The universal pathway to commodity structure upgrading in global trade evolution

Abstract

Some scholars have proposed that economies grow by upgrading the commodities they produce and export. Product space theory holds that the export structure is determined by a country’s factor endowment and technological level, proposing the revealed comparative advantage (RCA) to analyze the specialization on commodities for countries. With the development of global trade, the network of multilateral trade relations has become increasingly hierarchical and complex. It would, therefore, be valuable to identify general patterns across countries of upgrading the commodity structure in the evolution of their participation in global trade. This paper shows that a typical pattern of change in the dominant types of foreign trade occurs when an economy has grown to a certain scale. With economic development, the advantages of high-technology commodities in trade gradually become more prominent.

A. Appendix

1.1 The comparative advantage function

$F_{mn,i}$ is the trade flow from country/region m to n for commodity i. $RCA_{mn,i}$ is the comparative advantage of commodity i in all trade from countries m to n as (Hong et al. 2020),

$$\begin{aligned} RCA_{mn,i}=\frac{F_{mn,i}}{\sum _{j}F_{mn,j}}/\frac{\sum _{p,q}F_{pq,i}}{\sum _{p,q,j}F_{pq,j}}. \end{aligned}$$

A commodity is called characteristic or significant if $RCA_{mn,i}>1$ (Balassa 1965; Hidalgo et al. 2007; Hong et al. 2020). The trade flow $F_{mn,i}$ is proportional to the gross national product (GDP) of the two countries as $Y_m$ and $Y_n$, such as

$$\begin{aligned} \begin{aligned} F_{mn,i}&=Y_{i0}Y_m^{\beta _i}Y_n^{\alpha _i}=Y_{i0}(Y_mY_n)^{\beta _i}Y_n^{(\alpha _i-\beta _i)}\\&=Y_{i0}(N_{mn})^{\beta _i}Y_n^{(\alpha _i-\beta _i)}. \end{aligned} \end{aligned}$$

Then, we obtain

$$\begin{aligned} \begin{aligned} RCA_{mn,i}&\sim \frac{Y_{i0}(N_{mn})^{\beta _i}Y_n^{(\alpha _i-\beta _i)}}{\sum _{j}F_{mn,j}\sum _{p,q}Y_{i0}Y_p^{\beta _i}Y_q^{\alpha _i}} \\&=\frac{(N_{mn})^{\beta _i}Y_n^{(\alpha _i-\beta _i)}}{\sum _{j}F_{mn,j}\sum _{p,q}Y_p^{\beta _i}Y_q^{\alpha _i}}. \\ \end{aligned} \end{aligned}$$

For most industries, $\beta _i>(\alpha _i-\beta _i)$. For most countries, $N_{mn}=(Y_mY_n)>>Y_n$, and the change in Y is relatively small. Then,

$$\begin{aligned} RCA_{mn,i}\sim \frac{(N_{mn})^{\beta _i}}{\sum _{j}F_{mn,j}\sum _{p,q}Y_p^{\beta _i}Y_q^{\alpha _i}}. \end{aligned}$$

Fig. 6

a The lognormal distribution of $N_{mn}$ in 2015. b In the area of relatively small or large ${\varvec{N}}$, the distribution function is linear under double logarithm coordinates. Here black dots are empirical data, and red or blue ones are fitting curves (color figure online)

Figure 6 shows the lognormal distribution of $N_{mn}$ in 2015, which is consistent with the conclusion of some researchers (Hausmann and Hidalgo 2011). For ${\varvec{N}}\in [10^{16},10^{19}]$ and ${\varvec{N}}\in [10^{23},10^{26}]$, we can approximately define that $P( {\varvec{N}})\propto {\varvec{N}}^{-\gamma }$. Based on ${\varvec{N}}^{ \varvec{\beta }}\propto P( {\varvec{N}}) {\varvec{N}}^{ \varvec{\beta }}\propto {\varvec{N}}^{-\gamma } {\varvec{N}}^{ \varvec{\beta }}$, then

$$\begin{aligned} \sum _{pq} Y_{p}^{\beta _i}Y_q^{\alpha _i}\simeq & {} \int _{N_{\mathrm{min}}}^{N_{\mathrm{max}}} P( {\varvec{N}}) {\varvec{N}}^{ \varvec{\beta }} {\varvec{Y}}^{( \varvec{\alpha }- \varvec{\beta })}\mathrm{d} {\varvec{N}}\\\simeq & {} \int _{N_{\mathrm{min}}}^{N_{\mathrm{max}}} {\varvec{N}}^{ \varvec{\beta }-\gamma } {\varvec{Y}}^{( \varvec{\alpha }- \varvec{\beta })}\mathrm{d} {\varvec{N}}\\\simeq & {} \frac{N_{\mathrm{max}}^{( \varvec{\beta } -\gamma +1)}Y_{\mathrm{max}}^{( \varvec{\alpha }- \varvec{\beta })}-N_{\mathrm{min}}^{( \varvec{\beta }-\gamma +1)}Y_{\mathrm{min}}^{( \varvec{\alpha }- \varvec{\beta })}}{ \varvec{\beta } -\gamma +1}.\\ \end{aligned}$$

Based on the trade gravity model, this paper ignores the influence of geographical distance and simplifies the parameters. Here is the hypothesis of $\sum _j F_{mn,j}\sim N_{mn}$, then

$$\begin{aligned} \begin{aligned} \varvec{RCA}&\sim {\varvec{N}}^{ \varvec{\beta }-1}\frac{ \varvec{\beta } -\gamma +1}{N_{\mathrm{max}}^{ \varvec{\beta } -\gamma +1}Y_{\mathrm{max}}^{ \varvec{\alpha }- \varvec{\beta }}-N_{\mathrm{min}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{min}}^{ \varvec{\alpha }- \varvec{\beta }}}\\&= {\varvec{N}}^{ \varvec{\beta }-1}F( \varvec{\alpha }, \varvec{\beta }). \end{aligned} \end{aligned}$$

(5)

1.2 Changes in comparative advantage

• Changes with ${\varvec{N}}$.

  $$\begin{aligned} \begin{aligned} \frac{\partial \varvec{RCA}}{\partial {\varvec{N}}}\sim ( \varvec{\beta }-1) {\varvec{N}}^{ \varvec{\beta }-2}F( \varvec{\alpha }, \varvec{\beta }). \end{aligned} \end{aligned}$$

  (6)

  $\varvec{\beta }^*=1$. when $\varvec{\beta }>1$, $\frac{\partial \varvec{RCA}}{\partial {\varvec{N}}}>0$; $\varvec{\beta }<1$, $\frac{\partial \varvec{RCA}}{\partial {\varvec{N}}}<0$; and $\varvec{\beta }=1$, $\frac{\partial \varvec{RCA}}{\partial {\varvec{N}}}=0$.

• Changes with $\varvec{\alpha }$.

  $$\begin{aligned} \frac{\partial \varvec{RCA}}{\partial \varvec{\alpha }}\sim -\frac{ {\varvec{N}}^{ \varvec{\beta }-1}F^2( \varvec{\alpha }, \varvec{\beta })}{ \varvec{\beta }-\gamma +1}C( \varvec{\alpha }, \varvec{\beta }), \end{aligned}$$

  (7)
  $$\begin{aligned} \begin{aligned} C( \varvec{\alpha }, \varvec{\beta })=\,&N_{\mathrm{max}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{max}}^{( \varvec{\alpha }- \varvec{\beta })}\log Y_{\mathrm{max}}\\&-N_{\mathrm{min}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{min}}^{( \varvec{\alpha }- \varvec{\beta })}\log Y_{\mathrm{min}}. \end{aligned} \end{aligned}$$

  Here $\varvec{\alpha }^*$ cannot fall within the range obtained by empirical data. Therefore, in the later analysis, we assume that $\varvec{\alpha }$ is the average value of $\alpha _i$.

• Changes with $\varvec{\beta }$.

  $$\begin{aligned} \begin{aligned}&\frac{\partial \varvec{RCA}}{\partial \varvec{\beta }}\sim \frac{F( \varvec{\alpha }, \varvec{\beta }) {\varvec{N}}^{ \varvec{\beta }-1}}{ \varvec{\beta }- \varvec{\gamma }+1}\\&\times [( \varvec{\beta }- \varvec{\gamma }+1)\log {\varvec{N}}+1-F( \varvec{\alpha }, \varvec{\beta })D( \varvec{\alpha }, \varvec{\beta })],\\ \end{aligned} \end{aligned}$$

  (8)
  $$\begin{aligned} \begin{aligned} D( \varvec{\alpha }, \varvec{\beta })&\simeq N_{\mathrm{max}}^{( \varvec{\beta }-\gamma +1)}Y_{\mathrm{max}}^{ \varvec{\alpha }- \varvec{\beta }}\log N_{\mathrm{max}}\\&-N_{\mathrm{min}}^{( \varvec{\beta }-\gamma +1)}Y_{\mathrm{min}}^{ \varvec{\alpha }- \varvec{\beta }}\log N_{\mathrm{min}}. \\ \end{aligned} \end{aligned}$$

  ${\varvec{N}}^*=10^{(\frac{F( \varvec{\alpha }, \varvec{\beta })D( \varvec{\alpha }, \varvec{\beta })-1}{ \varvec{\beta }-\gamma +1})}$ In 2015, for the first critical point, $Y_{\mathrm{max}}\approx 10^{8.5}$, $Y_{\mathrm{min}}\approx 10^{7}$, and $N_{\mathrm{max}}\approx 10^{19}$, $N_{\mathrm{min}}\approx 10^{16}$, $\gamma =0.234$, when $\varvec{\beta }^*= 1$, and $\varvec{\alpha }=0.570$, ${\varvec{N}}_1^*=10^{18.4}$. When ${\varvec{N}}> {\varvec{N}}_1^*$, $\frac{\partial \varvec{RCA}}{\partial \varvec{\beta }}>0$; ${\varvec{N}}< {\varvec{N}}_1^*$, $\frac{\partial \varvec{RCA}}{\partial \varvec{\beta }}<0$; ${\varvec{N}}= {\varvec{N}}_1^*$, $\frac{\partial \varvec{RCA}}{\partial \varvec{\beta }}=0$. For the second critical point, $Y_{\mathrm{max}}\approx 10^{13}$, $Y_{\mathrm{min}}\approx 10^{11.5}$,$N_{\mathrm{max}}\approx 10^{26}$, $N_{\mathrm{min}}\approx 10^{23}$, $\gamma =1.516$, when $\varvec{\beta }^*= 1$, $\varvec{\alpha }=0.570$, and ${\varvec{N}}_2^*=10^{24.8}$. When ${\varvec{N}}> {\varvec{N}}_2^*$, $\frac{\partial \varvec{RCA}}{\partial \varvec{\beta }}>0$; ${\varvec{N}}< {\varvec{N}}_2^*$, $\frac{\partial \varvec{RCA}}{\partial \varvec{\beta }}<0$; ${\varvec{N}}= {\varvec{N}}_2^*$, $\frac{\partial \varvec{RCA}}{\partial \varvec{\beta }}=0$.

1.3 Analysis of critical points

Due to the complexity of $\varvec{\alpha }$, we mainly study the co-evolution of $\varvec{\beta }$ and ${\varvec{N}}$ from the perspective of export countries. ($\varvec{\beta }^{\mathrm{{*}}}=1, \varvec{N_1}^{\mathrm{{*}}}=10^{18.4}$) and ($\varvec{\beta ^{\mathrm{{*}}}}=1, \varvec{N_2}^{\mathrm{{*}}}=10^{24.8}$) are the critical points.

$$\begin{aligned} \begin{aligned} \frac{\partial ^2 \varvec{RCA}}{\partial {\varvec{N}}^2}=\,\,&( \varvec{\beta }-1)( \varvec{\beta }-2) {\varvec{N}}^{ \varvec{\beta }-3}\frac{ \varvec{\beta }-\gamma +1}{N_{\mathrm{max}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{max}}^{\ \varvec{\alpha }- \varvec{\beta }}-N_{\mathrm{min}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{min}}^{\ \varvec{\alpha }- \varvec{\beta }}}\\ \frac{\partial ^2 \varvec{RCA}}{\partial {\varvec{N}} \partial \varvec{\beta }}\simeq&\,\,\frac{( \varvec{\beta }-1)F( \varvec{\alpha }, \varvec{\beta }) {\varvec{N}}^{ \varvec{\beta }-2}}{ \varvec{\beta }-\gamma +1}(( \varvec{\beta }-\gamma +1)\log {\varvec{N}}+1\\&-F( \varvec{\alpha }, \varvec{\beta })D( \varvec{\alpha }, \varvec{\beta }))+N^{ \varvec{\beta }-2}F( \varvec{\alpha }, \varvec{\beta })\\ \frac{\partial ^2 \varvec{RCA}}{\partial \varvec{\beta } \partial {\varvec{N}}}=\,\,&\frac{ {\varvec{N}}^{ \varvec{\beta }-2}}{N_{\mathrm{max}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{max}}^{ \varvec{\alpha }- \varvec{\beta }}-N_{\mathrm{min}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{min}}^{ \varvec{\alpha }- \varvec{\beta }}}[( \varvec{\beta }-1)( \varvec{\beta }-\gamma +1)\log {\varvec{N}}\\&+(2 \varvec{\beta }-\gamma )-( \varvec{\beta }-1)F( \varvec{\alpha }, \varvec{\beta })D( \varvec{\alpha }, \varvec{\beta }))]\\ \frac{\partial ^2 \varvec{RCA}}{\partial \varvec{\beta }^2}=\,&\left[ \frac{ {\varvec{N}}^{ \varvec{\beta }-1}(\log {\varvec{N}}(N_{\mathrm{max}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{max}}^{ \varvec{\alpha }- \varvec{\beta }}-N_{\mathrm{min}}^{ \varvec{\beta }- \gamma +1}Y_{\mathrm{min}}^{ \varvec{\alpha }- \varvec{\beta }})-G( \varvec{\alpha }, \varvec{\beta }))}{(N_{\mathrm{max}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{max}}^{ \varvec{\alpha }- \varvec{\beta }}-N_{\mathrm{min}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{min}}^{ \varvec{\alpha }- \varvec{\beta }})}\right] \\&\times [\log {\varvec{N}}+\frac{D( \varvec{\alpha }, \varvec{\beta })(( \varvec{\beta }-\gamma +1)(\log ( \varvec{\beta }-\gamma +1)+\log ( \varvec{\alpha }- \varvec{\beta })))}{N_{\mathrm{max}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{max}}^{ \varvec{\alpha }- \varvec{\beta }}-N_{\mathrm{min}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{min}}^{ \varvec{\alpha }- \varvec{\beta }}}\\&-F( \varvec{\alpha }, \varvec{\beta })(N_{\mathrm{max}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{max}}^{ \varvec{\alpha }- \varvec{\beta }}(\log N_{\mathrm{max}}+\log Y_{\mathrm{max}}))\\&+F( \varvec{\alpha }, \varvec{\beta })(N_{\mathrm{min}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{min}}^{ \varvec{\alpha }- \varvec{\beta }}(\log N_{\mathrm{min}}+\log Y_{\mathrm{min}}))]\\ G( \varvec{\alpha }, \varvec{\beta })=\,&N_{\mathrm{max}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{max}}^{ \varvec{\alpha }- \varvec{\beta }}(\log N_{\mathrm{max}}+\log Y_{\mathrm{max}})\\&-N_{\mathrm{min}}^{ \varvec{\beta }-\gamma +1}Y_{\mathrm{min}}^{ \varvec{\alpha }- \varvec{\beta }}(\log N_{\mathrm{min}}+\log Y_{\mathrm{min}}) \end{aligned} \end{aligned}$$

This is a saddle point since the Hessian matrix is an HRM with one positive eigenvalue and one negative eigenvalue. Therefore, ($\varvec{\beta }^{\mathrm{{*}}}=1, \varvec{N_1}^{\mathrm{{*}}}=10^{18.4}$) and ($\varvec{\beta }^{\mathrm{{*}}}=1, \varvec{N_2}^{\mathrm{{*}}}=10^{24.8}$) are minimums in one direction but maximums for the other direction with

$$\begin{aligned} HMR=\begin{bmatrix} \frac{\partial ^2 \varvec{RCA}}{\partial {\varvec{N}}^2}=0 &{} \frac{\partial ^2 \varvec{RCA}}{\partial {\varvec{N}} \partial \varvec{\beta }}>0\\ \frac{\partial ^2 \varvec{RCA}}{\partial \varvec{\beta } \partial {\varvec{N}}}>0 &{} \frac{\partial ^2 \varvec{RCA}}{\partial \varvec{\beta }^2}<0\\ \end{bmatrix} \end{aligned}$$

Figure 7 shows the two saddle points with ($\varvec{\beta }^{\mathrm{{*}}}=1, \varvec{N_1}^{\mathrm{{*}}}=10^{18.4}$) in (a) and ($\varvec{\beta }^{\mathrm{{*}}}=1, \varvec{N_2}^{\mathrm{{*}}}=10^{24.8}$) in (b).

Fig. 7

Two saddle points in 2015, with $\varvec{RCA}$ are minimums in one direction but maximums in the other direction

1.4 Relationship between $\varvec{RCA}$ and trade attractiveness

From 1995 to 2015, for commodities, the relationship between $\varvec{RCA}$ and trade attractiveness ${\varvec{N}}$ showed a relatively stable law as below (Fig. 8).

Fig. 8

Relationship between $\varvec{RCA}$ and trade attractiveness ${\varvec{N}}$ during 1995–2015

1.5 Trade structure upgrading and GDP

Figure 5c shows the linear relationship between export country’s trade structure and GDP $\log Y_m$, and we try to analyze the principle by derivation, for export country m, based on $N_{mn}>>Y_n$ and $\beta _i>\alpha _i-\beta _i$,

$$\begin{aligned} \begin{aligned} F_{mn,i}&=Y_{i0}Y_m^{\beta _i}Y_n^{\alpha _i}=Y_{i0}(N_{mn}^{\beta _i})Y_n^{(\alpha _i-\beta _i)}\\&\approx Y'_{i0}(N_{mn}^{\beta _i})\\ \Rightarrow \beta _i&\sim \frac{\log (F_{mn,i})}{\log N_{mn}}\\ \Rightarrow \frac{<\beta _{\mathrm{export}}>_m}{<\beta _{\mathrm{import}}>_m}&=\sum _{i\in i_s}\sum _n \frac{\log (F_{mn,i})}{\log N_{mn}}/ \sum _{j\in j_s}\sum _n \frac{\log (F_{nm,j})}{\log N_{mn}},\\ \end{aligned} \end{aligned}$$

where $i_s$ is the collection of commodities which are significant in the trade from m to n, with $RCA_{mn,i}>$1; and $j_s$ is the collection of commodities which are significant in the trade from n to m, with $RCA_{nm,j}>$1.

$$\begin{aligned} \frac{<\beta _{\mathrm{export}}>_m}{<\beta _{\mathrm{import}}>_m}&=\sum _{i\in i_s}\sum _n \frac{\log (F_{mn,i})}{\log N_{mn}}/ \sum _{j\in j_s}\sum _n \frac{\log (F_{nm,j})}{\log N_{mn}}\\&\sim \sum _{i\in i_s}\sum _n \frac{\log (Y_m^{\beta _i} Y_n^{\alpha _i})}{\log N_{mn}}/ \sum _{j\in j_s}\sum _n \frac{\log (Y_n^{\beta _j} Y_m^{\alpha _j})}{\log N_{mn}}\\&=\sum _{i\in i_s}\sum _n \frac{\log (Y_m^{\alpha _i} Y_n^{\alpha _i}Y_m^{\beta _i-\alpha _i})}{\log N_{mn}}/ \sum _{j\in j_s}\sum _n \frac{\log (Y_n^{\alpha _j} Y_m^{\alpha _j}Y_n^{\beta _j-\alpha _j})}{\log N_{mn}}\\&=\sum _{i\in i_s}\sum _n (\alpha _i+\frac{(\beta _i-\alpha _i)\log Y_m}{\log N_{mn}})/ \sum _{j\in j_s}\sum _n (\alpha _j+\frac{(\beta _j-\alpha _j)\log Y_n}{\log N_{mn}})\\&=\frac{n\sum _{i\in i_s}\alpha _i+\log Y_m\sum _{i\in i_s}\sum _n\frac{\beta _i-\alpha _i}{\log N_{mn}}}{n\sum _{j\in j_s}\alpha _j+\sum _{j\in j_s}\sum _n\frac{(\beta _j-\alpha _j)\log Y_n}{\log N_{mn}}}. \end{aligned}$$

$N_{mn}>>Y_n$ and we suppose that $\frac{\log Y_n}{\log N_{mn}} \approx c_{N}$. The variation range of $\varvec{\alpha }$ and $\varvec{\beta }$ is also relatively small, then,

$$\begin{aligned} \begin{aligned} \frac{<\beta _{\mathrm{export}}>_m}{<\beta _{\mathrm{import}}>_m}&\sim \frac{n\sum _{i\in i_s}\alpha _i+\log Y_m\sum _{i\in i_s}\sum _n\frac{\beta _i-\alpha _i}{\log N_{mn}}}{n\sum _{j\in j_s}\alpha _j+c_N}\\&=\frac{c_{i_s}+c'_{i_s}\log Y_m}{c'_{j_s}}=\frac{c_{i_s}}{c_{j_s}}+\frac{c'_{i_s}}{c_{j_s}}\log Y_m. \end{aligned} \end{aligned}$$

(9)

$c_N, c_{i_s},c'_{i_s},c_{j_s}$ can be approximated as constants. For exporters, equation (9) shows the approximate linear relation of trade structure upgrading ($\frac{<\beta _{\mathrm{export}}>_m}{<\beta _{\mathrm{import}}>_m}$) and its GDP $\log Y_m$, which is consistent with the statistical results of the empirical data in Fig. 5c.