Leontief’s Input-Output Model Ashley Carter

MA 405 Section 002

Input-Output Model

Definition. Leontief’s Input-Output Model is a representation in modern economic theory that exemplifies the interrelationship between various sectors, or industries, of the economy; the model consists of n sectors producing n products (“Application to economics”)

Definition. A sector is an area of the economy in which different corporations share a similar product or service (“Sector”)

Wassily Leontief(August, 1906 – Feb., 1999)

American economist with citizenship in the Soviet Union and the United States

Nobel Prize Winner for the input-output model in 1973

Divided economy into 500 industries for the purpose of his work

Professor at Harvard Created and Managed the Institute

for Economic Analysis at New York University

(“Wassily Leontief”)

Goals of the Model

Equalize the total amount of goods produced and the total demand for the good being produced

Ensure no goods go unused Make economic predictions Can be adjusted to different settings depending

on the amount of economic sectors involved

(Rincon)

Linear Algebra Concepts We Will Use

Matrix Arithmetic (See 1.3) Identity Matrix Reduced Row Echelon Form (See 1.2) Matrix Augmentation (See 1.1) Nonnegative Matrices (See 6.8)

(Chapters and Sections coincide with the Linear Algebra: Eighth Edition by Steven Leon)

Nonnegative Matrices Definition. An nxn matrix A with real entries is

said to be nonnnegative if aij ≥ 0 for each i and j and positive if aij > 0 for each i and j. Similary, a vector x = (x1 , x2 , . . . , xn)T is said to be

nonegative if each xi ≥ 0 and positive if each xi > 0.

Leontief’s Input-Output Models are an applications of nonnegative matrices. It does not make sense for a sector to produce negative

output. (Leon)

Two Types of Models Open Model

The open model assumes that each sector must produce enough output to meet no only the input requirements of other industries but also the market demand.

Closed Model The closed model assumes that each sector must

produce enough output to meet the input requirements of only the other industries and itself, and thus, the market demand is ignored.

(Leon)

Consumption Matrix

The Consumption Matrix C is an nxn matrix that represents the units of input needed per unit of output.

Entry cij represents the output required from sector i in ordered to produce one unit of output in sector j.

(Daddel)

Production and DemandVectors

The Production Vector x represents the output produced by the sectors. Suppose there are n sectors. Then x = [x1 , x2 , . . . , xn]T. And xi would represent units of output of sector i.

The Final Demand Vector d represents the value of goods or services demanded. Assume there are still n sectors. Then,

d = [d1 , d2 , . . . , dn]T. And di represents the demand for output from sector i. (barnyard.syr.edu)

The Input-Output Model’s Equation Using the consumption matrix C, the final demand

vector d, and the production vector p, the following equation can be utilized:

p = Cp + d

For the closed model, d would be equal to 0. Thus, p = Cp. For now, we will focus on the open model and return to the closed model later.

(mavdisk.mnsu.edu)

Solving the Open Model Equation Suppose we have constructed our consumption matrix C

and we know the demand d. Then we can solve for how much output we need to satisfy the demand

p = Cp + d Ip = Cp + d Ip – Cp = d

( I – C )p = d We can augment the matrix ( I – C ) with our demand vector

d, such that we create (( I – C )|d) Now we can find the reduced row echelon form of (( I – C )|d) to find the amount of production needed to satisfy the demand d.

(Rincon)

Open Model Conditions

The entries of C have two important properties:

(i) cij ≥ 0 for each i and j. Hence C is a nonnegative matrix.

(ii) cj 1 = Σ cij < 1 for each j.

(Note: 1 denotes the 1-norm, or the matrix norm.)

(Daddel)

n

i = 1

Unique Nonnegative Solution for the Open Model

We want to show that (I – C)p = d has a unique nonnegative solution. To do so, we need to know, for any mxn matrix B,

We also need to know that the 1-norm satisfies the following multiplicative properties:

(Leon)

(1)

(2)

Unique Nonnegative Solution for the Open Model (cont.)

If the consumption matrix C is an nxn matrix satisfying both properties (i) and (ii), then it follows from (1) that C 1 < 1. Furthermore, if λ is any eigenvalue of C and x is an eigenvector belonging to λ, then

And thus,

Therefore, 1 is not an eigenvalue of C. It follows that I – C is nonsingular. Therefore, the system has the unique solution p = (I – C)-1 d.

(Leon)

Unique Nonnegative Solution for the Open Model (cont.)

Now we need to show that the solution p = (I – C)-1 d is nonnegative. To do so, we will show (I – C)-1 is nonnegative. Recall the multiplicative property (2) introduced on a previous slide. From this, we can say

Since C 1 < 1, it follows that Cm 1 0 as m ∞ and thus, Cm approaches the zero matrix as m approaches infinity because

(I – C)(I + C + . . . + Cm) = I – Cm+1 (Leon)

Unique Nonnegative Solution for the Open Model (cont.)

It follows that I + C + . . . + Cm = (I – C)-1 – (I – C)-1 Cm+1

As m ∞,(I – C)-1 – (I – C)-1 Cm+1 (I – C)-1

and thus, the series I + C + . . . + Cm converges to (I – C)-1 as m ∞. By property (i), the series is nonnegative for each m, and thus, (I – C)-1 is nonnegative. Since d and (I – C)-1 are both nonnegative, p must be nonnegative. (Leon)

Example of the Open Model Suppose we have four sectors: mining, coal,

farming, and steel. Thus, in this case, the consumption matrix will be a 4x4 matrix. The production vector and demand vector will both be 4x1 matrices.

Let our demand mining, coal, farming, and steel be 70, 50, 80, and 40 respectively. Then, we can create the demand vector d:

d = (70, 50, 80, 40)T

Example (cont.) Suppose we are given the following data

representing inputs consumed per unit of output:Purchased From:

Mining Coal Farming Steel

Mining .1 .6 0 .1Coal .2 .1 .4 .2Farming .3 0 .2 .3Steel .2 .2 0 .3

We can interpret the first column as the amount of dollars needed from each sector to produce one dollar of mining: $0.10 of mining, $0.20 of coal, $0.30 of farming, and $0.20 of steel.

Example (cont.) We can now construct our consumption matrix.

Let C =

Then (I – C) =

Example (cont.) Now, we can create the augmented matrix ((I – C)|d):

Therefore, p = (313, 312, 306, 236)T.(Values have been rounded to the nearest whole number.)

rref

Example (cont.) We used Reduced Row Echelon Form to solve for p

using the augmented matrix ((I – C)|d). We could have solved for p by multiplying the

matrices (I – C)-1 and d. Both ways yield the same result.

p = (I – C)-1 d = (313, 312, 306, 236)T

The solution p can be interpreted as follows: the sectors mining, coal, farming, and steel should respectively produce $313, $312, $306, and $236 of their products to meets their respective demands.

Closed Input-Output Model Total output from the jth industry should equal the total input from

that industry since demand is zero. Two conditions: cij ≥ 0 Σcij = 1, j = 1, . . . , n

Market demand is ignored. Therefore,p = Cp Cp – p = 0 (C – I)p = 0

Since each column of C’s entries has a sum of one, the row vectors of (C – I) add to zero. Therefore, (C – I) is singular. Thus, one is an eigenvalue of C, and p has a nontrivial solution.

Note that p, in the closed model, can have multiple correct values since it is an eigenvector.

ni = 1

(Leon)

Closed Model Example Assume we have a consumption matrix such that

C =

We want to find a solution for p such that (C – I)p = 0 where p is an eigenvector of C and λ= 1 is an eigenvalue of C.

Closed Model Example (cont.)

C – λI =

But we know λ= 1, so C – 1I =

It follows that the eigenspace is span(13, 2, 16)T. Therefore, p equals any multiple of the vector (13, 2, 16)T.

Works Cited“Application to Economics: Leontief Model.” Web. <www.math.ksu.edu>.Daddel, Alli. Leontief Input Output Model. 19 Sept. 2000. Web. <www.math.ucdavis.edu>.Leon, Steven. Linear Algebra with Applications. 8th ed. Upper Saddle River, NJ: Pearson Education Inc., 2010. Print.Leontief Input-Output Model in Economics. Web. <mavdisk.mnsu.edu>.Leontief Input Output Model: Lecture 32. Web. <barnyard.syr.edu>.Rincon, Maria. The Leontief Input-Output Model. 2009. Web. <web.csulb.edu>.“Sector.” Investopedia. Web. <www.investopedia.com>.“Wassily Leontief.” Library of Economics and Liberty. 2008. Web. <www.econlib.org>.