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INPUT-OUTPUT ANALYSIS, LINEAR PROGRAMMING AND THE OUTPUT MULTIPLIER

Lars Brink and Bruce McCarI*

The output multiplier is frequently used in input-output studies and its deriva- tion has been shown in elementary textbooks [9]. The primary purpose of this note is to show how the output multiplier can be derived as a shadow price in the linear programming formulation of an economic problem. This formula- tion differs from the commonly presented linear programming formulation [3, 41. On the basis of this formulation, a more general use of input-output data in conjunction with other economic data is proposed and discussed.

Basic Input-Output Theory

Input-output analysis is based on the transactions matrix, T [9, p. 81. In this matrix, each row shows the distribution of an industry’s output to other indus- tries and to final demand. Each column shows the magnitude of an industry’s use of inputs from the other industries. From the transactions matrix the technical coefficient matrix, A, is derived. Each element in the technical coeffi- cient matrix is calculated as

where i is the row subscript standing for the output from producing industry i, j is the column subscript standing for the use of input, by consuming industry j, and Tj is the total output of sector j.

The input-output problem consists of predicting the output response of different industries as the final demand on the industries change. The output response of one industry is measured as both the direct production required to meet its final demand and the indirect production required to support produc- tion required by other industries. Input-output analysis involves finding X, the vector of output levels for the industries in the transactions matrix, when a matrix A and a vector Y, the final demand for output from each industry, are given. In other words, the fundamental equation of input-output analysis is :

X = Y + A X (1) in which X vanes as Y varies and A is constant. This relation says that the outfiut of each industry, X, equals the sum of its direct uses in final demand, Y,

* Agriculture Canada and Purdue University, respectively. Journal Paper No. 6886, Purdue Agricultural Experiment Station. The research reported in this paper was completed under Project No. 1788 of the Purdue Agricultural Experiment Station, Purdue University, W. Lafayette, Indiana.

Canadian Journal of Agricultural Economics 25(3), 1977

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and the indirect uses in intermediate production, AX. The solution is obtained by rewriting as:

and solving:

where I is the identity matrix. The ( I - A) matrix is the Leontief matrix and (I - A1-I is the Leontief inverse. Each element rij of (I - A) -l indicates the total production directly and indirectly required from industry i for each unit of delivery by industry j to h a 1 demand.' The column sums of (I - A) -I are the output multipliers Rj = fri j (also called sector multipliers or business multipliers). An output multiplier specifies the total production required from all industries to produce one unit of final output in industry j.

( I - A ) X = Y

X = (I - A)-I Y

Linear Programming and Input-Output Analysis

In applying the mechanics of input-output analysis so far no reference has been made to the linear programming theory underlying the analysis. The next step is to demonstrate how the output multiplier is obtained from a linear program- ming problem with an explicitly stated objective function.

Let a linear programming problem be

maximize (1 1 . . . 1 1)X (2) subject to (I - A) X I Y

x r o (3)

The problem is to maximize the value of the sum of outputs from all indus- tries under the constraint that the output from each industry does not exceed the use of that output in final demand and as input to other industries. When the inequality is strict, the interpretation of the constraint is that final demand cannot be satisfied. In this case, the Leontief inverse does not exist.

Slack activities are added to provide equality constraints in the model. The objective function coefficients of the slack activities are zeros and can be omitted:

(4)

I

maximize (1 1 . . . 1 1)X subject to (I - A) X + IS = Y

x, s 2 0. The problem is now to determine which activities, out of activities X and S,

1 Such an interpretation of (I - A)-' arises as follows. The direct output required for satisfaction of final demand is Y (or IY). The input required for the production of IY is AY. The input required for the production of AY is A T , i.e., output of A"Y requires input of A""Y. Since all elements of A are between zero and one, it is true that l& A" = 0. Hence, the quantity I + A + A' + As + ... equals (I - A)-'. The total production required to satisfy final demand is X = IY + AY + A Y + A3Y ... = ( I + A + A' + A' ...) Y = (I - A)-lY. It is thus seen that an element r l l of (I - A)-' consists of production requirements from an infinite chain of input-output relationships.

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will be basic. If the information that goes into the model, i.e., A and Y, is realistic, only the activities X will be basic. The significance of all of the real activities, X, and none of the slack activities, S, being basic is that all the actual industries from the transactions table will produce positive output in the model.

The following argument explains why the only basic activities in the optimal solution will be the real activities X. All elements of the A matrix are between zero and one. This means that only the diagonal elements of the (I - A ) matrix can be positive in the first iteration of the solution procedure. The off- diagonal elements of (I - A) are non-positive. Also, the coefficient matrix, I, for the slack activities in expression ( 5 ) includes only non-negative elements. The coefficients in the matrix in each successive iteration of the solution pro- cedure have the same characteristics as in the first iteration, i.e., the pivoting operations performed in the solution procedure do not change the characteris- tics. The reason for this is that the (I - A) matrix in the first iteration has principal minors that are all positive. This is a necessary and sufficient condition for the existence of a solution to this type of problem [7].

Since the activities X contribute to the objective of maximizing the sum of their levels, and the activities S do not, it will be desirable to enter the activities X into the solution instead of the activities S . The non-positive coefficients for the activities X have the effect of relaxing the constraints. At the same time, the positive diagonal elements of these activities have the effect of depleting the constraining resource. An even larger depleting effect would be obtained by entering the activities S , since they have positive diagonal coefficients that are larger than the diagonal coefficients of activities X. It is thus clear that only activities X will enter the solution, Furthermore, all of the activities X will be in solution. This is because the solution to the non-degenerate linear program- ming problem requires as many positive activities as there are constraints.

In solution, when all activities X are basic, these basic activities must appear in the constraints premultiplied by the identity matrix, i.e., IX [ S , ch. 71. In order to obtain the IX expression in the constraints, equation ( 5 ) is premul- tiplied by the basic inverse ( I - A ) --i. The constraints then appear as

( 6 ) IX + (I - A)-'S = (I - A)-'Y

Obviously, the procedure requires that the input-output system is open, since otherwise the basis inverse (the Leontief inverse) does not exist. Now using the argument from above, the slack activities S must be at zero level in the solution. In other words, to keep the resemblance with reality, the matrices A and Y must be such that the equality in ( 3 ) holds.

Thus the equation ( 6 ) becomes I X = X = ( I - A ) - ' Y

This is the same solution as the one obtained from ( 1 ) , the fundamental equa- tion in input-output analysis.

One possible objection to the linear programming procedure used here is that the input-output problem can be solved easily without resorting to con- strained maximization. There are as many rows as there are columns in the ( I - A ) matrix, and hence, there exists only one unique solution, regardless of the optimizing objective. This has been pointed out in, for example [ l , p.

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5721, [3, p. 821 and [4, p. 2281. However, the advantage of viewing the prob- lem as an optimization problem becomes clearer in the discussion below. There it is indicated that the linear programming formulation allows enough flexibility to account for more complicated problems than the simple problem studied in input-output analysis.‘

The Input-Output Objective

The objective (4) in the linear programming problem is to maximize the value of the sum of gross outputs from all industries. This measure of output includes intermediate production, and should of course be distinguished from gross national product. It is important to note that the use of input-output analysis, based on the relation ( 1 ) , implicitly assumes that maximization of the value of gross output from all industries takes place in the economy. This is seen by the following argument.

The shadow price in the solution of the primal linear programming problem are the activity levels in the solution of the dual problem. These activity levels are obtained by multiplying the row vector of original objective function co- efficients in the primal problem by the inverse of the basis in the solution [5 , Ch. 81. In the case of the problem in equations (4) and ( 5 ) , the objective function coefficients form a row of ones, and the basis inverse is (I - A)- l . Thus, the shadow prices form a row vector, D, obtained as

D = ( 1 1 1 ... l l ) ( I - A ) - ’

The row vector of ones simply sums each column of (I - A) -I . In the section on input-output theory above, a column sum of (I - A ) -I, i.e., Rj, was defined as the output multiplier. Hence, by formulating the input-output problem in a particular linear programming format, the output multipliers are obtained as the shadow prices in the solution of the p r ~ g r a m . ~

Input-output analysis is sometimes advocated as being policy neutral [9, p. 102, 11, p. 2, p. 1951. However, the alleged neutrality of the technique when applied to multiplier analysis is a misconception, because of the implicit as- sumption made regarding the maximization objective.

The economic problem presented here differs from the problems solved by Dorfman, Samuelson and Solow [4] and Chenery and Clark [3]. In [3, p. 861, the objective is to maximize output, measured as national product, or to mini- mize cost. In addition, data on the use of primary factors is necessary to formu- late the constraints. In order to satisfy final demand for commodities, produc- tion and final demand are equalized. The slack activities, however, refer to the use of primary factors. Thus, the shadow prices of the slack activities in the 2 Another reason why it may be advantageous to use the linear programming approach

in input-output studies is that many computing centers have linear programming algorithms that are more accurate’than their inversion algorithms.

3 The dual of the problem in (2) and ( 3 ) is interpreted as the minimization of the value of the weighted sum of final demands, subject to the constraint that the weights used in this valuation impute a value of at least one to each unit of output in every sector. These weights are the output multipliers. These dual constraints mean that the output multiplier for each sector is at least one unit larger than the value imputed to inputs in that sector.

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Chenery and Clark formulation are not identical to the output multiplier as used in many input-output studies. The same is true in the discussion in [4, p. 2281, where the objective function is seen as any positively weighted sum which is to be maximized.

A formulation similar to the one presented in this note can be found in the book by Makower [8]. However, the application is not carried through to the problem of input-output analysis. Makower mentions the significance of the shadow price when the problem is maximize the sum of total output, subject to resource constraints and equalization of supply and demand, but the output multiplier as used in input-output analysis is not mentioned.

Discussion and Suggestions

It can now be seen how various types of multiplier can be derived in the linear programming framework. Various parts of the final demand vector, such as capital formation, government purchases, exports, etc., may be broken out from final demand and included as endogenous industries. Correspondingly, the maximization includes these additional activities. It is also easy to see that the objective function coefficients do not necessarily have to be zero or one. Differ- ent industries may be given different weights by assigning different values to the objective function coefficients.

The transactions table is the foundation of input-output analysis. The in- formation contained in the transactions table may be augmented by additional information about the problem under study. By leaving the restricted format of input-output analysis, both kinds of information can be incorporated into a systematic solution procedure by using linear programming. For example, it may be desired to study the effects of exogenous capacity limitations in some industries.4 This is, of course, easily achieved by adding additional constraints to the problem in equations (2 ) and ( 3 ) . Another problem in input-output analysis is the assumption of one industry - one output. This assumption may be relaxed in the linear programming framework by introducing several ac- tivities producing the same output but with different technical coefficients. Used in conjunction with capacity limitations, as is well known, it is thus possible to model production functions exhibiting decreasing returns to scale. It is also obvious that the next step in the use of input-output tables is the quadratic input-output problem. In the quadratic input-output framework it is possible to account for the relation between prices and output levels [6 ] . Additional discussion of the use of input-output data in the general linear programming problem is found in [3] and [4], as well as in any mathematical programming text.

The relation between the output multiplier and the implicit maximization of the sum of the gross outputs was shown above. However, other multipliers from input-output analysis are perhaps more often used in policy decisions, e.g., income and employment multipliers. It is therefore of interest to derive the implicit objective function that underlies these more descriptive multipliers. It is possible to formulate a linear programming problem utilizing input-output

4 An application of this procedure is presented in [lo].

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data, such that income multipliers are obtained as penalty costs. Similarly, using additional employment data, the employment multiplier is obtained. However, the economic interpretation of a linear programming problem result- ing in these multipliers is not as clear as the problem that gives the output multiplier. The natural question is then: why do input-output analysts use income and employment multipliers? The answer may be that these descriptive multipliers are useful for the purposes of input-output analysts and provide information relevant to their problems. This would indicate that the mathe- matical input-output technique by itself may not actually solve the problem to which input-output analysts traditionally address themse1ves.j The collection of multipliers presently in use may be an attempt by empiricists to obtain economically meaningful interpretations to the conceptually unclear results.

REFERENCES

1 Allen, R. G. D., Mathemntical Economics, Second Edition, London: Macmillan & Co., Ltd., 1959.

2 Candler, Wilfred and Michael Boehlje, “Use of Linear Programming in Capital Budgeting With Multiple Goals,” American Journal of Agricutlural Economics, 53 : 325-330, May, 1971.

3 Chenery, Hollis B. and Paul G. Clark, Interindustry Economics, New York: Wiley & Sons, Inc., 1959.

4 Dorfman, Robert, Paul A. Samuelson, and Robert M. Solow, Linear Programming and Economic Analysis, New York: McGraw-Hill Book Company, 1958.

5 Hadley, G., Linear Programming, Reading, Massachusetts: Addison-Wesley Book Company, 1962.

6 Harrington, David Holman, “Quadratic Input-Output Analysis: Methodology for Empirical General Equilibrium Models,” unpublished Ph.D. Dissertation, Purdue University, 1973.

7 Hawkins, David and Herbert A. Simon, “Note: Some Conditions of Macroeconomic Stability,” Econometrico, 17: 245-248, July-October, 1949.

8 Makower, H., Activity AnaIysis and the Theory of Economic Equilibrium, London: Macmillan & Co., Ltd , 1957.

House, Inc., 1965. 9 Miernyk, William H., The Elements of Input-Output Analysis, New York: Random

10 Penn, I. B., Bruce A. McCarl, Lars Brink, and George D. Irwin, “Modelling and Simulation of the U.S. Economy with Alternative Energy Availabilities,” American Journal of Agricultural Economics, 58: 663-671, November, 1976.

11 Richardson, Harry W., Input-Output and Regional Econortlics, New York: John Wiley & Sons, Inc., 1972.

5 The traditional problem in input-output analysis is taken to be the exploration of the structural interrelationships in an economy and the evaluation of impacts of a change in the structure. The practice of deriving income and employment multipliers in addition to the output multiplier may indicate that input-output analysts are concerned with multidimensional goals rather than the single goal of output maximization. In a linear programming framework such multidimensional goals can be handled in a more formal way through the use of multiple objective functions (see e.g., [2]).