International trade with sequential production

Abstract

We develop a tractable general equilibrium model of international trade with firm heterogeneity and sequential production, based on the partial equilibrium model of Antràs and Chor (Econometrica 81:2127–2204, 2013). The length of supply chains is endogenous, and in the autarky equilibrium more productive final-good producers are served by longer supply chains. International trade in final goods magnifies the differences in the length of supply chains between firms, with exporting firms being served by longer supply chains, and non-exporting firms being served by shorter supply chains than in autarky. We show that, for given model parameters, the gains from trade are larger than in the canonical Melitz model with Pareto-distributed productivities, since production along a sequential supply chain in the Antràs–Chor model is subject to external increasing returns to scale.

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Notes

  1. 1.

    Given the process of ongoing globalisation, our first model prediction is in line with Wang et al. (2017), who show that over the period 2000–2014 “production length”, the number of stages in a production chain, has increased on average in a dataset comprising 44 countries and 56 sectors. A detailed discussion of this line of research can be found in WTO (2017, ch. 2). The prediction is also compatible with the result in Johnson and Noguera (2017) that for a large set of countries the ratio of value added trade to gross trade, an inverse measure of the role played by production chains in international trade, fell by ten percentage points over the period from 1970 to 2009. The second prediction of our model is in line with data presented in Figure 1 of Antràs et al. (2017), which shows for US firms a strongly positive correlation between firm sales and the number of source countries. Also in line with the prediction of our model, Spray (2017) finds, using an instrumental variables specification and firm-level data for Uganda, that firms in his data set that start exporting increase the number of their suppliers by 12% on average.

  2. 2.

    Fally and Hillberry (2018) endogenise the length of supply chains in general equilibrium in the context of a model that uses a Coasian setting for the nature of frictions. In their model, the mass of production stages is fixed, but the scope of suppliers, i.e. the mass of production stages performed by a each supplier, is determined in general equilibrium. This implies the number of suppliers in a supply chain, which is the measure of Fally and Hillberry (2018) for the length of a supply chain. In contrast to Fally and Hillberry (2018), in the Antràs–Chor model only the final-good producer is endogenous in scope, while all non-integrated suppliers engage only in a single production stage.

  3. 3.

    There are parallels of this result to Caliendo and Rossi-Hansberg (2012) and Chen (2017) who find, in models with production hierarchies, that exporting increases the number of layers in the hierarchy.

  4. 4.

    This idea is analogous to firm-level assets “that are applied in one part of the firm can also be applied in another part” (Navaretti et al. 2004); or to managers who “are able to solve a wider range of the problems their team confronts in production” (Antràs et al. 2006).

  5. 5.

    With equal labour input l(i) in all production stages, output would become \(M^{s+1}l\), while the standard CES technology without the added term would imply an output equal to \(M^{1/\alpha }l\). For other applications of our more flexible CES technology in international trade, see Ethier (1982) and Egger et al. (2015). In Antràs and Chor (2013), the returns to specialisation are equal to \((1-\alpha )/\alpha \), and our production function collapses to theirs if we set \( s=(1-\alpha )/\alpha \).

  6. 6.

    As noted above, the production function we use collapses to the standard CES function used by Antràs and Chor (2013) for \(s=(1-\alpha )/\alpha \). The condition for the existence of a non-degenerate equilibrium with multiple supply chains then becomes \(\rho <\alpha \), which is the condition for the substitutes case. Hence, a benefit of the more flexible CES function we use is that it allows us to also look at the sequential complements case in general equilibrium.

  7. 7.

    Substituting in Eq. (10) for M from Eq. (14), we get \(r(\varphi _1)/r(\varphi _2)=(\varphi _1/\varphi _2)^{(s\varepsilon +1)\rho /(1-\rho )}=(\varphi _1/\varphi _2)^{\varepsilon }\).

  8. 8.

    It is easily checked that \(k>\varepsilon \) is equivalent to \(s<1/\rho -(k+1)/k\). This condition implies our earlier constraint \(s<1/\rho -1\), which is necessary for a non-degenerate equilibrium of our model irrespective of the assumed productivity distribution.

  9. 9.

    The required steps are as follows. First, one can use Eq. (27) to substitute for \(M_x(\varphi ^*_x)\) in Eq. (26). Combining the resulting expression with Eq. (15) gives a relationship between \( M_d(\varphi ^*_x)/M_d(\varphi ^*_d)\) and \(\varphi ^*_x/\varphi ^*_d\). Substituting for \(M_d(\varphi ^*_x)/M_d(\varphi ^*_d)\) from Eq. (14) yields the desired result.

  10. 10.

    In an online supplement, we derive an expression for the cutoff ratio \(\varphi ^*_d/\varphi ^*_a\) that holds independently from the assumption made on the productivity distribution. Using simulations with commonly used parameter values, we then show that our conclusion that gains from trade are larger in the Antràs–Chor–Melitz model than in Melitz (2003) also holds under the respective assumptions of a Weibull distribution and a lognormal distribution for final-good producer productivities.

  11. 11.

    A detailed derivation can be found in Appendix A-1 of Alfaro et al. (2018) , and it is also contained in Appendix D below. Notice that in our model the length of an individual supply chain is M rather than the normalised length, 1. Unlike in Alfaro et al. (2018), we assume that suppliers all have the same productivity \(c\left( m\right) \), normalised to 1, and that there is no asymmetry \(\psi \left( m\right) \) in the marginal product of different inputs’ investment, thus \(\psi \left( m\right) =c\left( m\right) =1\) in our model.

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Acknowledgements

We thank Davin Chor, Hartmut Egger, Jens Wrona as well as participants at the European Trade Study Group Annual Conference and the Göttingen Workshop in International Economics for very helpful comments and discussions.

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Correspondence to Udo Kreickemeier.

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Appendices

Appendix A

In this Appendix, we show how to compute average profits \(\overline{\pi }\) in the open economy. We can write

$$\begin{aligned} \overline{\pi }=\overline{\pi }_{d}+\chi \overline{\pi }_{x} \end{aligned}$$

where \(\overline{\pi }_{d}\) are average profits made by all firms in their home market, and \(\overline{\pi }_{x}\) are average profits made by exporting firms in their export market, which can be calculated as

$$\begin{aligned} \overline{\pi }_{x}&=\int _{\varphi _{x}^{*}}^{\infty }\pi _{x}\left( \varphi \right) \frac{g\left( \varphi \right) }{1-G\left( \varphi _{x}^{*}\right) }\hbox {d}\varphi \\&=\varTheta \theta \beta \left( 1-\beta \right) ^{\frac{\rho }{1-\rho }}\varOmega A \left[ M_{x}\left( \varphi _{x}^{*}\right) \right] ^{\frac{s\rho }{ 1-\rho }}\left( \varphi _{x}^{*}\right) ^{\frac{\rho }{1-\rho }}-f_{x} \end{aligned}$$

Using Eqs. (27), (28), and the exporter indifference condition

$$\begin{aligned} \beta \left( 1-\beta \right) ^{\frac{\rho }{1-\rho }}\varOmega A(\varphi _{x}^{*})^{\frac{\rho }{1-\rho }}\left[ M_{d}\left( \varphi _{x}^{*}\right) \right] ^{\frac{s\rho }{1-\rho }}\left[ (1+\theta )^{\frac{ \varepsilon (1-\rho )}{\rho }}-1\right] =f_{x}, \end{aligned}$$

we can rewrite this expression as

$$\begin{aligned} \overline{\pi }_{x}&=\left[ \frac{\varTheta \theta (1+\theta )^{s\varepsilon } }{(1+\theta )^{\frac{\varepsilon (1-\rho )}{\rho }}-1}-1\right] f_{x} \\&=\varTheta \theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon }f-f_{x} \end{aligned}$$

The average domestic profits \(\overline{\pi }_{d}\) can be calculated as

$$\begin{aligned} \overline{\pi }_{d}&=\int _{\varphi _{d}^{*}}^{\infty }\pi \left( \varphi \right) \frac{g\left( \varphi \right) }{1-G\left( \varphi _{d}^{*}\right) }\hbox {d}\varphi \\&=\left( \varphi _{d}^{*}\right) ^{k}\beta \left( 1-\beta \right) ^{ \frac{\rho }{1-\rho }}\varOmega A\\&\quad \times \left\{ \int _{\varphi _{d}^{*}}^{\varphi _{x}^{*}}\left[ M_{d}\left( \varphi \right) \right] ^{\frac{s\rho }{ 1-\rho }}\varphi ^{\frac{\rho }{1-\rho }}g\left( \varphi \right) \hbox {d}\varphi +\int _{\varphi _{x}^{*}}^{\infty }\left[ M_{x}(\varphi )\right] ^{\frac{ s\rho }{1-\rho }}\varphi ^{\frac{\rho }{1-\rho }}g\left( \varphi \right) \hbox {d}\varphi \right\} -f\\&=\left( \varphi _{d}^{*}\right) ^{k}\beta \left( 1-\beta \right) ^{ \frac{\rho }{1-\rho }}\varOmega A\\&\quad \times \left\{ \begin{array}{c} \int _{\varphi _{d}^{*}}^{\infty }\left[ M_{d}\left( \varphi \right) \right] ^{\frac{s\rho }{1-\rho }}\varphi ^{\frac{\rho }{1-\rho }}g\left( \varphi \right) \hbox {d}\varphi \\ +\left[ (1+\theta )^{s\varepsilon }-1\right] \int _{\varphi _{x}^{*}}^{\infty }\left[ M_{d}(\varphi )\right] ^{\frac{s\rho }{1-\rho }}\varphi ^{ \frac{\rho }{1-\rho }}g\left( \varphi \right) \hbox {d}\varphi \end{array} \right\} -f \\&=\left( \varphi _{d}^{*}\right) ^{k}\beta \left( 1-\beta \right) ^{ \frac{\rho }{1-\rho }}\varOmega A\int _{\varphi _{d}^{*}}^{\infty }\left[ M_{d}\left( \varphi \right) \right] ^{\frac{s\rho }{1-\rho }}\varphi ^{\frac{ \rho }{1-\rho }}g\left( \varphi \right) \hbox {d}\varphi \\&\quad +\left[ (1+\theta )^{s\varepsilon }-1\right] \xi ^{-k}\left( \varphi _{x}^{*}\right) ^{k}\\&\quad \times \left\{ \beta \left( 1-\beta \right) ^{\frac{ \rho }{1-\rho }}\varOmega A\int _{\varphi _{x}^{*}}^{\infty }\left[ M_{d}(\varphi )\right] ^{\frac{s\rho }{1-\rho }}\varphi ^{\frac{\rho }{ 1-\rho }}g\left( \varphi \right) \hbox {d}\varphi \right\} -f \\&=\varTheta f+\left[ (1+\theta )^{s\varepsilon }-1\right] \xi ^{-k}\varTheta \frac{ f_{x}}{\left[ (1+\theta )^{\frac{\varepsilon (1-\rho )}{\rho }}-1\right] }-f\\&=\varTheta f+\left[ (1+\theta )^{s\varepsilon }-1\right] \xi ^{-k}\varTheta \xi ^{\varepsilon }f-f \end{aligned}$$

Substituting for \(\overline{\pi }_{d}\), \(\overline{\pi }_{x}\) and \(\chi \), we finally get

$$\begin{aligned} \overline{\pi }&=\overline{\pi }_{d}+\chi \overline{\pi }_{x} \\&=\varTheta f+\left[ (1+\theta )^{s\varepsilon }-1\right] \varTheta \xi ^{\varepsilon -k}f-f+\xi ^{-k}\left[ \varTheta \theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon }f-f_{x}\right] \\&=\left( \varTheta -1\right) f+\left[ (1+\theta )^{s\varepsilon }-1+\theta (1+\theta )^{s\varepsilon }\right] \varTheta \xi ^{\varepsilon -k}f-\xi ^{-k}f_{x} \\&=\left( \varTheta -1\right) f +\left[ (1+\theta )^{s\varepsilon +1}-1\right] \varTheta \xi ^{\varepsilon -k}f-\xi ^{-k}f_{x} \\&=\left( \varTheta -1\right) f +\left[ (1+\theta )^{\frac{\varepsilon (1-\rho ) }{\rho }}-1\right] f\varTheta \xi ^{\varepsilon -k}-\xi ^{-k}f_{x} \\&=\left( \varTheta -1\right) f +f_{x}\xi ^{-\varepsilon }\varTheta \xi ^{\varepsilon -k}-\xi ^{-k}f_{x} \\&=\left( \varTheta -1\right) f +\varTheta \xi ^{-k}f_{x}-\xi ^{-k}f_{x} \\&=\left( \varTheta -1\right) \left( \chi f_{x}+f\right) \\&=\overline{\pi }_{a}\left( 1+\chi \frac{f_{x}}{f}\right) , \end{aligned}$$

where the last line is Eq. (29).

Appendix B

In this Appendix, we provide the proof of Proposition 1. Combining Eqs. (30) and (32) gives

$$\begin{aligned} \frac{M_{x}(\varphi )}{M_{a}(\varphi )}=(1+\theta )^{\frac{\varepsilon (1-\rho )}{\rho }}\left( 1+\chi \frac{f_{x}}{f}\right) ^{-\frac{\varepsilon }{k}} \end{aligned}$$

Rewriting this expression and substituting for \(\chi \) gives

$$\begin{aligned} \frac{M_{x}(\varphi )}{M_{a}(\varphi )}=\left[ B(\theta ,f_x,f)\right] ^{-\frac{ \varepsilon }{k}} \end{aligned}$$

where

$$\begin{aligned} B (\theta ,f_{x},f )\equiv \left( 1+{\normalsize \theta }\right) ^{-\frac{\left( 1-\rho \right) k}{\rho }}\left[ 1+\left( \left( 1+\theta \right) ^{^{\frac{ \varepsilon (1-\rho )}{\rho }}}-1\right) ^{\frac{k}{\varepsilon }}\left( \frac{f_{x}}{f}\right) ^{1-\frac{k}{\varepsilon }}\right] \end{aligned}$$

From Eq. (28), we have

$$\begin{aligned} \frac{f_{x}}{f}>\left( 1+\theta \right) ^{\frac{ \varepsilon (1-\rho )}{\rho }}-1, \end{aligned}$$

and therefore

$$\begin{aligned} B (\theta ,f_{x},f )&<\left( 1+{\normalsize \theta }\right) ^{-\frac{\left( 1-\rho \right) k}{\rho }}\left[ 1+\left( \left( 1+\theta \right) ^{^{\frac{ \varepsilon (1-\rho )}{\rho }}}-1\right) ^{\frac{k}{\varepsilon }+1-\frac{k}{\varepsilon }}\right] \\&=\left( 1+{\normalsize \theta }\right) ^{\frac{\left( 1-\rho \right) }{\rho }\left( \varepsilon -k\right) } \end{aligned}$$

Since \(k>\varepsilon >0\), this implies \(B (\theta ,f_{x},f )<1\), and therefore \(M_{x}(\varphi )/M_{a}(\varphi )>1\), which proves Proposition 1.

Appendix C

In this Appendix, we show how to derive for our model the share of expenditure on domestically produced goods. Using the solution for \(\overline{\pi }_{x}\) from Appendix A, as well the symmetry of countries, we know that the average revenue earned by an importer—and therefore the expenditure falling on the average importer—is \( \overline{r}_{x}=\frac{1}{\beta }\varTheta \theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon }f \). Use \(\lambda _{x}\) to denote the share of expenditure going to imported products. Then we have

$$\begin{aligned} \lambda _{x} =\frac{\overline{r}_{x}\chi N}{L} =\frac{\varTheta \theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon }f\chi \beta L}{\beta L \left( 1+\chi \frac{f_x}{f}\right) \varTheta f} =\frac{\theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon }\chi }{1+\chi \frac{f_{x}}{f}} \end{aligned}$$

The share of domestic expenditure follows as

$$\begin{aligned} \lambda _{d} =1-\lambda _{x} =\frac{1+\chi \left[ \frac{f_{x}}{f}-\theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon }\right] }{1+\chi \frac{f_{x}}{f }}, \end{aligned}$$

and we can show that if \(s>0\),

$$\begin{aligned} \frac{f_{x}}{f}-\theta (1+\theta )^{s\varepsilon }\xi ^{\varepsilon } =\frac{ f_{x}}{f}-\frac{f_{x}\theta (1+\theta )^{s\varepsilon }}{f\left[ (1+\theta )^{\frac{\varepsilon (1-\rho )}{\rho }}-1\right] } =\frac{f_{x}}{f}\left[ \frac{(1+\theta )^{s\varepsilon }-1}{(1+\theta )^{s\varepsilon +1}-1}\right] >0 \end{aligned}$$

Therefore

$$\begin{aligned} \lambda _{d}>\frac{1}{1+\chi \frac{f_{x}}{f}} \end{aligned}$$

The gains from the trade would be underestimated if we use the share of domestic expenditure as a sufficient statistic, conditional on observed trade data.

Appendix D

In this Appendix, we provide derivations for the results presented in Sect. 3. Solving program (34), we obtain the optimal choice of labour employment:

$$\begin{aligned} l\left[ m,\beta \left( m\right) \right] =\left[ \left( 1-\beta \left( m\right) \right) \rho \left( A^{1-\rho }\varphi ^{\rho }M^{\left( s+1-1/\alpha \right) \rho }\right) ^{\frac{\alpha }{\rho }}r\left( M,m,\varphi \right) ^{\frac{\rho -\alpha }{\rho }}\right] ^{\frac{1}{ 1-\alpha }} \end{aligned}$$

Plugging this expression into \(r^{\prime }\left( M,m,\varphi \right) \) delivers the differential equation

$$\begin{aligned} r^{\prime }\left( M,m,\varphi \right)&=\frac{\rho }{\alpha }\left( A^{1-\rho }\varphi ^{\rho }M^{\left( s+1-1/\alpha \right) \rho }\right) ^{\frac{\alpha }{\rho \left( 1-\alpha \right) }}\\&\quad \times r\left( M,m,\varphi \right) ^{\frac{\rho -\alpha }{\rho \left( 1-\alpha \right) }}\left[ \rho \left( 1-\beta \left( m\right) \right) \right] ^{\frac{\alpha }{1-\alpha }}, \end{aligned}$$

and we can verify that the solution to this differential equation is given by:

$$\begin{aligned} r\left( M,m,\varphi \right)&=A\left( \varphi M^{s+1-1/\alpha }\right) ^{ \frac{\rho }{1-\rho }}\left( \frac{1-\rho }{1-\alpha }\right) ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }}\\&\quad \times \rho ^{\frac{\rho }{ 1-\rho }}\left[ \int _{0}^{m}\left( 1-\beta \left( i\right) \right) ^{\frac{ \alpha }{1-\alpha }}{\text{ d }}i\right] ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }} \\&=\varOmega A\left( \varphi M^{s+1-1/\alpha }\right) ^{\frac{\rho }{1-\rho }} \left[ \int _{0}^{m}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{ 1-\alpha }}{\text{ d }}i\right] ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }} \end{aligned}$$

Differentiating \(r\left( M,m,\varphi \right) \) and substituting into \( \int _{0}^{M}\beta \left( i\right) r^{\prime }\left( M,i,\varphi \right) di\), we can compute operating profits as:

$$\begin{aligned} \pi _{op}=&\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) } \varOmega A\left( \varphi M^{s+1-1/\alpha }\right) ^{\frac{\rho }{1-\rho }}\\&\quad \times \int _{0}^{M}\beta \left( i\right) \left( 1-\beta \left( i\right) \right) ^{ \frac{\alpha }{1-\alpha }}\left[ \int _{0}^{i}\left( 1-\beta \left( u\right) \right) ^{\frac{\alpha }{1-\alpha }}du\right] ^{\frac{\rho -\alpha }{\alpha \left( 1-\rho \right) }}di \end{aligned}$$

Following the analysis in Antràs and Chor (2013) and Alfaro et al. (2018), we know that for a final-good producer \(\varphi \) served by a supply chain of length M, there exist thresholds \(m^*_c \in \left( 0,M\right] \) and \(m^*_s\in \left( 0,M\right] \) separating the stages that are outsourced from those that are integrated in the complements case and the substitutes case, respectively. With sequential complements, all production stages before \( m^*_c\) are outsourced, with the remainder being integrated, while in the case of sequential substitutes all stages before \(m^*_s\) are integrated, with the remainder being outsourced.

Consider first the complements case. Here, we can write the operating profit \( \pi _{op}\) as

$$\begin{aligned} \pi _{op}=\varOmega A\left( \varphi M^{s+1-1/\alpha }\right) ^{\frac{\rho }{ 1-\rho }}\int _{0}^{M}\beta \left( i\right) \frac{\partial \left( \left[ \int _{0}^{i}\left( 1-\beta \left( u\right) \right) ^{\frac{\alpha }{1-\alpha }}du\right] ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }}\right) }{\partial i}di, \end{aligned}$$

in which

$$\begin{aligned}&\int _{0}^{M}\beta \left( i\right) \frac{\partial \left( \left[ \int _{0}^{i}\left( 1-\beta \left( u\right) \right) ^{\frac{\alpha }{1-\alpha }}du\right] ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }}\right) }{\partial i}di \\&\quad =\beta _{o}\left( 1-\beta _{o}\right) ^{\frac{\rho }{1-\rho }}m_{c}^{\frac{ \rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }}\\&\qquad +\beta _{v}\left( \begin{array}{c} \left[ m_{c}\left( 1-\beta _{o}\right) ^{\frac{\alpha }{1-\alpha }}+\left( M-m_{c}\right) \left( 1-\beta _{v}\right) ^{\frac{\alpha }{1-\alpha }}\right] ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }} \\ -\left[ m_{c}\left( 1-\beta _{o}\right) ^{\frac{\alpha }{1-\alpha }}\right] ^{\frac{\rho \left( 1-\alpha \right) }{\alpha \left( 1-\rho \right) }} \end{array} \right) \end{aligned}$$

Taking the first-order condition with respect to \(m_{c}\), we find:

$$\begin{aligned} \left( \beta _{v}-\beta _{o}\right) \left( 1-\beta _{o}\right) ^{\frac{\rho }{1-\rho }}&=\beta _{v}\left[ \left( 1-\beta _{o}\right) ^{\frac{\alpha }{ 1-\alpha }}-\left( 1-\beta _{v}\right) ^{\frac{\alpha }{1-\alpha }}\right] \\&\quad \times \left( \left( 1-\beta _{o}\right) ^{\frac{\alpha }{1-\alpha }}+\frac{ M-m_{c}^{*}}{m_{c}^{*}}\left( 1-\beta _{v}\right) ^{\frac{\alpha }{ 1-\alpha }}\right) ^{\frac{\rho -\alpha }{\alpha \left( 1-\rho \right) }} \end{aligned}$$

From this, we obtain

$$\begin{aligned} \frac{m_{c}^{*}}{M}=h_{c}^{*}=\left\{ 1+\left( \frac{1-\beta _{o}}{ 1-\beta _{v}}\right) ^{\frac{\alpha }{1-\alpha }}\left[ \left( \frac{1-\frac{ \beta _{o}}{\beta _{v}}}{1-\left( \frac{1-\beta _{o}}{1-\beta _{v}}\right) ^{-\frac{\alpha }{1-\alpha }}}\right) ^{\frac{\alpha \left( 1-\rho \right) }{ \rho -\alpha }}-1\right] \right\} ^{-1}, \end{aligned}$$

The threshold in the substitutes case can be derived in an analogous way, and we get

$$\begin{aligned} \frac{m_{s}^{*}}{M}=h_{s}^{*}=\left\{ 1+\left( \frac{1-\beta _{v}}{ 1-\beta _o}\right) ^{\frac{\alpha }{1-\alpha }}\left[ \left( \frac{\frac{ \beta _{v}}{\beta _o}-1}{\left( \frac{1-\beta _{v}}{1-\beta _o}\right) ^{- \frac{\alpha }{1-\alpha }}-1}\right) ^{\frac{\alpha \left( 1-\rho \right) }{ \rho -\alpha }}-1\right] \right\} ^{-1} \end{aligned}$$

Substituting for \(m_{k}^{*}\) with \(k=c,s\) in Eq. (35), we can write total revenue along a supply chain as

$$\begin{aligned} r\left( M,\varphi \right)&=\varOmega A\left( M^{\left( s+1-1/\alpha \right) }\varphi \right) ^{\frac{\rho }{1-\rho }}\\&\quad \times \left[ \int _{0}^{h_{k}^{*}M}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{1-\alpha } }di+\int _{h_{k}^{*}M}^{M}\left( 1-\beta \left( j\right) \right) ^{\frac{ \alpha }{1-\alpha }}dj\right] ^{\frac{\rho \left( 1-\alpha \right) }{\left( 1-\rho \right) \alpha }} \\&=\varOmega A\left( M^{\left( s+1-1/\alpha \right) }\varphi \right) ^{\frac{ \rho }{1-\rho }}\\&\quad \times \left[ h_{k}^{*}M\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{1-\alpha }}+\left( M-h_{k}^{*}M\right) \left( 1-\beta \left( j\right) \right) ^{\frac{\alpha }{1-\alpha }}\right] ^{\frac{\rho \left( 1-\alpha \right) }{\left( 1-\rho \right) \alpha }} \\&=\varOmega A\left( M^{s}\varphi \right) ^{\frac{\rho }{1-\rho }}\left[ h_{k}^{*}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{ 1-\alpha }}+\left( 1-h_{k}^{*}\right) \left( 1-\beta \left( j\right) \right) ^{\frac{\alpha }{1-\alpha }}\right] ^{\frac{\rho \left( 1-\alpha \right) }{\left( 1-\rho \right) \alpha }} \end{aligned}$$

with \(\beta \left( i\right) =\beta _{o}\) and \(\beta \left( j\right) =\beta _{v}\) in the complements case, and with \(\beta \left( i\right) =\beta _{v}\) and \(\beta \left( j\right) =\beta _{o}\) in the substitutes case.

From the expression of \(r\left( M,m,\varphi \right) \), we have

$$\begin{aligned} r^{\prime }\left( M,m,\varphi \right)&=\frac{\rho \left( 1-\alpha \right) }{ \alpha \left( 1-\rho \right) }\varOmega A\left( \varphi M^{s+1-1/\alpha }\right) ^{\frac{\rho }{1-\rho }}\\&\quad \times \left[ \int _{0}^{m}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{1-\alpha }}{\text{ d }}i\right] ^{\frac{\rho -\alpha }{\alpha \left( 1-\rho \right) }}\left( 1-\beta \left( m\right) \right) ^{ \frac{\alpha }{1-\alpha }}, \end{aligned}$$

and rewriting this expression in terms of \(h=m/M\) gives

$$\begin{aligned} r^{\prime }\left( h,M,\varphi \right) =&\frac{\rho \left( 1-\alpha \right) }{ \alpha \left( 1-\rho \right) }\varOmega A\varphi ^{\frac{\rho }{1-\rho }}M^{ \frac{1-\left( s+1\right) \rho }{\rho -1}}\left( 1-\beta \left( j\right) \right) ^{\frac{\alpha }{1-\alpha }}\\&\quad \times \left[ h_{k}^{*}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{1-\alpha }}+\left( h-h_{k}^{*}\right) \left( 1-\beta \left( j\right) \right) ^{\frac{\alpha }{1-\alpha } }\right] ^{\frac{\rho -\alpha }{\alpha \left( 1-\rho \right) }} \end{aligned}$$

if \(h>h_{k}^{*}\) and

$$\begin{aligned} r^{\prime }\left( h,M,\varphi \right)&= \frac{\rho \left( 1-\alpha \right) }{ \alpha \left( 1-\rho \right) }\varOmega \varphi ^{\frac{\rho }{1-\rho }}M^{ \frac{1-\left( s+1\right) \rho }{\rho -1}}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{1-\alpha }}\\&\quad \times \left[ h_{k}^{*}\left( 1-\beta \left( i\right) \right) ^{\frac{\alpha }{1-\alpha }}\right] ^{\frac{\rho -\alpha }{\alpha \left( 1-\rho \right) }} \end{aligned}$$

if \(h<h_{k}^{*}\). Since both suppliers’ profits and wage costs are proportional to \(r^{\prime }\left( h,M,\varphi \right) \), we have

$$\begin{aligned} \frac{M\left( \varphi _{1}\right) }{M\left( \varphi _{2}\right) }=\left( \frac{\varphi _{1}}{\varphi _{2}}\right) ^{\varepsilon }, \end{aligned}$$

and therefore the length of supply chains is increasing in \(\varphi \) with the same constant elasticity as in the benchmark model. Plugging the expression of \(m_{k}^{*}\) with \(k=c,s\) into the final-good producer’s operating profits \(\pi _{op}\), we can show that \(\pi _{op}=\varGamma _{k}\left( \beta _{v},\beta _{o},\alpha ,\rho \right) \varOmega A\left( M^{s}\varphi \right) ^{\frac{\rho }{ 1-\rho }} \) with \(k=c,s\), in which \(\varGamma _{k}\left( \beta _{v},\beta _{o},\alpha ,\rho \right) \) collects \(\beta _{v}\) and \(\beta _{o}\). The profit of a final-good producer with productivity \(\varphi \) is given by

$$\begin{aligned} \pi \left( M,\varphi \right) =\varGamma _{k}\left( \beta _{v},\beta _{o},\alpha ,\rho \right) \varOmega A\left( M^{s}\varphi \right) ^{\frac{\rho }{1-\rho }} -f \end{aligned}$$

In analogy to the benchmark model, suppliers’ ex ante average profits along a supply chain are given by \(\left( 1-\alpha \right) \left[ r\left( M,\varphi \right) -\pi _{op}\right] /M\), which are equalised across supply chains in equilibrium:

$$\begin{aligned}&\frac{\left( 1-\alpha \right) \left( \varLambda _{k}-\varGamma _{k}\right) \varOmega A \left[ M\left( \varphi _{1}\right) ^{s}\varphi _{1}\right] ^{\frac{\rho }{ 1-\rho }}}{M\left( \varphi _{1}\right) }\\&\quad =\frac{\left( 1-\alpha \right) \left( \varLambda _{k}-\varGamma _{k}\right) \varOmega A\left[ M\left( \varphi _{2}\right) ^{s}\varphi _{2}\right] ^{\frac{\rho }{1-\rho }}}{M\left( \varphi _{2}\right) }=f_{m} \end{aligned}$$

The zero cutoff profit condition \(\pi _{op}=f\) follows as

$$\begin{aligned} \varOmega A\left[ M\left( \varphi ^{*}\right) ^{s}\varphi ^{*}\right] ^{ \frac{\rho }{1-\rho }}=\frac{f}{\varGamma _{k}} \end{aligned}$$

Immediately,

$$\begin{aligned} M\left( \varphi ^{*}\right) =\frac{f\left( 1-\alpha \right) \left( \varLambda _{k}-\varGamma _{k}\right) }{f_{m}\varGamma _{k}} \end{aligned}$$

and the average length of supply chains follows as \(\overline{M}=\varTheta M\left( \varphi ^{*}\right) \). As in the benchmark model, the cutoff productivity \(\varphi ^{*}\) is given by

$$\begin{aligned} \varphi ^{*}=\left[ \frac{\left( \varTheta -1\right) f}{f_{e}}\right] ^{ \frac{1}{k}} \end{aligned}$$

The average revenue of final-good producers equals \(\varTheta f \varLambda _{k}/\varGamma _{k}\), the number of final-good producers is given by

$$\begin{aligned} N=\frac{\varGamma _{k}L}{\varLambda _{k}\varTheta f}, \end{aligned}$$

and the total number of suppliers follows as

$$\begin{aligned} \overline{M}N=\frac{\left( 1-\alpha \right) \left( \varLambda _{k}-\varGamma _{k}\right) }{\varLambda _{k}f_{m}}L. \end{aligned}$$

Using the zero profit condition for final-good producers, welfare per worker is given by

$$\begin{aligned} W=P^{-1}=\left( \frac{\varGamma _{k}\varOmega L}{f}\right) ^{\frac{1-\rho }{\rho }} \left[ M\left( \varphi ^{*}\right) \right] ^{s}\varphi ^{*} \end{aligned}$$

In the open economy, two additional costs are involved. An exporting firm has the total revenue that is a multiple \(1+\theta \) of its domestic revenue. International trade has no effect on the threshold relative position \(h_{k}^{*}\). In analogy to Eq. (26), we rewrite the indifference condition for the marginal exporting firm as:

$$\begin{aligned}&\left( 1+\theta \right) \varGamma _{k}\left( \beta _{v},\beta _{o},\alpha ,\rho \right) \varOmega A\left[ M_{x}\left( \varphi _{x}^{*}\right) ^{s}\varphi _{x}^{*}\right] ^{\frac{\rho }{1-\rho }}\\&\quad -\varGamma _{k}\left( \beta _{v},\beta _{o},\alpha ,\rho \right) \varOmega A\left[ M_{d}\left( \varphi _{x}^{*}\right) ^{s}\varphi _{x}^{*}\right] ^{\frac{\rho }{1-\rho } }=f_{x} \end{aligned}$$

Similarly, we can also show that the cross-regime SIC implies

$$\begin{aligned} \frac{M_{x}\left( \varphi \right) }{M_{d}\left( \varphi \right) }=\left( 1+\theta \right) ^{\frac{\varepsilon \left( 1-\rho \right) }{\rho }}, \end{aligned}$$

exactly as in the benchmark model. As a consequence, we find that many results from this model are still valid in the extension discussed here. In particular, the share of exporting firms \(\chi \) is the same as before, and hence the gains from trade \(W/W_{a}\) in this extension are the same as in our benchmark model.

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Kreickemeier, U., Qu, Z. International trade with sequential production. Econ Theory 69, 1101–1125 (2020). https://doi.org/10.1007/s00199-019-01190-y

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Keywords

  • Trade liberalisation
  • Sequential production
  • Supply chains
  • General equilibrium
  • Gains from trade

JEL Classification

  • F12
  • F15