Inter-regional input-output and multiregional input-output

Input-Output Model and its Application

3.8 Varieties of additional concepts on input-output modeling

3.8.3 Inter-regional input-output and multiregional input-output

What we have seen are the single region/nation I-O model in which we classify one region as endogenous and all other regions as exogenous. As no nation can exist without varying degrees of trade with the RoW, there are trade activities of exports and imports in the transactions table. If a researcher is interested in details of flows of inputs and outputs, there are two methods to capture the details of regional linkages.

3.8.3.1 Inter-regional input-output

In the inter-regional input-output (IRIO) model, which is also referred to as the Isard model as Isard developed this structure in 1951, each region ’ s industrial sectors over the other region ’ s industrial sectors are shown in the form of submatrices. At the transactions table level, which we can use Z to represent a combined transactions table. To make it simple, we can use two regions, north region (N) and south region (S). The IRIO model is useful when industrial sectors of those regions have substantial linkages or interdependences.

Z Z Z

Z Z

⎡ ^NN_SN ^NS_SS

⎣⎢

⎢

⎤

⎦⎥

⎥

The Z matrix with two regions consists of four submatrices, starting from upper left submatrix Z ^NN , in which industrials sectors in N purchases goods and services from those in the same N, and lower-right submatrix Z ^SS , in which industrials sectors in S purchases goods and services from those in the same S. Those are two submatrices where production activities form the same regions are captured.

The lower-left submatrix Z ^SN shows that industrials sectors in N purchases goods and services from those in S region, capturing inter-regional purchase of inputs across the regional border. Similarly the upper-right submatrix Z ^NS shows that industrials sectors in S purchases

goods and services from those in the N, capturing inter-regional purchases of inputs on the opposite direction.

In a table structure, IRIO model ’ s transaction table appears as Figure 3-13 .

Each purchase of input can be traced back to selling sectors in either regions in this model, assuming free movements of goods and services without restrictions. This structure will be applicable to two regions where there are no physical border controls and goods and services move to each other freely. State borders, or borders between two cities, would be similar to this structure.

3.8.3.2 Multiregional input-output

In the multiregional input-output (MRIO) structure, we still see two regions but the purchase of inputs from other regions will be captured by the formal structure of trade accounts. We will follow the presentation shown in the Miller and Blair (1985) , which has been widely con- sidered as a standard textbook of I-O for graduate students of regional science.

3.8.3.2.1 Trade table

In the MRIO structure, inter-regional flows of goods and services are first captured by the trade table. We will study two regions, east and west. z_i^EW Denotes the flow of goods i from east to west, irrespective of the exact purchasing sector in the west, which can be intermedi- ate goods purchaser (i.e. industry) in the west or final demand consumer in the west. We can create a shipment table as shown in Table 3-18 .

The first column in Table 3-18 can be added up to create the total amount and can be expressed as follows;

T_i^W¹z_i¹¹z_i²¹z_iⁱ¹

Then, each element in column is divided by the total of T_i^W1 , to create coefficients rep- resenting the proportion of all of goods i from east used in the west. We will have an inter- regional trade coefficient, c_i^EW .

North AG

z11 z21NN z31 z11 z21SN z31

z12 z22 z32 z12 z22 z32

z31 z32 z33 z31 z32 z33

NorthSouth

MNF Serv AG

MNF Serv AG MNF Serv

South AG

z11 z21NS z31 z11 z21SS z31

z12 z22 z32 z12 z22 z32

z31 z32 z33 z31 z32 z33 MNF Serv

Figure 3-13 Inter-regional input-output model structure.

Notes: AG, agricultural sector; MNF, manufacturing sector; Serv, service sector.

I N P U T-O U T P U T M O D E L A N D I T S A P P L I C AT I O N 89

Table 3.18 Shipment table for multiregional input-output structure for commodity i . Receiving region

Shipping region W1 W2 Wj

E1 z¹¹_i z¹²_i z¹³_i

E2 z²¹_i z²²_i z²³_i

Ei z³¹_i z³²_i z³³_i

c z

c c c c

iEW iEW iW

EW EW EW EW

nEW

1 2 3

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

The column vector shows the proportion of goods that came from the region E to the region W. The next step is to construct a diagonal matrix, for the sake of calculations from the previous column vector. The new matrix will be a square matrix of n n with all the elements with zeros except the diagonal line (from upper left to lower right), which is the diago- nally transposed elements of the previous column vector.

Cˆ

c c

EW EW

nEW

1 2

0 0 0

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

⎥⎥⎥

⎥⎥

The circumflex ( ^ ) above C is a sign noting that column vector was transformed into a diagonal square matrix. Now that we have shown how to capture the inter-regional trade flows between two regions, there could be a similar matrix showing intraregional flows within the same region. There could be a matrix Cˆ^WW with elements

c_i^WW z_i^WW/T_i^W which shows the portion of goods (or services) i that had been produced in the region W and used within the region W. Again trade coefficients are shown along the diagonal line in the square matrix.

Cˆ c

c c

WW WW

nWW

1 2

0 0 0

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

⎥⎥⎥

⎥⎥

⎡

⎣⎢

⎢

⎤

⎦⎥

⎥

⎡

⎣⎢

⎢

⎤

⎦⎥

A a a

a a A a a

a a

E E E

E E W W W

W W

¹¹ ¹²

21 22

11 12

21 22⎥⎥

⎡

⎣⎢

⎢

⎤

⎦⎥

⎥

⎡

⎣⎢

⎢

⎤

⎦⎥

ˆ ˆ ⎥

C c

c C c

EW EW

EW WW WW

¹ WW 2

1 2

0 0

Then in multiregional structure, we will use the matrix

Cˆ A c a c a

c a c a

EW W

EW W EW W

¹ ¹¹ ¹ ¹²

2 21 2 22

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

which makes a contrast with the interregional modeling structure where we used the A^NS . In the same manner, instead of using A^NN , which was used in the interregional modeling, MRIO uses

So, in the Cˆ A c a

c a

WW W WW W

WW W

¹ ¹¹

2 21

1 12

2 22

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥ MRIO modeling, if we have:

A A

A C C C

C C X X

E W

EE EW

WE WW

0 W

⎡

⎣⎢

⎢

⎤

⎦⎥

⎥

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥ ⎡

⎣⎢

⎢

⎤

⎦⎥

⎥

ˆ ˆ

and

Y Y

⎡ W

⎣⎢

⎢

⎤

⎦⎥

⎥

This can be shown as ( I CA ) X CY , which can be rewritten as ( I CA ) ¹ CY X . The MRIO model can be useful either when you do not have enough detailed data for two regions ’ trade patterns or when two regions have a formal border control over goods and services.

Now, let us use some numbers to see how the MRIO works. For the simplicity, let us use a 2 2 matrix for the region east and west.

A^E 0 2 0 18 A^W 0 3 0 25

0 15 0 29 0 22 0 21

. .

⎡

⎣⎢

⎢

⎤

⎦⎥

⎥

⎡

⎣⎢

⎢

⎤

⎦⎥

⎥

ˆ .

. ˆ .

C^EE 0 7 0 C^EW C^WE . 0 0 4

0 2 0 0 0 3

0 3 0 0 0 6

⎡

⎣⎢

⎢

⎤

⎦⎥

⎥

⎡

⎣⎢

⎢

⎤

⎦⎥

⎥

⎡

⎣⎢

⎢⎢

⎤

⎦⎥

⎥

⎡

⎣⎢

⎢

⎤

⎦⎥

ˆ . ⎥

C^WW 0 8 0. 0 0 7

Now we assume that there would be an increase of $100 in final demand for the output of sector 1 in the region east. How can we calculate the impact on the total output for both regions east and west? Now consider the basic structure of MRIO and construct the matrices accordingly, so that you can calculate the necessary processes. In Figure 3-14 the processes are shown in the order. Namely:

1 create a combined A-matrix of two regions;

2 construct trade coefficients matrices;

3 multiply the combined trade coefficients matrix by the combined A-matrix to create CA;

4 create a combined final demand column vector to multiply C, so that you will have CY;

5 after deducting CA from the appropriate identity matrix, inverse the result to create a combined inverse matrix, so that the result can be multiplied by the combined column vector of

I N P U T-O U T P U T M O D E L A N D I T S A P P L I C AT I O N 91

A^E ^0.2_0.3 ^0.18_0.25 A^W ^0.15_0.22 ^0.29_0.21

0.7 0

0 0.4 C^EW ^0.2₀ ⁰_0.3

0.3 0

0 0.6 C^WW ^0.8₀ ⁰_0.7

Y^E ¹⁰⁰₀ Y^W ⁰₀ A ^0.2_0.3 ^0.18_0.25

0 0

0.15 0.29 0.22 0.21

C ^0.70 ⁰0.4

0.3 0

0 0.6

0.2 0

0 0.3

0.8 0

0 0.7

CA ^0.140.12 ^0.1260.1 0.06 0.054 0.18 0.15

0.03 0.058 0.066 0.063 0.12 0.232 0.154 0.147

CY ^0.70 ⁰0.4

0.3 0

0 0.6

0.2 0

0 0.3

0.8 0

0 0.7

100 Y

CY 0

0 0

70 0 30 0

70 CY

Delta-X 0

100 184.78 0

87.719 17.433 49.17 30.453

(I CA) _0.12^0.86 ^0.126_0.9

0.06 0.054 0.18 0.15

0.03 0.058 0.0660.063 0.88 0.232 0.154 0.853

(I CA)¹CY 1.2202 0.1952 0.1987 1.1679 0.1811 0.1575 0.3251 0.2750

0.0769 0.1183 0.1174 0.1317 1.2164 0.3548 0.2565 1.2845

1 0

0 1

0 0

1 0

0 1

(1)

(2)

(3)

(4)

(5)

I-Matrix

Cˆ^EE

Cˆ^WE ˆ

Figure 3-14 Calculation process of multiregional input output model for type-I output multiplier.

CY to generate change in total output not only for the region East, where the final demand occurred, but also for the region West, to be in line with the equation ( I CA ) ¹ CY X .

In Figure 3-14 , all the processes are shown step-by-step so that you can follow the calculation by using MS-Excel.

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