# A nonsurvey multiregional input–output estimation allowing cross-hauling: partitioning two regions into three or more parts

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## Abstract

This paper describes a nonsurvey method for estimating multiregional trades without eliminating cross-hauling, when a national biregional input–output table is available. Domestic outflows are assigned by interpolating the biregional trades on the basis of the gravity ratio between the origin and the destinations, with parameters estimated from an earlier survey on interregional transactions. The method is then applied to evaluate multiregional industrial waste disposal and landfill attributed to consumption in the city of Nagoya. Three-regional input–output tables with and without cross-hauling are estimated by partitioning the biregional table between Aichi prefecture and the rest of Japan.

### JEL Classification

C67 D57 R15## 1 Introduction

Regional economic impact is well assessed by relevant local area multipliers. Multiregional input–output (MRIO) analysis may be one of the primary models for incorporating regional interdependencies into an input–output framework. In Chenery-Moses-type MRIO model, the intra-regional coefficient matrices are located along the main diagonal, while another cross-regional trade matrix functions to incorporate the cross-regional effects (Hewings and Jensen 1987; Oosterhaven and Polenske 2009). It is said that MRIO models have an advantage over Isard-type interregional (IRIO) models, as they are able to use data that are more available (Polenske and Hewings 2004).^{1} Despite the simplification in the MRIO models, however, collecting real data of cross-regional trades is very costly.

Thus, preceding papers with nonsurvey approaches employed location quotients (LQ) as the primary reference to cross-regional trades (see e.g., Isserman 1977). With this approach, the domestic outflows and inflows (cross-regional trades) are estimated independent of the other figures such as the regional control totals, final demand, and imports and exports in the multiregional table. While LQ techniques are convenient to use, they also have some limitations; these techniques inevitably eliminate *cross-hauling* in cross-regional trades.^{2} Without cross-hauling, as suggested by many articles such as Roginson and Miller (1988), there is the risk of underestimating the regional propagation effect.

Consistent cross-regional trades can be estimated using given regional input–output tables. If we set the estimates on all the regional figures (including regional imports and exports) besides net regional trades, we can restrict to some extent the degrees of freedom in cross-regional trades estimation, as done in the commodity balance (CB) approach. In such cases, biproportional matrix reconciliation techniques using reference regional trades can be applied (see e.g., Lahr 2004; Canning and Wang 2005). However, the estimate will not include cross-hauling unless the reference domestic trades include cross-hauling. One-way method is to apply gravity trade flow models (Olson 1972) that permit cross-hauling. Regression-type gravity models (e.g., Begg 1985) have been applied and have shown to produce results by and large close to the survey data (Riddington et al. 2006). Kronenberg (2010) estimated biregional trades using the Leontief-Strout-type exact solution nonsurvey method that allows cross-hauling, which was then discussed in Flegg and Tohmo (2011).

Meanwhile, there are requests that the given multiregional transactions be disaggregated into smaller regions, such as in the case of Japan, rather than obtaining all cross-regional trades among the given regions. In other words, our objective in the study is to decompose one of the regions of a given cross-hauled MRIO table into two to obtain a more detailed MRIO table while maintaining the given structure of regional transactions. In such a context, however, the approaches mentioned above could spoil the original measured transactions. Thus, we consider an approach that requires less involvement. We partition the regional outflows into disaggregated regions where the inflows are determined accordingly. Further, for reallocating outflows, we use the gravity ratios.

For our empirical study, we use the biregional table of Aichi prefecture and the rest of Japan, 2005, and apply the calculation in order to disaggregate Aichi into Nagoya and the rest of Aichi, thus producing a three-region multiregional table. For evaluating multiregional transactions, we call on the gravity ratios that rule the outflow split between regions, in addition to meeting the entire commodity balance; the gravity ratios can be obtained from the market sizes and the distances between regions with the aid of the gravity parameters, which we estimate using the reference nine-region multiregional table of Japan. This method hence allows to partition a cross-hauled biregional table into three or more parts in an arbitrary manner, while being consistent with the original regional transactions. We proceed to use this three-region table for the analysis of industrial waste and landfill, which are attributed to the exogenous consumption boost in Nagoya.

The rest of the paper is organized as follows. In the next section, we introduce models with and without cross-hauling for partitioning biregional input–output models. Section 3 describes the estimation of gravity parameters of regional trades using survey data, along with the population-weighted distances across the regions. In Sect. 4, we introduce data on regional industrial waste generation and perform a multiregional analysis with and without cross-hauling. Section 5 concludes the paper.

## 2 The model

### 2.1 Regional partitioning

We now partition formula (1) into region \(i\) and the rest of the nation. In this event, \({\mathbf x }\) is divided into its proportion using the number of a shipments and employees. As for the final demand \({\mathbf f }_i\), we use the value-added (row) vector for nonhousehold expenses; for households, we may divide in proportion to the number of households; we may divide household expenses in proportion to the number of households, and government expenses, in proportion to the expenses of local governments. As for foreign imports \({\mathbf m }_i\) and exports \({\mathbf e }_i\), we may consult on the regional data, at least in the case of Japan, or divide them in proportion to market sizes such as total outputs and total domestic final use. We note that \({\mathbf A }_i\) should be estimated separately if possible, but we may assume that it is the same as the nationwide \(\mathbf A \) matrix if there is no other way.

*net*domestic inflows \(\mathbf s _i\) as follows:

^{3}

### 2.2 Cross-regional trades

Let us now verify the number of unknowns and equations. The unknowns are \(\mathbf t _{ij} \, (i, j = 1,\ldots ,R)\) while omitting the intra-regional transactions \(i=j, \mathbf h _i \, (i = 1,\ldots ,R)\) and \(\mathbf n _j \, (j = 1,\ldots ,R)\), which total to \(R^2 + R\) unknown variables.^{4} On the other hand, independent equations are (3) and (5), total to \(3R-2\). Hence, we must specify the system further in order to set all the unknown variables. In what follows, we presuppose that cross-hauled transactions in one region is available. For this region \(R\), we know \(\mathbf h _R\) and \(\mathbf n _R\). Thus, there are \(3R-2\) independent equations with \(R^2+R-2\) unknowns so that we will need \(R^2-2R\) more independent equations to specify the domestic trades. We will use the gravity ratio described in the next subsection to obtain these equations.

### 2.3 Multiregional outflow ratio

### 2.4 Three-region case

*without*cross-hauling as long as there are three-regions or less. If there is no cross-hauling, every region is either a domestic importer or an exporter; that is, we must have

Under condition (7), there will be \(R-1\) independent equations and at most \((2R^2-1+(-1)^R)/8\) unknowns in this case.^{5} The number of independent equations and the unknowns will necessarily coincide only when \(R \le 3\). This feature is also mentioned in Begg (1985). Hence, we can now estimate cross-regional trades with and without cross-hauling for the three-region models in this framework. We will accordingly compare the propagation effects later.

## 3 Estimation of gravity parameters

### 3.1 Distances between regions

We use the nine-region multiregional table of Japan to estimate the gravity parameters for Eq. (9) below. Prior to carrying out the regression, we ought to have the distances between regions, that is, \(d_{ij}\) for all regions \(i\) and \(j\). In this study, we use the population-weighted distances as described below.

Population-weighted distances (day)

Region | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

1 Hokkaido | ||||||||

2 Tohoku | 0.68 | |||||||

3 Kanto | 0.84 | 0.17 | ||||||

4 Chubu | 0.99 | 0.33 | 0.16 | |||||

5 Kinki | 1.09 | 0.43 | 0.29 | 0.13 | ||||

6 Chugoku | 1.22 | 0.56 | 0.42 | 0.26 | 0.16 | |||

7 Shikoku | 1.28 | 0.62 | 0.48 | 0.33 | 0.19 | 0.13 | ||

8 Kyushu | 1.40 | 0.75 | 0.60 | 0.45 | 0.33 | 0.14 | 0.31 | |

9 Okinawa | 3.25 | 2.60 | 2.45 | 2.30 | 2.18 | 2.05 | 2.16 | 1.86 |

### 3.2 Gravity parameters

^{6}There are 53 sectors total while the table excludes three sectors with unobserved trades, namely, rental housing, public service, and others. We consider the observations for eight regions, and for each region, the observations are given by a combination of the number of regions sans the origin and a pair of regions (\(_7\text{ C}_2\)); as such, the total number of observations is \(8 \times \, {}_7\text{ C}_2 = 168\) (ad extremum).

Estimation of gravity parameters for each sector

Sectors (\(c \in C\)) | \(\gamma \) | \(-\delta \) | \(R^2\) | \(N\) |
---|---|---|---|---|

Agriculture, fishery, and forestry | \(0.949 (13.11)^{***}\) | \(1.208 (15.79)^{***}\) | 0.694 | 168 |

Mining | \(0.386 (2.02)^{**}\) | \(1.058 (5.30)^{***}\) | 0.318 | 168 |

Coal oil and natural gas | \(1.691 (3.06)^{**}\) | \(-6.824 (-5.14)^{***}\) | 0.732 | 9 |

Food and beverages | \(0.828 (27.06)^{***}\) | \(0.915 (21.21)^{***}\) | 0.892 | 168 |

Textile industry products | \(0.662 (13.92)^{***}\) | \(0.211 (2.51)^{**}\) | 0.700 | 150 |

Apparel and textile products | \(1.163 (15.91)^{***}\) | \(0.583 (5.78)^{***}\) | 0.692 | 168 |

Lumber and furniture | \(1.296 (21.13)^{***}\) | \(0.782 (10.72)^{***}\) | 0.814 | 168 |

Pulp and paper | \(1.140 (18.31)^{***}\) | \(0.858 (11.00)^{***}\) | 0.787 | 168 |

Printing and binding | \(1.047 (12.61)^{***}\) | \(1.023 (7.34)^{***}\) | 0.612 | 168 |

Basic chemical products | \(0.566 (10.04)^{***}\) | \(0.910 (8.35)^{***}\) | 0.717 | 162 |

Synthetic resin | \(0.747 (23.14)^{***}\) | \(0.374 (4.61)^{***}\) | 0.863 | 147 |

Final chemical products | \(0.649 (19.39)^{***}\) | \(0.446 (6.67)^{***}\) | 0.781 | 168 |

Pharmaceutical products | \(0.495 (12.78)^{***}\) | \(0.205 (3.37)^{***}\) | 0.592 | 168 |

Petroleum and coal products | \(0.835 (6.21)^{***}\) | \(1.703 (8.14)^{***}\) | 0.534 | 168 |

Plastic products | \(0.499 (16.55)^{***}\) | \(0.848 (12.77)^{***}\) | 0.797 | 168 |

Ceramic and clay products | \(1.016 (14.79)^{***}\) | \(0.958 (10.52)^{***}\) | 0.757 | 168 |

Iron and steel | \(0.634 (14.72)^{***}\) | \(0.759 (7.34)^{***}\) | 0.727 | 168 |

Nonferrous metal | \(0.745 (16.46)^{***}\) | \(0.504 (4.79)^{***}\) | 0.743 | 168 |

Metal products | \(0.699 (13.02)^{***}\) | 0.847 (9.05) | 0.710 | 162 |

General machinery | \(0.643 (22.98)^{***}\) | \(0.126 (2.07)^{**}\) | 0.827 | 168 |

Office and service machinery | \(0.608 (12.01)^{***}\) | \(-0.047 (-0.33)\) | 0.531 | 141 |

Industrial electrical equip. | \(0.751 (22.53)^{***}\) | \(0.505 (7.38)^{***}\) | 0.820 | 162 |

Other electrical machinery | \(0.920 (22.18)^{***}\) | \(-0.072 (-0.92)\) | 0.783 | 162 |

Consumer electric equip. | \(0.589 (13.15)^{***}\) | \(0.088 (0.93)\) | 0.588 | 150 |

Telecommunication equip. | \(0.721 (19.16)^{***}\) | \(0.275 (3.88)^{***}\) | 0.744 | 168 |

Computers and related devices | \(0.926 (6.28)^{***}\) | 0.192 (0.80) | 0.281 | 112 |

Electronic components | \(0.763 (15.36)^{***}\) | 0.083 (0.63) | 0.654 | 168 |

Passenger cars | \(0.634 (25.92)^{***}\) | 0.073 (1.41) | 0.865 | 126 |

Other automobiles | \(0.170 (4.18)^{***}\) | \(-0.198 (-1.69)^{*}\) | 0.095 | 136 |

Auto parts and accessories | \(0.667 (20.34)^{***}\) | \(0.431 (3.13)^{***}\) | 0.775 | 162 |

Other transportation equip. | \(0.705 (10.42)^{***}\) | \(0.675 (6.48)^{***}\) | 0.528 | 168 |

Precision machinery | \(0.779 (15.75)^{***}\) | \(0.319 (3.79)^{***}\) | 0.658 | 168 |

Other manufactured products | \(0.866 (16.65)^{***}\) | \(0.335 (3.79)^{***}\) | 0.684 | 168 |

Renewables recovery | \(0.540 (4.69)^{***}\) | \(0.812 (3.55)^{***}\) | 0.378 | 156 |

Construction | \(0.630 (7.39)^{***}\) | \(0.446 (4.16)^{***}\) | 0.350 | 168 |

Electric power | \(2.121 (10.82)^{***}\) | \(1.247 (5.62)^{***}\) | 0.590 | 162 |

Gas and heat supply | \(0.330 (4.90)^{***}\) | \(0.712 (5.30)^{***}\) | 0.352 | 162 |

Water and waste processing | \(-0.214 (-2.37)^{**}\) | \(0.815 (7.09)^{***}\) | 0.365 | 151 |

Commerce | \(0.547 (32.66)^{***}\) | \(0.966 (31.75)^{***}\) | 0.932 | 168 |

Finance and insurance | \(0.892 (11.34)^{***}\) | \(-0.295 (-2.62)^{***}\) | 0.433 | 168 |

Real estate | \(0.402 (3.57)^{***}\) | \(1.077 (6.07)^{***}\) | 0.261 | 168 |

Transportation | \(0.938 (25.53)^{***}\) | \(0.858 (15.60)^{***}\) | 0.892 | 168 |

Other communications | \(0.881 (15.80)^{***}\) | \(1.087 (12.69)^{***}\) | 0.762 | 168 |

Information services | \(1.103 (13.97)^{***}\) | \(1.108 (7.00)^{***}\) | 0.667 | 151 |

Education and research | \(0.934 (15.68)^{***}\) | \(0.685 (8.35)^{***}\) | 0.744 | 168 |

Health care and social security | \(2.318 (20.25)^{***}\) | \(2.129 (17.72)^{***}\) | 0.862 | 157 |

Advertising | \(0.530 (12.72)^{***}\) | \(2.208 (23.87)^{***}\) | 0.838 | 168 |

Rental and leasing services | \(0.352 (11.53)^{***}\) | \(0.810 (14.86)^{***}\) | 0.735 | 168 |

Other office services | \(0.663 (13.10)^{***}\) | \(1.028 (12.53)^{***}\) | 0.715 | 168 |

Consumer service | \(1.067 (15.89)^{***}\) | \(1.834 (18.63)^{***}\) | 0.820 | 168 |

The estimates are fairly satisfactory, except for some sectors presenting signs opposite to the expected direction. For Coal oil and natural gas, a very small sample size representing the fact that domestic production and thus transactions are nearly absent, or, if any, being very specific. Similarly, parameters on the distance variables for office and service machinery, and other automobiles sectors may have been affected by specific factory locations. On the other hand, water and waste processing is well preferred in an underpopulated region; hence, the parameter on the demand variable should have a negative sign.

## 4 Application

### 4.1 Multiregional table for Aichi

For our empirical study, we use MRIO analysis to estimate industrial waste and final landfill resulting from the change in consumption patterns of Nagoya citizens. Specifically, we investigate how much final landfill is propagated owing to a 10 % proportional increase in the final demand bundle of Nagoya.^{7} For this purpose, we first prepare a three-region multiregional table for Nagoya (region 1), the rest of Aich (region 2), and the rest of Japan (region 3) by partitioning the available biregional table between Aichi (regions 1 and 2) and the rest of Japan (region 3). Then, we use the wastes disposal table for different regions by different types of wastes in order to calculate the change in total landfill of industrial wastes during our sample period in Nagoya. Thus, we use the change in the exogenous final demand in Nagoya (region 1) and calculate the regional propagation effects using Eqs. (11) and (12).

In partitioning Aichi’s table (APG 2010), we use Nagoya’s share of production for the control total in each sector, while we use the same input coefficient matrix for both regions.^{8} For the final demand, we use the value-added (row) vector for nonhousehold expenses; for households we divided in proportion to the number of households; for government expenses we divided in proportion to the expenses of local governments. For fixed capital formation, we use the national capital coefficients with respect to the final output. As for imports and exports, we use the survey data for Nagoya.

Cross-regional trades are estimated using the model described earlier, with gravity ratios estimated by the population-weighted distances among three-regions, namely, \(d_{12} = d_{21} = 0.028\) [day], \(d_{23} = d_{32} = 0.345\) [day], and \(d_{13} = d_{31} = 0.347\) [day]. As mentioned earlier, we naturally prepare two tables, that is, with and without cross-hauling, since there are just three-regions. As for the sectors that do not have cross-hauling in the biregional table, we assume to not have cross-hauling in the partitioned table also.

### 4.2 Multiregional analysis

### 4.3 Results

Exogenous change \(\varDelta {f}\) [MJPY], propagation effects \(\varDelta {x}\) [MJPY], and waste generation coefficients \({G}\) [Ton/MJPY]

Sectors | Region 1 | Region 2 | Region 3 | ||||
---|---|---|---|---|---|---|---|

\(\varDelta f_1\) | \(\varDelta x_1\) | \(G_1\) | \(\varDelta x_2\) | \(G_2\) | \(\varDelta x_3\) | \(G_3\) | |

Agriculture | 634 | 25 | 2.29 | 582 | 6.47 | 1,315 | 8.52 |

Mining | \(-26\) | 1 | 2.78 | 10 | 1.40 | 155 | 13.92 |

Construction | 12,815 | 12,378 | 1.22 | 1,676 | 1.37 | 540 | 1.21 |

Food | 4,138 | 1,738 | 0.21 | 113 | 0.16 | 3,897 | 0.42 |

Drinks and feeds | 1,860 | 91 | 0.81 | 499 | 0.09 | 1,920 | 0.27 |

Textiles | 60 | 94 | 0.54 | 24 | 0.52 | 61 | 0.44 |

Clothing | 833 | 61 | 0.09 | 6 | 0.08 | 158 | 0.05 |

Timber | 25 | 142 | 0.49 | 23 | 0.27 | 191 | 0.62 |

Furniture | 141 | 118 | 0.06 | 14 | 0.07 | 194 | 0.11 |

Pulp and paper | 68 | 189 | 1.58 | 335 | 1.56 | 914 | 4.49 |

Publishing and print | 23 | 471 | 0.21 | 48 | 0.19 | 585 | 0.17 |

Chemical products | 648 | 436 | 1.10 | 546 | 0.33 | 2,492 | 0.61 |

Oil and coal products | 1,158 | 35 | 0.00 | 168 | 0.15 | 2,532 | 0.10 |

Plastic products | 173 | 147 | 0.08 | 352 | 0.06 | 1,066 | 0.11 |

Rubber products | 84 | 30 | 0.11 | 50 | 0.20 | 252 | 0.11 |

Leather products | 228 | 2 | 0.00 | 3 | 0.00 | 136 | 0.14 |

Ceramic soil products | 92 | 129 | 0.76 | 257 | 0.77 | 816 | 1.39 |

Iron and steel | 58 | 260 | 0.47 | 372 | 1.69 | 2,268 | 1.71 |

Nonferrous products | 16 | 137 | 0.81 | 126 | 0.14 | 508 | 0.49 |

Metal products | 155 | 604 | 0.29 | 261 | 0.21 | 1,376 | 0.19 |

General machinery | 2,372 | 2,565 | 0.10 | 442 | 0.07 | 204 | 0.06 |

Electric machinery | 4,116 | 386 | 0.10 | 2,275 | 0.05 | 2,255 | 0.09 |

Cars and trucks | 2,575 | 788 | 0.11 | 2,850 | 0.14 | 1,057 | 0.07 |

Precision machinery | 578 | 172 | 0.00 | 20 | 0.02 | 402 | 0.06 |

Other products | 672 | 157 | 0.25 | 88 | 0.21 | 634 | 0.12 |

Electric power | 902 | 950 | 0.02 | 505 | 0.97 | 1,344 | 0.63 |

Gas and heat | 280 | 505 | 0.02 | 13 | 0.12 | 50 | 0.06 |

Water | 237 | 614 | 0.05 | 31 | 1.66 | 153 | 19.08 |

Transportation | 3,171 | 4,142 | 0.02 | 659 | 0.02 | 3,515 | 0.01 |

Commerce | 13,022 | 17,273 | 0.03 | 298 | 0.04 | 1,837 | 0.02 |

Services | 39,534 | 50,127 | 0.02 | 1,596 | 0.02 | 4,981 | 0.01 |

Unclassified | 16,281 | 18,199 | 0.00 | 3,596 | 0.00 | 2,618 | 0.00 |

Total | 106,920 | 112,965 | 17,840 | 40,425 |

Overall effects in industrial wastes \(\varDelta {W}\) [Ton]

| Region 1 | Region 2 | Region 3 | |||
---|---|---|---|---|---|---|

Generated | Landfill | Generated | Landfill | Generated | Landfill | |

Ash | 71 | 35 | 73 | 30 | 172 | 32 |

Sewage | 3,782 | 386 | 1,188 | 75 | 13,480 | 674 |

Oil/fat | 286 | 13 | 62 | 3 | 241 | 7 |

Acid | 47 | 4 | 15 | 1 | 263 | 19 |

Alkaline | 317 | 10 | 58 | 2 | 166 | 14 |

Plastic | 819 | 241 | 146 | 39 | 396 | 128 |

Paper | 234 | 17 | 92 | 8 | 144 | 9 |

Wood | 575 | 66 | 85 | 10 | 181 | 14 |

Fiber | 9 | 0 | 2 | 1 | 2 | 0 |

Residue | 155 | 12 | 28 | 2 | 515 | 18 |

Rubber | 6 | 6 | 2 | 1 | 4 | 2 |

Metal | 1,092 | 65 | 513 | 11 | 707 | 40 |

Glass | 297 | 71 | 93 | 19 | 327 | 108 |

Tailing | 172 | 15 | 576 | 22 | 2,263 | 197 |

Rubble | 11,372 | 845 | 1,780 | 136 | 690 | 33 |

Manure | 58 | 0 | 3,771 | 0 | 11,158 | 167 |

Carcass | 0 | 0 | 0 | 0 | 25 | 4 |

Dust | 108 | 9 | 522 | 61 | 1,532 | 241 |

Total | 19,399 | 1,794 | 9,004 | 421 | 32,268 | 1,707 |

Comparison of propagation effects with and without cross-hauling

| Exogenous \(\varDelta {f}\) [MJPY] | With cross-hauling | Without cross-hauling | ||
---|---|---|---|---|---|

Propagation | Landfill | Propagation | Landfill | ||

\(\varDelta {x}\) [MJPY] | \(\varDelta {W}\) [ton] | \(\varDelta {x}\) [MJPY] | \(\varDelta {W}\) [ton] | ||

Region 1 | 106,920 | 112,965 | 1,794 | 122,352 | 2,012 |

Region 2 | 0 | 17,840 | 421 | 22,780 | 526 |

Region 3 | 0 | 40,425 | 1,707 | 25,806 | 1,062 |

Total | 106,920 | 171,230 | 3,923 | 170,938 | 3,600 |

Finally, in Table 5, we compare the propagation effects, as well as the landfill abatement effects, with and without cross-hauling. The exogenous change in the final demand, as mentioned earlier, is an increase of about JPY 107 billion in total. The propagation effects are essentially identical in terms of total propagation (JPY 171 billion) in both cases, while its distribution among regions differs in the two cases. That is, propagation in region 1 is greater without cross-hauling while that in region 3 is less in the same case. As the inner propagation (region 1) is greater than outside regions in both cases and as waste and landfill coefficients are smaller inside and greater outside, differences in the regional propagations will not enhance the differences in the final landfill, but we may still observe differences. This study shows that unless cross-hauling is used, there exists the risk of underestimating final landfill.

## 5 Concluding remarks

In this paper, we proposed a nonsurvey method for estimating multiregional trades without eliminating cross-hauling, when a national biregional input–output table is available. Domestic outflows are assigned by interpolating the biregional trades on the basis of the gravity ratio between the origin and the destinations, with parameters estimated from a detailed survey on multiregional trades. The method is then applied to evaluate cross-regional industrial waste and final landfill propagation. We compiled three-region MRIO tables, including Nagoya, with and without cross-hauling by partitioning the biregional table of Aichi that includes Nagoya, and the rest of Japan. Although the propagation effects in monetary terms for the two cases (with and without cross-hauling) coincide in total, they have different distributions among regions such that different regional characteristics of industrial waste processing lead to differences in assessing the overall landfill abatement, initiated by an artificial consumption boost in Nagoya.

## Footnotes

- 1.
For definitions of MRIO and IRIO, we rely on Polenske (1995).

- 2.
There are many variations to the LQ techniques, those proposed by Round (1972), Morrison and Smith (1974), Flegg and Webber (2000), for example, but all without cross-hauling. Richardson (1985), and, more recently, Gallego and Lenzen (2009) give a thorough review of the regional input–output framework.

- 3.
We mean that there are \(R - 1\) independent vector equations of dimension \(C\), where \(C\) denotes the number of sectors.

- 4.
We have \(R^2-R\) unknowns for the interregional transaction vectors and \(2R\) for the outflow and inflow vectors.

- 5.
The proof is standard and is therefore omitted.

- 6.
We checked the robustness of the ordinary specification of the gravity model in the Appendix.

- 7.
Nagoya is the largest city in Aichi prefecture and the fourth-largest city in Japan.

- 8.

## Notes

### Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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