## Interindustrial model and national account statistics

### a. Introduction

The MacroEconomic Calculator is based on a Leontief Open Interindustry Model introduced after the second world war. As all economic models, the model used is a simplified explanation of the real economic system. It assumes that the economy can be divided in to a set of sectors, the intermediate production/service sectors, reflecting the different products and services delivered by the economic system and one sector, the final use sector, being served by the others. The final use includes consumption, capital fornation, changes in valuables and inventories and exports.

The Figure give a graphical picture how the flows among the sectors and to the final use are conceived. The delivery of products/services from an intermediate sector, requires in general inputs from a set of the other sectors. The total production of a production/ service sector is thus flowing into the production processes of other production sectors and into the different categories of final use.

The model used for the computations of the app is is expressed by a set of mathematical equations. These equations is based on the assumption that the input required by any intermediate sector from another can be expressed as a fixed proportion of the output of the receiving sector. This is obviously a critical assumption and its validity is best for situations near the economic status reflected by the national account statistics used for implementing the model. The model used includes 60 intermediate sectors.

The flows among the sectors and to the final use sector are measured in NOK for the year 2011. The data used are prepared and published by Statistics Norway.

### b. Model

The model can be summarized in a more formal manner as follows. Let b[i,j] express the value of the flow from sector i to sector j, c[i] the flow received by the final use sector from intermediate sector i, and t[i] the total output flow from sector [i], all for i,j=1...60 in the observation year.

The flow representations can be organized by matrix notation and operations in a flow system:

(1) B+C = T

It is assumed that the required inputs to an intermediate sector are constant proportions of the output of the sector:

(2) a[i,j] = b[i,j]/t[j]    (i, j = 1, .. 60)

Representing the set of constants by a matrix A, the flow system can be substituted by:

(3) A*T+C = T

which can be rearranged to:

(4) T = (inv) [I-A]*C

(inv) [I-A] symbolizes the inverted [I-A] matrix, and I is the diagonal unity matrix.

It is interesting to note that when this matrix inversion were first carried out on Norwegian data , it required many hours of computation by means of the most powerful computing equipment available compared with a fraction of a second on a desktop computer today. It should also be noted that the inversion is only required for building the model. The MacroEconomic Calculator only include the inverted matrix to perform the computation (4). The equations in (4) permit the computation of the total production vector, T, required to satisfy a wanted final use vector, C, by a simple matrix vector multiplication.

Assume that the compensation to employees, e[i], in each sector is also proportionally related to the production:

(5) e[i] = k[i]*t[i]    (i = 1, .. 60)

where k[i] is the compensation coefficient associated for sector i, and let K be the row vector of the coefficients. The product of K and T will then be an estimate of total compensation for employment correspomding to a production vector T: :

(6) E = K*T

Just with the same reasoning, we can assume that any level of production, t[i], in a sector will require a certain amount of imports v[i], and an import coefficient h[i] can be computed:

(7) h[i ]= v[i]/t[i]    ( i = 1. .. 60)

The total amount of imports required for a given production vector is then determined by the vector product:

(8) V = H*T

Like the assumption (2), the assumptions (5) and (7) are at best realistic in the neighborhood of the situation reflected by the statistics on which the coefficients are based.

### c. Statistics

The statistical data used to compute the model coefficients A, K and H as well as for the 2011 base values, are all from the report ESA95 Questionnaire 1750 - Symmetric input-output table at basic prices, (industry*industry), for 2011 to EUROSTAT from Statistics Norway.

The figures of the official statistics have been modified slightly to fit the 60 sectors used by the model. Using the official final use data for 2011 in the model, the resulting total production values for the 60 supplying sectors therefrore deviate from the official production values. These computed totals for 2011are referred to as the 2011 imputed production values and used as a basis for comparison for production totals computed from hypothetical final uses.

### c. Examples

Assume that the export of salmon is expected to increase with 300 Million NOK compared with level of 2011 while all other activities remain as they were in 2011. We add 300 to the finale use of products from sector R03 Fish and other fishing products; aquaculture products; support services to fishing and obtain the hypothetical final use of 25 286. Table 1 reports a total production of the sector which amounts to 52101 Mill. NOK or about 0.6% increase in total production. The considered increase in export does not seem to have any significant effect on the production of other sectors.

The average age of the population of Norway is increasing, and so does the population above 70 years. This trend is expected to continue and the need for care of old people will increase. Assume that this will require an addition use of 120000 Mill. NOK equally divided between sector R86 Human health services and sector R87_88 Social work services in 2030. If the number of additional persons above 70 years in the popultion in 2030 is 1 Mill., this corresponds to about 120000 NOK/person.What will this require of the Norwegian economy expressed in 2011 NOK?

The production increases wanted in the 2 sectors are 37% and 41% in R86 and R87_88, respectively. The effect on the total production in all sectors is 160000 Mill. NOK indicating that the 2 sectors will require increased input from other sector corresponding to 40000 Mill. NOK. These repercusion effects are distributed to a number of sectors of which R95 Repair services of computers and personal and household goods and R96 Other personal services have relative increases of almost 5%. Obviously the 2 sectors studied depend on repair services and other personal services.

Back