The global propagation of the US–China trade war

Abstract

This paper examines upstream propagations of the US–China tariff war in 2018–2019 through the lens of exports from 32 countries to China. Building an industry-country specific measure of input–output linkages with China, we obtain new empirical evidence that the US tariffs on Chinese imports had a significant adverse effect on third countries by dampening Chinese demand for foreign inputs. A one standard deviation rise in this upstream shock leads to a decline in the growth rate of exports to China by 2.6 percentage points. A quantification exercise shows that the upstream propagation of US tariffs has incurred an average GDP loss of 0.05 percent for these countries. Taiwan and Korea, key suppliers to China, were most severely hurt by this vertical effect. Firm-level analysis using a panel of Korean manufacturers lends further support to the importance of this vertical linkage channel.

Appendices

More statistics and results

Table 6 Exports values and US vertical effect, by country

Fig. 6

Dynamic specification for US–China tariffs on sales growth of Korean firms. Figure plots regressions with full time dummies interacted with each tariff shock on the growth rate of sales for Korean firms. Shaded areas indicate 95% confidence intervals

Theoretical framework for vertical effects

This appendix describes a simple theoretical framework to illustrate how US–China tariff changes are associated with China’s demand for third countries’ intermediate inputs. Then, we derive the empirical measures of each vertical shock. The main purpose is to show that the vertical effect due to US tariffs on Chinese imports, the focus of our research, is robust to inclusions of alternative vertical channels related to Chinese tariffs on the US. The framework is in a partial equilibrium setting and focuses on the short-run impacts of the US–China tariffs. For simplicity, we assume one representative firm in each industry-country pair.

1.1 Technology

Consider a representative Chinese firm in tradable industry i that uses labour and multiple imported intermediate inputs to produce a single differentiated product in the following Cobb–Douglas production function:⁴⁶

$$\begin{aligned} Y_{i} = A_{i} L_{i}^{\alpha _{i}} \bigg (\prod _{j=1}^{N} \prod _{c=1}^{N_{c}} {X_{i,j(c)}}^{\gamma _{i,j(c)}} \bigg ) \end{aligned}$$

(B1)

where $A_{i}$ denotes firm productivity which is exogenously given, $L_{i}$ is labour and $X_{i,j(c)}$ indicate foreign imported input varieties j from origin country c that firm i uses, respectively. N and $N_{c}$ denote the numbers of industries and origin countries, respectively. Constant returns to scale implies $\alpha _{i} + \sum _{c}^{N_{c}} \sum _{j}^{N} \gamma _{i,j(c)} = 1$. Total production cost is⁴⁷:

$$\begin{aligned} TC_{i} = w L_{i} + \textstyle \sum _{j} \textstyle \sum _{c} \tau _{j(c)} v_{j(c)} X_{i,j(c)} \end{aligned}$$

(B2)

where w is nominal wage and $v_{j(c)}$ is the unit producer price of foreign intermediate input j(c) in the importer’s currency and $\tau _{j(c)}$ is the Chinese ad valorem tariff imposed on input j from origin country c. Cost minimization yields marginal cost as:

$$\begin{aligned} c_{i} = \frac{w^{\alpha _{i}}}{A_{i} \text{\O}mega _{i}} {\bigg [\prod _{j=1}^{N} \prod _{c=1}^{N_{c}} \big ( \tau _{j(c)} {v_{j(c)}} \big )^{\gamma _{i,j(c)}} \bigg ]} \end{aligned}$$

(B3)

where $\text{\O}mega _{i} = \alpha _{i}^{\alpha _{i}} \prod _{j=1}^{N} \prod _{c=1}^{N_{c}} ({\gamma _{i,j(c)}}^{\gamma _{i,j(c)}})$ is a collection of technology parameters.

1.2 Profit maximization

The Chinese firm sells its product to home and foreign markets. We assume that each firm faces monopolistic competition in each market and consumers have a CES preference over differentiated products with elasticity of substitution between products common across destinations ($\sigma$). Then, the residual demand faced by a firm i in home and foreign market d is, respectively:

$$\begin{aligned} Q_{i}^{H} = \bigg (\frac{p_{i}}{P} \bigg )^{-\sigma } E Q^{d}_{i} = \bigg (\frac{\tau _{i}^{d} p_{i}^{d}}{{P_{d}}} \bigg )^{-\sigma } E_{d} \end{aligned}$$

(B4)

where E and $E_{d}$ are demand shifters and P and $P_{d}$ are price indices for home and foreign market d. $p_{i}$ and $p_{i}^{d}$ are the prices set by Chinese firm i for home (H) and foreign market d. $\tau _{i}^{d}$ is the tariff imposed on Chinese firm i’s exports by importing country d. Profit maximization in each market yields the common optimal price:

$$\begin{aligned} p_{i} = p_{i}^{d} = \frac{\sigma }{\sigma -1} c_{i} \end{aligned}$$

(B5)

Market clearing for firm i’s output yields:

$$\begin{aligned} Y_{i} = Q^{H}_{i} + \textstyle \sum _{d}^{} Q^{d}_{i} \end{aligned}$$

(B6)

1.3 Foreign input demand

Let us turn to Chinese demand for non-US imported inputs. Combining Eqs. (B3)-(B6) together with the first-order condition for $X_{i,j(c)}$ in the cost minimization problem yields

$$\begin{aligned} \ln X_{i,j(c)} = \ln \gamma _{i,j(c)} - \ln v_{j(c)} + \ln c_{i} + \ln \bigg [Q^{H}_{i} + \textstyle \sum _{d}^{} Q^{d}_{i} \bigg ] \end{aligned}$$

(B7)

To focus on the short-run impact of tariffs, we assume that the terms related to firm technology ($A_{i}, \alpha _{i}, \gamma _{i,j(c)}$ and $\text{\O}mega _{i}$) and macroeconomic factors (w, E, $E_{d}$, P, $P_{d}$, $\forall d$) are not affected by the changes in tariffs between the US and China. We further assume that the unit producer prices of foreign inputs ($v_{j(c)}$) also remain unchanged. Total differentiation of Eq. (B7) with respect to US tariffs on China $\tau _{i}^{US}$ and China’s retaliatory tariffs on US inputs $\tau _{j(US)}^{CN}$ leads to:

$$\begin{aligned} d\ln X_{i,j(c)} = \underbrace{-\sigma \psi _{i}^{US} d\ln \tau _{i}^{US}}_{\text {Vertical effect due to US tariff}} + \underbrace{(1-\sigma ) {\textstyle \sum }_{j} \gamma _{i,j(US)} d\ln \tau _{j(US)}}_{\text {Vertical effect due to China tariff}} \end{aligned}$$

(B8)

where $\psi _{i}^{US}=\frac{Q^{US}_{i}}{Q^{H}_{i} + \sum _{d}^{} Q^{d}_{i}}$ denotes the fraction of Chinese firm i’s total output exported to US markets and $\gamma _{i,j(US)}$ is the cost share of input j sourcing from the US.

For non-US foreign input supplier j(c) ($\forall c \ne US$), the total demand change from across all Chinese tradable industries, with $X_{j(c)} = \sum _{i} X_{i,j(c)}$, can be expressed as:

$$\begin{aligned} d\ln X_{j(c)}&= \textstyle \sum _{i} \theta _{i,j(c)}^{F} d\ln X_{i,j(c)} \nonumber \\&= \textstyle \sum _{i} \theta _{i,j(c)}^{F} \Bigg [-\sigma \psi _{i}^{US} d\ln \tau _{i}^{US} + (1-\sigma ) \textstyle \sum _{j} \gamma _{i,j(US)} d\ln \tau _{j(US)} \Bigg ] \end{aligned}$$

(B9)

where $\theta _{i,j(c)}^{F}=\frac{X_{i,j(c)}^{F}}{\sum _{i} X_{i,j(c)}^{F}}$. Equation (B9) shows that the vertical linkage effect associated with US tariffs ($d\ln \tau _{i}^{US}$) is definitely negative as a demand-side effect. The sign of the vertical effect related to Chinese tariffs ($d\ln \tau _{j(US)}^{CN}$, $\forall j$) is determined by $(1-\sigma )$ which is negative as long as $\sigma >1$. To interpret, there are two different channels through which Chinese tariffs on US inputs could affect its demand for other foreign inputs. First is a “substitution effect”. The higher US input prices due to Chinese tariffs will result in substitution of the US imports into the other countries’ inputs. This is conceptually identical to the trade diversion effect mentioned in the text, except that the substitution effect in this section holds for intermediate inputs only. Second is that, due to complementarity between different inputs, higher US input prices driven by Chinese tariffs increase the production cost of Chinese firms. The resulting demand fall and profit loss of Chinese producers would eventually lead to a fall in China’s demand for every input, not only US inputs. This negative “production cost effect” is more pronounced if consumers are more price-elastic (higher $\sigma$).

So far, we assumed that the price index in Chinese market (P) remains unchanged. We may go one step further by relaxing this assumption and allow Chinese retaliatory tariffs on US imports of final goods in industry i ($\tau _{i(US)}$) to shift Chinese price index ($P = \big [p_{i}^{1-\sigma } + \textstyle \sum _{c} (\tau _{i(c)} p_{i(c)})^{1-\sigma }\big ]^{\frac{1}{1-\sigma }}$) upward, thereby strengthening the price competitiveness of Chinese producers in their home market. And this will lead to an increase in foreign input demand by Chinese producers. Incorporating this additional effect yields:

$$\begin{aligned} d\ln X_{j(c)} =&\textstyle \sum _{i} \theta _{i,j(c)}^{F} \Bigg [-\sigma \psi _{i}^{US} d\ln \tau _{i}^{US} + \sigma \psi _{i}^{H} \eta _{i(US)} d\ln \tau _{i(US)} \nonumber \\&+ (1-\sigma ) \textstyle \sum _{j} \gamma _{i,j(US)} d\ln \tau _{j(US)} \Bigg ] \end{aligned}$$

(B10)

where $\psi _{i}^{H}=\frac{Q^{H}_{i}}{Q^{H}_{i} + \sum _{d}^{} Q^{d}_{i}}$ is the fraction of Chinese industry i’s total output sold in Chinese home market and $\eta _{i(US)}$ is the US market share in Chinese markets of that industry i. The additional term ($\sigma \psi _{i}^{H} \eta _{i(US)} d\ln \tau _{i(US)}$) captures the potential gains in home market for Chinese producers due to Chinese tariffs on US final goods.

1.4 Linking theory to data

To build the empirical counterparts of each vertical shock from US–China tariffs that are consistent with Eq. (B10), we exploit the industry-level IO tables described in the text. The underlying assumption is that individual firms do not deviate systematically from the aggregate input-output structure of the industries to which they belong.

Two $\mathbf {N x N}$ matrices are constructed. First is $\theta ^{\mathbf{F}}_{\mathbf{c}}$, which is obtained by dividing China’s intermediate imports from each origin country c for each input-output industry pair by China’s total intermediate import of a given input industry from the origin country. $\gamma _{\mathbf{US}}$ is a matrix for the cost share of each US input in each Chinese industry’s total output. Both $\theta ^{\mathbf{F}}_{\mathbf{c}}$ and $\gamma _{\mathbf{US}}$ build on the origin country-specific import matrix that is constructed by disaggregating the sector-level World IO table proportionally to the industry-level Chinese detailed IO table.

$$\begin{aligned} \theta ^{\mathbf{F}}_{\mathbf{c}} = \begin{bmatrix} \theta _{1,1(c)}^{F} &{} \theta _{1,2(c)}^{F} &{} \cdots &{} \theta _{1,N(c)}^{F} \\ \theta _{2,1(c)}^{F} &{} \theta _{2,2(c)}^{F} &{} \cdots &{} \theta _{2,N(c)}^{F} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \theta _{N,1(c)}^{F} &{} \theta _{N,2(c)}^{F} &{} \cdots &{} \theta _{N,N(c)}^{F} \end{bmatrix} \quad \gamma _{\mathbf{US}} = \begin{bmatrix} \gamma _{1,1(US)} &{} \gamma _{1,2(US)} &{} \cdots &{} \gamma _{1,N(US)} \\ \gamma _{2,1(US)} &{} \gamma _{2,2(US)} &{} \cdots &{} \gamma _{2,N(US)} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \gamma _{N,1(US)} &{} \gamma _{N,2(US)} &{} \cdots &{} \gamma _{N,N(US)} \end{bmatrix} \end{aligned}$$

Table 7 Testing for alternative vertical shocks

We use $\psi ^{\mathbf{US}}$, $\psi ^{{\mathbf {H}}}$ and $\eta ^{\mathbf {US}}$ to denote $\mathbf {N x 1}$ vectors for $\psi _{i}^{US}$, $\psi _{i}^{H}$ and $\eta _{i}^{US}$, constructed using Chinese detailed IO for 2012 and trade share between the US and China for 2016. Likewise, $\tau _{\mathbf {CN,t}}^{\mathbf {US}}$, $\tau _{\mathbf {J(US),t}}^{\mathbf {CN}}$ and $\tau _{\mathbf {US,t}}^{\mathbf {CN}}$ are vectors of US tariffs on China at time t, Chinese tariffs on US intermediates and on US non-intermediates within each CSIC industry, respectively. Then, based on Eq. (B10), we build three distinct $\mathbf {N x 1}$ vectors of vertical shocks as follows:

$$\begin{aligned}&{\textit{Vertical Shock from US tariffs}} = \theta ^{{\mathbf {F}}}_{{\mathbf {c}}}*(\psi ^{\mathbf {US}} \circ \varvec{\Delta } \tau _{\mathbf {CN,t}}^{\mathbf {US}}) \end{aligned}$$

(B11)

$$\begin{aligned}&{\textit{Vertical Shock from CN tariffs on US inputs}} = \theta ^{{\mathbf {F}}}_{{\mathbf {c}}}*(\gamma _{\mathbf {US}} \circ \varvec{\Delta } \tau _{\mathbf {J(US),t}}^{\mathbf {CN}}) \end{aligned}$$

(B12)

$$\begin{aligned}&{\textit{Vertical Shock from CN tariffs on US final goods}} = \theta ^{{\mathbf {F}}}_{{\mathbf {c}}}*(\psi ^{{\mathbf {H}}} \circ \eta ^{\mathbf {US}} \circ \varvec{\Delta } \tau _{\mathbf {US,t}}^{\mathbf {CN}}) \end{aligned}$$

(B13)

where $\circ$ denotes element-wise multiplication. The formula (B11) is a vector expression of the benchmark measure 1 of the US vertical shock in the text except that we use the US share in the China’s total export of industry i as $\psi _{i}^{US}$ (not the US share in China’s total output of that industry) based on the discussion in Sect. 4.1 of the main text. (B12) and (B13) express the newly constructed measures of vertical shocks related to Chinese tariffs on US imports of intermediates and final goods, respectively.

Table 7 shows the results including two additional vertical channels of (B12) and (B13), along with the US vertical effect (B11). These two alternative shocks related to Chinese retaliatory tariffs were not statistically significant. The US vertical effect, on the other hand, remains highly significant with little changes in magnitude across different specifications. These confirm that the US vertical shock is the major channel in propagations of the US–China tariff shocks to third countries at the current level of aggregation and the time horizon.