Chapter 8

8.1 Introduction

This section develops an economic model from the accounting tables for Nova Scotia. The process parallels that used in the traditional regional format presented in Chapter 4. It differs in that we now shift from a world in which regional economists collect their own data into one requiring vast teams of statisticians. As seen in the previous chapter, much new information and detail is now available. We need more assumptions to reduce it to a manageable format.

As the literature associated with the Canadian and Atlantic Provinces input-output studies suggests, and as demonstrated in the appendix to this chapter describing the mathematics of the U.S. system, the model could be phrased in terms of commodities. The procedures are similar, however, and the reader is referred to the earlier literature for further details. Here we do a simple sketch.

Essentially, we manipulate the new system of accounts until the number of variables in a set of simultaneous equations describing the economy equals the number of equations. The system differs slightly from the traditional square regional matrices in that now we have not only more commodities (and so more equations) than industries but also two equation systems to face! We must devise some way to redress this imbalance.

While the commodity-by-industry format traditional to the Canadian and Nova Scotia input- output systems opens up a variety of possibilities for forming this model, the approach employed here develops a solution to the model in terms of industries rather than commodities. This means that, while the economy is described initially in terms of sets of equations representing the tables of commodity-by-industry flows and origins, we must convert them to regional flows (through application of the constant-imports assumption) and then reduce the equations to a solvable form in industry-by-industry dimensions on the basis of the fixed market-share assumption. Finally, using the constant-technology assumption, we can solve the system conventionally.

8.2 The data

We start with two sets of tables defining the economy: The commodity flows table and the commodity origins table. Both are described in the previous chapter and are the sources for the use and make matrices used below.

8.3 Technical conditions

Now say that final demands have been estimated for some future date and that we wish to identify the effect of this demand on the Province. What are the gross outputs of industries in Nova Scotia at that time? It is obvious that the system of flow equations is not prepared for solution: in the flows table, there are 11 equations, one for each commodity, and 82 variables of which only 11, the final demands, have preassigned values. Neither is the system of origins, or make equations, which consists of 5 equations and 66 unknowns.

The minimum requirement for solution of this system is that the number of equations equals the number of unknowns. One task, therefore, is to reduce the number of unknowns. Further, the system is expressed in terms of both commodity outputs and industry inputs, while our stated goal is to determine the effect of final-demand changes on industries. So a second task is to express the system in terms of industry variables alone. This conversion will both reduce the number of equations and, more importantly, permit us to establish a familiar equilibrium condition. The third and final task before solution is that of reducing the flows matrix from a statement of production technology to one of regional trade. We proceed in reverse order.

8.3.1 The constant-imports assumption

Few of the recent regional input-output studies in this format have included survey-based, cell- specific import details. As a result, regional (or provincial, in this case) trade flows must be estimated by assumption. We know imports for each commodity in the system (Table 0.2) and we know total industry domestic final demands. And that is all, so the best way to estimate regional trade is to assume that each industry purchases imported commodities in constant proportions:

k = mk/(jukj + ek) (8-1)

where k is the import coefficient for commodity k , mk is an element in m, the imports vector, and the denominator is the sum of intermediate and local-final demands for commodity k. Provincial flows of commodity k can thus be estimated as

Pkj = ukj*(1 - k) (8-2)

where uij is the purchase of commodity k by industry j . This yields a set of equations for sales of locally produced commodities such as the following:
P11 + P12 + ... + P1m + e1 + x1 = q1 (8-3)

P21 +

P22 + ... + P2m + e2 + x2 = q2
Pn1 + Pn2 + ... + Pnm + en + xn = qn

where Pkj is local sales of commodity k to industry j, ek is sales of commodity k to final demand, xk is export sales of commodity k, qk is total local production of the commodity, and n and m are the number of commodities and industries, respectively. In this example, n is 11 and m is 5.

The results of this process are reported in Table 8.1, the aggregated provincial flows table, showing purchases of commodities produced by local industries. Note that it is very similar to Table 7.1, the commodity flows table. It differs in the values of commodity purchases, which are now only local in origin. It also contains a row of imports (the external transfers row). The industry outputs and final demands remain the same.

(Note here that imports were estimated at the detailed, 602-commodity, worksheet level using uniform import coefficients for each commodity. These detailed worksheets permit a level of realism not attainable when estimates of imports are made at an aggregated level.)

8.3.2 The constant market-share assumption

The matrix reported in the commodity origins table (Table 7.2) provides data necessary for converting from a commodity sales dimension to an industry one. If each entry (vjk) in the make matrix is divided by the appropriate total commodity output, the resulting coefficients can be used as weights with which to aggregate commodity rows in the provincial flows matrix. Called "domestic market-share coefficients", these weights are computed as

djk = vjk/ qk (8-4)

where djk is the market share of commodity k produced by industry j, and qk is the total domestic output of commodity k . The domestic market-share matrix, D, is comprised of these djk.

Now, assuming that market shares remain constant, the provincial flows matrix P and the vector h representing local final demand can be aggregated to industry-by-industry dimensions by matrix multiplication:

Z = D*P (8-5)

h = D*(e + x) (8-6)
This process means that each element zij in Z is a sum of weighted commodity sales,

zij = k (dik*Pkj ) (8-7)
and that sales are in proportion to the industry product mixes. The provincial final-demand

Table 8.1 Aggregated commodity-by-industry provincial flows, Nova Scotia, 1984

IMAGE imgs/chap0801.gif

IMAGE imgs/chap0802.gif

matrix, presented as the summed vector e and the commodity exports vector x may be similarly aggregated, yielding the vector h.

Table 8.2 reports domestic market-share coefficients as computed from the aggregated make matrix. That the transpose of this table times the provincial flows matrices yields Table 8.3, the provincial interindustry matrices, may be verified through application of equation 8-7.

This process has yielded an industry-by-industry provincial flows table and has prepared the system for solution in terms of industry outputs. But the number of variables with unknown values, 30, is still in excess of the number of equations, 5. The equation system now takes the following form:

z11 + z12 + ... + z1n + h1 + = t1 (8-8)
z21 + z22 + ... + z2n + h2 + = t2
. . .. . . = .
. . .. . . = .
. . .. . . = .
zm1 + zm2 + ... + zmn + hm + = tm

where ti represents industry sales.

8.3.3 The constant-technology assumption

Now if we assume that industries continue to purchase inputs from other industries in proportion to their purchases in 1984, the number of variables can be reduced to equality with the number of equations. On the basis of this assumption, a set of provincial production coefficients is computed, defined as:

aij = zij/gj (8-9)

where aij is the proportion of purchases from local industry i by industry j. The direct- requirements matrix, A, is composed of these coefficients. For the aggregated system, this matrix is reported in Table 8.4.

Note that equation 8-9 can be rewritten as: zij = aij*gj(8-10)

If the aij remain reasonably stable over time, aij * gj can be substituted for zij in the equation system 8-8 with the needed result:

a11*g1 +

a12*g2 + ... + a1m*gm + h1 = t1
a21*g1 + a22*g2 + ... + a2m*gm + h2 = t2
. . ... . . .
. . ... . . .
. . ... . . .

am1*g1 +

am2*g2 + ...+ amm*gm + hm = tm (8-11)

Table 8.2 Domestic market-share coefficients, Nova Scotia, 1984

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Table 8.3 Provincial interindustry transactions, Nova Scotia, 1984

IMAGE imgs/chap0804.gif

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Table 8.4 Direct requirements per dollar of gross output, Nova Scotia, 1984

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The system is now reduced to a set of 5 simultaneous equations in 10 unknowns and can easily be reduced to solvable dimensions by imposing the traditional equilibrium condition.

8.3.4 Equilibrium condition: supply equals demand

Equilibrium occurs when anticipated supply equals demand, or when the gross outputs of an industry equal its sales. So the condition is that

gj = t j (8-12)

(The condition could also be stated in terms of commodity outputs and demands.)

8.3.5 Solution: the total-requirements table

The response of an economy in moving to another equilibrium position when faced with a change in demand can be seen in the solution to the set of simultaneous equations established in 8- 11. In terms of matrix algebra, this system may be rewritten as:

A*g' + h' = g' (8-13)

where A is the matrix of provincial production coefficients (aij) and g' is a column vector of gross industry outputs. h' is a column vector of final demands for industry outputs (from equation 8-6.)

The solution is analogous to that of common algebra in its formulation. Grouping the g' terms yields:

g' - A*g' = h' (8-14)

while factoring produces:

(I - A)*g' = h' (8-15)

Multiplying both sides of this equation by the inverse of (I - A) produces a solution in terms of X':

(I - A)-1* (I - A) *g' = (I - A)-1 * h''


g' = (I - A)-1 * h' (8-16)

Table 8.5 Total requirements (direct, indirect, and induced), Nova Scotia, 1984

IMAGE imgs/chap0807.gif

Table 8.6 Industry-output multipliers, Nova Scotia, 1984

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The inverse, (I - A)-1, is called a "total-requirements table" and shows the direct and indirect effects of a change in final demand. Table 8.5 reports such a table for the aggregated system closed with respect to households . Table 8.6 converts these numbers to "multipliers," properly recognizing the status of households as recipient of flows of final incomes.

Now, we are at the same stage as completed in chapter 5, and can revert to its descriptive elements. Only a few comments remain on economic change in the new system.

8.4 Economic change in commodity-by-industry models

As discussed in chapter 5, economic change can take two forms in input-output analysis: structural change or change in final demand. Changes in final demand are traced as discussed. Structural change can be treated in a slightly different manner now. The following comments amend slightly the points made in chapter 5.

Structural change normally manifests itself in changes in provincial production coefficients (direct requirements) but there are a number of ways in which it can occur. Let us look at these in terms of the coefficients in the model.

First, a change in technology could occur. This would affect the relations underlying the use matrix contained in the commodity flows table; that is, it would change the relevant technical production relations of the system. Such a change could involve a shift from oil to coal for industrial fuel needs or a shift from glass bottles to cans in the beverage industry. This is admittedly a restrictive interpretation of technological change in that it only involves changes in current flows. Changes in capital intensity, or technological changes which essentially affect the man-machine relationship, are more difficult to trace through an input-output system. The initial impact of new construction or equipment expenditures may be traced as a change in final demand. And, if the increase in output associated with a change in capacity is sold to a final-demand sector, especially as exports, its total effect may be traced through the system. But many of the effects of capital accumulation on economic activity are transmitted through other means. Technological changes in the broad sense are related to the dynamic questions of economic growth. The empirical resolution of these questions involves far more than a static input-output model, and, in fact, far more than any dynamic economic model in current use.

Second, a change in import patterns, or a change in mij, might occur. The discovery of domestic oil or the entry of a major producer of plastic containers might substantially reduce imports of these commodities and thus increase the provincial flows. A program of import substitution might increase provincial flows, delaying the inevitable leakage of money flows from the economy and increasing the multiplier effect of export activity.

Third, alterations in the product mix of local producers may change domestic market shares in commodity outputs. These coefficients determine the commodity content of industry sales in the provincial interindustry matrix and thus represent one final, probably minor, cause of changes expressed in terms of the existing plant structure.

Fourth, new plants in existing industries may enter the provincial economy. A new plant would have the effect of altering production coefficients and import patterns in its industry to the extent that its technology and market areas differ from that of established producers. The significance of the alteration would depend on the size of the new plant relative to the rest of the industry. But only in exceptional cases should the impact of the structural change be more important than the impact of its sales pattern.

Fifth, a completely new industry might appear in the province. This should normally be considered part of the new-plant alternative discussed above.

In summary, structural change in input-output models is primarily a matter of changes in technical production coefficients, in domestic market-share coefficients, or in import coefficients.

8.5 Study questions

  1. Contrast the commodity-by-industry input-output model with the industry-by-industry input- output model.
  2. In Illustration 8-1, provide the details in the "manipulation to solution" from
    g = D(Bg + e + x - (Bg + e) to
    g = (I - D(I - )B)
    -1D((I - )e + x).

Illustration 8.1 The simple commodity-by-industry input-output model for a region

Symbol definitions

^ on a vector produces a diagonal matrix with values from the vector on the diagonal and zeroes elsewhere.

e is a vector of the sum of domestic final demands for commodities.

g is a vector of the values of industry outputs.

i is a unit (summation) vector containing all ones (Premultiplication by i sums columns; postmultiplication sums rows).

i as a subscript counts producing industries.

k as a subscript counts commodities.

j as a subscript counts buying industries.

q is a vector of the values of locally produced commodity outputs.
x is a vector of commodity exports.
m is a vector of commodity imports.
y is a vector of household incomes from industries
B is a commodity-by-industry matrix showing for each industry the commodity use per dollar of industry output. It is the production-coefficients matrix.
D is an industry-by-commodity matrix showing for each commodity the proportion of the total output of that commodity produced in each industry. It is the market-share matrix.
I is the identity matrix (î)
U is a commodity-by-industry matrix showing for each industry the amount of each commodity used.
V is an industry-by-commodity matrix showing for each commodity the amount produced by each industry.
is a diagonal matrix of commodity imports expressed as proportions of local industry and final demands. It is the import-coefficients matrix.

Definitions or identities:

Industry outputs = Sum of commodities produced

g = Vi (1)

Commodity supply = Commodity output + imports

qs = q + m (2)

Commodity demand = Sum of intermediate demands, final demands and exports

qd = Ui + e + x (3)

Behavioral or technical assumptions:

Constant production coefficients

B = U -1 or Ui = Bg (4)

(elements of commodity-by-industry B are bij = uij/gj, so uij = bijgj)

Constant market-share coefficients

D = V-1 or g = Dq (5)

(elements of industry-by-commodity D are dji = vji/qi , so vji = dijqi

Constant import coefficients

= m(Ui + e)-1 or m = (Bg + e) (6)

(elements of commodity-by-commodity diagonal matrix of import coefficients are

mk/(jukj + ek))

Equilibrium condition:

Commodity supply = Commodity demand
qs = qd, or
q + m = Ui + e + x (7)

Solution by substitution:
Problem: given final demands (e' and x'), reduce the number of unknowns to equal the number of equations.

Substitute production (4) and import coefficients (6) into the equilibrium condition (7) expressed in terms of q:

q = Bg + e + x - (Bg + e)

Multiply by D to eliminate q and express in terms of g (industry outputs) using equation (5):

Dq = D(Bg + e + x - (Bg + e))

g = D(Bg + e + x - (Bg + e))

Expand, regroup, and manipulate to solution in terms of g:

g = DBg + D(e + x) - D (Bg + e))

g = DBg + De + Dx - D Bg - D e

g = DBg - DBg +De - De + Dx

g = D(I - )Bg + D((I - )e + x)

g - D(I - )Bg = D((I - )e + x)

(I - D(I - )B)g = D((I - )e + x)

(I - D(I - )B)-1(I - D(I - )B)g =

(I - D(I - )B)-1D((I - )e + x)

g = (I - D(I - )B)-1D((I - )e + x)

Note in coordination with text:

B is total production coefficients, or proportional purchases without regard to location of production, expressed in commodity-by-industry terms.

(I - )B is regional production coefficients, or proportional purchases of locally produced outputs, expressed in commodity-by-industry terms.

D(I - )B is regional production coefficients expressed in industry-by-industry terms.

(I - D(I - )B)-1 is the inverse, the solution, or the total requirements matrix equivalent to that in the simple solution presented earlier.

D((I - )e + x) is a vector of final demands for locally produced commodities and exports expressed (through premultiplication by D) in industry terms.

Output multipliers:

gi/xj = rij , where rij is an element of R=(I - D(I - )B)-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:

yi/xj = rij*yi/gi), where yi is household income earned in industry i.

Each of these partial income multipliers shows the change in household income in industry i caused by a change in exports by industry j. Their sum over i is the total income multiplier for industry j.


This summary is a variation on the U.S. model (described in the following appendix) to include imports and to yield an industry-by-industry model. It also includes elements of the description of the Canadian system from several sources such as (Statistics Canada 1976). Both sources follow the standard United Nations format and symbols.

8.6 Appendix 1 The mathematics of the United States input- output model

The following mathematics is taken from the documentation of the 1987 U.S. Interindustry Study. It follows the standard United Nations commodity-by-industry system symbolically and logically.(5)


September 1, 1993
The following are definitions:
q is a column vector in which each entry shows the total amount of the output of each commodity.
U is a commodity-by-industry matrix in which the column shows for a given industry the amount of each commodity it uses, including Noncomparable imports (I-0 80) and Scrap, used and secondhand goods (I-0 81). I-0 81 is designated below as scrap.
^ is a symbol that, when placed over a vector, indicates a square matrix in which the elements of the vector appear on the main diagonal and zeros elsewhere.
i is a unit (summation) vector containing only l's; î is the identity matrix (I).
e is a column vector in which each entry shows the total final demand purchases for each commodity from the use table (table 2).
g is a column vector in which each entry shows the total amount of each industry's output, including its production of scrap.
V is an industry-by-commodity matrix in which the column shows for a given commodity the amount produced in each industry. V has columns showing only zero entries for noncomparable imports and for scrap. The estimate of V is contained in columns 1 - 79 of the make table (table 1) plus columns of zeros for columns 80 and 81.
h is a column vector in which each entry shows the total amount of each industry's production of scrap. The estimate of h is contained in column 81 of the make table. Scrap is separated to prevent its use as an input from generating output in the industries in which it originates.
B is a commodity-by-industry matrix in which entries in each column show the amount of a commodity used by an industry per dollar of output of that industry. Matrix B is derived from matrix U.
D is an industry-by-commodity matrix in which entries in each column show, for a given commodity (excluding scrap), the proportion of the total output of that commodity produced in each industry. D is referred to as the market share matrix.
p is a column vector in which each entry shows the ratio of the value of scrap produced in each industry to the industry's total output.
W is an industry-by-commodity matrix in which the entries in each column show, for a given commodity, the proportion of the total output of that commodity produced in each industry adjusted for scrap produced by the industry. This matrix is the transformation matrix.

The following are identities:

q = Ui + e (1)

g = Vi + h (2)

The following are assumptions:

Inputs are required in proportion to output and the proportions are the same for an industry's primary and secondary products (the industry technology assumption); then: U = B (3)

Each commodity (other than scrap) is produced by the various industries in fixed proportions (the market shares assumption); then:

V = D (4)

Scrap output in each industry is proportional to total output of the industry; then:

h = g (5)

The model expressed in equations (1) through (5) thus involves three constants (B, D, p) and six variables (U, V, h, e, q, g). The model solution is derived as follows:

Substituting (3) into (1) gives:

q = Bg + e (6)

Substituting (4) into (2) gives:

g - h = Dq (7)

Substituting (5) into (7) and solving for g:

g - g = Dq (I - p)g = Dq

g = (I - ) -1Dq (8)

Let (I - ) -1D = W, then (8) becomes

g = Wq (9)

Substituting (9) into (6) and solving for q:

q = BWq + e

(I - BW)q = e

q = (I - BW)-1e (10)

Substituting (10) into (9) gives:

g = W(I - BW) -1e

(I - BW)-1 is the commodity-by-commodity total requirements matrix, giving commodity output required per dollar of each commodity delivered to final users.(6)

W(I - BW)-1 is the industry-by-commodity total requirements matrix, giving the industry output required per dollar of each commodity delivered to final users.(6)

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