Matrix Inversion

3.5 Input-Output Models

(This topic is also in Section 3.5 in Finite Mathematics and Finite Mathematics & Applied Calculus)

Some On-line Resources for This Topic:

To understand this tutorial you need to first understand matrix multiplication and matrix inversion.

Here, we look at an application of matrix algebra developed by Wassily Leontief in the middle of this century. He won the Nobel prize in economics in 1973 for this work. The application involves analyzing national and regional economies by looking at how various parts of the economy interrelate.

Here is a simple example, similar to Example 1 in Section 3.4 of Finite Mathematics and Finite Mathematics and Applied Calculus. : Think of the US economy of a country or a region as being composed of various sectors, or groups of one or more industries. For example, consider two specific sectors: the crude petroleum and natural gas sector (Sector 1: crude) and the petroleum refining and related industries (Sector 2: refining). Both produce a commodity: the crude sector produces crude petroleum and natural gas, and the refining sector produces refined petroleum and related products. By one unit of these products product, we shall mean $1 million worth of that product.

To analyse the use of the products of one sector by another, we use an input-output table:

	To	crude	refining
From	crude	1,600	72,000
From	refining	0	4,000
Total Output		40,000	100,000

We read the table as follows:

1,600 units of crude products were used in the production of crude products.
72,000 units of crude products were used in the production of refined products.
0 units of refined products were used in the production of crude products.
4,000 units of refined products were used in the production of refined products.
A total of 40,000 units were produced by the crude sector, and 100,000 units were produced by the refined sector.

To use to given data, we obtain the technology matrix by computing how much of each sector went into the production of one unit of each product: we divide each entry by the total for its column:

Technology matrix

	0.04	0.72
	0	0.04

We got these as follows:
a₁₁ = units of crude to produce one unit of crude. We are told that 1,600 million units of crude were used to produce 40,000 million units of crude. Thus, to produce one unit of crude, 1,600/40,000 = 0.04 units of crude were used, and so a₁₁ = 0.04
a₁₂ = units of crude to produce one unit of refined. a₁₂ = 72,000/100,000 = 0.72
a₂₁ = units of refined to produce one unit of crude. a₂₁ = 0/40,000 = 0
a₂₂ = units of refined to produce one unit of refined. a₂₂ = 4,000/100,000 = 0.04.

We can use the technology matrix to answer some interesting questions. For instance:

Suppose there is an external demand for 30,000 units of crude and 300,000 units of refined products. How much much be produced by each sector in order to meet the demand?

Q Duh! Just produce 30,000 units of crude and 300,000 units of refined. That's a no-brainer, right?
A Wrong! You need to take into account the fact that there are internal usages for these products in order for the sectors to function. That is what the technology matrix is telling us; for instance, to produce one unit of refined product, we need 0.72 units of crude and 0.04 units of refined. Thus, we need to take these internal consumption figures into account when computing the amount that each sector should produce. This is done as follows.

Finding the Production Necessary to Meet a Given Demand

To meet an external demand of D, the economy must produce X, where X is the production vector and satisfies the matrix equation

X = AX + D.

Explanation

The demand vector D is a column vector containing the number of units demanded from each sector. For instance, in our question above,

D =

	30,000		Demand for Sector 1 products
	300,000		Demand for Sector 2 products

The production vector X is the column vector showing the quantities that each sector must produce to meet the external demand.

We can solve the above equation for the production vector to obtain: X = (I - A)^-1D (provided (I-A) is invertible).

Example
Use the On-Line Matrix Algebra Tool or some other method for the following steps. Enter all answers accurate to 4 decimal places or in fraction form.

With

	0.04	0.72
	0	0.04

(I - A)^-1

Let us go back to the original equation X = AX + D and use it to answer a question based on the input-output table we have been studying:

	To	crude	refining
From	crude	1,600	72,000
From	refining	0	4,000
Total Output		40,000	100,000

Notice that production totals were already given in the table, so we know that

X =

	40,000		Actual Production by Sector 1
	100,000		Actual Production by Sector 2

Given the actual production figures above, what was the external demand for the products of each sector? We will answer it in several steps.

Q The correct formula for D is ...

	D = (A-I)X			D = (I-A)X			D = X(A-I)
	D = X(I-A)

Q The fact that the external demand for crude oil products was negative indicates that:

	The given production output results in a surplus of the stated quantity of crude oil, wich is thus available for export.
	The given production output cannot meet a positive external demand; the stated quantity of crude products must be imported.
	The supply of crude products exceeded the demand by the stated amount.
	The demand for crude products exceeded the supply by the stated amount.
	The given problem is unrealistic; demand and production figures cannot be negative in real life situations.

Let us continue with the issue of crude and refined products. The figures we have been using in the input-output table are quite close to the actual figures for the US economy in 1977. In other words, the answer you just obtained above for the demand are quiet accurate, and show that, in 1977, the US was extremely dependent on imported crude oil products. Now suppose that you were a senator in 1977 and wished to propose a program for domestic production of crude products in order to make the US entirely self-sufficient, whle continuing to meet the current demand for refined products. In other words, you will need to answer the following question:

Q The current entry in D for crude products is -33,600, and you wish to adjust production in order to change that figure to zero. How much more crude oil will need to be produced?
A To answer this question, we can use the same formula X = (I - A)^-1D as before, modified as follows (see the text for the justification):

^-1

Change in production = (I - A)^-1

change in demand

Let A be as above. Current imports of crude oil amount to 33,600 units, and you wish to adjust production in order to change that figure to zero. How much more crude oil will need to be produced? (answer should be accurate to the nearest 100 units -- no commas, please!)

Q How come, if only 33,600 units of crude oil products are bing imported, the economy needs to produce more than that just to avoid having to import any?
A That is because the production of crude oil products actually uses up some additional crude in the process. Here is the matrix (I - A)^-1 in decimal form:

(I - A)^-1

	1.04167	0.78125
	0	1.04167

Look the (1,1) entry tells us that approximately 1.04 units of crude need to be produced for every additional one unit demanded, explaining the exact value of the answer above. Similarly, the (1,2) entry tells us that approximately 0.78 units of crude need to be produced to meet each additional one unit of refined products demanded, and so on.

To learn more about the entries of (I - A)^-1, consult Section 3.5 of Finite Mathematics or Finite Mathematics and Applied Calculus Also, you should now be ready to try some of the exercises in Section 3.5.

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