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Math 547

Research Project

Minju Kim

Leontief Input-Output Model

(Application of Linear Algebra to Economics)

Introduction

Professor Wassily Leontief started input-output model with aquestion, “what level of output should each of the n industriesin an economy produce, in order that it will just be sufficient tosatisfy the total demand for that product?” Leontief Input-output analysis which was developed by Professor WassilyLeontief in the 1930’s is a method used to analyze therelationships between sectors in an economy. These sectors areinterdependent on the other sectors in the economy. In order toproduce something, each sector needs to consume of its own

output and some of output from the other sectors. He developed the models to model economiesusing empirical data. He divided U.S. economy into 500 economic sectors and described theinterdependence between sectors with input-output matrices. With input-output model, it becamepossible to determine the total output of industries that must be produced to obtain a givenamount for final demand. By using the Leontief Input-output Model, it is possible to findproduction levels which will meet the demands of all sectors inside and outside of that economy.On October 18 in 1973, Wassily Leontief won Nobel Prize in economy for this work in this area.This analysis has been used extensively in economic production planning and in developingcountries. Also, by looking at the Leontief Input Output Model, it is possible to tell whether aneconomy is productive or non-productive.

Assumptions for the Input-Output Model

Since Leontief input-output model normally can have a large number of industries and it will bequite complicated. For a simplification, the following assumptions are adopted

1) Each industry produce only one homogeneous commodity 2) Each industry uses a fixed input ratio for the production of its output

3) Production in every industry is subject to constant return to scale (constant returns toscale means k-fold change in every input will result in an exactly k-fold change in output)

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Using of Linear Algebra for the Model

The Leontief model represents the economy as a system of linear equations. To find P(production vector) in terms of d (demand vector), we will solve sets of linear equations. Suchequations are naturally represented using the formalism of matrices and vectors. We will solvelinear equations with matrix algebra. In matrix algebra, we will use matrix inverses and matrixmultiplication. Also, to find a solution, we will use Gaussian-elimination technique. To decide ifthe economy is productive, we will use the Hawkins-Simons conditions.

The Open Leontief Model

There is a Closed Leontief Model where no goods leave or enter the economy. However, in real economic world, it does not happen very often. Normally, a certain economy has outside demandfrom like government agencies. Therefore, we will use the Open Leontief Model. In Open Leontief Model, there are industries in an economy. Each industry has a demand for products from other industries (internal demand). Also, there are external demands from outside. We will find a production level for the industries that will satisfy both internal and external demands.

Consider there are n interdependent industries (or sectors): S1, S2,…..,Sn

Let mij: the number of units produced by industry Si to produce one unit of industry Si Pk: the production level of industry Sk

mijpj: the number of units produced by industry Si and consumed by industry Sj di: demand from the ith outside industryThen, total number of units produced by industry Si, pi= p1mi1 + p2mi2 + … + pnmin + di

From this, we can get linear equations,

p1 = m11p1 + m12p2 + … + m1npn +d1

p2 = m21p1 + m22p2 + … + m2npn + d2

: : : : :

pn = mn1p1 + mn2p2 + … + mnnpn +dn

We can have matrix A and vectors P, and d,

( ) ( ) ( )

A = [ () ( ) (

) ], P= [ ], and d=[ ]

( ) ( ) ( )

We can write above linear equations as P = AP + d

Matrix A is called input-output matrix or consumption matrix. A consumption matrix shows the quantity of inputs needed to produce one unit of a good. The rows of the matrix represent the

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producing sector of the economy. The columns of the matrix represent the consuming sector of the economy. The entry mij in consumption matrix represent what percent of the total production value of sector j is spent on products from sector i. d is the demand vector. Demand vector d represents demand from the non-producing sector of the economy. Vector P represents the total amount of the product produced.

P = AP + d ↔ [ ] = [ ] + [ ]

To solve this linear system,

P = AP + d → (I – A)P = d → P = (I-A)-1d

If consumption matrix A and demand vector d have nonnegative entries, and if consumptionmatrix A is economically feasible, then the inverse of the matrix (I-C) exists and the productionvector P has nonnegative entries and has the unique solution for the model. We call matrix A isproductive in this case.

The Open Leontief Model with Real Data

To help understanding how the Open Leontief Model works, I have a real data to explain.

Agriculture Manufacturing Services Open SectorAgriculture 34.69 4.92 5.62 39.24

Manufacturing 5.28 61.28 22.99 60.02Services 10.45 25.95 42.03 130.65

Total Gross Output 84.56 163.43 219.03<Exchange of Goods and Services in the U.S. for 1947 (in billions of 1947 dollars)>

By dividing each column of a 3 X 3 table by the Total Gross Output for sectors, we can get theconsumption matrix from the table.

In this data, open economy consists of three industries: Agriculture, Manufacturing, Services.These three industries depend upon each other. To produce $1 of Agriculture, Agriculture mustpurchase $0.4102 of its own production, $0.0624 of Manufacturing, and $0.1236 worth ofServices. To produce $1 worth of Manufacturing, it needs $0.0301 of Agriculture, $0.3783 ofManufacturing, and $0.1588 of Services. To produce $1 worth of Services, Services industrymust buy $0.0257 of agriculture, $0.1050 of Manufacturing, and $0.1919 of Services. There is anexternal demand of $39.24 worth of Agriculture, $60.02 worth of Manufacturing, and $130.65worth of Services. We can find the production level of each three industries with the OpenLeontief Model to satisfy both internal and the external demands.

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Solution for Real Data using the Open Leontief Model

The input-output matrix (or consumption matrix) of the economy isA = [ ] .

Matrix A is showing relationships of inputs consumed per unit of sector output.External demand for the economy is d = [ ]

P = (I – A)-1dI – A = [ ]

To find out (I – A)-1, first we need to know if inverse of (I-A) exists.In 3x3 matrix, B = [ ].

There exists an inverse of matrix B detB = b11b22b33+b21b32b13+b31b12b23-b11b32b23-b31b22b13-

b21b12b33 0, and it is

B-1 = [ ]

Since det(I-A)=0.5898*0.6217*0.8081+0.9376*0.8412*0.9743+0.8764*0.9699*0.8950-0.5898*0.8412*0.8950-0.8764*0.6217*0.9743-0.9376*0.9699*0.8081=0.115752≠0, there existsan inverse of matrix (I-A).(I – A)-1 = [ ]

P = (I – A)-1d = [ ] [ ] = [ ]

Therefore, the total output of the Agriculture must be $82.40. The total output for theManufacturing must be $138.85. The total output for the Services sector is $201.57.

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Or we can get a production vector using the Gauss elimination method.

• Invertibility

An n x n matrix C is invertible if (and only if) rref (C) = In or, equivalently, if Rank(C) = n

[ ] =

[ ] ↔[ ]

↔[ ] [ ]

3 x 3 matrix (I-A) is invertible since rref (I-A)= I3 and Rank(I-A)= 3. So, invertible matrix of

matrix I-A exists. Therefore, we can get (I – A)-1 = [ ]

Which gives P = (I – A)-1d = [ ] [ ] = [ ]

Characteristics on Consumption Matrices A in Open Leontief Model

In the Closed Leontief Model where no goods leave or enter the economy, consumption matriceswould have columns adding to one. However, in the Open Leontief Model, the sum of columns in consumption matrix must be less than 1. In a real data used above, a consumption matrix A =[ ]. We can check that the sums of each column are less than 1. (The

sum of first column= 0.4102+0.0624+0.1236=0.5962 < 1, the sum of second column= 0.0301+0.3783+0.1588=0.5672 < 1, the sum of third column= 0.0257+0.1050+0.1919=0.3226 <1) Since sums of each column represent the partial input cost incurred in producing a dollar’s worth of some commodity. If the sum is greater than or equal to $1, production will not be economically justifiable.

Coefficients of the consumption matrix must be positive. In order to meet demand, there arecertain restrictions in Open Leontief Model. First, the equation that is being solve is

P = (I – A)-1d. If an inverse does not exist, then it is impossible to solve for the production vector. Also, a positive production vector is necessary. Because the demand vector is always positive, multiplying the demand by the inverse of I-A needs to result in a positive production vector. For this, (I-A)-1 needs to be a positive definite matrix. With basic economic knowledge, an increase

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∆d > 0 in final demand in equation P = (I – A)-1d should result in an increase ∆P > 0 in totaloutput. Therefore, if the matrix (I – A)-1 is not positive, the logic is violated.

Being a positive definite matrix ((I – A)-1) assures that the economy can meet any given demand. When this happens, consumption matrix A and the economy are called productive.

Is the Economy Productive?

Now, we know that existence of positive definite matrix ((I – A)-1) tells us consumptions matrix A and the economy are productive. To check if the economy is productive, we will try to find outthat inverse of matrix (I – A) is a positive definite matrix. For this, we will use the Hawkins-Simons conditions.

• The Hawkins-Simons conditions say,

→ If all the principal leading minors of a matrix are positive, then an inverse exists and is nonnegative.

The principal leading minors of a matrix are set of determinants from sub-matrices of a certain matrix. In the Open Leontief Input-Output Model, they come from (I – A). The principal leadingminors start with the determinant of the entry, which is left after every row except the first is omitted and every column except the first is omitted. The second principal leading minor excludes every row past the second and every column past the second. Until the determinant of the entire matrix is taken, this pattern needs to continue.

For example, in a matrix ,

[ ]

The first principal leading minor: the determinant of a11, or │a11│. The second principalleading minor: | |. The third principal leading minor: | |. It will

continue until the last principal leading minor that is the determinant of the matrix.

If all these principal leading minors are positive, a matrix is invertible and positive definite. Also,it means that a production vector P satisfies any demand and the economy is productive.

1- Examples of Productive Economies Let’s suppose that there is consumptions matrix A in an open economy, A=[ ]. We

can check that the sums of the columns are less than 1. It means that the industries require fewinputs to make output and most output will be sent to satisfy an outside demand.

7I – A= [ ]. The first principal leading minor is 0.9 and it is positive. The second

leading minor is 0.9*0.9-0.8*0.9=0.09 and it is positive. The third leading minor is 0.9*0.9*0.9+0.8*0.6*0.7+0.7*0.9*0.8-0.7*0.9*0.7-0.9*0.6*0.8-0.8*0.9*0.9=0.048 and it is positive number. We check that all of principal leading minors are positive. So, we can know that I – A is invertible and positive by Hawkins-Simons conditions. Therefore, it means that itcan meet any demand and the economy is productive.

Approaches to Analysis: Multipliers

If there is change in final demand, how does it affect to total output or total factor use? Multiplieranalysis is widely used to analyze the impact of changes in final demand on total output or total factor use.

Let’s assume that there is a change in final demand (∆d). So, the final demand is changed d to d+ ∆d. The ∆d can be positive, zero, or negative. We can get (P + ∆P) = A(P + ∆P) + (d + ∆d), which is sum of P= AP+d and ∆P = A∆P+∆d. Solving for ∆P, we get ∆P=(I-A)-1∆d.

Since the matrix (I – A)-1 is positive, if ∆d > 0, then ∆P > 0. Because industries on an economy depend on each other, the change of final demand of one commodity will cause a change in output. For example, if there is a positive change of final demand of commodity i, while all otherfinal demand of commodities remains same, cause increase of production. Therefore, all industries have to increase their production and increase in factor used can be obtained.

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References

1) The Leontief Input-Output Model. Web. Retrieved from http://www.personal.psu.edu/kes32/MichiganClasses/math217/Worksheets2/leontief.pdf

2) Kallem, Nicholas. Input-Output Analysis with Leontief Models. Web. Retrieved from http://home2.fvcc.edu/~dhicketh/LinearAlgebra/studentprojects/spring2006/nicholaskallem/Leontief%20project.htm

3) Chiang, Alpha. Leontief Input-Output Models. Web. Retrieved from http://www.docstoc.com/docs/129503308/From-Chapter-5-Alpha-Chiang-Fundamental-Methods-of-Mathematical

4) Duchin, Faye. Rensselaer Polytechnic Institute. Department of Economics. Web.Retrieved from http://www.economics.rpi.edu/workingpapers/rpi0610.pdf