zeta }\cdot \frac{q-1}{q}\cdot \exp\! \left (-1\right )\cdot \left (1-\alpha \right )^{\frac{1}{\alpha }}\nonumber\\ &&\quad\cdot \left [\left (P_{N}^{i}\right )^{\frac{1}{\alpha }}\cdot l_{N}^{i}\cdot (L_{N}^{i})^{1-\xi }-\left (P_{R}^{i}\right )^{\frac{1}{\alpha }}\cdot l_{R}^{i}\cdot (L_{R}^{i})^{1-\xi }\right ], \end{eqnarray}

(47) \begin{eqnarray} \frac{\dot{Q_{N}^{h}}}{Q_{N}^{h}}-\frac{\dot{Q_{R}^{h}}}{Q_{R}^{h}} & = & \left [q^{\left (\frac{1-\alpha }{\alpha }\right )}-1\right ]\cdot \frac{\beta }{\zeta }\cdot \frac{q-1}{q}\cdot \exp\! \left (-1\right )\cdot \left (1-\alpha \right )^{\frac{1}{\alpha }}\nonumber\\ &&\qquad\cdot \left [\left (P_{N}^{h}\right )^{\frac{1}{\alpha }}\cdot l_{N}^{h}\cdot (L_{N}^{h})^{1-\xi }-\left (P_{R}^{h}\right )^{\frac{1}{\alpha }}\cdot l_{R}^{h}\cdot (L_{R}^{h})^{1-\xi }\right ] \end{eqnarray}

4.2. Steady-state results

Taking into account, the aggregate expenditures in the final good are given by $Y=P_{N}Y_{N}+P_{R}Y_{R}$ from the profit maximization problem of the producer of aggregate output, considering that aggregate expenditures in machines and R&D activities are the sum of aggregates in both sectors already derived in equilibrium in the previous sections, $X\equiv X_{N}+X_{R}$ and $E\equiv E_{N}+E_{R}$ , and that assets in the economy are the present value of the patent of all producers of machines, that is, that $a=\sum _{s=N,R}\int _{0}^{J}V_{s}^{h}(k,j,t)dj+\int _{J}^{1}V_{s}^{i}(k,j,t)dj$ we can prove that in equilibrium the aggregate flow constraint of households can be expressed as $Y=C+X+E$ —see the respective proof in Appendix A.5. Since $Y$ , $X$ , and $E$ are all multiples of the quality indexes $Q_{N}^{h}$ , $Q_{N}^{i}$ , $Q_{R}^{h}$ , and $Q_{R}^{i}$ , the aggregate flow constraint implies that consumption $C$ is also a constant multiple of these variables, which implies that the path of all relevant variables outside the steady state depends on the path of the different quality indexes. At the end of transitional dynamics, the economy reaches the steady state, which is unique and stable, and all relevant macroeconomic variables grow at the same constant rate $g^{\ast }$ :

(48) \begin{equation} g^{\ast }\equiv \left (\frac{\dot{Q_{s}^{i}}}{Q_{s}^{i}}\right )^{*}=\left (\frac{\dot{Q_{s}^{h}}}{Q_{s}^{h}}\right )^{*}=\left (\frac{\dot{Y}}{Y}\right )^{*}=\left (\frac{\dot{X}}{X}\right )^{*}=\left (\frac{\dot{E}}{E}\right )^{*}=\left (\frac{\dot{C}}{C}\right )^{*}=\frac{r^{\ast }-\rho }{\theta }. \end{equation}

The uniqueness of the steady state is guaranteed by the uniqueness of the interest rate. To take into account the stability, it must be considered (42), (43), (44), (46), and (47). The dynamics of the Economy can be characterized by a two-dimensional dynamic system in detrended variables such as $\frac{Q_{s}^{i}}{Q_{s}^{h}}$ (or $\frac{Q_{N}^{i}}{Q_{R}^{i}}$ or $\frac{Q_{N}^{h}}{Q_{R}^{i}}$ or $\frac{Q_{N}^{i}}{Q_{R}^{h}}$ or $\frac{Q_{N}^{h}}{Q_{R}^{h}}$ ) and $\frac{C}{Q_{s}^{i}}$ (or $\frac{C}{Q_{s}^{h}}$ ) that has a recursive structure since the dynamics of $\frac{Q_{s}^{i}}{Q_{s}^{h}}$ depends, exclusively, on itself. For example, considering an economy starting out of the steady state, where $\frac{\dot{Q_{s}^{i}}}{Q_{s}^{i}}\gt \frac{\dot{Q_{s}^{h}}}{Q_{s}^{h}}$ , we will prove that, over $t$ , $\frac{\dot{Q_{s}^{i}}}{Q_{s}^{i}}\gt 0$ and $\frac{\dot{Q_{s}^{h}}}{Q_{s}^{h}}=0$ . Bearing in mind this situation, it is easy to perceive that $\overline{v}_{s}\gt \overline{v}_{s}^{*}$ which, in turn, implies $\frac{\dot{P}_{s}^{i}}{P_{s}^{I}}\lt \frac{\dot{P}_{s}^{h}}{P_{s}^{h}}$ . Thereby, $\frac{P_{s}^{i}}{P_{s}^{h}}$ is declining until

(49) \begin{equation} \left (\frac{P_{s}^{i}}{P_{s}^{h}}\right )^{*}=\left [\frac{l_{s}^{i}}{l_{s}^{h}}\cdot \left (\frac{L_{s}^{i}}{L_{s}^{h}}\right )^{1-\xi }\right ]^{-\alpha }\text{,} \end{equation}

attenuating the rate at which $\frac{Q_{s}^{i}}{Q_{s}^{h}}$ is increasing. In this sense, even with $\frac{\dot{Q_{s}^{i}}}{Q_{s}^{i}}\gt \frac{\dot{Q_{s}^{h}}}{Q_{s}^{h}}$ the difference between both equilibrium paths of technological knowledge, $\frac{\dot{Q_{s}^{i}}}{Q_{s}^{i}}-\frac{\dot{Q_{s}^{h}}}{Q_{s}^{h}}$ , is decreasing until approaches the steady state, where $\frac{\dot{Q_{s}^{i}}}{Q_{s}^{i}}=\frac{\dot{Q_{s}^{h}}}{Q_{s}^{h}}$ . The argument to exhibit the convergence to the steady state if $\frac{\dot{Q_{s}^{i}}}{Q_{s}^{i}}\lt \frac{\dot{Q_{s}^{h}}}{Q_{s}^{h}}$ is similar—see, for example, Acemoglu and Zilibotti (Reference Acemoglu and Zilibotti2001). Thus, the economy converges and remains in a steady state, if no other exogenous changes occur.

We now show the calculation of the variables of interest in the steady state and, for each one, a Proposition is presented in the face of an eventual “shock.”

From (23) and (49), we have that:

(50) \begin{equation} \left (\frac{Q_{s}^{i}}{Q_{s}^{h}}\right )^{*}=\frac{l_{s}^{i}}{l_{s}^{h}}\cdot \left (\frac{L_{s}^{i}}{L_{s}^{h}}\right )^{1-2\xi }\text{.} \end{equation}

In possession of (50), one can use (20) to determine

(51) \begin{equation} \overline{v}_{s}^{*}=\left [1+\frac{l_{s}^{i}}{l_{s}^{h}}\cdot \left (\frac{L_{s}^{i}}{L_{s}^{h}}\right )^{1-\xi }\right ]^{-1}\text{.} \end{equation}

Hence, the “race” between automation and relocation is won by automation if the level of domestic medium-skilled workers exceeds the number existing abroad—that is, if the “room” for automation exceeds the “room” for relocations—and if the absolute advantage of domestic medium-skilled workers outweighs the absolute advantage of the same type of workers abroad.

Therefore, from (32) and (50), the (domestic) steady-state skill premium and the inter-country wage inequality in favor of the domestic country are, respectively:

(52) \begin{equation} \frac{w_{N}^{i}}{w_{N}^{h}}=\frac{l_{N}^{i}}{l_{N}^{h}}\cdot \left (\frac{L_{N}^{i}}{L_{N}^{h}}\right )^{-\xi }\:\textit{and}\:\frac{w_{R}^{i}}{w_{R}^{h}}=\frac{l_{R}^{i}}{l_{R}^{h}}\cdot \left (\frac{L_{R}^{i}}{L_{R}^{h}}\right )^{-\xi }. \end{equation}

To compute the steady-state wage polarization, we start noting that in (46) and (47), $\left (\frac{\dot{Q_{N}^{i}}}{Q_{N}^{i}}\right )=\left (\frac{\dot{Q_{R}^{i}}}{Q_{R}^{i}}\right )$ and $\left (\frac{\dot{Q_{N}^{h}}}{Q_{N}^{h}}\right )=\left (\frac{\dot{Q_{R}^{i}}}{Q_{R}^{i}}\right )$ , which implies that

(53) \begin{equation} \left (\frac{P_{N}^{i}}{P_{R}^{i}}\right )^{*}=\left [\frac{l_{N}^{i}}{l_{R}^{i}}\cdot \left (\frac{L_{N}^{i}}{L_{R}^{i}}\right )^{1-\xi }\right ]^{-\alpha } \,\,\text{and}\,\,{\left (\frac{P_{N}^{h}}{P_{R}^{i}}\right )^{*}}={\left [\frac{l_{N}^{h}}{l_{R}^{i}}\cdot \left (\frac{L_{N}^{h}}{L_{R}^{i}}\right )^{1-\xi }\right ]^{-\alpha }}, \end{equation}

which, by equating, respectively, with the expressions that correspond to the relative price of $L_{s}^{i}$ task, obtained bearing in mind (23), that is, $\left (\frac{P_{N}^{i}}{P_{R}^{i}}\right )^{*}=\left (\frac{P_{N}^{i}}{P_{R}^{i}}\right )$ and $\left (\frac{P_{N}^{h}}{P_{R}^{i}}\right )^{*}=\left (\frac{P_{N}^{h}}{P_{R}^{i}}\right )$ , makes it possible to obtain $\left (\frac{Q_{N}^{h}}{Q_{R}^{h}}\right )^{*}$ , $\left (\frac{Q_{N}^{i}}{Q_{R}^{i}}\right )^{*}$ and $\left (\frac{Q_{N}^{h}}{Q_{R}^{i}}\right )^{*}$ which is given by:

(54) \begin{eqnarray} \left (\frac{Q_{N}^{i}}{Q_{R}^{i}}\right )^{*} & = & \left (\frac{\chi _{N}}{\chi _{R}}\right )^{\epsilon }\left (\frac{l_{N}^{i}}{l_{R}^{i}}\right )\left (\frac{L_{N}^{i}}{L_{R}^{i}}\right )^{1-2\xi }\left [\frac{l_{R}^{h}\cdot \left (L_{R}^{h}\right )^{1-\xi }+l_{R}^{i}\cdot \left (L_{R}^{i}\right )^{1-\xi }}{l_{N}^{h}\cdot \left (L_{N}^{h}\right )^{1-\xi }+l_{N}^{i}\cdot \left (L_{N}^{i}\right )^{1-\xi }}\right ]^{1-\epsilon \alpha +\alpha }, \end{eqnarray}

(55) \begin{eqnarray} \left (\frac{Q_{N}^{h}}{Q_{R}^{i}}\right )^{*} & = & \left (\frac{\chi _{N}}{\chi _{R}}\right )^{\epsilon }\left (\frac{l_{N}^{h}}{l_{R}^{i}}\right )\left (\frac{L_{N}^{h}}{L_{R}^{i}}\right )^{1-2\xi }\left [\frac{l_{R}^{h}\cdot \left (L_{R}^{h}\right )^{1-\xi }+l_{R}^{i}\cdot \left (L_{R}^{i}\right )^{1-\xi }}{l_{N}^{h}\cdot \left (L_{N}^{h}\right )^{1-\xi }+l_{N}^{i}\cdot \left (L_{N}^{i}\right )^{1-\xi }}\right ]^{1-\epsilon \alpha +\alpha }. \end{eqnarray}

Thus, bearing in mind (33) and (34) as well as (54) and (55) enable us to determine, respectively,

(56) \begin{eqnarray} \left (\frac{w_{N}^{i}}{w_{R}^{i}}\right )^{*} & = & \left (\frac{\chi _{N}}{\chi _{R}}\right )^{\epsilon }\left (\frac{l_{N}^{i}}{l_{R}^{i}}\right )\left (\frac{L_{R}^{i}}{L_{N}^{i}}\right )^{\xi }\left [\frac{l_{R}^{h}\cdot \left (L_{R}^{h}\right )^{1-\xi }+l_{R}^{i}\cdot \left (L_{R}^{i}\right )^{1-\xi }}{l_{N}^{h}\cdot \left (L_{N}^{h}\right )^{1-\xi }+l_{N}^{i}\cdot \left (L_{N}^{i}\right )^{1-\xi }}\right ]^{1-\epsilon \alpha +\alpha }, \end{eqnarray}

(57) \begin{eqnarray} \left (\frac{w_{N}^{h}}{w_{R}^{i}}\right )^{*} & = & \left (\frac{\chi _{N}}{\chi _{R}}\right )^{\epsilon }\left (\frac{l_{N}^{h}}{l_{R}^{i}}\right )\left (\frac{L_{R}^{i}}{L_{N}^{h}}\right )^{\xi }\left [\frac{l_{R}^{h}\cdot \left (L_{R}^{h}\right )^{1-\xi }+l_{R}^{i}\cdot \left (L_{R}^{i}\right )^{1-\xi }}{l_{N}^{h}\cdot \left (L_{N}^{h}\right )^{1-\xi }+l_{N}^{i}\cdot \left (L_{N}^{i}\right )^{1-\xi }}\right ]^{1-\epsilon \alpha +\alpha }. \end{eqnarray}

It is also worth noting that as $\epsilon \rightarrow 0$ (high complementarity) and $\alpha \rightarrow 1$ (low importance of machines in production), the lower would be the relevance of reallocations (offshoring) for wage polarization. Moreover, with low-scale effects $\xi \rightarrow 0$ , wage polarization would be greatly driven by productivity differences related both to reallocations and automation. In particular, in that case, wage polarization in favor of domestic (nonroutine) high(low)-skilled workers is positively affected by an increase in domestic (nonroutine) high(low)-skilled absolute productivity and decrease in domestic (routine) medium-skilled absolute productivity, $l_{N}^{i}$ . Moreover in the specific case when sectors are gross substitutes ( $\epsilon \geq 1$ ), wage polarization in favor of domestic (nonroutine) high-skilled workers is always positively affected an increase in foreign (routine) medium-skilled absolute productivity, $l_{R}^{h}$ , and domestic (nonroutine) high-skilled absolute productivity, $l_{N}^{i}$ , and never by a decrease in domestic (nonroutine) low-skilled absolute productivity, $l_{N}^{h}$ , and domestic (routine) medium-skilled absolute productivity, $l_{R}^{i}$ . Also wage polarization in favor of domestic (nonroutine) low-skilled workers is always positively affected by an increase in foreign (routine) medium-skilled absolute productivity, $l_{R}^{h}$ , and domestic (nonroutine) low-skilled absolute productivity, $l_{N}^{h}$ , and never by a decrease in domestic (nonroutine) high-skilled absolute productivity, $l_{N}^{h}$ , and domestic (routine) medium-skilled absolute productivity, $l_{R}^{i}$ .

Bearing in mind (42) and (48), the stable and unique steady-state economic growth rate is—considering, as example, the case in which labor is type $h$ :

(58) \begin{equation} g^{\ast }=\frac{\left [q^{\left (\frac{1-\alpha }{\alpha }\right )}-1\right ]\left \{ \left [\frac{\beta }{\zeta }\cdot \left (\frac{q-1}{q}\right )\cdot \exp\! \left (-1\right )\cdot \left [\left (1-\alpha \right )\cdot P_{s}^{h}\right ]^{\frac{1}{\alpha }}\cdot l_{s}^{h}\cdot (L_{s}^{h})^{1-\xi }\right ]-\rho \right \} }{\left [q^{\left (\frac{1-\alpha }{\alpha }\right )}-1\right ]\theta +1}\text{,} \end{equation}

where $\left (P_{s}^{h}\right )^{*}=\left (P_{s}\right )^{*}\cdot \exp\! \left (-\alpha \right )\cdot \left (\overline{v}_{s}^{-\alpha }\right )^{*}$ and $\left (P_{s}\right )^{*}$ is determined in Appendix A.6.

Hence, from (58), economic growth depends positively on automation—in line with, for example, Zeira (Reference Zeira1998), Acemoglu and Restrepo (Reference Acemoglu and Restrepo2017), and Graetz and Michaels (Reference Graetz and Michaels2018)—and also on relocations—in line with the empirical studies performed by, for example, Li and Liu (Reference Li and Liu2005), Baharumshah and Almasaied (Reference Baharumshah and Almasaied2009), Wang (Reference Wang2007), Krammer (Reference Krammer2010), Cuadros and Alguacil (Reference Cuadros and Alguacil2014), and Su and Liu (Reference Su and Liu2016).

5.1. Data and calibration strategy

To carry out the quantitative exercises, it is required to calibrate several parameters and exogenous variables, previously exhibited on (56), (57), and (58). The values of most parameters were selected based on the literature on related models. However, crucial parameters and exogenous are data-based, following a thorough quantitative assessment.

First, we focus on the parameters based on the literature. In order to calibrate the elasticity of substitution between sectors, we followed Afonso, et al. (Reference Afonso, Lima and Sequeira2022) where $\epsilon =0.50$ while for scale effects, the value of $\xi$ will be assumed to be $0.8$ , following the idea that in modern economies we may have positive but small-scale effects (e.g., Sequeira et al. Reference Sequeira, Gil and Afonso2018). For the labor share, the work carried out by Jones et al. (Reference Jones, Manuelli and Rossi1993) was considered whereas $\alpha =0.64$ , which in turn allowed us to calculate the constant quality upgrade, $q=2.78$ , since $q=\frac{1}{1-\alpha }$ . Moreover, the value for the learning-by-past domestic R&D, measured by the obsolescence of (past) investments in R&D, was chosen to take into account Afonso (Reference Afonso2012), $\beta =2$ , and the cost of complexity, measured through the Economic Complexity Index (ECI) developed by the Harvard Growth Lab’s Country Rankings, where, for the USA, $\zeta =1.64$ . Additionally, the rate time of preference and the inverse of the inter-temporal elasticity of substitution were attained based on Arrow (Reference Arrow1999) and Croix and Delavallade (Reference De la Croix and Delavallade2009), respectively, where $\rho =0.015$ and $\theta =0.5$ .

Second, we focus on data-based exogenous variables. The relative importance of the routine sector was measured as a division between the gross value added of four manufacturing subsectors which account for more than 80% of all industrial robots—food products (C10–12), fossil fuels, chemicals and pharmaceuticals (C19–21), rubber, plastic and mineral products (C22–C23), metal products (C24–C25), computer, electronic and electrical equipment (C26–27), machinery and equipment (C28), and automotive (C29–C30)—and the gross value added of all industries.Footnote ¹⁹ The average value for this specific period (2003–2020) was subsequently calculated, resulting in $\chi _{R}=0.087$ . Similarly, bearing in mind the remaining industries and manufacturing subsectors not mentioned previously as the numerator, the relative importance of the nonroutine sector was computed, leading to $\chi _{N}=0.913$ . Furthermore, the labor performed in both the domestic economy ( $L_{N}^{i}$ , $L_{N}^{h}$ , $L_{R}^{i}$ )—high-skilled, low-skilled, and medium-skilled in the USA—and foreign economy ( $L_{R}^{h}$ )—medium-skilled in Mexico—was measured by the number of employees in the respective skill level, through the “employment by sex, age and education (in thousands)” dataset, based on the ISCED, as previously detailed in Section 2.Footnote ²⁰ Subsequently, an arithmetic average was computed, leading to $L_{N}^{i}=0.966$ , $L_{N}^{h}=1.502$ , $L_{R}^{i}=0.662$ , and $L_{R}^{h}=2.499$ . Lastly, we calculated the productivity by each labor type.Footnote ²¹ Our purpose is to collect information regarding the level of education that is mostly employed in each occupation, rather than the minimum education required. We adopted the evidence produced by O^*NET that provides the level of schooling commonly undertaken in each job.Footnote ²² When the occupations in the Bureau of Labor Statistics (BLS) database do not coincide with those in the O^*NET, we used the information available in the “typical education needed for entry.” Then, to categorize each detailed occupation by industry, we utilized the database released by the BLS “May 2021 National industry-specific occupational employment and wages estimates” to allocate the percentage of employees of each job in each industrial sector. In a later step, we retrieved the data, from the BLS (together with the Standard Occupational Classification system), for the number of employees by detailed occupation. Therefore, by obtaining the detailed jobs and employment data by detailed occupation, we were able to match both databases. It was possible to attain the amount and percentage of workers employed in each occupation, by education level in each industry. Consequently, bearing in mind the number of hours worked and value added by industry, it became feasible to obtain the corresponding hours worked and value added across each educational level within each sector.Footnote ²³ That said, absolute productivity by skill level in the USA was measured using the ratio of the two aforementioned variables, yielding $l_{N}^{i}=82.339$ , $l_{N}^{h}=45.678$ , and $l_{R}^{i}=64.218$ . Lastly, to determine the absolute productivity of medium-skilled workers in Mexico, we decided to use the value of the parameters and variables indicated throughout the subsection in the expression (52) which refers to inter-country wage inequality in favor of the domestic country. Thus, in order to gather data on the wages of domestic and of foreign medium-skilled workers, we used the database “Average monthly earnings of employees by sex and occupation (in thousands),” based on the ISCO and the respective level of qualification, produced by the ILO, resulting in $\frac{w_{R}^{i}}{w_{R}^{h}}=7.942$ . Therefore, we acquired the value for absolute productivity foreign medium-skilled labor, which is given by $l_{R}^{h}=26.667.$ Table 2 presents each parameter and variable with its corresponding value, description, and source.

Table 2. Parameters and variables calibrated

Moreover, given the values of the variables and parameters of interest,— $\chi _{R}$ , $L_{R}^{i}$ , $L_{R}^{h}$ , $l_{R}^{i}$ , and $l_{R}^{h}$ —we calculate the range of values for each will fall within an interval of $20\%$ below and $20\%$ above the mean, disposed in Table 3, which will be used for examining fluctuations in the mentioned key variables.

Table 3. Range of values for the variables and parameters of interest

5.2. Results and discussion

Bearing in mind the steady-state expressions for the crucial variables, —(54), (55), (56), and (57)—the calibration and the range of values, we visually analyze the behavior of each critical variable in order to confirm the theoretical conclusions.

The relative distribution parameters have the potential to contribute to biases inherent in the production process, which can interfere with technological-knowledge advances. Thus, if the relative distribution parameter tends in favor of the routine sector, it suggests that the economy attributes a higher importance to this sector, even before any technological changes. Nevertheless, as technology improves, it can reinforce this existing tendency. This specific impact on the technological-knowledge bias will depend on the values assigned to these parameters and their interaction with technological advances, expressed in the equations previously obtained in steady state, which ultimately lead to changes in the distribution of income and the allocation of resources.

Examining the effect of a shift in the relative importance of the routine sector on the key variables, $\left (\frac{Q_{N}^{i}}{Q_{R}^{i}}\right )^{*}$ and $\left (\frac{Q_{N}^{h}}{Q_{R}^{i}}\right )^{*}$ , a rise in this parameter conducts, in the first place, a decrease in the relative importance of the nonroutine sector and, results, subsequently, in an improvement in technological knowledge oriented toward tasks performed by medium-skilled domestic workers specialized in routine tasks.Footnote ²⁴ In particular, a positive shift in the importance of the routine sector by $0.0348pp$ causes a bias of technological knowledge in favor of routine tasks performed by domestic medium-skilled workers, compared to nonroutine tasks carried out by high-skilled domestic workers, by $26.026\%$ —see Figure 2a. Similarly, the same fluctuation biases technological knowledge in favor of routine tasks undertaken by domestic medium-skilled workers, compared to nonroutine tasks performed by low-skilled domestic workers, by $24.825\%$ —see Figure 2b.

Figure 2. Effect of variations in the routine sector relevance, $\chi _{R}$ , on the inter-sector technological-knowledge gap biased in favor of tasks produced by domestic (nonroutine) high- and low-skilled workers, $\left(\frac{Q_{N}^{i}}{Q_{R}^{i}}\right)^{*},$ and $\left(\frac{Q_{N}^{h}}{Q_{R}^{i}}\right)^{*}$ .

Furthermore, variations in both automation, experienced through changes in the amount of domestic medium-skilled labor and its productivity in carrying out routine tasks, and offshoring, perceived as fluctuations in the supply of foreign medium-skilled labor and its productivity in carrying out relocated routine tasks, conduct significant transformations in technological progress. These modifications in turn lead to a transformation of the intermediate goods used by workers, causing changes in the competitiveness of each task and sector. As a result, these factors have a notable impact on economic growthFootnote ²⁵ and wages variations.

Thus, an upward shift in the amount of the labor performed by both domestic and foreign (routine) medium-skilled employees on the key variables, $\big (\frac{w_{N}^{i}}{w_{R}^{i}}\big )^{*}$ and $\big (\frac{w_{N}^{h}}{w_{R}^{i}}\big )^{*}$ , induce a rise in the wage polarization in favor of domestic (nonroutine) high- and low-skilled workers.Footnote ²⁶ More specifically, a positive fluctuation in the domestic and foreign (routine) medium-skilled labor amount by $0.2648pp$ and $0.9996pp$ generates a wage polarization increase toward domestic (nonroutine) high-skilled workers of $49.269\%$ and $3.8269\%$ , respectively—see Figure 3a and b. Along the same lines, the same variations induce a wage polarization rise in favor of domestic (nonroutine) low-skilled employees of $49.244\%$ and $3.8260\%$ , sequentially—see Figure 4a and b. Regarding the absolute productivity of domestic and foreign (routine) medium-skilled employees in both key variables, these changes lead, respectively, to a decline and a rise in the wage polarization toward domestic (nonroutine) high- and low-skilled laborers.Footnote ²⁷ In particular, a positive change in the absolute productivity of domestic and foreign (routine) medium-skilled workers of $25.687pp$ and $10.667pp$ , respectively, causes a decrease and an increase of $5.920\%$ and $20.420\%$ in the wage polarization in the upper tail, that is, in the relative domestic (nonroutine) high-skilled workers—see Figure 3a and b. Additionally, a similar shift induces, respectively, a drop and growth of $5.918\%$ and $16.419\%$ in the wage polarization toward domestic (nonroutine) low-skilled employees—see Figures 4a and b.

Figure 3. Effect of variations in the exogenous variables that capture automation, $L_{N}^{i}$ and $l_{N}^{i}$ , and reallocations, $L_{N}^{h}$ and $l_{N}^{h}$ , on the wage polarization in favor of domestic (nonroutine) high-skilled workers, $\left(\frac{w_{N}^{i}}{w_{R}^{i}}\right)^{*}$ .

Figure 4. Effect of variations in the exogenous variables that capture automation, $L_{N}^{i}$ and $l_{N}^{i}$ , and reallocations, $L_{N}^{h}$ and $l_{N}^{h}$ , on the wage polarization in favor of domestic (nonroutine) low-skilled workers, $\left(\frac{w_{N}^{h}}{w_{R}^{i}}\right)^{*}$ .

In summary, as we have noticed, the “race” between automation and offshoring is won by automation if there are positive shocks in the supply of domestic and foreign medium-skilled labor. In other words, the earnings of high- and low-skilled employees are more susceptible to shifts in response to automation. In turn, the “race” between automation and offshoring is won by offshoring if there are improvements in the absolute productivity of domestic and foreign medium-skilled workers. In simpler terms, when the variables associated with offshoring are affected by fluctuations, both high- and low-skilled workers are more prone to experience shifts in their gains.