SAM Multipliers: Their Decomposition, Interpretation and Relationship to Input-Output Multipliers

Transcription

1 Research Bulletin XB SAM Multipliers: Their Decomposition, Interpretation and Relationship to Input-Output Multipliers ~ Washington State University - College of Agriculture and Home Economics Research Center

2 TABLE OF CONTENTS Introduction Structure of the SAM The SAM Accounts Production Accounts Consumption Accounts Accumulation Accounts Trade Accounts and the Treatment of Imports The Construction of SAM Models SAM Coefficients Decomposing SAM Multipliers: Interactions Among Sections of the Economy Rationale The Impact of a Shock, Round-by-Round The Algebra of Multiplier Decomposition A Reduced Version of the SAM A Full Version of the SAM The Stone Decomposition The Tables of Decomposed Multipliers--Stone and Pyatt and Round The Economic Interpretation of the Effects of Exogenous Shocks The Own, Direct or Intragroup Effects The Open Loop and Extragroup Effects Closed Loop Effects SAM and Input-Output Multipliers: Implications for Closure of Regional Models.. 31 Using Value-Added as a Proxy for Income Using Earnings as a Proxy for Income Implication of Findings References Appendix Reducing the Number of Sectors in a SAM First Reduction Second Reduction

3 LIST OF TABLES 1. Aggregate SAM for the United States, 1982 (import-ridden) Aggregate SAM for the United States, 1982 (imports purged) Matrix of normalized expenditure shares (S) The SAM inverse matrix Round-by-round approximation of the SAM inverse Reduced aggregate SAM, nine sectors Pyatt and Round decomposition Stone decomposition Stone decomposition in percentages The reduced SAM with households combined The reduced SAM: Value-added closure Reduced SAM: Earnings/personal income closure Al. Reducing the SAM ABOUT THE AUTHORS David Holland is a professor of Agricultural Economics and Peter Wyeth is an associate professor, Agricultural Economics and Marketing, both at Washington State University, Pullman, Washington. Research was conducted under project number 798.

4 SAM MULTIPLIERS: THEIR DECOMPOSITION, INTERPRETATION AND RELATIONSHIP TO INPUT-OUTPUT MULTIPLIERS by David Holland and Peter Wyeth INTRODUCTION This paper focuses on the social accounting matrix and economic models constructed from social accounting matrices. This topic has an increased relevance and importance now that the IMPI.AN system provides for the construction of social accounting matrices at both the regional and national level, thus facilitating the linkage of social accounting and inputoutput accounts in model building (Alward et al., 1989). A major advantage of social accounting accounts over input-output accounts is the improved modeling of linkage between income generation and the distribution of income as identified in the social accounting matrix. The result is more accurate model closure when households are treated as an endogenous variable in fixed price multisector models. This paper has three parts. The first part reviews the flow of income as it is captured in a social accounting matrix (SAM) representative of the U.S. economy. From this matrix a simple economic model is constructed and SAM multipliers are calculated. The role of imports is reviewed and economic interpretation of the multipliers is explained. The second part deals with the algebra of SAM multiplier decomposition and the results of both Stone, and Pyatt and Round decompositions is discussed. Although there is extensive literature on SAM models, SAM multipliers, and their decomposition (Pyatt and Round, 1985; Robinson and Roland-Holst, 1987), it is difficult to fully understand because of the complexity of SAM models used to illustrate the decompositions. By working with a simple and highly aggregated SAM model that is similar to an input-output model with endogenous households, the presentation of basic issues should be more clear, especially to economists familiar with input-output analysis. Of particular interest is the relationship between the structure of SAM linkage, the algebra of multiplier decomposition, and the economic interpretation of the decomposition. We conclude with a discussion and comparison of SAM coefficients and input-output (IO) Leontief coefficients when households are treated as endogenous in both models. This illustrates the distinctions between SAM and IO models more clearly as well as points out the implications of our work on model closure concerning the households endogenous model constructed from IO accounts. 1

5 STRUCI'URE OF THE SAM The SAM Accounts The basic structure of a SAM is derived from the National Income and Product Accounts. Table 1 gives a highly aggregated schematic version of the U.S. SAM based on data provided by Economic Engineering Associates (Adelman and Robinson, 1986). Major categories that appear for both the rows and columns of the SAM are production, consumption, accumulation, and trade accounts. These main accounts are broken down into several subaccounts. Although the specification of subaccounts for any given SAM vary considerably, the major accounts in Table 1 are common to all SAMs. Production Accounts Production accounts are composed of production activities and factors of production. Activities use commodities in the form of goods and services to produce commodities. For the version of the SAM in Table 1, separate use and make accounts from a more disaggregated SAM have been combined into activity accounts. All production activities in the U.S. input-output accounts have been aggregated into the three activity accounts represented in Table 1. The factors of production accounts relate to the primary factors used by society in the production process. They are often referred to as the value-added accounts, which are used extensively in input-output analysis. Traditionally they comprise land, labor, and capital. The factor accounts are paid by activities when production takes place. Referring to Table 1, it is possible to read across the activities row to determine total commodity demand. It is composed of commodities consumed by activities in production, household consumption, government consumption, investment, and exports. consumption of commodities by activities is referred to as intermediate demand and is used to form the technical requirements matrix. The activities columns show expenditures on inputs used in the production process, value-added payments to primary factors, and indirect taxes paid to government. In a SAM, rows represent receipts and columns represent expenditures. Each account is represented by a row and column. Row totals equal column totals for a given account. Value-added consists of payments for labor, returns to capital, and indirect business taxes. The sum of capital and labor inputs used in production must equal gross domestic production at factor cost. The sum of all factor payments comprises gross factor incomes. These payments, along with indirect business taxes, are in turn redistributed to what are called institutional accounts in the value-added columns (columns 4-6). The rows and The 2

6 Table 1. Aggregate SAM for the United States, 1982 (Import-Ridden). ACTIVITIES Agriculture 2 Ag.-Related Ind. 3 Other Industries --ACilVfTIES-- --VALUE-ADDED- ---INSITfUTIONS/HOUSEHOLDS--- --EXOGENOUS-- TOTAL Wage Agric. Ag.-Rel. Other Labor Cap. lnd.tax Prop. Ent. IAOHH M4HH H2HH CapAc. Gvt. RoW Work A C {) VALUE-ADDED 4 Labor ----V Capital 6 Indirect Taxes w INSTS/HSLDS 7 Wage Work 8 Proprietors 9 Enterprises 1 Low4%HH 11 Medium 4% HH 12 High 2% HH ---- y H o {) {) {) EXOGENOUS 13 Capital Account 14 Government 15 Rest of World TOTAL Row Totals May Not Equal Column Totals Due To Rounding. A = Interactivity Flows V =Activity/Value-Added Flows Y = Distribution of Value-Added to Institutions C = Household Consumption Flows H = Distribution of Institutional Income to Households

7 columns for factors of production sum to gross factor income and must equal each other, so that all income received by a given factor account is distributed to institution accounts. Institution accounts receive factor income from the value-added accounts and distribute it to other institutions that generate final demand: the government, household, and capital (saving) accounts. The enterprises institution represents incorporated business and receives income in the form of returns to capital and depreciation allowances. This institution pays part of these returns back to households in the form of dividends, interes~, and rent. Depreciation and retained earnings are the basis for the enterprises contribution to the capital or savings row. The wage work institutional account receives income from labor and distributes it to households. Additional labor income is paid to government in the form of social security payments. The proprietors' institutional account represents sole proprietor types of business. It receives income from the capital factor account and distributes it to households. Consumption Accounts The consumption accounts consist of households (rows and columns 1-12) and government (row and column 14), and are a major component of the final demand accounts. The columns for the accounts of households (columns 1-12) sum to gross expenditures and consist of household expenditures on goods and services, payments of direct taxes, as well as savings and gross transfers abroad. The rows for households (rows 1-12) represent gross receipts from labor, proprietors' income, receipts for capital earnings from enterprises, receipts from government transfers, and earnings from abroad. Gross household receipts must equal gross household expenditures. Household income in the U.S. SAM is distinguished according to the size distribution of income. Accumulation Accounts Accumulation accounts record capital investment and change in stocks in the column (13) and savings from households, enterprises, and government as well as the balance of foreign trade on capital account in the row (13). The saving from enterprises, households, and government accounts are all combined into one row which shows the source of capital payments. Investment is financed by savings of domestic institutions and foreign financing through the balance of payments, such that gross capital receipts and capital payments equate. Trade Accounts and the Treatment of Imports Trade accounts show U.S. economic interactions with the rest of the world. There are two separate trade accounts, one representing outflows of goods and services (exports) 4

8 and inflows of money; the other representing inflows of goods and services (imports) and outflows of money. The trade row (15) shows the outflows of revenue to other countries from the purchase of imports and transfers to abroad from institutions. The trade column (15) shows the inflows of revenue from other countries from the purchase of U.S. exports. Once again, gross payments abroad must equal gross current receipts from abroad, with the required balance of payments represented in the capital account row in the rest of the world column (cell 3, 15). The commodity rows (1-3) in Table 1 show the absorption of goods and services by major components of the economy. The flows represented in these rows are import-ridden. In other words, the flows of commodities represent sales of both domestic production and imports. At the bottom of the activity columns, imports by commodity are identified. The column total represents gross total outlay for the activity as well as for imports of the commodity produced by that activity. In this way the sum of each column is equal to the sum of each row for rows 1-3. As a result of this treatment of imports in the U.S. SAM, it is possible to calculate the technical coefficients by simply normalizing the column totals for the activities block. The resulting technical coefficients aij represent the production recipe from both domestic and imported sources for any activity. The SAM inverse coefficients in an economic model based on these technical coefficients would show the necessary..t.q1sl supply response (imports plus domestic production) for a shock in an exogenous variable. By making the assumption that domestic commodities are perfect substitutes for imported commodities, the SAM coefficients could be interpreted as indicating the required change in domestic output (more will be said on model construction later). A more conventional modeling approach would treat domestic supply and imported supply as imperfect substitutes. In this case, the domestic component of the technical coefficient, r ij, would show the amount of domestic input per unit of output. In order to construct an economic model of this kind, it is necessary to first purge the commodity rows of their import content. The material balance equations may be written as follows: (1) X+ M = V + F + E. 5

9 where: X = domestic supply M = imported supply V = intermediate demand F = domestic final demand E = export demand. In order to construct an economic model of the domestic economy under the assumption that imports are not perfect substitutes for domestically produced goods, it is necessary to purge imports from V and F. A common approach to dealing with this problem is to assume that imports are used in only domestic production and are not directly embodied in exports. The second assumption is that imports are absorbed in both intermediate and final demand (except exports) in a given proportion for every commodity. This proportion may be found by the following equation: where: and: I; = Import supply ratio--the ratio of imported supply to domestic supply X; = domestic output of commodity i E; = exports of domestic commodities i M; = imports of commodity i d; = domestic demand for domestic production of commodity i divided by total commodity supply for commodity i less exports of commodity i X; = domestic production of commodity i A diagonal matrix D of the d; parameters is then used to purge the material balance equations of imports as follows: and Then we may write the domestic supply and demand balance equations as: 6

10 (3) X=Vd+Fd+E. The result of purging imports from the activity rows in the SAM are presented in the SAM depicted in Table 2. Imports now appearing in the rest of the world row are classified by sector of destination rather than by sector of origin as was the case in Table 1. TilE CONSTRUCTION OF SAM MODELS To move from a set of social accounts to a SAM model requires that additional assumptions be made (Adelman and Robinson, 1986). A common approach in Type I input-output models is to use the fixed coefficients assumption. Under this assumption the elements in each column of the interindustry accounts are divided by the respective column total resulting in a table of technical coefficients. These coefficients are assumed to represent the production functions of the firms represented by each sector. By assuming that firms respond to changes in demand according to the parameters of the ftxedproportion production function, a model can be specified as a system of simultaneous linear equations. The model can then be solved to yield coefficients through which changes in final demand are translated into changes in each sector's supply (Miller and Blair, 1985). As in the input-output model, supply is assumed to always meet demand. Similar assumptions are needed when creating a SAM model. Since the SAM model includes a more comprehensive view of the circular flow of income than a standard inputoutput model, it requires the ftxed coefficients assumption to extend to the coefficients of all the endogenous accounts. The ftxed coefficients assumption, which in interindustry input-output models is a ftxed technology assumption, now must include the assumption that various household expenditure coefficients are ftxed when household variables are treated as endogenous. In input-output accounts (IO), only the interindustry linkages are formally specified. The linkage between household income and household spending is not defined nor is the linkage between government revenues and government spending or the linkage between saving and investment. The identification of these linkages in SAM accounts permits industry /household linkages to be specified with the same precision that interindustry linkages are specified in the IO model. The result is that in SAM models, household, government, and investment variables may be more accurately treated as endogenous variables. For purposes of this paper, only households are treated as endogenous. Our intent is to encourage a connection to a similar type of IO model (Type II). In order to construct a SAM model, assumptions similar to the fixed coefficients assumption for the input-output model must be made. The result is that in addition to the ftxed technical coefficients of the 7

11 Table 2. Aggregate SAM for the United States, 1982 (Imports Purged). ACTIVITIES Agriculture 2 Ag.-Related Ind. 3 Other Industries --AcnYmES -- -VALUE-ADDED- INSITIUilONS/HOUSEHOLDS EXOGENOUS-- TOTAL Agric. Ag. Rel. Other Labor Cap A Ind. Tax 6 Wage Work Prop. Ent. IAOHH M4HH H2HH CapAc. Gvt c RoW VALUE-ADDED 4 Labor 5 Capital 6 Indirect Taxes ----V INSI'S/HSLDS 7 Wage Work 8 Proprietors 9 Enterprises 1 Low4% HH 11 Medium 4% HH 12 High 2% HH ---Y H EXOGENOUS 13 Capital Account 14 Government 15 Rest of World TOTAL 196.o Row Totals May Not Equal Column Totals Due To Rounding. A = Interactivity Flows V = Activity/Value-Added Flows Y = Distribution of Value-Added to Institutions C Household Consumption Flows H = Distribution of Institutional Income to Households

12 1 model, the distribution of nominal income between wages and profits are assumed to be fixed. Also, the distribution of wage and profit income to households is given in fixed proportions as is the average tax and savings rates of households, and the sectoral composition of household consumption (columns 1-12, Table 3). The result of treating households as endogenous is a partitioned SAM, (the exogenous columns and rows are excluded) shown as follows: Activities Value-Added s = Endogenous Institutions where: S = matrix of SAM direct coefficients (12,12) A = matrix of technical coefficients (3,3) V = matrix of value-added (VA) coefficients (3,3) Y = matrix of VA distribution coefficients (6,3) C = matrix of expenditure coefficients (3,6) H = matrix of institutional and household distribution coefficients ( 6,6). The supply and demand balance equations can then be written as: (4) s [n. [:J where: X = vector of sector supply (3, 1) V = vector of value-added by categories (3, 1) Y = vector of household incomes ( 6,1) ex = vector of exogenous commodity demand (3,1) ev = vector of exogenous value-added (3, 1) ey = vector of exogenous household incomes ( 6,1) The (I - S) matrix can then be inverted to specify a matrix equation that expresses levels of sectoral supply, value-added, and household income as a function of exogenous variables. This yields: 9

13 Table 3. Matrix of Normalized Expenditure Shares (S). Agric. Ag.-Rel. 2 Other 3 Labor 4 Cap. 5 Ind. Tax 6 Wage Work 7 Prop. 8 Ent. IAOHH M4HH H2HH CapAc. Gvt. RoW Agriculture 2 Ag.-Related Ind {) Other Industries Labor 5 Capital 6 Indirect Taxes Wage Work 8 Proprietors 9 Enterprises 1 Low4% HH 11 Medium 4% HH 12 High 2% HH {) {) {).1 13 Capital Account 14 Government 15 Rest of World {) {) {) Total 1 1

14 (5) = where (/- St 1 represents the matrix of SAM inverse coefficients. SAM Coefficients The matrix of SAM direct coefficients is displayed in Table 3 while the matrix of SAM inverse coefficients is displayed in Table 4. Since we are treating households as endogenous, the interpretation of SAM inverse coefficients remains the same, as would be the case in a Type II IO model. Consider the coefficients in column 1 of Table 4. In response to an increase of $1 billion in final demand for agricultural output, an additional output of $.41 billion from agriculture is generated as well $1.98 billion from agriculturally related sectors and $.49 billion from other industries. Payments to labor are increased by $.94 billion, payments to capital by $.73 billion and indirect taxes paid are increased by $.15 billion. If we wish to know how households will be impacted, the SAM framework contains the linkage between the payment of wage and profit income and the distribution of that income. The income of low income households is estimated to increase by $.14 billion, medium income households will experience an increase of $.5 billion, and high income households will receive an increase of $.58 billion. An example of the improvement in multiplier interpretation that is offered with SAM models comes from the coefficients in the household columns. Consider an increase in government spending in the form of a government transfer to low income households of $1 billion, financed by government borrowing. The linkage between government transfers and households is missing in the IO accounts, so an IO model cannot address a policy shock to a household income class. As a result of this shock, agricultural output will increase by $.11 billion, agriculturally related output will increase by $2.64 billion and other industry output will increase by $.65 billion. Payments to labor will increase by $1.8 billion, payments to capital will increase by $.57 billion and indirect taxes will increase by $.16 billion. Perhaps most interesting of all, income to low income households will increase by $.14 billion above the original injection, medium households will experience an increase of $.53 billion and high income households will receive an increase of $.59 billion. In the previous example we could have assumed that the increased government borrowing crowded out private investment from the financial capital market and reduced 11

15 Table 4. The SAM Inverse Matrix. Agric. Ag.-Rel. Other Labor Cap. Ind. Tax Wage Work Prop. Ent. IAOHH M4HH H2HH Agriculture o.4n Ag.-Related Ind Other Industries n9 4 Labor Capital o.3n Indirect Taxes o.12n N 1 7 Wage Work TI Proprietors Enterprises Low4% HH Medium 4% HH High 2% HH

16 investment purchases from other industries by a like amount. The policy shock would then include the government transfer change and the investment change simultaneously. Rationale DECOMPOSING SAM MULTIPLIERS: INTERACfiONS AMONG SECfiONS OF THE ECONOMY The matrix of SAM multipliers derived in the previous section shows what the impact of an external shock on any given sector of the economy will ultimately be, after all the repercussions have worked themselves out. What it permits, for example, is a comparison of how the economy looks before and after a change in economic policy concerning tax rates or public investment, or an alteration in some other external condition such as the level of export demand. As far as policy decision makers are concerned, this may be as much as they want to know, but from the viewpoint of understanding the process of economic adjustment to these external shocks, the information provided by these multipliers alone is limited. There is more than one way of analyzing the SAM multipliers. We will demonstrate two. First, the process by which the multiplier effects accumulate round-by-round will be examined. Second--following a procedure developed by Pyatt and Round (1985)--we will derive three submultipliers, each of which alone calculates shock effects as they travel through subsections of the total matrix. The result is a multiplicative decomposition of the SAM inverse coefficients derived in Table 4. A variation on this decomposition provided by Stone (1985) will also be described. The Impact of a Shock, Round-by-Round Suppose the demand for agricultural exports goes up by $1 billion. This exogenous change constitutes the first round of increased expenditure. In the second round this money, all $1 billion of it, is spent by agriculture in proportions given by the first column of normalized expenditures given in Table 3. This tells us that agriculture will spend $.2471 within its own sector, $.3566 on agriculturally related industries, $.346 on other industries, and so on. In the third round of spending, each sector now increases its spending by the amount of the boost it has just received from agriculture according to the expenditure coefficients in Table 3, providing a further increment in demand for every other sector. The total increase in demand this time for agriculture is made up of $(.2471 x.2471) billion from itself, $(.234 x.3566) from agriculturally related industries, and $(.51 x.346) billion from other industries, which added together gives $.696. This figure is entered in the (1,1) 13

17 cell of the Round 3 matrix in Table 5. The increased expenditure on agriculturally related industries is $(.3566 x.2471) from agriculture, $(.2843 x.3566) from agriculturally related industries themselves, and $(.2349 x.346) from other industries, totalling $.1976 billion. While households spend money on commodities produced by these sectors, as shown by the household columns in the S matrix, they have not yet received an increase in income from the change in agricultural exports and therefore do not yet raise their expenditure. This delayed impact on households is a consequence of the accounting structure of the SAM. In Round 4 the funds available for expenditure are those in column 1 of the table for the third round, and they are allocated in the same way, that is, according to the normalized expenditure shares in the S matrix. Though, this time the coefficients are applied to the figures in the Round 3 matrix. For example, expenditure on agriculture is $(.2471 x.696) from agriculture, $(.234 x.1976) from agriculturally related industries, and $(.51 x.54) from other industries, totalling $.221 billion (see cell (1,1) of the Round 4 matrix). The reason we confine ourselves to the agriculture column is that the original shock was to agriculture. Shocks to other sectors would shift our attention to other columns. This process can be carried on ad infinitum, at least mathematically, but with the coefficients declining at each round because some of the expenditure leaks out of the system with each round. Note that the accumulated round-by-round effect is written algebraically as: ( 6) I + S + S 2 + S 3 + S 3... S n = (I - S) -l and that this sum is a well-known approximation for the expression (I- St 1 (see Miller and Blair, 1985, pp ). How quickly the numbers diminish to insignificance depends on what proportion payments to exogenous variables (leakages) are of total payments and on the structure of the accounts that pass added value to the consuming institutions (households and government). In the case of the relatively self-contained U.S. economy, the leakages are relatively small and the SAM accounts fairly extensive, so it takes many rounds (over ten) before the ripple effects diminish into insignificance. Most other national economies are much more open. Imports and exports are proportionally more important to them. Regional economies within countries, such as states or counties, are substantially more open and far fewer rounds are needed to exhaust all appreciable impact of a given external shock. While most round-by-round effects diminish in size, the number of nonzero coefficients in the agriculture column lengthens with each successive round. This is a 14

18 Table 5. Round-by-Round Approximation of the SAM Inverse. 1 Agriculture 2 Ag.-Related Ind. 3 Other Industries 4 Labor 5 Capital 6 Indirect Taxes 7 Wage Work 8 Proprietors 9 Enterprises 1 Low 4% HH 11 Medium 4% HH 12 High 2% HH Agric. Ag.-Rel. Other Labor Cap. lnd.tax Wage W. Prop. Ent. I.AOHH M4HH H2HH Agric. Ag.-Rel. Other Labor Cap Agriculture Ag.-Related Ind Other Industries Labor Capital Indirect Taxes Wage Work Proprietors Enterprises Low 4% HH Medium 4% HH High 2% HH lnd.tax Wage W. Prop. Ent I.AOHH M4HH H2HH Round 5 (S 4 ). Agric. Ag.-Rel. Other Labor Cap Agriculture Ag.-Related Ind Other Industries Labor Capital Indirect Taxes Wage Work Proprietors Enterprises Low 4% HH Medium 4% HH.822 o.615 o.om o 12 High 2% HH lnd.tax Wage W. Prop. Ent I.AOHH M4HH H2HH O.OD

19 Round 6 (S 5 ). Table 5. Continued. Agric. Ag.-Rel. Other Labor Cap. Ind.Tax Wage W. Prop. Ent. L4HH M4HH H2HH Agriculture ()()()() Ag.-Related Ind ()()()() Other Industries ()()()() Labor ()()()() Capital ()()()() Indirect Taxes ()()()() Wage Work ()()()() Proprietors ()()()() Enterprises ()()()() Low 4% HH Medium 4% HH High 2% HH Round 7 (S 6 ). Agric. Ag.-Rel. Other Labor Cap. lnd.tax Wage W. Prop. Ent. L4HH M4HH H2HH Agriculture ()()()() Ag.-Related Ind ()()()() Other Industries ()()()() Labor ()()()() Capital ()()()() Indirect Taxes ()()()() Wage Work ()()()() Proprietors ()()()() Enterprises ()()()() Low 4% HH ()()()() Medium 4% HH ()()()() High 2% HH ()()()()

20 consequence of the way the SAM is set up. In Round 3, the three rows for nonhousehold institutions, which had contained only zeros in Round 2, are now full of positive elements. Similarly, Round 4 shows that the household rows of the agriculture column have filled, showing that the labor, sole proprietor, and incorporated enterprise institutions have "distributed" income to households. All this follows from the nature of matrix multiplication, but also falls in neatly with the flow of income in the SAM. Income reaches nonhousehold institutions via the valueadded block, and then is passed on to the household accounts. The entire matrix does not fill up until Round 7. As a result of the structure of SAM accounts, the feedback from production to consumption and back to production takes a number of rounds to work out and coefficients of higher rounds are not uniformly lower than coefficients from previous rounds. For example, the Agric.-Ag.-Related industry coefficient in Round 5 is larger than it is in Round 4 (Table 5). While the round-by-round process represented by equation (6) is illustrative of the flow found through the economy, in reality the spreading process depicted is quite artificial. What happens in the real world is that households receive their income directly from industries. The value-added accounts and nonhousehold institutions are accounting conventions to help record income moving out of the production sectors and into the institutions that provide final demand. While a certain time path will characterize a given economy's reaction to economic shocks, that path is not likely to be well represented by the adjustment process characterized in equation ( 6). The Algebra of Multiplier Decomposition Rather than begin with the mathematics necessary to derive the decomposed multipliers for the SAM model, we will work up to it in stages through simpler versions of the same SAM. The reason for this is not that the mathematics are difficult, but that simpler cases allow us to bring out interesting points about the decomposition of multipliers and the associated derivation process. A Reduced Version of the SAM In reality, households receive their income directly from industries and the simplest SAM could omit the value-added and institutional sectors (income distribution accounts) without loss of realism. Table 6 is an example of such a table. The industry accounts are now shown as paying the household, capital, and government accounts directly. Note that 17

21 the column totals of the original SAM of Table 2 have been preserved. 1 In this case the block form of endogenous variables for SAM previously presented would be reduced to this: [t ~] Table 6. Reduced Aggregate SAM, Nine Sectors. Agric. Ag.-Rel. Other IAOHH M4HH H2HH CapAc. Gvt. RoW Total Agriculture Ag.-Related Ind Other Industries Low4% HH Medium 4% HH High 2% HH Capital Account Government Rest of World Total It is the upper left partition, designated SR, that provides the technical coefficients incorporated in the SAM inverse. In order to pursue the question of how the endogenpus components of the model interact, this partition can be split into two matrices, one containing the blocks on the diagonal and the other the off-diagonal blocks, giving sr = Q + R, where: Q= [~ g] R = [t ~] 1 The reduction of the SAM was performed with a program called MATS produced by Berkeley Economic Advising and Research. The principles employed in the calculations are described in the Appendix. 18

22 We can now rework the algebra of the previous section making use of this decomposition of the SR matrix: [:] = sr [:] [:] (7) = Q [:] + R [:] + [:] = (/ - Q)-1 R [:] + (/ _ Q)-1 [:] where: x = vector of sector supply (3, 1) y = vector of household income (3, 1) ex = vector of exogenous sector demand (3, 1) ey = vector of exogenous household income (3, 1) (8) [; J = r [; J. <I - Qr 1 [: J where: Multiply through (8) by T and then substitute for T [;], also in (8): T [;] = T 2 [;] + T(l - Q)-1 [:] (9) [;] = Tz [;] + T(I _ Q)-1 [:] + (/ _ Q)-1 [:] (/ _ T2) [;] = (/ + 1)(/ _ Q)-1 [:] 19

23 (1) [xy] =<I- r 2 r 1 <I + n(j- Qr 1 [exey] where: M3 = (I- y2yl M2 = (I+ T) Ml = (I- Qtl As the matrix Q is block diagonal, so must M 1 be also: What this matrix contains are the own, intragroup, or direct effects multipliers. If there is an exogenous shock to an industry, the multipliers in the (I- At 1 block show how the effects will be passed on to other industries. Effects transmitted to or from sectors in other partitions are excluded. The (I- AY 1 block is, in fact, the Leontief matrix of interindustry multipliers and, in this simple SAM, there is no other block of direct effects multipliers. There are no own effects between household sectors because transfers of income do not take place directly between the household groups (Table 6). The M 2 matrix has quite a different form: M2= [ I H (I - AI)-1 C] It provides what are known as the ex~ragroup, indirect or open loop multipliers. When a sector is affected by an external shock, these multipliers show those effects that are transmitted to other blocks and end there, not fed back to the sector where they originated. These are one-way, outward effects. Any impact on the originating block is excluded, shown by the fact that the diagonal blocks of the M 2 matrix are identity matrices. Suppose, for example, that there is an increase in export demand for agricultural produce. The open loop impact of this on households is given by the multipliers in the H 2

24 partition (unaltered from the sr matrix of normalized expenditure coefficients). In this simple formulation the open loop effects can go no further. The (1,2) block, (I- At 1 C, is important if there is a shock to a household sector, such as a change in government transfers. The multipliers in this partition will then show how the consumption of agricultural and other commodities will be affected, in response to the increased household income. Finally, the M 3 multipliers show those effects that proceed outwards from the block where they originate and then feed back to it. These are known as the intergroup, cross or closed loop multipliers. As a shock must start and end in the same group of sectors, this matrix, like M 1, is block-diagonal. In this matrix, each of the two diagonal partitions contain the same elements, though not in the same order. These are the inverse submatrix from M 1 and the two off-diagonal blocks (Hand C) from the Q matrix. None of them is included more than once? The economic interpretation of this is that the exogenous change has travelled out from the original block which first felt the shock, through all the other blocks and back to where it started. Suppose there is an increase in export demand on certain commodities. This affects household income, which affects consumption, which in turn feeds back to a further change in the demand for commodities. The income-induced increase in supply is represented in the upper left block. Note that the bottom right partition has nonzero elements in it even though the sr matrix showed only zeros. This is because one household sector can be indirectly affected by an exogenous shock to another household sector even though there are no direct transfers between them. The linkage is from households/ consumption to increased supply in the production sectors and back to increased income to households. A Full Version of the SAM We return now to the SAM as it was first presented (4). Clearly, with only 15 sectors, this is a greatly aggregated version of the 528 production sector model from which it was derived. However, in terms of the flow of income among institutions it is more 2 These elements are in a different order and we are dealing in matrix algebra, so the figures within the partitions are not the same. 21

25 where x = [f] ex = [ ~] refer to equation (4) and equation (8) which we multiplied detailed than the reduced version. It differs from the reduced version in that not only the nonhousehold institutions, but also the value-added sectors are explicitly included. S = Q + R would now become: [; ~ ~ l = [~ l [ ~ ~ ~ ~ ~] Y H OH Y This change results in an increase in the number of off-diagonal partitions in R and therefore must also change the form of the expressions for M 2 and M 3 (The expression for M 1 remains (I- Qt 1 because Q is still a diagonal matrix). If we used the forms found in (1) for M 2 and M 3, we would find that these matrices do not have the off-diagonal and diagonal characteristics which they had before and which they must have to make economic sense. The remedy is to expand the equation (8) twice instead of once. We had (11) X = Tx + (I - Qr 1 ex through by T and then substituted for Tx in the same equation, obtaining To expand a second time, we multiply through (12) by T and again substitute for Tx in (11). (13) x = T 3 x + T 2 (I - Qr 1 ex + T(I - Qr 1 ex + (I - Qr 1 ex = (I - T3rl (I + T + T2) (I - Qrlex So M 3 = (I - T 3 )- 1 M 2 = (I + T + T 2 ) Ml = (I - Q)-1 22

26 In matrix form: (/ - A)-1 C(I - H)-1 y I (/ - H)-1 y [/- (/- A)-1 C(I- H)-1 YV]-1 M3 = [I-V(l-A)- 1 C(I-H)- 1 Y]- 1 [ The own effects multipliers in M 1 are necessarily unchanged because, by definition, they ignore the presence of other blocks and refer to effects only within themselves. The M 2 and M 3 matrices still have the same basic form as before, but the shocks must travel yet longer routes to work themselves out. In a shock to an activities sector, the (3,1) partition of M 2 tells us that there would be an open loop effect on value-added, hence through to distribution institutions, and finally to household income. The V (1,1) block of M 3 says that the closed loop effect follows exactly the same route, but with the added final stage of an effect fed back to the consumption sectors and from there back to activities. In fact, comparing corresponding columns in M 2 and M 3 shows that the path travelled by the shock in M 3 is a continuation of that in M 2, where the extension is whatever is necessary to close the loop? 3 Anyone comparing the matrices here with those in the Pyatt and Round article will see differences in the expressions for various partitions. This is because Pyatt and Round, and also Stone, order the rows and columns of their SAMs differently, e.g., factors of production come first instead of industrial activities, and household comes second instead of last. 23

27 The Stone Decomposition 4 Stone (1985) proposed an additive variation of the decomposition developed by Pyatt and Round, in which the decomposition becomes additive rather than multiplicative. Using Pyatt and Round's notation for the matrices of decomposed multipliers, the expression for Stone's version is: If we denote Stone's three submultipliers as N 1, N 2, and N 3, they are: N 1 = M 1 : Own or intragroup effects (identical to Pyatt and Round's) N 3 = M~1 - M 1 : Closed loop or intergroup effects (diagonal) N 2 = M,pl~ 1 - M~1 : Extragroup effects (off-diagonal matrix) Note that Stone's expression form reduces to M,p1~1 which reverses Pyatt and Round's ordering ofm 2 andm 3, but is otherwise the same. This makes no difference to the matrix of full multipliers, M, but when the closed loop and extragroup effects are calculated, Stone's ordering must be preserved if the resulting matrices are to retain their meaning. 5 Conceptually, Stone's decomposition is simpler than Pyatt and Round's. The (diagonal) matrix of intragroup effects is subtracted from the full matrix, (I- St 1. What is left behind in the diagonal blocks, Stone calls the intergroup (own) effects. Everything off the diagonal blocks he calls extragroup effects, which are no different from the off-diagonal multipliers in the full matrix, (I- St 1 For this reason it is better to use Stone's term and 4 In Stone's 1985 article, he uses precisely the same notation as Pyatt and Round, but he refers to Pyatt and Round's M 2 as M 3 and to their M 2 as M 3 Here we use M 2 and M 3 the same way Pyatt and Round do. 5 The full matrix, M, is unaffected because (a) M 2 and M 3 are still both ahead of M 1 in the order of multiplication, and (b) M,p1 3 = M~2 due to the placement of their nonzero and zero elements, which in turn follows from the way the original SAM is set up. If Stone's order of multiplication is not preserved when calculating the matrices of decomposed multipliers and N 3, the matrix of closed loop multipliers, is taken to be (M 3 - I)M,pi 1, it will not be a diagonal matrix, which it must be to illustrate the concept of a shock returning to the same block from which it started. The form of N 2, the matrix of open loop multipliers, will not look obviously wrong, but the coefficients will be incorrect. In this respect the algebra of the Stone decomposition in the MATS program is believed to be incorrect. 24

28 not call them open loop multipliers. They are no longer the "pure" open loop effects because they have been multiplied by both the intragroup and the closed loop effects. 6 The Tables of Decomposed Multipliers--Stone and Pyatt and Round The Economic Interpretation of the Effects of Exogenous Shocks We have been talking of exogenous shocks to sectors and their effects without close attention to what form the shocks themselves might take in practice. To remedy this, we turn to tables derived from the aggregate SAM for the U.S. Where the industrial sectors are concerned, there is no difficulty. The shock could come from a change in final demand either for exports, government purchases or private investment. All three of these are commonly regarded in macro-models as "injections," i.e., as determined outside the system. Their effects should be read down the columns of the sectors to which the shocks first occurred. Households can be affected by either increasing or decreasing their incomes. This can be done directly by changing government transfers to households, such as welfare or social security payments. A change in personal income tax rates would also have direct effects, but it would not constitute a shift in demand in the same way that a rise in transfer payments would. Rather than being a change in injections, an alteration in direct taxes paid by households would constitute a change in leakages, for these payments are found along the bottom of the SAM in the government row. Any increase or decrease in such taxes would change these entries. Hence the normalized expenditure coefficients would be altered, and the SAM and the matrix of full multipliers would have to be recalculated to reflect the change in direct taxes. Precisely analogous reasoning applies to alterations in tax rates applying to the profits of incorporated enterprises (wage work and sole proprietors do not pay taxes in their roles as institutions, but in their roles as households). A change in the minimum wage rate is directed at labor. Households will be affected, but indirectly via the value-added sector. Moreover, the impact is not simply to raise the value-added going to labor by the amount of the change times the number of workers being paid the minimum. There will be an effect on numbers employed and also on the value-added going to capital and perhaps on the prices of the goods and services produced. If the product prices are relatively unaffected, the short-run consequence of an increase in the minimum wage would be to change the distribution of income between labor 6 Similar reasoning might lead to questioning the use of the term closed loop effects for Stone's N 3 matrix, but as the two diagonal matrices alone are involved in calculating N 3, it still seems accurate to use the same term as Pyatt and Round, simply distinguishing between multiplicative and additive effects. 25

29 and capital in favor of labor. This would mean recalculation of the SAM direct coefficient and the multiplier matrix. Given this, is there any practical use for the multipliers in the columns for valueadded and nonhousehold institutions if no exogenous changes can be attributed directly to them? The answer is no, in general, unless separate accounting rows can be introduced to distinguish value-added for selected sectors. 7 However, the information referring to these sectors in the original SAM is of central interest because it shows how income is distributed among firms, households and government, and this information is incorporated into the multipliers for the industrial and household sectors. While calculating the multipliers for these sectors, multipliers for value-added and institutions emerge as a by-product. If the government wanted, for example, to gift $1, each year to all sole proprietors, then the multipliers in that column would be useful to trace the impact of such a policy. In the absence of such arbitrary action their practical significance remains latent. The Own. Direct or Intragroup Effects The matrices for the Pyatt and Round decomposition are found in Table 7 and those for the Stone version in Table 8. 8 The own effects matrix is the same for both decompositions. Within it, except for the (1,1) interindustry partition, the 3 x 3 blocks or sub-blocks on the diagonal are identity matrices. This is a result, already noted, of the fact that there are no direct transfers within the value-added, institutional and household groups, only transfers from one group to the next. However, for the purposes of multiplier decomposition, the institutional and household groups are put together in the bottom right partition, (3,3) so there are some transfers within this block even though they are not strictly "own" effects. What the M 1 matrix tells us is that if there were an exogenous increase in the demand for agricultural goods of $1 billion, interindustry transactions alone would induce increases in output of $1.35 billion for agriculture, $.72 billion for agriculturally related industries, and $.16 billion for other industries. On the other hand, if there were shocks 7 There are instances, as Adelman and Robinson show for value-added, where the accounts can be more narrowly defined so that government action can have a differential effect on payments to factors. 8 The Pyatt and Round decomposition was performed with the MATS program published by Berkeley Economic Advising and Research. The Stone decomposition was derived from that using Lotus Our interpretation of the Stone decomposition results m very different matrices from those calculated in the MATS program. 26

30 Table 7. Pyatt and Round Decomposition. The Own Effects Matrix, Ml = inv(i- Q) Agriculture Ag.-Related Ind. Other Industries Labor Capital Indirect Taxes Wage Work Proprietors 9 Enterprises 1 Low4% HH 11 Medium 4% HH 12 High 2% HH Agric. Ag.-Rel. Other Labor Cap o.4n o.256 o o ()()()().()()()().()()()() 1.()()()().()()()().()()()() lnd.tax WageW. Prop. Ent ()()()().()()()() 1.()()()() IAOHH M4HH H2HH ()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()() 1.()()()().()()()().()()()().()()()() ()()()().()()()().()()()() ()()()() }.()()()().()()()() ()()()().()()()() 1.()()()() 1.()()()().()()()().()()()().()()()() 1.()()()().()()()() Open Loop Effects Matrix, M2 = I + T + (T x 7) Agric. Ag.-Rel. Other Labor Cap Agriculture Ag.-Related Ind Other Industries Labor Capital Indirect Taxes.186 Wage Work.829 Proprietors.271 Enterprises.231 Low 4% HH.283 Medium 4% HH.769 High 2% HH ()()()() ()()()() ()()()() lnd.tax WageW. Prop. Ent ()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()() 1 1 IAOHH M4HH H2HH The Closed Loop Effects Matrix, M3 = inv(l- T x T x 7). 1 Agriculture 2 Ag.-Related Ind. 3 Other Industries 4 Labor 5 Capital 6 Indirect Taxes 7 Wage Work 8 Proprietors 9. Enterprises 1 Low4% HH Agric. Ag.-Rel. Other Labor Cap. lnd.tax WageW. Prop. Ent. IAOHH M4HH H2HH ()()()() 11 Medium 4% HH 12 High 2% HH o.3no ()()()() 1.()()()() 1.()()()().()()()().()()()().()()()() 1.()()()().()()()().()()()().()()()() 1.()()()().()()()().()()()().()()()() ()()()().()()()().()()()() ()()()().()()()().()()()()

31 The Own Effects Matrix, Ml = inv(i - Q). 1 Agricultwe 2 Ag.-Related Ind. 3 Other Industries 4 Labor 5 Capital 6 Indirect Taxes 7 Wage Work 8 Proprietors 9 Enterprises 1 Low4% HH 11 Medium 4% HH 12 High 2% HH Table 8. Stone Decomposition. Agric. Ag.-Rel. Other Labor Cap ()()()().()()()().()()()() 1.()()()().()()()().()()()() Extragroup Multipliers, N2 = (M2 -I) M3Ml. Ind. Tax Wage W. Prop. Ent ()()()().()()()() 1.()()()() 1.()()()().()()()().()()()().()()()() 1.()()()().()()()().()()()().()()()() 1.()()()() IAOHH M4HH H2HH ()()()()..()()()() 1.()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()().()()()() ()()()() 1.()()()().()()()() ()()()().()()()() 1.()()()() Agric. Ag.-Rel. Other Labor Cap. lnd.tax Wage W. Prop. Ent. IAOHH M4HH H2HH Agricultwe Ag.-Related Ind. Other Industries Labor Capital Indirect Taxes ()()()().()()()().()()()() Wage Work 8 Proprietors 9 Enterprises 1 Low4% HH 11 Medium 4% HH 12 High 2% HH ()()()() ()()()() ()()()() ()()()() ()()()() ()()()() Closed Loop Multipliers, N3 = (M3 - I) Ml. 1 Agriculture Agric Ag.-Related Ind Other Industries Labor 5 Capital 6 Indirect Taxes 7 Wage Work 8 Proprietors 9 Enterprises 1 Low 4% HH 11 Medium 4% HH 12 High 2% HH Ag.-Rel Other Labor Cap. Ind. Tax Wage W. Prop. Ent. IAOHH M4HH H2HH ()()()() ()()()() ()()()()

32 to household income in the form of government transfers, there would be no own effects at all. The Open Loop and Extra~roup Effects If there is an exogenous shock to a sector, these matrices tell us what the impact will be on other sectors outside that block. Stone's extragroup multipliers are simply the offdiagonal elements in the matrix of full multipliers, and so are generally larger and can never be smaller than Pyatt and Round's open loop multipliers. The differences between the two versions are by no means uniform. For example, the relative effect of a shock to agriculture is greater on labor than on capital according to the Stone extragroup effects (.9418 versus.732), but the opposite occurs in Pyatt and Round's open loop effects (only.958 for labor but.232 for capital). In other cases the relative impacts are similar between the two versions. For instance, both tell us that an exogenous shock to agriculture will affect indirect taxes, sole proprietors and low income households less than the other sectors in their blocks. Pyatt and Round's open loop effects are all in the matrix M 2, whereas Stone's extragroup effects are M~#/ 1 - M#{ 1, so where labor and capital are concerned, the open loop effects are dominated by the intragroup and closed loop effects, but the contrary holds for indirect taxes, proprietors and enterprises. 9 Table 9 shows the open loop as a proportion of the extragroup effects. They are considerably less than 5% in the columns for the industrial sectors. The size of an open loop multiplier by itself may or may not tell how much of an impact a shock to an industry is likely to have on other sectors. This impact will depend very much on how the open loop effects interact with other "diagonal" effects (both intragroup and closed loop). Closed Loop Effects In the Pyatt and Round closed loop matrix, M 3, the 1 's before the decimal points on the diagonal denote the original exogenous shocks and the figures after the decimal points 9 The contrast between the two types of decomposition is most obvious when we consider the effects which a shock to any of these last three institutions would have on the activity an4 value-added sectors. In the Pyatt and Round matrix there are no open loop effects at all on the activity or value-added sectors, but in Stone's decomposition the extragroup effects here are substantial. The economic interpretation of this difference makes good sense. If any one of the three nonhousehold institutional sectors were somehow affected by an exogenous change, the only way the activity and value-added sectors could be affected is via effects on household income. This would differ from a shock to a household sector, which would have open loop effects on activity output and value-added because consumption spending would be immediately affected. 29

33 Table 9. Stone Decomposition in Percentages. Open Loo~ as a Percenta~e of Extra~ou~ Multi~liers. Agric. Ag. Rel. Other Labor Cap. Ind. Tax WageW. Prop. Ent. IAOHH M4HH H2HH Agriculture Ag.-Related Ind Other Industries Labor Capital Indirect Taxes Wage Work Proprieton; Enterprises Low4%HH Medium 4% HH High2%HH Own Effects as a Percenta~e of Full SAM Multi~liers. Agric. Ag.-Rel. Other Labor Cap. Ind. Tax WageW. Prop. Ent. IAOHH M4HH H2HH Agriculture Ag.-Related Ind Other Industries Labor... 5 Capital... 6 Indirect Taxes... 7 Wage Work Proprietors Enterprises Low4%HH Medium 4% HH High 2%HH Stone's Closed Loo~ Effects as a Percenta~e of Full SAM Multipliers. Agric. Ag.-Rel. Other Labor Cap. Ind. Tax WageW. Prop. Ent. IAOHH M4HH H2HH Agriculture Ag.-Related Ind Other Industries Labor Capital Indirect Taxes Wage Work Proprietors Enterprises Low4%HH Medium 4% HH High 2%HH

34 measure the induced portion of the closed loop effect. If we concentrate on the latter and look at the industrial block we find that the agricultural sector shows the least amount of induced effect and the agriculturally related industries show the most. The same block in the Stone decomposition, where the 1 's before the decimal are now part of the induced effect and cannot be disregarded, shows, in this case, the same relative magnitudes. 1 The fact that Stone's own or intragroup effects and closed loop effects add up to the full multipliers in the diagonal blocks of the (/ - St 1 matrix can be exploited in a further calculation in which the closed loop effects are each expressed as a percentage of the full multiplier, as shown in Table 9. The 1's on the diagonals of the matrices of intragroup and full multipliers have been subtracted so that the results refer only to induced effects. The figures show that if an exogenous shock affects agriculture, nearly all of the total impact on that sector (86.9% of it) will be generated within the same partition, i.e., from interactions among industries, and only 13.1% will be fed back to it from effects it has on sectors in other partitions (institutions and households). Agriculturally related industry, on the other hand, will benefit more from feedback effects from nonindustrial sectors than from changes induced by interindustry feedback. Sixty-three percent of the SAM inverse coefficient in the agriculture column and the ag.-related row is accounted for by induced spending effects. The additive simplicity of the Stone decomposition and its ease in examining induced effects makes it more commonly used than the Pyatt and Round decomposition. SAM AND INPUT-OUTPUT MULTIPLIERS: IMPLICATIONS FOR CLOSURE OF REGIONAL MODELS In regional (state and substate) input-output tables, the flow of income to households and other institutions detailed in the institutional accounts in a SAM is missing. As a result, regional models treating households as endogenous, a fairly common practice, have a problem with accurate specification of household income (the household row) and the corresponding income base from which consumption and saving decisions are made. The SAM represented in Table 1 returns to the reduced SAM presented in Table 6. It makes an aggregation so that the household accounts in Table 6 are collapsed into one 1 one nonzero elements in the other two diagonal partitions are precisely the same in both versions of the decomposition for the middle block. However, where there are zeros in the bottom right block of the Pyatt and Round matrix in the columns for the nonhousehold institutions, there are positive numbers in Stone's. If one considers how these zeros in the one are converted to nonzeros in the other and in the process of matrix multiplication, it becomes apparent that the households spread the effects of shocks to the institutional sectors, via the chain of consumption, production, and value-added distribution. 31

35 row and column (Pyatt, 1986). As a result, the household, government and capital accounts appear as they would in a conventional input-output table if it were possible to trace the payment of value-added in each sector to the household, government or capital accounts of final destination. Note that households are not represented by the sum of labor and capital payments. The income flow to households includes wages and salaries less social insurance contributions, plus proprietor income and capital income not retained by corporations. This capital income is returned to households by enterprises in the form of interest, dividends and rent. Indirect business taxes accrue to government as does a portion of income paid to labor and capital. Households also receive income transfers from government and net income from abroad. The resulting sum of the household row is equal to personal income. In the household column, personal income taxes, consumption, and saving are the main flows accounting for the "expenditures" of personal income. The resulting SAM multipliers treating households as endogenous are presented in Table 1. The SAM multipliers for this model provide the target against which other limited information forms of model closure will be compared. Using Value-Added as a Proxy for Income A commonly used approach to closing a regional input-output model with respect to households is to simply use total value-added as the estimate of household receipts and as the base on which household saving and consumption decisions are made (Bourque, 1987). The results of closing the model in this way are presented in Table 11. All of the estimates of sector payments to households are considerably overestimated and, as a result, the estimates of household income per unit of sector output are overestimated (row 4, Table 11). The household column technical coefficient estimates are underestimated relative to the corresponding SAM estimates in Table 1. This results from the fact that total value-added is a much larger figure than personal income, the income base in the SAM. In other words, consumption per unit of income is underestimated in the value-added closure. The Leontief coefficients in the interindustry portion of the model are for most coefficients substantially overestimated relative to the same coefficients in the SAM model (Table 11). The range of the overestimate is between 2% and 41% greater than the corresponding SAM coefficient. This results from using value-added as the proxy for payments to households. However, the real damage from the value-added approach shows up in the estimates of simple income multipliers (the household row). Here the overestimate is on the order of 8% to over 1%. This results from the overstatement of income to households that stems from using value-added as a proxy for household income. 32

36 Table 1. The Reduced SAM with Households Combined. Reduced Aggregate SAM, Household Sectors Combined. Agric. Ag.-Rel. Other Hslds. CapAc. Gvt. RoW Agriculture Ag.-Related Ind Other Industries Households Capital Account Government Rest of World Total Reduced Aggregate SAM, Normalized Expenditure Coefficients. Agric. Ag.-Rel. Other Hslds Agriculture Ag.-Related Ind Other Industries Households Capital Account Government Rest of World Total SAM Multipliers, Household Sector Endogenous. Agric. Ag.-Rel. Other Hslds Agriculture Ag.-Related Ind Other Industries Households

37 Table 11. The Reduced SAM: Value-Added Closure. Reduced Aggregate SAM, with Value-Added. Reduced Aggregate Sam, Expenditure Coefficients from Value- Aerie. 4-RA:l. Other lklds. Total Added Val.-Add. Apic. 4-RA:l. Other Hskls. 1 Agriculture RA:lated Ind lltlj.s Apiculture Other lndustrie& S 2 4 -RA:lated Ind Value-Added Other Industries Capital Aa:ount... 13S.SO 4 Value-Added GoYemment tlJ 7 Capital Account Rest of World Government Totals Used Rest of World True Household Total Total w.j>o. Full Multipliers from Value-Added. Percentage of "True" Multipliers. Agric:. 4-RA:l. Other Hslds. Agric. 4 -RA:l. Other Hslds Agriculture S Agriculture Ag.-RA:lated Ind Ag.-RA:Iated Ind Other Industries Other Industries tlJ Value-Added Value-Added

38 Using Earnings as a Proxy for Income With this approach, earnings (as defined by the Bureau of Economic Analysis, BEA), rather than value-added, is used as the proxy for income paid by each sector to households. In the household column, personal income (available from the BEA) rather than total earnings is used as the household income base (Bourque, 1987). The results from the earnings/personal income approach to model closure are summarized in Table 12. The household column for the endogenous portion of the model is identical to the household column in the SAM. The household row technical coefficients are underestimated by the earnings estimates, but are of roughly the same order of magnitude as the SAM income payments to households (Table 12). The result is that the Leontief earnings model coefficients are much closer to the true SAM model coefficients than was the case with the value-added closure (Table 12). Except for one income multiplier, all the Leontief coefficients in the earnings model are within 1% of the SAM model coefficients. Implication of Findings The problem of closure of regional Type II input-output models has long been recognized. Some authors have suggested that since the earnings closure underestimates the true income consumption linkage and the value-added closure overestimates the income part of that same linkage, that the average of coefficients obtained from the two closure approaches would provide the best estimate of true coefficients. The evidence presented in this paper doesn't support this argument. The upward bias in the value-added closure was much larger than the downward bias in the earnings closure. Averaging the two estimates would result in greater upward bias than just using the results of the earnings approach. The results of this study suggest that a reasonable approach to the issue of model closure in the case of Type II regional input-output models is to use estimates of sector earnings in the household row. Personal income (obtained from BEA, REIS data) for the region should be used as the base on which the consumption coefficients in the household column are calculated. Models based on the value-added proxy will result in a large upward bias in Leontief coefficients, especially in the simple income multipliers and should not be used if the earnings-personal income alternative is available. Researchers with the ability to generate an IMPLAN-based SAM of the region are faced with a different set of problems. How can one integrate the information from the regional SAM into the IMPLAN-based input-output accounts? In a model with endogenous households, the main question is how should the transboundary flows of labor and capital flows be related to regional household consumption behavior? 35

39 Table 12. Reduced SAM: Earnings/Personal Income Oosure. Reduced Aggregate SAM, with Earnings. Apic:. Ac--Rcl Other Hlkls s 1 Apiculture S Ac--Rclated Ind Other Industries S.S Eaminp S S98.9 s Other VA 1S.14 9(172J) Capital Account... 13S.SO 8 GoYemment Rest of World S4.94 Total Reduced Aggregate SAM, Expenditure Coefficients from Earnings. Total Apic:. 4-Rcl. Other Hlkls. Eamp s 1 Apiculture Ac--Rclated Ind Other Industries Eaminp s Other VA Capital Account GoYemment Rest of World Total Vl '1 Eaminp: Labor VA plus Proprietors Eaminp prorated thus: Agric. 3%, Ac--Rcl. 1%, Other 6%. Percentage of "True" Multipliers. Apic:. Ag.-Rcl. Other Hslds s 1 Agriculture S 2 Ac--Rclated Ind Other Industries S Eaminp Percentage of "True" Multipliers. Agric. AI--Rel. Other Hslds s 1 Apiculture S 2 Ac--Rclated Ind Other Industries S Eaminp

40 Consider capital flows first. The work done for this paper with the national SAM shows that most of the capital income generated by enterprises does not support household consumption and saving, but instead flows to savings by enterprises in the form of depreciation allowances and retained corporate earnings. In addition, at the regional level it is likely that much of the regional capital base will be owned by households outside the region. Much of the regional capital payments made to households will accrue to households outside the region. For both of these reasons a reasonable approach to household closure with information in the IMPIAN SAM is to assume that all regionally generated other property income that accrues to households goes to households outside the region. Other property income from outside the region that accrues to households in the region, as indicated by the BEA, should be included in the row representing household income in the SAM accounts. This outside capital income will affect household consumption and saving behavior. The same treatment does not extend to proprietors' income. It seems more reasonable to assume that those firms generating proprietors' income are owned by households in the region as well as households that commute into the region to work. As a result, that income is assumed to accrue to regional households according to labor commuting patterns as identified in Journey to Work data from the Census of Population. This brings us to the issue of labor income and the issue of trans boundary flows. At the regional level, trans boundary flows of labor income can be fairly substantial due to labor commuting to work across the regional boundary. The IMPIAN SAM estimates transboundary flows of labor income, but only on a net basis. The required estimates should be on a gross in and a gross out basis. For simple regions it is possible to obtain accurate estimates of these flows by referring to Journey to Work data as well as estimates of gross trans boundary earnings flows available from the BEA Since the estimate of net flows is not a good approximation to gross flows, the transboundary earnings estimates in the IMPIAN SAM should not be used to approximate labor income leakages. Instead, independent estimates of labor leakages must be developed. The importance of this will depend on the size and nature of the region. Other things equal, the larger the region the smaller should be the problem with labor transboundary flows. An example of how information from the IMPIAN SAM and IO accounts was combined to build a household endogenous input-output model may be found in a paper prepared for the 1992 annual meetings of the Western Agricultural Economic Association (Holland, Weber and Waters, 1992). 37

41 REFERENCES Adelman, 1., and S. Robinson. "U.S. Agriculture in a General Equilibrium Framework: Analysis with a Social Accounting Matrix." Amer.l Agr. Econ., 68(1986): Alward, G., E. Siverts, D. Olsen, J. Wagner, D. Suef, and S. LindaU. Micro Implan Users Manual. Dept. of Agricultural and Applied Economics, U. of Minnesota, St. Paul, Bourque, P. J. The Washington State Input-Output Study for Graduate School of Business Administration, U. of Washington, Seattle, March Holland, D., B. Weber, and E. Waters. "Modeling Economic Linkage Between Core and Periphery Regions: The Portland Oregon Trade Area." Paper presented at the 1992 annual meeting of the Western Agricultural Economics Association, Colorado Springs, CO, July Miller, R. E., and P. D. Blair. Input-Output Analysis. Englewood Cliffs, NJ: Prentice-Hall, Pyatt, G. "Collapsing SAM's: A Technique with Particular Relevance for the Computation of Tax Incidence." Review of Income and Wealth Pyatt, G., and J. I. Round, eds. Social Accounting Matrices: A Basis for Planning. Washington, DC: World Bank, Robinson, S., and D. Roland-Holst. Modelling Structural Adjustment in the U.S. Economy: Macroeconomics in a Social Accounting Framework. Working Paper No. 44, Dept. of Agricultural and Research Economics, U. of California-Berkeley, Stone, R. "The Disaggregation of the Household Sector in the National Accounts." In Social Accounting Matrices: A Basis for Planning, G. Pyatt and J. I. Round, eds., pp Washington, DC: World Bank,

42 APPENDIX REDUCING THE NUMBER OF SECI'ORS IN A SAM The objective is not only to reduce the number of sectors, but to maintain the accounting integrity of the SAM. Flows must still make economic sense and the total flows for all sectors remaining in the SAM must remain unchanged. For example, eliminating the value-added sectors should show income going straight to nonhousehold institutions (labor, sole proprietors and incorporated enterprises). The totals in the industry columns will not change because these sectors are not paying out any more or less than they did before, but there are entries now in the institution rows. Similarly, the institutions are receiving the same total amounts as before, but now from the industrial instead of the value-added sectors. The following method of calculation uses these general principles. 11 To illustrate, the original "full" SAM and the two reduced versions are reproduced here. There is one stage of reduction between R (identical to Table 2) and R1, in which the value-added sectors were taken out, and another between R1 and R2 that removed the institutional sectors (Table A1). First Reduction In the first stage, flows must be reallocated from the value-added sectors directly to wage work, proprietors and enterprises institutions ($1612.9, $111.5 and $834.77) and to government ($251 and $258). The $1612 represents the total labor value-added from all industries distributed to wage work as an institution. The best assumption for prorating this figure among the industries is that the proportion for each industry in total labor (value-added) also holds for wage work (institution), i.e.: for agriculture: ag.-related: other: ( ) X = ( ) X = ( ) X = Similar reasoning holds for the $111.5 billion and the $ billion. Each of them is a flow from capital, so the same proportions are used to prorate both, e.g.: 11 The MATS program used to make the reductions in this paper may have followed other steps, for there is more than one way to make the calculations, but the basic approach is probably the same. 39

43 Aggregate SAM for the United States, 1982 (Imports Purged). Aaric Ai--ReL Other Labor Cap. Appendix Table 1. Reducing the SAM. Ind. Tax Wqe Work Prop. Ent. LAOHH M.WHH H2HH Cap.Ac. Ovt. RoW S 1 Agriculture Ac.-Rclated Ind ~ < Other Industries Labor Capital ~ Indirect Taxes Wage Work Proprietors Enterprises 834.n Low4% HH Medium 4% HH ~ 12 High 2% HH Capital Account Government Rest of World Total

44 Rl. Reduced Aggregate SAM, 12 Sectors. Appendix Table 1. Continued. Agric. Ag.-Rel. Other Wage W. Prop. Ent. L4HH M4HH H2HH CapAc. Gvt. RoW Agriculture Ag.-Related Ind Other Industries Wage Work Proprietors Enterprises Low4%HH Medium 4% HH High 2% HH Capital Account Government Rest of World Total R2. Reduced Aggregate SAM, Nine Sectors. Agric. Ag.-Rel. Other IAOHH M4HH H2HH CapAc. Gvt. RoW Agriculture Ag.-Related Ind Other Industries Low4% HH Medium 4% HH High 2% HH Capital Account Government Rest of World Total

45 from agriculture to proprietors: ( ) x = 5.32 agriculture to enterprises: ( ) x = Moving down to the government row, the allocation of the $ need not be calculated, for it is already given in Row 6, indirect taxes. The figures in these rows, which add up to $258.76, are simply moved down to the government row. The $ must, however, be prorated in the same way as the other flows and the resulting figures added to the indirect taxes. The proportions used are, again, those each industry accounts for in total labor (value-added), for the $ is also a flow from this sector. E.g.: from agriculture to government: [( ) x ] = 6.17 Second Reduction This involves more lengthy calculations, although the principles are the same. In R1 a block of nine figures (rows 4-6, columns 4-6) show flows from institutions to households. These must be reallocated so that the flows are from activities or industries directly to households. The amount that low 4% HH receive from agriculture is made up of flows from labor, proprietors and enterprises, thus: from wage work: proprietors: enterprises: ( ) X = 1.47 ( ) X 9.72 =.46 ( ) X 8.61 = The $338.5 of savings going from the enterprise sector to the capital account is allocated in the same proportions in which funds flow from the activities to enterprises. E.g.: from agriculture to capital: ( ) x = 17.4 The $6.66 of taxes flowing from enterprises to the government is similarly apportioned, and is added to the figures already in the government row under the activities columns, for example: for ag.-related ind.: [ ) x 6.66] = 8.89 Finally, $53.25 is transferred from government to enterprises (see column 11 in R2) which is distributed among households, capital and government itself according to the proportions in the enterprise column. In R2 these transfers go directly from government 42

46 to households, capital and government in proportion to the flow of $888.1 to these same institutions. The prorated figure is then added to the flows already in the government column. So, for instance: from government to government: [( ) x 53.25] =