2. A general equilibrium model of vertical specialization

Consider a small open economy that produces two final goods, denoted by $X$ and $Y$ . Good $X$ uses capital and intermediate good $M$ as inputs with Cobb–Douglas production technology $X=AK^\alpha M^{1-\alpha }$ . Good $Y$ uses unskilled labor and skilled labor as inputs with production technology $Y=S_Y^\beta L_Y^{1-\beta }$ , where $S_Y$ is the usage of skilled labor and $L_Y$ denotes the unskilled labor employed by sector $Y$ . Sector $M$ uses a continuum of services or components $Z=[0,1]$ to produce the intermediate good with costless assembly technology. Finally, let service $z$ be produced both at home country and abroad using skilled labor with Ricardian production technology. In particular, let Ricardian unit labor demand be $a_S(z), \forall z\in Z$ , where $a_S(1)=1$ . For any $z\in Z$ , let $\delta (z)\equiv a^*_S(z)/ a_S(z)$ , where an asterisk denotes foreign variables in the remainder of the paper. We assume that $\delta '(z)\lt 0, \forall z\in Z$ . Define $\tilde{z}\in Z$ such that $\delta (\tilde{z})=w_S/w^*_S$ , where $w_S$ denotes skilled wage rate. Therefore, all $z\in [0,\tilde{z}]$ will be produces at home and all $z\in (\tilde{z}, 1]$ will be produced in the rest of the world. Then, given our setup, the total skilled labor whose service will be assembled in $M$ , denoted by $S_M$ , can be given by

(1) \begin{equation} S_M=M\int _0^{\tilde{z}}a_S(z)dz \end{equation}

We maintain full employment of labor, implying that $L_Y=\bar{L}$ and:

(2) \begin{equation} S_M+S_Y=\bar{S} \end{equation}

where $\bar{L}$ and $\bar{S}$ are constant endowments of unskilled and skilled labor, respectively. The equilibrium price of intermediate good $M$ is given by

(3) \begin{equation} p_m(\tilde{z})=B(\tilde{z})w_S+[1-B(\tilde{z})]w_S^* \end{equation}

where $B(\tilde{z})\equiv \int _0^{\tilde{z}} a(z)dz$ is the share of home-made components $z\in Z$ in $M$ .Footnote ⁵ By our small open economy assumption, prices of $X$ and $Y$ , denoted by $p_x$ and $p_y$ , as well as $w_s^*$ are all given. Hence, $w_s$ and $w$ will be determined with $p_x=p_y=w_s^*=1$ by appropriate choice of units. Note also that, following the definition of $\delta$ and equation (3), we have $p_m(\tilde{z})\lt 1$ .Footnote ⁶

Profit maximization implies the demand for intermediate as $M=[(1-\alpha )X/p_m(\tilde{z})$ . Hence, equilibrium output of $X$ for any amount of capital can be obtained as

(4) \begin{equation} X=\left (A (1-\alpha )^{1-\alpha }\right )^{\frac{1}{\alpha }}\left (\frac{1}{p_m(\tilde{z})}\right )^{\frac{1-\alpha }{\alpha }} K \end{equation}

Using the above derived demand for $M$ and equation (4), we can obtain instantaneous equilibrium quantity of $M$ for any given level of capital as

(5) \begin{equation} M=\tilde{A} K\tilde{p}_m^{-\frac{1}{\alpha }} \end{equation}

where $\tilde{A}\equiv [A (1-\alpha )]^{1/\alpha }$ and $\tilde{p}_m\equiv p_m(\tilde{z})$ for notational simplicity. Therefore, it follows from equations (1) and (5) that the equilibrium level of employment for skilled labor, used in production of domestic components $[0, \tilde{z}]$ for any given level of capital, can be given as

(6) \begin{equation} S_M(w_s)=\frac{\tilde{A}KB(\tilde{z})}{\left (B(\tilde{z})w_s+[1-B(\tilde{z})]\right )^{\frac{1}{\alpha }}} \end{equation}

Differentiating equation (6), it can be shown that:

\begin{equation*} \frac {\partial S_M}{\partial w_s}=\tilde {A}K \frac {B'(\tilde {z})\frac {d\tilde {z}}{dw_s}-\frac {1}{\alpha }[B(\tilde {z})]^2[\tilde {p}_m]^{-1}}{p_m^{\frac {1}{\alpha }}} \end{equation*}

where we have used $\left (d [B(.)w_s+(1-B^*(.))]/d\tilde{z}\right )\left (d\tilde{z}/dw_s\right )=0$ , since $d [B(.)w_s+(1-B^*(.))/$ $d\tilde{z}=0$ due to the envelope theorem. Recall also that $B'(.)\gt 0$ and $d\tilde{z}/dw_s\lt 0$ . Hence, we conclude that $\partial S_m/\partial w_s\lt 0$ .

Next, consider sector $Y$ . Equilibrium in this sector implies:

(7) \begin{equation} S_Y(w_s, \bar{L})=\left (\frac{w_s}{\beta \bar{L}^{1-\beta }}\right )^{\frac{1}{\beta -1}} \end{equation}

where $\partial S_Y/\partial w_s\lt 0$ . Therefore, the market clearing condition for skilled labor can be rewritten as

(8) \begin{equation} S_M(w_s)+S_Y(w_s, \bar{L})=\bar{S} \end{equation}

which determines equilibrium $w_s$ , for any given level of $K$ , hence sectoral skilled-labor demand will be determined. Then, unskilled-labor market clearing condition determines unskilled wage, that is, $w=(1-\beta )[S_Y(w_s)]^\beta/ \bar{L}^\beta$ . Moreover, skilled–unskilled wage gap is given by

(9) \begin{equation} \frac{w_s}{w}=\frac{\beta \bar{L}}{(1-\beta )S_Y(w_s,.)} \end{equation}

implying that any change that leads to a decrease in demand for skilled labor in sector $Y$ will increase the skilled–unskilled wage gap. Clearly, capital accumulation has consequences on skilled–unskilled wage gap and on inequality. Particularly, equations (6) and (8) imply that an increase in capital will increase (decrease) the demand for skilled labor whose services are used in sector $M$ ( $Y$ ). Hence, we have the following result.

Thus, it is crucial to study capital accumulation. We shall consider this in the next section.

3. Economic growth and Inequality

Our model and its analysis in the previous section is for any given capital level. We shall now allow capital to be endogenously determined and grow over time for any given initial value. Let the representative consumer’s utility function be given by $u=u(c_t)$ , where $c_t$ denotes the consumption of Hicksian composite good at time $t$ and all neoclassical assumptions are maintained. Moreover, we also assume throughout the rest of the paper that $u$ exhibits constant inter-temporal elasticity of substitution. Our dynamic optimization problem can be written as

\begin{eqnarray*} \max _{\{c_t\}_{t=0}^\infty } \sum _{t=0}^\infty \xi ^t u(c_t)\\ s.t. \quad K_{t+1}-K_{t}=\frac{\tilde{A}K_t}{\phi (\bar{S}, K_t) }-c_t\\ K_0=\bar{K} \end{eqnarray*}

where $\phi (\bar{S}, K_t)\equiv (1-\alpha )\tilde{p}_m^{(1-\alpha )/\alpha }$ and $\xi =1/(1+\rho )$ is the discount factor and $\rho \gt 0$ is the discount rate. Recall that at a temporal equilibrium $w_s$ depends on $\bar{S}$ and $K_t$ , implying that $\tilde{p}_m$ also depends on $\bar{S}$ and $K_t$ . Bellman equation for this dynamic programming problem can be written as

(10) \begin{equation} v(K_t)=\max _{c_t} \{u(c_t)+\xi v(K_{t+1})\}+\lambda _t \left [\frac{\tilde{A}K_t}{\phi (\bar{S}, K_t) }-c_t-(K_{t+1}-K_{t})\right ]. \end{equation}

The first-order conditions for this problem can then be obtained as

(11) \begin{eqnarray} u'(c_t)=\lambda _t \end{eqnarray}

(12) \begin{eqnarray} \xi v'(K_{t+1})=\lambda _t \end{eqnarray}

(13) \begin{eqnarray} v'(K_t)=\lambda _t\left (\frac{\tilde{A}}{\phi }-\frac{K_t}{\phi ^2}\frac{\partial \phi }{\partial K_t}+1\right ) \end{eqnarray}

Rewrite equation (13) for $t+1$ , to get:

\begin{equation*} \xi v'(K_{t+1})=\xi \lambda _{t+1}\left (\frac {\tilde {A}}{\phi }\left [1-\frac {\varepsilon _{p_mk}(1-\alpha )}{\alpha }\right ]+1\right ) \end{equation*}

where $\varepsilon _{p_mk}$ is the elasticity of $p_m$ with respect to capital. It directly follows from equation (5) that $\varepsilon _{p_mk}=\alpha$ . Hence, by using equation (12), we can rewrite the above equation as

(14) \begin{equation} \frac{\alpha \tilde{A}}{(1-\alpha ) \tilde{p}_m^{\frac{1-\alpha }{\alpha }} }+1=(1+\rho )\frac{\lambda _t}{\lambda _{t+1}} \end{equation}

Then, given the constant intertemporal elasticity of substitution, it follows from equations (11) and (14) that:

(15) \begin{equation} \frac{\alpha \tilde{A}}{(1-\alpha ) \tilde{p}_m^{\frac{1-\alpha }{\alpha }} }+1=(1+\rho )(1+g)^\sigma \end{equation}

where $\sigma \equiv u''(.)/u'(.)$ is the (constant) inter-temporal elasticity of substitution and $g$ is the growth rate. Recall that $u'(c^*(k_t))/u'(c^*(k_{t+1}))=[c^*(k_{t+1}/c*(k_t))]^{\sigma }=(1+g)^\sigma$ , where $k$ is capital-skilled labor ratio. By solving equation (15), we obtain:

(16) \begin{equation} g(t)=\Delta \left [1+\frac{\alpha \tilde{A}}{(1-\alpha ) \tilde{p}_m^{\frac{1-\alpha }{\alpha }}}\right ]^{\frac{1}{\sigma }}-1 \end{equation}

where $\Delta \equiv [1/(1+\rho )]^{1/\sigma }$ . Recall that $\phi$ is monotonically increasing in $p_m$ . Hence, we have the following result.

To see this more clearly, let us consider an example where $u(c_t)=\ln c_t$ , that is, $\sigma =1$ . Then, it follows form equation (15) that approximately $g(t)\approx \alpha \tilde{A}/[(1-\alpha )\tilde{p}_m^{1-\alpha/\alpha }]-\rho$ . Therefore, growth rate crucially depends on the price of intermediate. Vertical specialization and trade opening in components will lower this price, resulting in higher temporal growth rate.

Now, turning to the catch-up hypothesis, we need to establish whether our small open developing economy experiences a greater reduction in price of the intermediate good as a result of vertical specialization and trade in components. To do this, first we have to derive autarky equilibrium price of $M$ , denoted by $p_m^a$ . It is evident from (3) that $p_m^a=w_S$ since $B(1)=1$ . Hence, we have to evaluate the change in skilled wage due to trade opening. Using $p^a_m=w$ and equations (6)–(8) as well as their equivalence in the rest of the world, we can show that at autarky we have:

(17) \begin{equation} \frac{\tilde{A}}{w^a_S}\bar{k}_S+\left (\frac{\beta }{w^a_S}\right )^{\frac{1}{1-\beta }}\bar{l}_S=1 \end{equation}

where $\bar{k}_s\equiv \bar{K}/\bar{S}$ and $\bar{l}_s\equiv \bar{L}/\bar{S}$ . Using equation (17) and its equivalence for rest of the world, recalling that $w_s^*=1$ , we obtain:

(18) \begin{equation} \frac{\tilde{A}}{w^a_S}\bar{k}_S+\left (\frac{\beta }{w^a_S}\right )^{\frac{1}{1-\beta }}\bar{l}_S= \tilde{A}\bar{k}_S^*+\beta ^{\frac{1}{1-\beta }}\bar{l}_S^* \end{equation}

Thus, it follows from equation (18) that $w_S^a\gt w^*_S=1$ if $\bar{k}_S\gt \bar{k}_S^*$ and $\bar{l}_S\gt \bar{l}_S^*$ . Hence, we have the following result.

The condition of the above proposition is sufficient for skilled wage to be higher in the home economy. However, it is not a necessary condition. The necessary condition is that one of these skilled labor intensity conditions to be met. As it is evident from equation (18), for $w_S\gt w_S^*$ under autarky, it must be the case that $\bar{k}_S\gt \bar{k}_S^*$ or $\bar{l}_S\gt \bar{l}_S^*$ . That is, killed labor must be scarce at home at least relative of the other primary production factors.

A crucial corollary to Proposition 3 follows from equation (3): $p_m^a\gt p_m^{a*}$ if home country is skilled labor scarce both relative to capital and unskilled labor. Suppose this sufficient condition is met. Then, home country will experience a bigger price drop for the intermediate good as a result of vertical specialization, global production fragmentation, and trade in fragments. That is, $dp_m=p_m^a-p_m(\tilde{z})\gt p_m^{a*}-p_m(\tilde{z})$ if home country is skilled-labor scarce both relative to capital and unskilled labor. This, in turn, implies from equation (16) that $g(t)\gt g^*(t)$ if home country is skilled labor scarce both relative to capital and unskilled labor. Hence, we have the following formal result that addresses the catch-up hypothesis.

This result is compatible with the observation of cross-country growth convergence (e.g., see Bal). A small developing country with skilled-labor scarcity grows faster than the developed world so that ultimately the cross-country per capita income converges.

We have already established in preceding section that a higher level of capital will increase the skilled–unskilled wage gap. Hence, this and Proposition 2 conclude the following important result on skill-unskilled wage inequality.

Intuition of this result deserves attention. Following proposition 2, vertical specialization and trade in components will increase growth rate, hence raises the capital level. As K accumulates and S does not expand at that rate, $w_s/w$ will go up in each group of countries. Again, this explains within country divergence that we observe from empirical evidence. This result contributes to the literature on rising inequality (e.g., Autor and Murnane, Reference Autor, D. and Murnane2003; Acemoglu and Restrepo, Reference Acemoglu and Restrepo2018).

While the effect of global production fragmentation and vertical specialization on wage inequality is generally unambiguous, its extent differs between home country and the rest of the world. This follows from the implication of catch-up hypothesis in our setup. The following result highlights the differential effects of vertical specialization on inequality.

As a final note, it must be emphasized that the results of our paper are robust to many alterations of the simple model we develop in the paper. Everything hinges on the price of $M$ , where a lower $p_m$ via trade increases the marginal product of capital. As long as trade in intermediates leads to a decline in the world price of $M$ , affecting the marginal product of capital, trade will lead to growth. One could bring in unskilled labor in production of $M$ and define comparative advantage appropriately to generate this result.

4. Parametrization

In this section, we outline the calibration strategy for our general equilibrium model, discussed in previous sections. The primary aim of this calibration is to serve as a heuristic tool for visualizing theoretical outcomes and gaining intuitive insights into the relationships and phenomena under study.

It is important to note that our approach is more qualitative in nature and not geared toward precise parametric estimation. To guide our calibration, we adhere to established conventions in the literature, sourcing parameter choices from existing research to maintain consistency with mainstream approaches. The chosen parameters are consistent with “less developed” country relative to the rest of the world which resonates the theoritical framework that we are proposing.

The most crucial element of our model is the labor demand in both the home country and the rest of the world, which represents the trade partner in our model, for each unit of good $M$ . To discipline $a_s(z)$ and $a_s^*(z)$ , we follow Eaton and Kortum (Reference Eaton and Kortum2002). In a multicountry context, they formulated $Z_i(j)$ as the amount of good $j$ that a bundle of inputs can produce in country $i$ . In our framework, this corresponds to the inverse of labor demand for each unit of good $M$ , expressed as $\dfrac{1}{a_s(z)}$ . We posit that the productivity distribution for each good $j$ in country i follows a Frechet distribution:

\begin{equation*} Pr(Z_i(j) \leq z) = F_i(z) = e^{-T_i z^{-\theta }} \end{equation*}

with $T_i \gt 0$ governing the location of the distribution or the state of the technology in $i$ , meaning that a higher $T_i$ implies a higher efficiency draw for good $j$ . We choose this parameter to capture the distinctions between the developed (rest of the world) and the developing country under focus. Specifically, we assign a smaller $T$ for the home country and a larger $T$ for the rest of the world. Parameter $\theta$ is responsible for dictating the variability in productivity; a higher $\theta$ leads to lower variability. Consistent with existing literature, we set a common $\theta$ for both locations. Consequently, we have two Frechet distributions that generate $a_s(z)$ and $a_s^*(z)$ for home country and the rest of the world, respectively.

Another set of parameters that we have in the model are $\overline{S}$ and $\overline{L}$ which represents the skilled-labor supply and unskilled-labor supply. The important aspect for our analysis is, in fact, the relative size of labor supply. therefore, we normalized the skilled-labor supply to 1 and set the unskilled-labor supply to 2.3 which is the average ratio of this ratio for low-income countries using world bank dataset.Footnote ⁷ Table 1 summarizes parameter choices employed in our model.

Table 1. Parameter values

Note: This table summarizes values of parameters with brief descriptions.

5. Exploratory experiments: from theory to visualization

We conducted a series of computational experiments to visualize the theoretical predictions made by our model. These experiments should be viewed as illustrative, providing qualitative insights into the effects of global production fragmentation, vertical specialization, and trade on economic growth and wage inequality.

Figure 1. Skilled–unskilled wage gap.

Note: This figure illustrates an expanding wage gap between skilled and unskilled labor as the level of capital investment rises, which is consistent with the outcomes presented in Proposition 1.

The first result in Section 2 reveals that, within the context of our small open economy, an increase in capital correlates with a widening skilled–unskilled wage gap. To visualize this relationship, we conducted model simulations over varying levels of capital, capturing the respective values of skilled wage, $\omega _s$ , and unskilled wage, $\omega$ . Figure 1 depicts the ratio of skilled to unskilled wages across these capital levels. As illustrated, the wage gap—measured as the ratio of $\dfrac{\omega _s}{\omega }$ – expands as capital increases.

Another key finding presented in Section 3 is that the vertical specialization and opening of international trade in components, along with the global expansion of supply chains, contribute to an increase in the temporal growth rate $g(t)$ . This is mediated through their impact on the price of the intermediate good $M$ . Specifically, vertical specialization and trade liberalization lead to a decline in the price $\tilde{p}_m$ , which in turn boosts the temporal growth rate both at the time of opening and in subsequent periods, compared to a no-trade scenario. To illustrate this inverse relationship between $\tilde{p}_m$ and $g(t)$ , we simulate our model across a range of $\tilde{p}_m$ values to calculate the corresponding growth rates. Figure 2 highlights this inverse relation, showing that an increase in the price of intermediate goods results in a reduced temporal growth rate. Consequently, any period in which the economy opens up to trade—and thus experiences a decrease in $\tilde{p}_m$ —will see a surge in temporal growth rate.

Figure 2. Temporal growth rate and price of intermediate good.

Note: The figure demonstrates the inverse correlation between the price of the intermediate good $\tilde{p}_m$ , which declines as the country engage in vertical specialization and trade, and the temporal growth rate, $g(t)$ .

A further set of findings elaborated in the previous section underscores the influence of skilled labor scarcity on a country’s response to vertical specialization and trade. To illustrate these outcomes, we simulate our model for two countries with identical levels of capital, unskilled-labor supply, and technology, but divergent levels of skilled-labor supply. Specifically, one of those countries has a skilled-labor supply twice as large as the other. The country with the smaller skilled labor supply serves as a proxy for higher skilled labor scarcity, relative to its counterpart. In the accompanying graphs, this country is labeled as ’Low-Skilled Labor Supply’ and is represented by a dashed blue line. Conversely, the country with the larger skilled labor supply signifies lower skilled labor scarcity and is labeled as “High-Skilled Labor Supply,” depicted by a solid red line in the graphs.

Figure 3. Autarky skilled-labor wage.

Note: The figure demonstrates that at autarky, the skilled labor wage is higher for the country which is more skilled-labor scare both relative to capital and unskilled labor.

Figure 3 displays the skilled-labor wages, $\omega _s$ in autarky—that is, in the absence of trade—across a range of capital levels for two countries, with a high-skilled labor supply and a low-skilled abor supply. As evident from the graph, the skilled labor wage is higher in the country with relative skilled labor scarcity. It is important to note that this scarcity is relative to both capital and unskilled labor by its setup.

Figure 4. Influence of skilled-labor supply on trade openness.

Note: Panel A compares the price of the intermediate good $M$ under autarky across a spectrum of capital levels for both high and low skilled-labor supply countries. The purple line illustrates how this price evolves when the country engages in trade. Panel B follows a similar structure, but focuses on representing the temporal growth rate across the same range of capital for both types of countries.

Skilled-labor intensity is one of the significant factors when considering the impact of trade and vertical specialization on an economy. In the absence of trade, that is, autarky, higher skilled-labor scarcity results in a relatively higher price for intermediate goods. Consequently, when economies open up to trade, the country with lower skilled-labor supply experiences a more substantial decline in the price of intermediate goods compared to its higher skilled-labor supply counterpart. This is demonstrated in Panel A of Figure 4, which compares $\tilde{p}_m$ under autarky and post-trade conditions for both types of countries. The gap between autarky and trade-induced prices is wider for the country with skilled-labor scarcity. This suggests that upon opening up to trade, such a country will experience a more pronounced increase in its temporal growth rate, as illustrated in Panel B of Figure 4. In essence, with vertical specialization and the global expansion of supply chains, a small economy with skilled-labor scarcity—relative to both capital and unskilled labor—will grow faster than the rest of the world.

Figure 5. Influence of skilled-labor supply on skilled-unskilled wage inequality.

Note: This Figure illustrates the evolution of the skilled–unskilled wage gap as capital increases, contrasting cases with high and low skilled-labor supply.

As previously outlined and illustrated in Figure 1, the skilled–unskilled wage gap expands as capital increases. Building on the discussion of the impact of skilled-labor scarcity, we show in section 3 that capital accumulation has a more pronounced effect on this wage gap, which is quantified as the ratio of skilled to unskilled-labor wages in a skilled-labor scarce country. Figure 5 contrasts the evolution of this wage gap as capital grows, comparing countries with high and low skilled-labor supply. The figure reveals that the skilled–unskilled wage inequality intensifies more significantly in the country experiencing skilled-labor scarcity as its capital stock expands.