BUS606: Operations and Supply Chain Management

We compute embodied services in manufacturing sectors using a method developed by Koopman, Wang and Wei and Wang, Wei, and Zhu that generalizes the vertical specialization measures proposed by Hummels, Ishii and Yi. Assume a world with G countries, in which each country produces goods in N tradable sectors. Goods and services produced in each sector can be consumed directly or used as intermediate inputs, and each country exports both intermediate and final goods to other countries. All gross outputs (X) produced by a country must be used as intermediate goods/services or as final goods/services (F), i.e.,

(3) $X_{i}=\sum_{j}^{G}\left(A_{i j} X_{j}+F_{i j}\right), \quad \mathrm{i}, \mathrm{j}=1,2, \ldots \mathrm{G}$

where $X_{i}$ is the $\mathrm{N} \times 1$ gross output vector of country $\mathrm{i}, F_{i j}$ is the $\mathrm{N} \times 1$ vector for final goods and services produced in country $\mathrm{i}$ and consumed in country $\mathrm{j}$, and $A_{i j}$ is the $\mathrm{N} \times \mathrm{N}$ input-output coefficient matrix, giving intermediate use in $\mathrm{j}$ of goods and services produced in $\mathrm{i}$.

The G-country, N-sector production, and trade system can be written as an inter-country input-output (ICIO) model in block matrix notation as follows.

(4) $\left[\begin{array}{c}X_{1} \\ X_{2} \\ \vdots \\ X_{G}\end{array}\right]=\left[\begin{array}{cccc}A_{11} & A_{12} & \cdots & A_{1 G} \\ A_{21} & A_{22} & \cdots & A_{2 G} \\ \vdots & \vdots & \ddots & \vdots \\ A_{G 1} & A_{G 2} & \cdots & A_{G G}\end{array}\right]\left[\begin{array}{c}X_{1} \\ X_{2} \\ \vdots \\ X_{G}\end{array}\right]+\left[\begin{array}{c}F_{11}+F_{12}+\cdots+F_{1 G} \\ F_{21}+F_{22}+\cdots+F_{2 G} \\ \cdots \cdots \\ F_{G 1}+F_{G 2}+\cdots+F_{G G}\end{array}\right]$

After rearranging, we have

(5) $\left[\begin{array}{c}X_{1} \\ X_{2} \\ \vdots \\ X_{G}\end{array}\right]=\left[\begin{array}{cccc}I-A_{11} & -A_{12} & \cdots & -A_{1 G} \\ -A_{21} & I-A_{22} & \cdots & -A_{2 G} \\ \vdots & \vdots & \ddots & \vdots \\ -A_{G 1} & -A_{G 2} & \cdots & I-A_{G G}\end{array}\right]^{-1}\left[\begin{array}{c}\sum_{j=1}^{G} F_{1 j} \\ \sum_{j=1}^{G} F_{2 j} \\ \vdots \\ \sum_{j=1}^{G} F_{G j}\end{array}\right]=\left[\begin{array}{cccc}B_{11} & B_{12} & \cdots & B_{1 G} \\ B_{21} & B_{22} & \cdots & B_{2 G} \\ \vdots & \vdots & \ddots & \vdots \\ B_{G 1} & B_{G 2} & \cdots & B_{G G}\end{array}\right]\left[\begin{array}{c}F_{1} \\ F_{2} \\ \vdots \\ F_{G}\end{array}\right]$

where $I$ is an $\mathrm{N} \mathrm{xN}$ identity matrix, and $B_{i j}$ denotes the $\mathrm{N} \times \mathrm{N}$ block Leontief inverse matrix, which is the total requirement matrix that gives the amount of gross outputs in producing country $i$ required for a one-unit increase in final demand in destination country $j$.

Let $V_{i}$ be the $\mathrm{N} \times 1$ direct value-added coefficient vector. Each element of $V_{i}$ gives the ratio of direct domestic value-added to gross output (exports) for country i at the sector level. This is equal to one minus the intermediate input share from all countries (including domestically produced intermediates):

(6) $\quad V_{i}=\left(u-\sum_{j=1}^{G} A_{j i} u\right)$

where $u$ is an $\mathrm{Nx} 1$ unit vector of 1 . Putting all $V_{\mathrm{i}}$ in the diagonal and denoting it with a hatsymbol $\left(\hat{V}_{i}\right)$, we can define a GN $\times$ GN matrix of direct domestic value-added coefficients for all countries as,

(7) $\hat{V}=\left[\begin{array}{cccc} \hat{V}_{1} & 0 & \cdots & 0 \\ 0 & \hat{V}_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \hat{V}_{G} \end{array}\right]$

Putting final demand in the diagonals, we can define another GN×GN matrix of all countries' final demand as

(8) $\hat{F}=\left[\begin{array}{cccc} \hat{F}_{1} & 0 & \cdots & 0 \\ 0 & \hat{F}_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \hat{F}_{G} \end{array}\right]$

Then the decomposition of value-added in final demand can be conducted by the following equation:

(9) $\hat{V B} \hat{F}=\left[\begin{array}{cccc}\hat{V}_{1} & 0 & \cdots & 0 \\ 0 & \hat{V}_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \hat{V}_{G}\end{array}\right]\left[\begin{array}{cccc}B_{11} & B_{12} & \cdots & B_{1 G} \\ B_{21} & B_{22} & \cdots & B_{2 G} \\ \vdots & \vdots & \ddots & \vdots \\ B_{G 1} & B_{G 2} & \cdots & B_{G G}\end{array}\right]\left[\begin{array}{cccc}\hat{F}_{1} & 0 & \cdots & 0 \\ 0 & \hat{F}_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \hat{F}_{G}\end{array}\right]$

where $\hat{V} B \hat{F}$ is a $GN \times G N$ square matrix that gives the estimates of sector and country sources of value-added in a country's total final demand. Each block matrix $\hat{V}_{i} B_{i j} \hat{F}_{j j}$ is an $\mathrm{N} \times \mathrm{N}$ square matrix, with each element representing the value-added from a source sector of a source country directly or indirectly used by an absorbing sector in a destination country's total final demand (both domestic and foreign). Because we assume that the same technology is used in the production meeting a country's domestic demand and foreign demand (exports), we use total final demand, which is the sum of domestic final demand and final export demand, to calculate embodied services ratios.

Based on equation (9), we create the following measure of domestic services input intensity in manufacturing sectors in country j:

(10) $\quad SII_{j}^{s m}=v_{j}^{s} b_{j j}^{s m} f_{j}^{m} / V A_{j}^{m}, \mathrm{j}=1,2, \ldots, \mathrm{G}$

where $v_{j}^{s} b_{j j}^{s m} f_{j}^{m}$, an element in equation $(10)$, refers to country j's domestic services (superscript

s) values embodied in country j's total final demand in the manufacturing sector (superscript $\mathrm{m}$ ); $V A_{j}^{m}$ is the total value-added created by the factors employed in the manufacturing sector of the absorbing country $j$ (or the manufacturing GDP in country $j$ ). $SII$ defined in formula (10) is a scalar if $s$ and $m$ refer to a specific services and manufacturing sector respectively, in a country $j$ in a given year. The numerator on the right-hand side of formula (10) refers to the value added contributed directly and indirectly by the factors employed in a services sector, while the denominator measures the value added contributed by factors employed in a manufacturing sector. Therefore, the denominator is not a part of the numerator and the $SII$ measure is not bounded by one, although it is always less than one in the data. It would be bounded by one if we used the gross manufacturing output in the denominator, and the $SII$ of one services sector would likely be negatively correlated to the $SII$ of other goods or services sectors, and so omitted variable bias can be a problem if we do not include all other sectors in our analysis. The strategy we adopt to measure $SII$ as in formula (10) can help us to avoid this problem and keep our specification simple.

It is tempting to use a country's own services input intensity $SII$ directly in the regression. But there are a number of issues with such a strategy. $SII$ of a country with underdeveloped services sectors (e.g., financial repression) may not be able to capture the required services input intensity along the manufacturing production possibility frontier. Hence, instead of using countries' own services input intensities, we use U.S. services input intensity for all the countries under the assumption that the U.S. is among the countries with the least financial and business services transaction costs and frictions. If inter-sectoral linkage is considered as a feature of the production technology, it should be the same across countries in the absence of services under-development. Adopting a similar strategy, Rajan and Zingales measure industries' dependence on external funds using only U.S. data for all countries covered by their analysis. Figure 5 shows a scatter plot of the domestic financial services input intensity in manufacturing against the business services input intensity for each of the WIOD countries in 2005. As we expect, U.S. embodied services ratios are among the highest for both financial and business services.

Another problem of using countries' own services input intensity is a potential endogeneity issue because a country's embodied services and services development can also be affected by its own manufacturing performance. For example, a country like India with comparative disadvantage in manufacturing may choose to specialize in services, which in turn will promote services development and reduce embodied services due to the weakness of the manufacturing sectors. When we use only U.S. embodied services, the feedback or reverse causality to the U.S. embodied services from other countries' manufacturing export RCA will be less a concern. In addition, we will drop U.S. observations from our regressions to further alleviate the endogeneity problem. Finally, as another justification for using U.S. measures, the U.S. is arguably one of the countries with the most reliable data. 

We will either use the time-varying U.S. services input intensities or take their averages over years. An advantage of the former measure is that it retains the time variations, while the later measure can smooth temporal fluctuations and hence is less sensitive to outliers. The variations in the U.S. services input intensities over the years are small for most of the WIOD sectors and some of the input-output data in the WIOD are filled in based on interpolation, so we will take the averaged measure as the benchmark and use the time-varying measure only as a robustness check. When average U.S. $SII_{i}$is used, this variable will drop out of regressions with sector or timevarying sector fixed effects. When time-varying sector fixed effects are used, $SII_{i t}$ will also be dropped. 

An important caveat is that even a measure based on U.S. data is still a proxy intending to capture the potential linkage between services and manufacturing sectors. A noisy measure, however, should create a bias against finding a significant effect of services intensity on manufacturing RCA. Should we be able to find a better measure, the effect is likely to be even stronger.

In our empirical analysis, we also use the share of foreign embodied services in the total embodied domestic and foreign services as follows (for country j):

(11) $forsh _{j}^{s m}=\sum_{i, i \neq j}^{G} v_{i}^{s} b_{i j}^{s m} f_{j}^{m} / \sum_{i=1}^{G} v_{i}^{s} b_{i j}^{s m} f_{j}^{m}$

The denominator in equation (11) sums $v_{i}^{s} b_{i j}^{s m} f_{j}^{m}$ over all source countries $\mathrm{i}=1,2, \ldots, \mathrm{G}$, including $\mathrm{j}$ itself, while the numerator leaves out country $j$'s own (domestic) embodied services.