Interregional price linkages of fossil-energy and food sectors: evidence from an international input–output analysis using the GTAP database

Abstract

This study examined the impacts of price changes in fossil-energy resources on food prices across the world. We approached this issue with the input–output (IO) price model, which is suitable for evaluating the price effects due to cost-push (supply driven) processes through transaction linkages of intermediate goods. We employ an international interregional IO table comprising 14 regions and 57 industrial sectors, which is compiled from the global trade analysis project (GTAP) 9 database. The results revealed: (1) interregional price linkages are particularly strong between a wheat-producing region and energy producers in the adjacent regions; and (2) the world oil sector appears to exert the strongest effects on mechanized sectors (wheat, plant fibers and fishing) and regions practicing advanced agriculture (East Asia, the US, and Russia). Furthermore, a comparison with empirical observations suggests that approximately one tenth of observed 2-year price changes in US wheat in 2004–2016 can be explained by the cost-push processes derived from the world oil price changes.

Appendix 1: Mathematical derivation of Eq. (6)

This appendix explains a mathematical derivation for Eq. (6) in the main text, on the basis of Tamamura (2014). Assume an economy with n goods (sectors) and one region for notational simplicity. We set to describe a situation in which the price of the nth good is exogenously increased from $p_{n}$ to $p_{n} + \Delta p_{n}$, which induces price increases in the other n − 1 goods in response, via cost-push processes, from $p_{i}$ to $p_{i} + \Delta p_{i} (i = 1, \ldots ,n - 1)$.

We begin by partitioning the input coefficient matrix A into four sub-matrices:

$$\varvec{A} = \left[ {\begin{array}{*{20}c} {\varvec{A}_{11} } & {\varvec{A}_{12} } \\ {\varvec{A}_{21} } & {\varvec{A}_{22} } \\ \end{array} } \right],$$

(8)

where

$$\varvec{A}_{11} = \left[ {\begin{array}{*{20}c} {a_{11} } & \cdots & {a_{1,n - 1} } \\ \vdots & \ddots & \vdots \\ {a_{n - 1,1} } & \cdots & {a_{n - 1,n - 1} } \\ \end{array} } \right],\;\varvec{A}_{12} = \left[ {\begin{array}{*{20}c} {a_{1n} } \\ \vdots \\ {a_{n - 1,n} } \\ \end{array} } \right],\;\varvec{A}_{21} = \left[ {\begin{array}{*{20}c} {a_{n1} } & \cdots & {a_{n,n - 1} } \\ \end{array} } \right],\;\varvec{A}_{22} = \left[ {a_{nn} } \right]$$

Notice when A is transposed, the corresponding sub-matrices become:

$$\varvec{A}^{\prime} = \left[ {\begin{array}{*{20}c} {\varvec{A}_{11}^{\prime} } & {\varvec{A}_{21}^{\prime} } \\ {\varvec{A}_{12}^{\prime} } & {\varvec{A}_{22} } \\ \end{array} } \right]$$

(9)

First, using these sub-matrices, we rearrange Eq. (4) with respect to goods 1 to n − 1:

$$\varvec{A}_{11}^{\prime} \varvec{p}_{1} + \varvec{A}_{21}^{\prime} p_{n} + \varvec{v}_{1} = \varvec{p}_{1}$$

(10)

where $\varvec{p}_{1}$ and $\varvec{v}_{1}$ are the vectors of price and value-added of goods 1 to n − 1, respectively.

When the price of the nth good is raised, the prices of the other goods are increased in response, as mentioned above, and the economy will reach another price equilibrium:

$$\varvec{A}_{{\text{11}}}^{\prime} \left( {\varvec{p}_{1} + \Delta \varvec{p}_{1} } \right) + \varvec{A}_{21}^{\prime} \left( {p_{n} + \Delta p_{n} } \right) + \varvec{v}_{1} = \varvec{p}_{1} + \Delta \varvec{p}_{1}$$

(11)

Subtracting Eq. (10) from (11) gives:⁸

$$\begin{aligned} \varvec{A}_{11}^{\prime } \Delta \varvec{p}_{1} + \varvec{A}_{21}^{\prime } \Delta p_{n} = \Delta \varvec{p}_{1} \; \Leftrightarrow \hfill \\ \Delta \varvec{p}_{1} = \left( {\varvec{I} - \varvec{A}_{11}^{\prime } } \right)^{ - 1} \varvec{A}_{21}^{\prime } \Delta p_{n} \hfill \\ \end{aligned}$$

(12)

Second, we define the transposed Leontief inverse matrix in four sub-matrices corresponding to Eq. (9):

$$\varvec{B}^{{\prime }} = \left( {\varvec{I} - \varvec{A}^{{\prime }} } \right)^{ - 1} = \left[ {\left( {\varvec{I} - \varvec{A}} \right)^{ - 1} } \right]^{{\prime }} = \left[ {\begin{array}{*{20}c} {\varvec{B}_{11}^{{\prime }} } & {\varvec{B}_{21}^{{\prime }} } \\ {\varvec{B}_{12}^{{\prime }} } & {\varvec{B}_{22} } \\ \end{array} } \right]$$

(13)

where the sub-matrices B₁₁ to B₂₂ are defined in correspondence with A₁₁ to A₂₂ in Eq. (8).

Multiplying $\left( {\varvec{I} - \varvec{A}^{{\prime }} } \right)$ and $\varvec{B}^{{\prime }}$ (Eq. 13) gives:

$$\left( {\varvec{I} - \varvec{A}^{{\prime }} } \right)\varvec{B} = \left[ {\begin{array}{*{20}c} {\varvec{I}_{{\varvec{n} - 1}} - \varvec{A}_{11}^{{\prime }} } & { - \varvec{A}_{21}^{{\prime }} } \\ { - \varvec{A}_{12}^{{\prime }} } & {\varvec{I}_{1} - \varvec{A}_{22} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varvec{B}_{11}^{{\prime }} } & {\varvec{B}_{21}^{{\prime }} } \\ {\varvec{B}_{12}^{{\prime }} } & {\varvec{B}_{22} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\varvec{I}_{{\varvec{n} - 1}} } & {\varvec{O}_{{\varvec{n} - 1}} } \\ {\varvec{O}_{{\varvec{n} - 1}}^{{\prime }} } & {\varvec{I}_{1} } \\ \end{array} } \right],$$

(14)

where I_n is the n-dimensional identity matrix and O_n is the n-dimensional zero vector.

Hence, we have:

$$\begin{aligned} \left( {\varvec{I} - \varvec{A}_{11}^{{\prime }} } \right)\varvec{B}_{21}^{{\prime }} + \left( { - \varvec{A}_{21}^{{\prime }} } \right)\varvec{B}_{22} = \varvec{O}_{{\varvec{n} - 1}} \; \Leftrightarrow \hfill \\ \varvec{B}_{21}^{{\prime }} /b_{nn} = \left( {\varvec{I} - \varvec{A}_{11}^{{\prime }} } \right)^{ - 1} \varvec{A}_{21}^{{\prime }} \hfill \\ \end{aligned}$$

(15)

Finally, inserting Eq. (15) into (12) gives:

$$\Delta \varvec{p}_{1} = \left( {\varvec{B}_{21}^{{\prime }} /b_{nn} } \right)\Delta p_{n}$$

(16)

This proves Eq. (6).