Chapter 4
THE LOGIC OF INPUT- OUTPUT MODELS


4.1 Introduction

The logic of input-output models is analogous to that of economic-base models, and requires only an extension of mathematical sophistication from simple algebra to matrix algebra. A piece of cake. The complexities of input-output models come in devising schemes to manage the massive data sets required to build them.

We proceed here to repeat the process evolved in Chapter 3 to develop the simplest of models, a plain, square, industry-by-industry model. In the next chapter, we will develop the multiplier system associated with this model. After this work on the model and its uses, we will return to the process of building an input-output system. By then we will have manipulated the system sufficiently to easily understand the more complex commodity-by-industry accounting system now in current use by the United States and recommended by the United Nations.(3)

4.2 The rationale for a model: analysis vs. description

While the transactions table describes the economy and yields interesting bits of information for a particular point in time, in itself it has no analytic content. That is, it does not permit us to answer questions concerning the reaction of the economy to change. Let the transactions table represent the economy in equilibrium and subject it to a shock, say an increase in tourism or a cutback in defense expenditures. When the repercussions of the shock have moved through the economy, what will be its new "equilibrium position?" In other words, which industries will be larger or smaller and whose income or employment will have changed? Such analysis requires an economic model, which we now proceed to construct.

4.3 Preparing the transactions table: closing with respect to households

As we shall see, it is important to include in the interindustry structure (Quadrant II) all economic activities which make buying decisions primarily on the basis of their incomes. These activities are called endogenous since their behavior is determined within the system. Other activities, such as federal government expenditures or exports, are based on decisions made outside the system and so are called exogenous activities. Activities which are labeled "industries" are normally considered endogenous and those which are labeled "final-demand sectors" are normally considered to be exogenous. But sometimes it is not so easy to classify activities.

The household sector is a case in point. While traditionally classified as a final-demand sector, it is frequently treated in regional economic models as an "industry." Households sell labor, managerial skills, and privately owned resources; they receive in return wages and salaries, dividends, rents, proprietors' income, etc. And to produce these resources, they buy food, clothing, automobiles, housing, services, and other consumer goods. Exceeded in total expenditures only by the manufacturing sector, the household sector is obviously a critical part of the Georgia economy or of any area economy, for that matter. So we move the household row and column into the interindustry part of the transactions table and treat households as another industry. The household sector becomes the sixth "industry" in the aggregated table repeated as revised in Table 4.1.


Table 4.1 Hypothetical interindustry transactions with endogenous household sector

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The state and local government sectors (included in "other final demand" and "other final payments" in our aggregated table) also are difficult to classify. While we leave them in the exogenous part of the table now, primarily for simplicity, they are occasionally included in the endogenous part of the table in detailed forecasting models.

4.4 The economic model

As is now familiar, an equilibrium model is based on three sets of relations: (1) definitions or identities, (2) technical or behavioral conditions, and (3) equilibrium conditions. A model thus uses a set of assumptions to extend a description of an economy so that it can be used to trace the effects of disequilibrating forces. In the case at hand, each set of relations can be easily identified.

4.4.1 Identities: the transactions table

The state transactions table as extended above provides our set of identities: it defines the economy for the base year. Now let's express these relations in simple algebra. Let xijbe the sales of industry i to industry j, yi the sales of industry i to final demand (ultimate consumers), and zi the total sales of industry, or total demand for the output of the industry. Then we can define the sales of Georgia industries in terms of the following equations:

x11+ x12 + x13+ x14 + x15 + x16+ y1 z1
x21 + x22 + x23+ x24 + x25 + x26+ y2 z2
x31 + x32 + x33+ x34 + x35 + x36+ y3 z3
x41 + x42 + x43+ x44 + x45 + x46+ y4 z4
x51 + x52 + x53+ x54 + x55 + x56+ y5 z5
x61 + x62 + x63+ x64 + x65 + x66+ y6 z6

This set of identities can be seen symbolically in the top six rows of Figure 4.1 and numerically in the five industry rows and in the household row (now "industry" six) of the transactions table (Table 0.1). Since we are now primarily concerned with Quadrant II, we have reduced Quadrant I to one column in these tables and we have dropped the various intermediate totals. In a more concise formulation, the equations may be summarized as:

zi j xij+ yi (4-1)

where the operator j sums sales by industry i over all industries j.

As can be seen, Figure 4.1 and the above set of equations differ in only two ways: (1) the arithmetic operators are implicit in the table; and (2) the table includes values for other final payments (vj) and imports (mj), completing the accounting framework.

While the above set of equations (identities) are our basic concern, a second set may reveal insight into our equilibrium problem. For the economy to be in a state of equilibrium, row and column totals must agree. (A simple fact of double-entry accounting.) We can now define total output, or supply, for industry j as qj , the sum of all intermediate purchases, local payments to factors of production, and imports:

qj ixij + vj + mj ,(4-2)

where i sums purchases by industry j over all industries i.

4.4.2 Technical conditions: the direct-requirements table

Now, recall that the final-demand vector (y) is exogenous. The y's are free to change, outside of our control. We wish to know the effects of such change on the economy as expressed by changes in output. It is obvious that little additional information can be gleaned from the transactions table. We have six equations and 48 variables, of which only six (the y's) now have assigned values. The minimum requirement for a solution to this system is that the number of equations equals the number of unknowns; therefore, we must reduce the number of unknown variables by 42.

To do this, we introduce a set of technical conditions. Assume that the pattern of purchases identified in the base year is stable. We can now define a set of values called "direct requirements," or "production coefficients:"

aij = xij /qj (4-3)

Table 4.2records these aij coefficients for the hypothetical model. We have simply divided each value in a column by the total inputs (output) of the industry represented in the column. These numbers show the proportions in which the establishments in each industry combine the goods and services which they purchase to produce their own products.


Figure 4.1: Algebraic transactions table

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Table 4.2: Hypothetical direct-requirements table

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Notice that we can define xij, the sales by industry i to industry j, in another way. It can be written as xij = aij*qj. That is, if the manufacturing sector purchases 6.1 percent of its inputs from the service sector (a53), and if manufacturing purchases a total of $14,162 million worth of inputs (q3), then its purchases from the service sector would have to amount to $862 million, or 6.1 percent of $14,162 million. If the proportions in which industries buy their inputs remain reasonably stable over time, then we can define purchases by industry i from industry j in the future as xij = aij*qj' As we shall see, this simple assumption solves our problem.

A second assumption is hidden within the definition of xij in that it is already defined as purchases from local industry i by local industry j. This is best seen by looking at the derivation of the regional production coefficients, aij . Each regional production coefficient is the product of a "technical production coefficient," pij , which shows the proportion of inputs purchased from industry i by industry j without regard to the location of industry i, and a "regional trade coefficient," rij, which shows the proportion of that purchase made in the region. Symbolically, the regional production coefficient is

aij = pij *rij

aij = (txij/qj )*(txij - mij )/txij (4-4)

Here, txij is purchases from industry i by industry j without regard to the location of industry i,and mij is imports of the products of industry i by industry j. This point is expanded upon both in the summary included in Illustration 4.1 and in the section below on economic change.

4.4.3 Equilibrium condition: supply equals demand

Note that along the way we have implicitly stated the equilibrium condition. This condition is that anticipated demand equals supply, or that the sales of an industry equal its gross output:

qj = zj (4-5)

for all industries. Over any long period of time in a market economy, it is irrational to produce more than is used and impractical to consume more than is produced. Under normal conditions, an economy faced with a change in demand will react by changing supply. When anticipations are fulfilled, the economy is in a state of equilibrium.

4.4.4 Solution to the system: the total-requirements table

We can now rewrite the equation system to meet the obvious conditions for its solution. Two substitutions are required: 1) since we assume the production coefficients are stable, aij*qj can be substituted for xij , and 2) since inputs must come back into balance with outputs, qj is substituted for zj. Thus,

a11q1'+a12*q2'+a13*q3'+a14*q4'+a15*q5'+a16*q6'+ y1'= q1'
a
21q1'+a22*q2'+a23*q3'+a24*q4'+a25*q5'+a26*q6'+ y2'= q2'
a
31*q1'+a32*q2'+a33*q3'+a34*q4'+a35*q5'+a36*q6'+ y3'= q3'
a
41*q1'+a42*q2'+a43*q3'+a44*q4'+a45*q5'+a46*q6'+ y4'= q4' a51*q1'+a52*q2'+a53*q3'+a54*q4'+a55*q5'+a56*q6'+ y5'= q5' a61*q1'+a62*q2'+a63*q3'+a64*q4'+a65*q5'+a66*q6'+ y6'= q6'

The prime applied to each variable indicates "future" value.

The power of our assumption that the technology of production is constant is now clear. With it, we have reduced the number of unknowns from 48 to six, the q's, and can proceed to solve the system and thus to determine the outputs of industries in our economy in the future.

Compressed, the system is expressed as

q' i jaij*q' j + y' i.(4-6)

A full explanation of the solution to this system can be easily expressed in matrix algebra and, in this case, is analogous to the one in simple algebra used with economic-base models. Say we wish to solve the following equation for q, where q represents the vector of qi' on the right side of the equation system above, A represents a matrix of the aij's on the left side of the system, and y represents the column vector of yi 's also on the left side of the above system:

q = A*q + y (4-7)

We subtract Aq from both sides of the equation,

q - A*q = y (4-8)

factor q from the terms on the left,

(I - A)*q = y (4-9)

and multiply both sides by the inverse of (I - A) to get

(I - A)-1*(I - A)*q = (I - A)-1*y,(4-10)

or, since a matrix multiplied by its inverse yields an identity matrix,

q = (I - A)-1*y, (4-11)

the solution for q in terms of y. Here, I is the identity matrix, which is the matrix equivalent to the number 1 and the exponent (-l) shows that the parenthetic expression is inverted, or divided into another identity matrix. The term (I - A) is sometimes called the "Leontief matrix" in recognition of Wassily Leontief, the originator of input-output economics; (I-A)-1, of course, is called the "Leontief inverse." A more descriptive title is "total-requirements table."
Table 4.3 shows the total-requirements matrix for the hypothetical economy. Each entry shows the total purchases from the industry named on the left for each dollar of delivery to final demand by the industry numbered across the top. This sentence is complex and should be read carefully. While the direct requirements matrix recorded purchases from the industries named at the left for each dollar's (or hundred dollars' if expressed in percentages) worth of output by the industry numbered across the top, this table reports all of the purchases due to delivery to final demand. As the title shows, the table records both direct and indirect flows.

Now, let's insert summing rows into the table and add another meaning to it. The result is shown here as Table 4.4 and renamed as a table of "industry-output multipliers". The key row is labeled "total industry outputs" and included the sum of industry outputs required for the industry named at the top to deliver one dollar's worth of output to final demand (that is, to export this amount). Thus, in exporting a dollar's worth of output the manufacturing sector would cause production by other industries amounting to $1.67. This seems magical if not impossible until you recall the lessons of economic-base theory. The total double- counts the values of outputs bought and rebought as materials move from one processor to another, acquiring more value each time. We will pursue this in a little more detail in the next chapter.

Since the household sector is not really an 'industry' but has been included to assure that all internal flows related to production are counted, we exclude it from the summation of industry outputs. Later, we will treat entries in the household row as "household-income multipliers."

For many years, the total of both industry outputs and household incomes in this total- requirements table was discussed as an "output multiplier." This was incorrect, however, and most analyst avoid this error. The sum in Table 4.4 is labeled as "total activity," but it has no real meaning beyond noting the rippling of money through the economy.


Table 4.3 Total requirements (with households endogenous)

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Table 4.4 Industry-output multipliers (with households endogenous)

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Now that we have developed the logic of a regional input-output model and can see that it is a means for tracing the effects on local industries of changes in the economy, let us go back and examine the effect of closing the model with respect to households. Recall that we have included the household sector as the sixth industry in the model. Under these conditions, the total-requirements table traces the flows of goods and services required to accommodate changes in final demand through all industries and through households as well. What if the household sector had been left in final demand? What if we had continued to treat it as exogenous to the system rather than endogenous?

Table 4.5 reports a total-requirements table which is based on a five-industry version of Table 4.2 the direct-requirements matrix. Examination of the column sums in the rows entitled "total output" in each table reveals the importance of the household sector in generating new activity in the economy.


Table 4.5 Total requirements (with households exogenous)

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Table 4.6 A comparison of output multipliers under different household assumptions

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Table 4.6 compares these tables. Just including the household sector in the inverted table leads to increases in output by the processing industries (1 through 5) of 27 to 44 percent. (When we include households as an industry and count the flows through it as "industry output," the percent increase in output rises from 66 to 111 percent of the flows based on a table excluding households.) As we shall see later in a more detailed discussion of multipliers, income flows induced by households are important to a regional input-output analysis, but we must be careful to distinguish increases in houshold incomes from increases in industry outputs.).

4.5 Economic change in input-output models

4.5.1 Causes vs. consequences of change

An input-output model is designed to trace the effects of changes in an economy which has been represented in an input-output table. Such models show the consequences of change in terms of flows of monies through an economy and in terms of incomes generated for primary resource owners. The models themselves do not show the causes of change; these causes are exogenous to the system.

Economic change as traced through an input-output model can take two forms: (1) structural change or (2) change in final demand. Changes in the economic structure of an area can be initiated in several ways. It can be through public investment in schools, highways, public facilities, etc., or it can be through private investment in new production facilities, or it can be through changes in the marketing structure of the economy. Changes in final demand are basically changes in government expenditure patterns and changes in the demands by other areas for the goods produced in the region.

4.5.2 Structural change

Structural change in an input-output context can be interpreted to mean it changes in regional production coefficients." In turn, this can be interpreted as either changes in technology or changes in marketing patterns or both. Let us see what this means in terms of the direct-requirements matrix, or the A matrix of our earlier discussion. Recall that aij is the proportion of total inputs purchased from industry i by industry j in the region. We can treat this regional production coefficient as the product of two other coefficients and write it symbolically as

aij = pij*rij.

The "technical production coefficient," pij , shows the proportion of inputs purchased from industry i by industry j without regard to the location of industry i, while the "regional trade coefficient," r ij shows the proportion of that purchase made locally.

A change in technology, or a change in pij, could be illustrated by a shift from glass bottles to metal cans by the soft-drink industry. But a change in location of purchase, or a change in rij, would be illustrated by a shift from metal cans made in another area to metal cans produced in our region.

The above changes are couched in terms of existing industries. Another way in which change can take place is through the introduction of new plants or even new industries. The introduction of a new plant into an existing industry has the effect of changing the production and trade patterns of the aggregated industry to reflect more of the transactions specific to the detailed industry of which the new plant is a member. For example, consider the manufacturing sector of our highly aggregated five-industry model. As presented, it reflects the combination of all manufacturing activities in the region. The introduction of new plants in the transportation-equipment industry would change the combination of purchases presently made in the manufacturing sector. The same statement might be made concerning the purchase pattern displayed by the transportation equipment industry if a new aircraft-producing plant were established (or an old one were to cease operation).

The addition of a completely new industry to the system means adding another row and column to the interindustry table to represent the new industry. This is done in a manner similar to that involved in closing the table with respect to households.

To account for structural changes which are caused by changes in technology or in marketing requires a revision of the interindustry flows table and is best accomplished when a biennial revision is made.

4.5.3 Changes in final demand

Accounting for structural changes in an input-output model requires substantial skill and familiarity with the mechanics of the model on the part of the analyst. This is not the case when accounting for the effects of changes in final demand. It can easily be accomplished with the inverse matrix, or the total-requirements table, Table 4.3 in this chapter or, more appropriately, with its detailed equivalent.

Two kinds of changes can be traced. One form is a set of long-run changes in the demands for the outputs of all industries. This set takes the form of a vector of predicted exogenous demands (the y' vector discussed above) and represents our best judgment of the export demands for the products of our industries in some later year. Using the formula

q' = (I - A)-1y'

we can easily derive projections of the expected gross outputs (q') of industries in the later year.

The other form which change in final demand might take is an assumed change in the final demand for the output of one industry. Say we wish to know the effect on the economy of a $100,000 change in the demand for floor coverings. We would simply go to the detailed tables and look for the column sum for the floor-covering industry in the total requirements matrix. Using the detailed table for our hypothetical economy (not shown here), this entry is 1.8094; multiplied by $100,000, it shows that these additional export sales of carpets would increase regional outputs by a total of $180,940. A look at the household row in that same column would have yielded a household-income coefficient of .4136, meaning that the additional carpet sales would have increased local household incomes by $41,360.

The example can be pursued on a more gross level by looking at Table 4.3 and assuming a $100,000 increase in the output of the manufacturing sector. The output multiplier in manufacturing is 1.67, meaning that the $100,000 change in export demand yields $167,000 in additional business to local firms. The household-income coefficient of .52 means that household incomes increase by $52,000. The differences between these figures and those in the above paragraph show the consequences of aggregation, which conceals a substantial amount of variation in the detailed tables.

We discuss the multiplier model in more detail in the next chapter.

4.6 Study questions

  1. Why is an interindustry table sometimes called an input output-table?
  2. Outline the form of an input-output table.
  3. Describe the four quadrants of an input-output table.
  4. How would you derive gross regional product from a regional input-output table?
  5. What is the rationale for constructing a model from an input-output table?
  6. Of what use is a regional input-output table in itself?
  7. Outline the steps involved in constructing an input-output model from a transactions table.
  8. How can the effect of structural changes be traced through an input-output model?
  9. Clearly show the relationship between regional input-output models and economic-base theory.
  10. Given the interindustry transactions matrix (X) and the gross outputs vector (z), calculate values for the final-demand (y), final-payments (v), and input (q) vectors.

    IMAGE imgs/chap0408.gif

  11. Using the system in question 10, calculate A and then estimate (I -A)-1 using determinants (to two digits).

Illustration 4.1 The simple input-output model

Definitions or identities:
Inputs = Sum of purchases from other local industries and from final-payment sectors q1= x11 + x21 + v1 + m1
q2= x12 + x22 + v2 + m2
or, in matrix terms,
q = XTi + v + m
where XT is a transpose and i represents the summing vector
Outputs = Sum of sales
to other local industries and final users
z1= x11 + x12 + y1 + e1
z2= x21 + x22 + y2 + e2
or, in matrix terms,
z = Xi + y + e
Behavioral or technical assumptions:
Constant production coefficients
pij = txij/qj
or txij = pijqj
Constant regional trade coefficients
rij = (txij- mij)/txij
Equilibrium condition:
Inputs = Outputs
qi= zi
Solution by substitution:
Problem: given final demands (y and e), reduce the number of unknowns to equal the number of equations.
Let aij = pij *rij.
Then aij = (txij/qj)*((txij- mij)/txij)
= (
txij- mij)/qj = xij/qj or
x
ij= aijqj
Substituting into the output equations,
z1= a11z1 + a12z2 + y1 + e1
z2= a21z1 + a22z2 + y2 + e2
Or, continuing in matrix terms,
z = Az + y + e
z - Az= y + e
(I - A)z = y + e
z = (I - A)
-1(y+e)
Output multipliers:
dzi/dej = rij , where rij is an element of R = (I - A) -1.
Each of these partial output multipliers shows the change in local output i associated with a change in exports by industry j. Their sum over i is the total output multiplier for industry j. Income multipliers:
incomei = Sum(rij*vi(h)/qi), where vi(h) is household income. Each of these income multipliers shows the change in household income caused by a change in exports by industry j.

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