## Abstract

Trade in intermediates, also known as unbundling of production, and trade in capital have become increasingly important in the world economy during the last 25 years. To jointly study these two phenomena, we develop a dynamic, factor-proportions model with trade in final goods, intermediates and capital where countries differ in their aggregate productivity levels (TFP). Our central result is to identify a novel channel whereby trade in intermediates generates a reallocation of capital across countries that exacerbates world inequality in both income and welfare. With unbundling, high-productivity countries sort into the production of capital-intensive intermediates. They increase their capital stock (via capital imports and accumulation), and, ultimately, their real wages. This exacerbates initial productivity differences across countries and increases world income inequality. We also show that income inequality rises with unbundling (i) even in the case of ex-ante identical countries (symmetry breaking), (ii) when emerging countries start participating in trade in intermediates and (iii) when a labor-saving technology (computerization) is introduced. For an empirically-motivated model parametrization, middle-income countries experience the largest output decline with unbundling.

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## Notes

We can reject the null of identical dispersion at a 5% level using the one-sided F-test for the ratio of variances. We find similar results if we define final goods as bundles of intermediates using direct requirements input-output tables (see Table A.1 in the online appendix).

Table B.1 (in the online appendix) provides evidence consistent with this pattern of specialization in intermediates. Moreover, it shows that it is quantitatively important. We find that, if the productivity of a country moved from the 25th to the 75th percentile, the rise in the value of exports of intermediates in the 75th percentile of capital intensity would be 40% larger than the increase in the 25th percentile. See also, among others, Baxter and Kouparitsas (2003), Hanson (2012) and Schott (2003a, b, 2004).

Online Appendix C provides an exact microfoundation that delivers this result as an endogenous outcome.

The online supplemental material A (available at the authors’ websites) considers the case in which only a fraction of intermediates can be traded.

Even though output inequality emerges in equilibrium, consumption (and welfare) remains the same across countries. The reason is that the asymmetric allocation of capital does not change factor prices. Thus, total factor payments remain the same in all countries.

We note that for the case in which countries have identical productivities, if we eliminated the varieties stage of production, we would have a standard (dynamic) Heckscher-Ohlin model. In this case, we can also derive the unique steady-state world income distribution (using the symmetry breaking equilibrium refinement). This is in contrast with the indeterminacy of equilibria in the case of a finite number of traded goods (e.g., Bajona and Kehoe 2010). We focus on the case of trade in intermediates because they are more heterogeneous in capital intensity.

In terms of techniques, the characterization of the unbundling equilibrium relies in solving for the equilibrium assignment in a similar manner to Matsuyama (2013). Our solution differs from the Ricardo-Roy assignment models as used in Costinot and Vogel (2010) and Grossman and Helpma (2014) because our production functions are not linear and factors of production are homogeneous.

The online appendix is available from the journal’s or authors’ websites.

The microfoundation is based on Jones (1995). There exists an innovation sector that produces varieties using the final good. Innovators sell patents to the producer of varieties and extract all the surplus of the producer, who has monopoly rights on the production of the variety. With this microfoundation, the Armington assumption is not needed as firms endogenously choose to produce different products. We also note that this particular microfoundation for \(\mu _{j}\) plays a simplifying role. However, the mechanism driving our results is independent of this microfoundation. We could carry \(\{ \mu _{j}\}_{j=1}^J\) as additional parameters or simply set \( \mu _j=1\) for all

*j*and obtain the same qualitative results.This formulation has been used, among others, by Dornbusch et al. (1980) in the context of Heckscher-Ohlin models.

This assumption is made to simplify the exposition of the results. In the next section, we show that it corresponds to the steady-state of the equilibrium without unbundling. Importantly, our results on changes of relative GDP per capita and welfare do not depend on this assumption. See also footnote 18.

If countries had different labor endowments, we would obtain the same qualitative results. In this case, all variables would need to be interpreted in per capita terms. In particular, \(\theta _{j}\) would be the number of varieties per capita (i.e., \(\mu _{j}=\kappa \theta _{j}L_{j})\). Under this assumption, we can compute the labor market clearing condition following the same steps as in the main text and check that the same exacerbation results hold (even though the exact value of the thresholds governing the endogenous selection of intermediates in the unbundling equilibrium would change). For example, we can use an analogous argument to the one below to show that unbundling increases real wage differences.

In Section A in the online supplemental material (available at the authors’ websites), we consider the intermediate case in which only a fraction \(\alpha \) of the intermediates is traded and perform comparative statics on \(\alpha \).

These results are a direct application of the Ramsey model, see Section A in Caselli and Ventura (2000). Note that \(V_{j}=\int _{0}^{\infty }\frac{1}{1-\gamma }c_{j}(t)^{1-\gamma }e^{-\rho t}dt=\int _{0}^{\infty }\frac{1}{1-\gamma }\left[ c_{j}(0)e^{\int _{0}^{t}\frac{r(v)-\rho }{\gamma }dv}\right] ^{1-\gamma }e^{-\rho t}dt\). Thus, \(\frac{V_{j}}{V_{i}}=\left[ \frac{c_{j}(0)}{c_{i}(0)} \right] ^{1-\gamma }\).

If the initial asset distribution were different, the trade balance would also be zero in the steady-state because \( b_{j}^{R}=\pi _{j}\) in the steady-state (but generically would be non-zero along the transition path).

We are implicitly assuming that each intermediate is produced only by one country, which is indeed true almost everywhere in equilibrium.

In the case that \(\theta _{1}=\theta _{2}=\cdots =\theta _{J}\), this is still an equilibrium but there exist other equilibria. See Sect. 3.1.

If we had not made the assumption that each country has an initial stock of capital proportional to its productivity, capital flows would be different than zero in the equilibrium without unbundling. In this case, the results discussed in this paragraph would simply imply that high productivity countries import relatively more capital in the unbundling equilibrium.

Section B in the online supplemental material (available at the authors’ website) shows that this result does not depend on the Cobb-Douglas assumption in the production function for intermediates. We show that it extends to a general CES production function.

Top-bottom inequality is the ratio between the 90th percentile of GDP per capita (PPP) and the 10th percentile. We choose to finish in 2008 to exclude the effects of the Great Recession. If we include all available information (up to 2014), the top-bottom GDP inequality is 27.1 in the post-1990 period. The same qualitative results hold if we define top-bottom inequality as the ratio between the 95th and 5th percentile.

We discuss in more detail this refinement concept in Section F of the online supplemental material (available at the authors’ websites).

The first derivative is proportional to \(j^{-1/2}-(j-1)^{-1/2}\), which is negative for \(j>1\). The second derivative is proportional to \( -j^{-3/2}+(j-1)^{-3/2}\), which is positive for \(j>1\).

World output with and without unbundling is \(Y^{\text {World}}=2\theta \left( BJ\right) ^{ \frac{1}{2}}\).

With two ex-ante identical countries

*H*and*F*, the threshold equilibrium is \(z_{1}=1-\sqrt{1/2}\). It is readily verified that this equilibrium implies that the relative world production share of country*H*becomes\(\left( \frac{s_{H}^{y}}{s_{F}^{y}}\right) ^{with}=\frac{ Z_{H}}{Z_{F}}=\frac{1-z_{1}}{z_{1}}=\frac{1}{\sqrt{2}-1}>1=\left( \frac{ s_{H}^{y}}{s_{F}^{y}}\right) ^{without}\).There exists an analogous equilibrium that corresponds to the permutation of

*H*and*F*, \(\theta _{F}=\theta _{H}+\varepsilon \), \(\varepsilon \rightarrow 0^{+}\), which corresponds to point*B*in the figure.Section A in the online supplemental material (available at the authors’ websites) shows that the symmetry breaking result also extends to the intermediate case in which only a fraction \(\alpha >0\) of intermediates are traded.

From the equilibrium assignment in the discrete case, we know that more productive countries specialize in capital-intensive (higher index

*z*) intermediates, \(z^{\prime }(j)<0\). Thus, \(\theta ^{\prime }(j)z^{\prime }(j)>0\). Rearranging (16),we find that*z*(*j*) is convex, as \( z^{\prime \prime }(j)=(1-z(j))^{-1}(\theta ^{\prime }(j)z^{\prime }(j)/\theta (j)+z^{\prime 2}(j))>0\).This fit is better than a “power law” where we regress the log of TFP or country GDP per capita on the log of the country ranking. In this case, the log-log regression yields an \(R^{2}\) of 0.86 and 0.69, respectively, see Table I.1 in the online appendix for the estimation results. We can also compute the solution of the differential equation for the power-law case, i.e., \(\theta (j) \propto j^{-\upsilon }\) for some \(\upsilon >0\). In this case, the solution is not more tractable than with an exponential. The choice of 1988 is given by our data source, Hall and Jones (1999), which report TFP data for this year. We choose this source because it roughly coincides with the increase in trade in intermediates documented in Fig. 1a.

We would obtain the same differential equation if we allowed the constant in equation (17) to be different from the exponential decay \(\lambda \), i.e., \(\theta (j)={\tilde{\lambda }} e^{-\lambda j}\). Using \({\tilde{\lambda }} =\lambda \) allows us to economize on notation in the computation of the Lorenz curves below.

To see this, rewrite the Lorenz curves in terms of the assignment

*j*(*z*), (18), so that \(L(z)^{without}=ze^{1-z}\) and \( L(z)^{with}=z\), and the result follows.In this case, the ratio \(s^{y,\text {with}}(j)/s^{y,\text {without}}(j)\) is a monotonically decreasing function, positive for

*j*lower than a threshold and negative thereafter. This means that there is a strict ranking in the percentage increase of world output share, being highest for the most productive country and decreasing thereafter.For the case of a discrete number of countries, denoting by \(\eta _{\omega }\) the amount of varieties produced by southern countries (i.e., \(\eta _{\omega }\)\(=\sum _{\omega ={\underline{j}}}^{J}\mu _{\omega })\), the trade balance of northern countries

*j*becomes$$\begin{aligned} \frac{\mu _{\omega }}{N}\left( Y-y_{\omega }\right) +Z_{\omega }\frac{N-\eta _{\omega }-\mu _{\omega }}{N} Y=\frac{N-\mu _{\omega }}{N}y_{\omega }+(1-Z_{\omega })\frac{\mu _{\omega }}{N}Y. \end{aligned}$$Rearranging, \(s_{\omega }^{before}=Z_{\omega }^{before}\left( 1-\frac{\eta _{\omega }}{N} \right) \). Taking the limit to a continuum of countries, the expression becomes (20).

This condition has been applied in other economic contexts, see Hopkins and Kornienko (2004) and the references therein. The normal, uniform and exponential distribution among other distributions satisfy this condition.

Moreover, they also show that a Monotone Likelihood Ratio (MLRP) order implies MPR. Thus, MPR is more stringent than first-order stochastic dominance but less stringent than MLRP.

Note that if \(\beta (z)=1,\) we obtain that \(j(z)=\lambda ^{-1}(z-\ln z-1)\) as in the baseline model. Also, note that, for simplicity, we are reporting the case in which the support of intermediates remains [0, 1]. Online Appendix H.2 discusses the case when \(\beta (z)\) takes the form of (21), in which the support changes with computerization.

Note that \(j(z,\chi )\) increases monotonically with an increase in \({\mathcal {I}} (z,\chi )\).

Using that \(\theta (j)\) follows an exponential decay (equation (17)), the threshold \( {\overline{j}}\) is \({\overline{j}}=\frac{\ln \left( \frac{\lambda _{1}}{\lambda _{2}}\right) }{\lambda _{1}-\lambda _{2}}\).

See online appendix H.3 for a derivation and further characterization of the change in the world output distribution.

Online Appendix E derives these aggregate production functions. We show that \(f(\theta _{j})\equiv 2\theta _{j}\), \(\alpha =1/2\), \(g(\theta _{j})\equiv \theta _{j}\left[ \exp \int _{z\in {\mathcal {Z}}_{j}} \ln \left[ \frac{\Delta _{j}^{1-z}}{\left( Z_{j}-\Delta _{j}\right) ^{z}}\right] dz \right] \) and \(\alpha _j\equiv Z_{j}-\Delta _j\). We also show that \(\alpha _j\) is increasing with the productivity of the country, which directly follows from Proposition 1.

This accounting exercise with heterogeneous capital shares has recently been done by Feenstra et al. (2015), who document sizable differences from traditional TFP accounting. For example, the fraction of the variance of GDP explained by the variance of inputs (in 2005) sharply increases when allowing for heterogeneous shares from 0.25 to 0.34.

We obtain similar results if we use long differences of 5-years or if we use the initial TFP level in 1990 as a regressor instead of \(\log (TFP)_{ct}\).

The proof is immediate and follows from the fact that our competitive equilibrium replicates the social planner allocation. If the social planner is allowed a thinner geographical allocation of inputs, aggregate output cannot decrease.

From the steady-state level condition of world capital and \({\dot{b}}_{j}(t)=0\), we can find the amount of assets hold by agents in country

*j*. However, there exists an indeterminacy in the composition of the asset portfolio of agents. This is inconsequential for our purposes given that the interest rate is equalized across countries and, thus, the composition of the asset portfolio does not affect income nor consumption.

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We thank Daron Acemoglu, Pol Antràs and Jaume Ventura for useful comments and suggestions, and Guido Lorenzoni, Kiminori Matsuyama, an associate editor and three anonymous referees for comments on earlier drafts. We have also benefited from comments in our presentations at Barcelona GSE Summer Forum, Bank of Spain, Berkeley, Calgary, CEMFI, CEPR-ESSIM, Chicago Fed Trade Conference, IMF, Nottingham, Stanford, UC3M and Universitat de Barcelona. First draft: July 2014. We thank Shekhar Tomar for his excellent research assistance. Basco acknowledges financial support from Fundación Ramón Areces. Mestieri thanks the ANR for their financial support while at TSE.

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## Appendices

### A Data Appendix: Construction of figure 1b

To construct the histogram for final goods we weight the capital share of each 6-digit NAICS by its final-use content. The final-use content is a weighted average of the different end-uses that each 6-digit NAICS has. The weight is computed as a function of the U.S. imports in 1990. Capital shares of each 6-digits NAICS are computed from the NBER CES Manufacturing database definition. The histogram for intermediates is computed in an analogous way. We use the end-use classification of Feenstra and Jensen (2012). Table A.1 in the online appendix reports additional measures of capital intensity dispersion. It also documents the same fact defining final goods as bundles of intermediates.

### B Proofs

This section presents the proofs to some of the claims in the paper. The rest can be found in the online appendix.

**Derivation of trade balance in steady state** The trade balance in the steady-state is \(TB_{j}=r(k_{j}-b_{j})\). From the Euler Equation, \(r=\rho \). Thus, from the demand of capital in country *j*, \( rk_{j}=\rho k_{j}=\left[ Z_{j}-\Delta _{j}\right] Y\). Lastly, from the evolution of asset holdings, \(b_{j}=\pi _{j}B=2\Delta _{j}B=2\Delta _{j}K=\Delta _{j}\frac{2}{2\rho }Y=\frac{\Delta _{j}}{\rho }Y\). Thus, \( TB_{j}=\left[ Z_{j}-2\Delta _{j}\right] Y\).

**Proof of claim on change in relative welfare** Note that \(\left[ \frac{c_{j}(0)}{c_{j+1}(0)}\right] ^{with}>\left[ \frac{ c_{j}(0)}{c_{j+1}(0)}\right] ^{without}\) if \(\frac{\Delta _{j}}{\Delta _{j+1} }>\frac{\mu _{j}}{\mu _{j+1}}=\frac{\theta _{j}}{\theta _{j+1}}\). This is true because, from the threshold equilibrium, equation (16), \( \frac{\Delta _{j}}{\Delta _{j+1}}=\left( \frac{\theta _{j}}{\theta _{j+1}} \right) ^{\frac{1}{1-z}}>\frac{\theta _{j}}{\theta _{j+1}}\) given that \(z<1\).

**Proof of claim on change in relative GDP** To prove this result, we use the equality \(\frac{\Delta _{j}}{\Delta _{j+1}}= \frac{Z_{j}}{Z_{j+1}}\frac{a_{j}}{a_{j+1}}\) where \(a_{j}\equiv 1-\frac{1}{2} (z_{j-1}+z_{j})\). First, notice that \(\frac{a_{j}}{a_{j+1}}<1\) since \(z_{j}\) is decreasing with *j*. Second, we know from the threshold equilibrium, equation (16), that \(\frac{\Delta _{j}}{\Delta _{j+1}}=\left( \frac{\theta _{j}}{\theta _{j+1}}\right) ^{\frac{1}{1-z}}>1\). These two statements imply that \(\frac{Z_{j}}{Z_{j+1}}>\frac{\Delta _{j}}{\Delta _{j+1} }\). This last inequality means that production inequality increases because \( \frac{Z_{j}}{Z_{j+1}}>\frac{\Delta _{j}}{\Delta _{j+1}}=\left( \frac{\theta _{j}}{\theta _{j+1}}\right) ^{\frac{1}{1-z}}>\frac{\theta _{j}}{\theta _{j+1} },\) where we have used the threshold equilibrium and that \(0<z<1\).

**Derivation of production shares as a function of the assignment of intermediates** To derive this expression note that \(s^{y,\text {without}}(z)=\lambda e^{-\lambda \left( -\frac{1+\ln \left( ze^{-z}\right) }{\lambda }\right) }=\lambda ze^{1-z}\). To express the output share with unbundling, note that \( s^{y,\text {with}}(j)=-\frac{dz}{dj}\Longleftrightarrow s^{y,\text {with}}(z)=- \frac{1}{\frac{dj}{dz}}\). Using that \(\frac{dj}{dz}=-\frac{1-z}{\lambda z},\) we have that \(s^{y,\text {with}}(z)=\frac{\lambda z}{1-z}\). The change in output share in terms of *j* is \(\Delta s_{j}^{y}=\frac{\lambda W(-\exp (-1-\lambda j))}{1+W(-\exp (-1-\lambda j))}-\lambda \exp (-\lambda j)\).

**Proof of claim that the change in world output share is negative for **\(z\in (0,{{\bar{z}}})\)**and positive for**\(z\in (\bar{z},1]\). Note that \(\Delta s^{y}(z)\) is continuous, increasing for *z*\(\in (1-3W(1/3),1]\) and decreasing otherwise. Moreover, \(\Delta s^{y}(0)=0\), \( \frac{d\Delta s^{y}}{dz}(0)<0\), \(\frac{d\Delta s^{y}}{dz}(1)=\infty \) and the result follows. We note in passing that \(\Delta s^{y}(z)\) is convex for all *z*.

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Basco, S., Mestieri, M. The world income distribution: the effects of international unbundling of production.
*J Econ Growth* **24**, 189–221 (2019). https://doi.org/10.1007/s10887-019-09164-4

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DOI: https://doi.org/10.1007/s10887-019-09164-4

### Keywords

- Trade in intermediates
- Unbundling
- International capital flows
- World income distribution
- Symmetry breaking