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Input/Output analysis - Shortcuts to life cycle data?

5. Introduction to IOA

Anne Merete Nielsen, 2.-0 LCA Consultants

National statistics monitor trade between different industry sectors. Such data can be used for a variety of analyses. In this chapter the basic definitions of Input/Output Analysis (IOA) are described.

5.1 What is an Input/Output-table?

An IO-table gives an overview of the trade in a national economy. It shows how products are being sold from producers to either be used by final consumers or to contribute to further production in other industry sectors. The buyers and suppliers on the market are grouped in production sectors and sectors for final use. The number of sectors and their definition vary from country to country.

Table 5.1 Look here!
The R-matrix. A simplified input-output table for Denmark, 1975 (Lihn Jørgensen, 1982). All figures in 1*10⁹ DKK.

Table 5-1 shows a simplified version of the IO-table for Denmark in 1975. Such a set of data is also called a matrix of requirements, in short the R-matrix. In the rows the total sales from each supply sector can be seen. The supply sectors are divided into a number of domestic production sectors, foreign production sectors (import) and primary production factors.

In the columns the input to the demand sectors is seen. The demand sectors are divided into domestic production sectors and different categories of final use. The domestic production sectors are identical to those found on the supply side. The categories of final use are consumption by private or public bodies, gross investments (investments minus depreciation) and export. Trade includes the profit earned by the industries. That is the reason why buying and selling prices adds up to the same value.

From Table 5-1 can be seen that Danish agriculture in 1975 sold goods worth 24 billion DKK in total. The majority of these goods were bought by Danish industry, that purchased an amount worth 15.7 billion DKK. 3.0 billion DKK of the remaining sales were purchased by Danish colleagues within the agricultural sector, and products worth 3.1 billion DKK were exported.

To simplify further calculations, the R-matrix can be normalised, thus showing how the average DKK sold from each sector is distributed. The normalised matrix is called an A-matrix. Table 5-2 shows an example in which we have grouped the sectors further, into just two separate sectors: agriculture and industry (based on the same data as Table 5-1).

Table 5.2
The A-matrix. IO-coefficients.

The A-matrix shows the input to each sector as a percentage of the total input. For example, it is seen from Table 5.2 that for each DKK spent by the industry sector, 0,11 DKK is spent on agricultural products, 0,67 DKK is spent on imported goods and wages, and the remaining 0,22 DKK remains within the sector, i.e. purchases from other producers within the industry.

IO-data in the form of A or R-matrices can be used to assess impacts from changes in production on e.g. environment or employment if the input of the relevant factors to the different production sectors are known. Such factor input can be presented in a Q-matrix as shown in Table 5-3.

Table 5.3
The Q-matrix. Direct input of factors (f) to production sectors (n). The figures shown are presented as an example only (not real data).

Table 5-3 shows the relationship between the production and the factor input to one sector. The inputs may be use of labour, energy, or resources, but data on release of emissions may also be entered in this way.

5.2 Beyond the simple matrices

However, the data presented in section 5.1 only show the effects in the first stage in an infinite cascade of productions. Production in one sector is based on inputs produced in all sectors. But the production of these inputs is based on inputs from all sectors. And these inputs again are based on inputs from all sectors, and so forth.

Figure 5.1
The direct demand for an industry product and the related, indirect demands for inputs from all sectors. Figure based on Treloar (1998).

5.3 Example: How much energy does it take to produce a product worth 1 DKK by Danish industry?

In this section a calculation is carried through to serve as illustration of the difference between the direct and indirect effects discussed above. In this example we try to answer the question: how much energy is used, directly and indirectly, when a product worth 1 DKK is produced by Danish industry?

In this example we stay in the two-sector economy as presented in Table 5.2 and Table 5-3. The total energy consumption can be calculated in two different ways, either by an approximation of an infinite number of additions or by means of matrix-calculation.

5.3.1 The easy math solution

The accumulated result, M, can be calculated as the sum of the energy consumptions in each stage of production:

M = Q + Q₁ + Q₂+ Q_{3 ....} + Q¥

Q is the direct energy consumption by industry. From Table 5-3 follows that Q is 107 J.

Q1 is the energy consumption to produce inputs at stage 1. The inputs come from either agriculture or industry. Table 5.2 shows that it takes 0,11 DKK of input from agriculture and 0,22 DKK of input from industry to produce the industry product.

Table 5-3 shows how much energy has been used to produce the inputs in agriculture respectively industry. Therefore Q₁ can be calculated:

Q₁ = 0,22*107 + 0,11*3 = 23,87

Q₂ is the energy consumption to produce inputs at stage 2. There are four paths for the inputs. Using data from the A and Q-matrix as above, Q₂ can be calculated:

Q₂ = 0,22²*107 + 0,22*0,11*3 + 0,11*0,13*3 + 0,11*0,16*107 = 7,18

Q₃ is the energy consumption to produce inputs at stage 3. There are eight paths for the inputs. Q₃ can be calculated:

Q₃ = 0,22³*107 + 0,22²*0,11*3 + 0,11*0,13*0,16*107 + 0,11*0,16*0,22*107 + 0,22*0,11*0,13*3 + 0,11*0,13*0,16*107 + 0,11*0,13*0,13*3 + 0,11*0,16*0,13*3 = 2,08

If a high level of accuracy is needed, the calculation can be continued to Q₄, Q₅ and further. For this example, we will end here:

M » 150

5.3.2 The matrix calculation

Basic mathematics teaches us that the infinite sum presented in section 5.3.1 can be calculated as a finite calculation by use of matrix calculation.

M = Q + Q₁ + Q₂+ Q_{3 ....} + Q¥ Û M = Q (1-A)^-1

Following the basic rules of matrix calculation, we obtain:

The above calculation shows the precise answer to our question: When Danish industry produces a product worth 1 DKK, agriculture will use 21,7 J and industry itself 142 J.

5.4 References

Lihn Jørgensen, N. (1982). Input-output modeller - en introduktion. Economic Institute, University of Copenhagen. Memo no. 108.

Treloar, G.J. (1998). A Comprehensive Embodied Energy Analysis. Australia: Faculty of Science and Technology, Deakin University.

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