Input-Output Model and its Application
3.5 Multiplier calculations in the input-output framework
3.5.1 Multiplier calculations
Caution is required with the term multiplier as there are various types. Unless you calculate them yourself, or know precisely which multiplier is being discussed, a mere comparison of extracted multipliers across results of different impact studies carried out by different researchers may be misleading. In this section you will learn how to calculate several types of multipliers.
3.5.1.1 Type-I multiplier: direct impacts and indirect impacts
Changes in final demand drive the whole economic system. Changes in final demand, as you saw, can be caused by changes the consumer patterns of domestic residents, firms, or
(IA)^1 Agriculture Manufacturing Services
1.75 Case 1
delta Y (IA)^1
2.40
1.66
‘for the development of the input-output method and for its application to important economic problems.’
8. This is the most challenging part. We will have a help from Excel.
(1) Highlight all the nine cells in the target matrix (where you want to write result of calculation, in our example, yellow ones)
(2) Type “MINVERSE()” and put the cursol within the ().
(3) Then, choose the 3 3 original matrix as an array. (in our exmaple, pink ones)
(4) Important to follow: While Hold down both Shift & Control Keys, hit the Enter key.
This is the Inverse matrix! Now you see the fascinations for industry analysis.
Leontief Inverse!
‘So, what is this stuff?
How can it be useful?’
WASSILY LEONTIEF 1973 Nobel Laureate in Economics
1.18 0.37 0.12 0.22 1.55 0.21 0.35 0.48 1.33 AG MNF Serv
(IA)^1 Agriculture 8
9
Manufacturing Services
1.18 0.37 0.12 1
0 0
Case 2 delta Y
0 1 0
Case 3 delta Y
0 0 1
delta X
delta X
delta X
1.18 0.22 0.35 0.37 1.55 0.48 0.12 0.21 1.33 0.22 1.55 0.21
0.35 0.48 1.33 AG MNF Serv
Figure 3-4 (Continued)
governments, or by the export of goods and services. In the impact studies environment, the change in final demand can be called direct impact, direct shock, direct effect, or initial impact because this is the exogenous shock that stimulates the entire economic system. When the shock is caused by a change in the final demand, the economy responds to it by producing a new level of total output through interindustry transactions in the regional economy.
In our case, shown in section 3.4.3, we gave a positive change of 1 in the final demand for agricultural goods, and found the resulting change in total output to be a positive 1.75. The net additional increase was only of 0.75, if we defined the net increase as the difference between the initial change in final demand and the resulting change in total output. The simple concept of output multiplier is shown as the change in total output to the change in final demand.
Output multiplier change in total output/change in fi X
Y nnal demand
(direct impact indirect impact)/direct impact
In our example, given the shock of 1 for the agricultural sector ’ s final demand, the total output resulted in 1.75, thus the output multiplier for the agricultural sector, in this par- ticular economy, is of 1.75. Direct impact was 1, and indirect impact was 0.75, so the out- put multiplier was 1.75/1 1.75 for the agricultural sector. An additional impact of 0.75 was generated as the economy ’ s response to the direct positive shock indirectly through the inter- dependency of the industrial sectors. This is why this additional response of the industrial sectors is called an indirect impact.
The combined effect of direct impacts and indirect impacts can be put in relative terms by standardizing the direct (i.e. initial) impact as 1 so that we can view the size of the resulting total output in perspective. How large the resulting impact would be in response to the initial impact, relatively, is the concept of the multiplier. In our case, a direct impact of 1 given to the agricultural sector generates an additional indirect impact of 0.75, so that the total impact becomes 1.75. Because the initial impact was 1, the size of the total impact was 1.75 fold larger than the direct (initial) impact.
The combined effect of direct impacts and indirect impacts can be called a type I multi- plier, which reflects the impact caused by the interdependency within the industrial sectors only. You will soon understand why I say ‘ only ’ .
Since we looked at the impact on output, we may call the result a type I output multi- plier, which gives more precise information on that we did. In type-I multiplier calculations, the I-O structure included only the industrial sectors as endogenous. Institutions, particularly households, are not included in the model – they remain exogenous. You may be puzzled by the sudden emergence of new terms, such as institutions and households; these are parts of the components we initially ignored, i.e. the final demand, in the I-O sample table.
You may have heard of other multiplier-associated words, such as induced impacts, induced shock, induced effect, etc. We will discuss induced impacts when we internal- ize the households into the model by turning them into an endogenous sector in the type II multipliers.
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 61
3.5.1.2 Type-II multipliers: addition of induced impacts by endogenizing households In addition to the direct and indirect impact caused by interindustry transactions in the I-O framework, we can internalize the households sector as if it were an additional industrial sector, at the bottom of the rows and at the end of the columns of the I-O table. The house- holds ’ row will then provide their goods and services (such as labor) into each industrial sector, in exchange for receipt of money (income), and the households ’ sector in the column will spend some part of its income to purchase output from the industrial sector, as neces- sary input to ensure its existence. This will generate additional monetary flow towards the interindustry table. While this still falls short of a complete inclusion of all economic trans- actions within a region or nation, including households in the I-O structure will yield extra impact, thanks to their additional purchasing activities. An example of the structure is shown in Table 3-9 .
If you look at the rows, you see the households row at the bottom as if it were another industrial sector. If you look at the columns, you can see HH (PCE). PCE stands for personal consumption expenditures, which means that households will spend some portion of their received income to purchase other industrial sector ’ s output. Because of the addition of another sector, the output multiplier would be higher.
Type-I multipliers are a group of multipliers that are based on the usage of the generic structure of the I-O model, without any other nonindustrial sector, while type-II multipliers are a group of multipliers that utilize the I-O, while including households as an additional quasi-industrial sector. Type I and type II distinguish the structure of the I-O being used to calculate several multipliers, as follows. Besides output multipliers, which can be calculated in a type I or type II environment (using the typical I-O or the I-O including households), there are other multipliers that can be calculated along type-I and type-II structures.
For the sake of simplicity, the following multipliers are explained in type-I structure, in which there are only endogenous industrial sectors.
3.5.1.3 Employment multipliers (type I)
Once you have the data for the number of workers employed in each industrial sector, together with the transaction table, you can calculate employment multipliers for each sector
Table 3-9 A-matrix with households as an additional industrial sector .
Standardized AG MNF Serv HH (PCE)
Agriculture 0.1 0.2 0.05 0.01
Manufacturing 0.1 0.3 0.1 0.04
Services 0.2 0.2 0.2 0.07
Households 0.11 0.12 0.2 0
Notes: AG, agricultural sector; HH, household; MNF, manufacturing sector; PCE, personal consumption expenditures; Serv, service sector.
( Figure 3-5 ). Given the employment-related data of the region/nation under study, you will be able to do the following:
1 Calculate employment per output, based on the data. In our example, let us say we use
$ million as a unit. You then find how many people are employed per $1 million in output.
2 From the result of your output multiplier calculations, divide the multiplier into direct effects and indirect effects. If you gave the positive shock of 1 to the agricultural sector only, the direct impact will be 1 for the agricultural sector, and 0 for other sectors. You can then subtract the direct impact from the column vector extracted from the Leontief inverse matrix. The remaining numbers shown in the column vector represent the indirect impact, as shown in Figure 3-6 .
I-O Transaction table Employment data
Agriculture Manufacturing Services Value added Total input
AG MNF Serv FD Total Output
Total Employment
Employment /Output
1 2 1 6 10
3 2 4 10
2 4 12 20
3 13 10 20 1
2 6 10
600 60
400 40
900 45
Figure 3-5 Input-output transaction table and employment data.
Notes: AG, agricultural sector; FD, final demand; MNF, manufacturing sector; Serv, service sector.
Indirect 0.18 0.22 0.35 Direct
1.00 0 0 Results of type I
output multiplier
Type I OP multp divided
Type I OP multp divided (2)
Impacts Agriculture Manufacturing Services
AG 1.18 0.22 0.35
Figure 3-6 Decompose the type-I output multiplier into direct and indirect impacts.
Notes: AG, agricultural sector; multp, multiplier; OP, output.
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 63
3 Multiply each industrial sector by the direct effects and indirect effects by the result of point (1).
4 Sum up the results of point (3) for employment direct effects and employment indirect effects respectively.
5 The results of point (4) will be calculated along the following equation:
(Employment direct effects employment indirect effects)/employment direct effects type I employment multiplier
The interpretation of this multiplier would be as follows. Given the increase in final demand for the agricultural sector, for $1 million, which initially creates 60 new jobs in the agricultural sector in the area under study, the economy will generate a total of 95.2 new jobs owing to the interdependency of the industrial sectors. In response to an initial increase of 1 job, the economy will generate an additional 0.59 jobs, totaling the new jobs created to be 1.59 jobs. The calculations for the other two sectors are shown in Figure 3-8 .
3.5.1.4 Income multiplier (type I)
Income multipliers can be calculated similarly to the way we calculated the employment mul- tiplier. In terms of actual calculations, you can use most of the same worksheet format and follow the steps you took with the employment multiplier. Once you have the data for the total income for each industrial sector, together with the transaction table, you can calculate
(5) Type I employment multiplier for agricultural sector(ab) / a (60.0035.20) / 60.001.59
Indirect 0.18 0.22 0.35 Direct
1.00 0 0 Results of type I
output multiplier
Type I OP multp divided
Type I OP multp divided (2)
Impacts Agriculture Manufacturing Services
AG 1.18 0.22 0.35
Emp/output 60 40 45
Emp/output 60 40 45
Direct emp # 60.00
0 0 60.00 (a)
Indirect emp # 10.74 8.73 15.72
35.20 (b) Direct emp impacts
Indirect emp impacts
(3)
(3) (4)
(4)
Figure 3-7 Type-I employment multiplier calculation processes for the agricultural sector.
Notes: AG, agricultural sector; emp, employment; Multp, multiplier; OP, output.
the income multipliers for each sector. Given the income-related data of the region/nation under study, you will be calculating the following:
1 Calculate the income per output from the separate data. In our example, let us say we use
$1 million as a unit. You then find how much total income is paid per $1 million of output, as shown in Figure 3-9 .
(5) Type I employment multiplier for manufacturing sector(ab) / a (40.0065.90) / 40.002.65
Indirect 0.37 0.55 0.48 Direct
0 1.00 0 Results of type I
output multiplier
Type I OP multp divided
Type I OP multp divided (2)
Impacts Agriculture Manufacturing Services
MNF 0.371 1.55 0.48
Emp/output 60 40 45
Emp/output 60 40 45
Direct emp # 0 40.00
0 40.00 (a)
Indirect emp # 22.27 22.01 21.62 65.90 (b) Direct emp impacts
Indirect emp impacts
(3)
(3) (4)
(4)
(5) Type I employment multiplier for services sector(ab)/a (45.0030.44) / 6.001.68
Indirect 0.12 0.21 0.33 Direct
0 0 1.00 Results of type I
output multiplier
Type I OP multp divided
Type I OP multp divided (2)
Impacts Agriculture Manufacturing Services
Serv 0.12 0.207 1.332
Emp/output 60 40 45
Emp/output 60 40 45
Direct emp # 0 0 45.00 45.00 (a)
Indirect emp # 7.21 8.30 14.93
30.44 (b) Direct emp impacts
Indirect Emp impacts
(3)
(3) (4)
(4)
Figure 3-8 Type-I employment multiplier calculation processes for the manufacturing and service sector.
Notes: emp, employment; MNF, manufacturing; Multp, multiplier; OP, output.
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 65
2 From the result of your output multiplier calculations, divide the multiplier into direct effects and indirect effects. If you gave the positive shock of 1 to the agricultural sector, the direct impact will be 1 for the agricultural sector, and 0 for the other sectors. You can then subtract the direct impact from the column vector extracted from the Leontief inverse matrix. The remaining numbers shown in the column vector are the indirect impacts as shown in Figure 3-10 , which actually is not different from the Figure 3-6 .
3 Multiply each industrial sector by the direct effects and the indirect effects by the result of point (1).
4 Sum up the results of point (3) for income direct effects and income indirect effects
respectively.
5 The results of point (4) will be calculated along the following equation:
(Income direct effects income indirect effects)/income direct effects type I income multiplier
This multiplier could be interpreted as follows: given the $1 million increase in final demand for the agricultural sector, which initially generates $0.3 million in additional income for the agricultural sector, the economy will generate a total of $0.56 million in addi- tional income for the workers, owing to the interdependency of the industrial sectors. Thus, in response to an initial increase in $1 income, the economy will generate additional $0.87 income, totaling a new income of $1.87. The calculations for the other two sectors are shown in Figure 3-12 .
3.5.1.5 Other value added multipliers
Other value added multipliers can be calculated using the same processes, and are subject to the availability of other social and economic data for the nation or region under study. They may appear to be extremely useful in making important decisions, such as choosing suitable types of tourism-related policies.
I-O transaction table Income data
Agriculture Manufacturing Services Value added Total input
AG MNF Serv FD Total output
Total income
Income /output
1 2 1 6 10
3 2 4 10
2 4 12 20
3 13 10 20 1
2 6 10
3 0.3
2.3 0.23
9 0.45
Figure 3-9 Input-output transaction table and income data.
Notes: AG, agricultural sector; FD, final demand; MNF, manufacturing sector; Serv, service sector.
Indirect 0.18 0.22 0.35 Direct
1.00 0 0 Results of type I
output multiplier
Type I OP multp divided
Type I OP multp divided (2)
Impacts Agriculture Manufacturing Services
AG 1.18 0.22 0.35
Figure 3-10 Decomposing type I into direct and indirect impact.
Notes: AG, agricultural sector; multp, multiplier; OP, output.
(5) Type I income multiplier for agricultural sector(ab) /a (0.30.25)/0.31.87
Indirect 0.18 0.22 0.35 Direct
1.00 0 0 Results of type I
income multiplier
Type I OP multp divided
Type I OP multp divided (2)
Impacts Agriculture Manufacturing Services
AG 1.18 0.22 0.35
Income/output 0.3 0.23 0.45
Income/output 0.3 0.23 0.45
Direct income 0.30 0 0 0.30 (a)
Indirect income 0.05 0.05 0.16 0.26 (b) Direct income impacts
Indirect income impacts
(3)
(3) (4)
(4)
Figure 3-11 Type-I income multiplier calculation processes for the agricultural sector.
Notes: AG, agricultural sector; multp, multiplier; OP, output.