Social Accounting Matrix Model and its Application
4.3 Structure of the social accounting matrix table
4.3.2 Matrix operations for social accounting matrix modeling
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For example, a Finnish employee is working on the Executive Floor at 6-star luxury hotel in Dubai, United Arab Emirates (UAE). The employee left behind her family members in Helsinki, to which she remits her income. Then on the SAM table of the UAE, her labor is put as required ingredients of the hotel sector, which is captured at F. Then when the factors gives the wage back to its owner, the employee, instead of putting all the wage into the domestic institutions as shown in W, she makes remittance of the money, which appears in the cell (4, 2), as if UAE hotel sector imported the factors. At the same time, if we consider the Finland’s SAM table, her remittance from abroad will be recorded at (2, 4) as if she exported her factors to overseas.
The cell where the column intersects with institutions (3, 4) is where the institutions’
receipt from outside of the region is recorded. This cell usually does not have large num- bers. Recall that households receive substantial income from factors only when they put their labors and capital into the factors, which is an exchange market for labor and capital. There is not much to receive if their resources (labor and capital) are not put into the economic system.
4.3.2.1 Row interpretation in transaction table
Let us consider production activities component, which is indeed the I-O component, show- ing the interindustry transactions and beyond. Looking at the first row of the production activities at agriculture sector’s output, the number would look like row vector of
[1 2 1 0 2 5 11] in Table 4-2.
This means that the agricultural sector sold total amount of goods of 1 to the agriculture sec- tor, 2 to the manufacturing sector, 1 to the services sector, 2 to institutions, and 5 to the others, such as exports, thus making the total output of 11.
To put the numbers into equation 3.1, (1 2 1) (2 5) 11
Intermediate goods final demands total outpput
Intermediate goods are sold to industrial sectors as necessary ingredients, or as inputs for those sectors. By looking at the row of the agriculture sector, you can see the destination of the agriculture sector’s outputs. In this case, total of 4 provide industrial sectors with interme- diate goods and the total of 7 goes to satisfy the final demands. In this table, total amount of transactions are recorded with the actual currency unit, such as dollars, so the table is called a transaction table. The final demand consists of 2 from the institutions (households) and 5 from the others. An example of this portion is final demand from outside of the study region (nation), which can be considered as exports from the study region. The agriculture sector shipped the goods outside of the study region, and received monetary inflow in exchange.
The cell where the agriculture sector’s output intersects with factors is zero. This is in line with the structure of SAM as we learned in Figure 4-1. It is the households, the owner of labor and capital, who has the money to purchase goods from the industry, and it is not the factors of production, which by structure will not be able to have final demands. This structure of having separate factors and institutions poses the difference from the extended I-O structure such as type-II structure in which households shown in a single column and in a single row are added as if it were another industrial sector in the I-O model.
4.3.2.2 Column interpretation in transaction table You see that the agricultural sector’s column in Table 4-2 has
1 1 2 4 0 3 11
⎡
⎣
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢⎢
⎢
⎤
⎦
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥⎥
⎥
This can be interpreted that there was the purchase by the agricultural sector of 1 from agricul- tural sector, of 1 from manufacturing sector, 2 from service sector, 4 from factors of p roduction
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(labor and capital), 0 from institutions, and 3 from others, thus making the total agricultural sector’s purchase of 11. In the I-O example that we studied in chapter 3, we saw only one row, which was value added, below the interindustry transactions rows. As was the case with I-O table, the numbers that you see in the column depicts all the required inputs, with the bottom number showing the total inputs for the sector in the period of 1 year. This can be likened to a recipe list all the required ingredients, i.e. the agricultural sector required 1 ingredient from the agricultural sector, 1 from the manufacturing sector, etc. The cell where the agriculture column intersects with institutions row is 0. This is in line with what we learned in Figure 4-1.
The column shows 3 at the intersection with others, and this can be imported goods from the outside of the study region.
Just like the I-O model, total output amount is equal to total input amount. In this case, the total output of 11 equals the total input of 11. In order to produce total output of 11, the agricultural sector required total inputs of 11 (total outputs total inputs).
4.3.2.3 Endogenous versus exogenous
In the I-O table, we followed a general rule that interindustry transactions remain endog- enous (inside of the model) and others were treated as exogenous (outside of the model).
This was clear and universal, leaving little opportunity for researchers’ discretion for incom- pliance. There was a notable exception for type-II multipliers simulations, where only the households sector was added as if it were the additional industrial sector. As for SAM, as we reviewed, there appears to be wider scope for researchers’ discretion (flexibility) as to which sectors should remain endogenous, and which sectors were made exogenous, because of lack of clearly shared uniform rules.
You may wonder what would happen if you take the complete SAM table by keeping all columns and rows endogenous and conduct series of matrix operations. That way, we may avoid ambiguous discussion regarding which columns and rows should be taken out as exogenous. Because it is a square matrix, however, there would be no solution for the inverse operation of the complete SAM-based matrix. In the SAM structure, researchers have to decide which activities are to be kept as exogenous, and transform the chosen ones from
Table 4-3 Transactions table with endogenous columns only.
AG MNF Serv Factors Institutions (HH)
AG 1 2 1 0 2
MNF 1 3 2 0 2
Serv 2 2 4 0 4
Factors 4 2 5 0 0
Institutions 0 0 0 10 1
Others 3 4 4 3 3
TOTAL 11 13 16 13 12
Notes: AG, agricultural sector; MNF, manufacturing sector; Serv, service sector.
endogenous to exogenous. In our example, we keep the factors of production and households in the institutions as endogenous and take all others as exogenous (Table 4-3).
Now you have interindustry, factors of production and institutions (households) columns only. In the sample, the matrix has 7 rows and 5 columns (a 7 5 matrix). In other words, we have interindustry square matrix (which means that numbers of rows and columns are the same, such as a matrix of 3 3), factors of production row (labor, and capital), institutions (households), Others, and the total input rows which consist of column sum of each column.
4.3.2.4 Standardization
Here, you standardize the required inputs in the transaction table by putting them in rela- tive terms along each column. The process is rather simple. You take each required inputs in each column to be divided by the column sum (total input). For example, let us take the agriculture sector’s column. The relative input from the agricultural sector to the agri- cultural sector would be calculated as 1 divided by 11 1/11 0.0909, the relative input from the manufacturing sector to the agricultural sector would be calculated as 1 divided by 11 1/11 0.0909, etc. Now, repeat the processes for all the other cells and columns.
Once all the calculations are complete, you see all the transaction amounts are converted into relative inputs to each column’s total input. What you see are the relative inputs to total inputs of each column in relative terms. After standardization, the table will appear as Table 4-4.
Before we consider steps of matrix operations, we us focus on how the concept we learned is reflected over the numbers in the SAM table.
4.3.2.5 Flows between production activities and factors
In Figure 4-1, we learned how the production activities required factors of production. That is captured in the factors’ row of Table 4-4.
Production activities (industrial sectors) require labor and capital from factors of produc- tion, which is like an exchange market for labor and capital. These are represented by arrows (3) and (4) in Figure 4-1. If we consider the agriculture sector column, for the total input of
$1.00, $0.36 of labor and capital are required. In exchange, the agriculture sector pays $0.36
Table 4-4 Standardized transactions matrix with endogenous columns only.
AG MNF Serv Factors Institutions (HH)
AG 0.0909 0.1538 0.0625 0 0.1667
MNF 0.0909 0.2308 0.1250 0 0.1667
Serv 0.1818 0.1538 0.2500 0 0.3333
Factors 0.3636 0.1538 0.3125 0 0
Institutions 0 0 0 0.7692 0.0833
Others 0.2727 0.3077 0.2500 0.2308 0.2500
TOTAL 1.0000 1.0000 1.0000 1.0000 1.0000
Notes: AG, agricultural sector; MNF, manufacturing sector; Serv, service sector.
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for every input of $1.00 (which means every output of $1.00). In comparison, the input struc- ture of manufacturing sector is less labor intensive. Note that production activities (industrial sectors) require no inputs from institutions, thus all zeros are recorded.
Now look at factor’s column of Table 4-4. There are no inputs from production activities (industrial sectors) or factors of production. The cell where the factors column intersects with the institutions row is the reflection of arrows (5) and (6) in Figure 4-1. Institutions (roughly households or you) put their endowments into the exchange market for labor and capital (arrow 5 in Figure 4-1: the institutions row to factors column here), and in exchange factors pay monetary rewards back to the owner of those labor and capital, institutions, which you see as money flow from factors column to institutions row (the arrow 6 in Figure 4-1).
Imagine that you have a car and a driving license, and decide to earn some income at a pizza delivery shop. You, as institutions, decide to use labor (labor of a person who has a driv- ing license and is willing to do deliveries) and capital (a car) in the exchange market (factors market) to be utilized by the production activities (in this case, the delivery of the hot pizza).
In return, factors pays money to you (your salary and costs for providing your car is shown in 0.7692). Recall the money flows from columns to rows in exchange for flow of goods, services, and other inputs such as labor and capital from rows to columns. So what would happen next?
After the institutions (households) receive the monetary flow, they are back to the con- sumption mode as shown in the institutions column. With the newly received income, you want to visit restaurants (purchase from services sector), buy a new car produced in the same nation (purchase from manufacturing sector), buy your favorite apples (purchase from agriculture sector), or ask for a 1-day cleanup of your messy room from a cleaner (purchase of services from other institutions). You may purchase goods made in some other foreign nations to be imported (import to be shown as purchase from others).
The institutions column is where the institutions spend their money to have their final demand for goods and services satisfied either by the three domestic industrial sectors, insti- tutions (interinstitutional transfer) or imported goods and services, which is shown at the intersection with others.
The required processes are the same as in the case for the I-O simulations. They are indeed the same steps used in the I-O tables in Chapter 3; namely:
(i) Standardization (reviewed above), (ii) Creating standardized matrix,
(iii) Create an appropriate identify matrix,
(iv) Subtract the standardized matrix from the identify matrix, (v) Calculating the inverse of the result of the above.
They will be shown together from standardization of the SAM transaction table to the inverse matrix (Figure 4-3).
In this case, creation of the inverse matrix of 5 5 will require understanding of the rel- evant matrix computations in the MS-Excel, which were covered in the I-O case.
After you obtain the inverse matrix, you can sum up the column under each industrial sector just as we did with the I-O inverse matrix table. This time, we will make two calcula- tions per each column. One is the sum of the numbers in the production activities only, and the other is the total sum of the column. if we take the agriculture sector’s column, the first one is 1.335 0.409 0.693, and the second one is (1.335 0.409 0.693) 0.765 0.642.
The results will be 2.437 and 3.843. The additional effect, the difference between two num- bers, is attributed to the assumption that the factor payment passed back to institutions (wage paid to household) are assumed to stimulate additional sets of consumption (i.e. increase in final demand). This is the basic concept of induced effect of the SAM output multipliers. Care is needed when it comes to applying them to impact analyses (see following section).
AG 0.0909 0.0909 0.1818 0.3636 0
0.1538 0.2308 0.1538 0.1538 0
0.0625 0.1250 0.2500 0.3125 0
MNF Serv
AG MNF Serv Factors Institutions
Factors 0 0 0 0 0.7692
0.1667 0.1667 0.3333 0 0.0833 Institutions (HH)
AG 1.335 0.409 0.693 0.765 0.642
0.396 1.555 0.631 0.580 0.487
0.315 0.462 1.781 0.742 0.623
MNF Serv
AG MNF Serv Factors Institutions
Factors 0.330 0.404 0.683 1.396 1.171
0.429 0.525 0.888 0.514 1.523 Institutions (HH) AG
0.909 0.091 0.182 0.364
0
0.154 0.769 0.154 0.154
0
0.063 0.125 0.750 0.313
0
MNF Serv
AG MNF Serv Factors Institutions
Factors 0 0 0 1.000 0.769
0.167 0.167 0.333
0 0.917 Institutions (HH) AG
1 0 0 0 0
0 1 0 0 0
0 0
1 0
0 0
MNF Serv
AG MNF Serv Factors Institutions
Factors 0 0 1 0
0 0 0 0 1 Institutions (HH)
Figure 4-3 Series of matrix computations from the standardized transactions matrix with endogenous columns to the inverse matrix.
Notes: AG, agricultural sector; MNF, manufacturing sector; Serv, service sector.
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