A. Brief Background

Let us consider an economy consisting of n industries, each producing only one commodity needed by the others and possibly by itself. For example, suppose we have coal, steel, and auto, which are interrelated in a way described by a 3 x 3 matrix as follows.

Let cij be the dollar amount of the ith commodity needed to produce 1 dollar's worth of the jth commodity. Suppose also it takes 0.30 dollars of coal to produce 1 dollar's worth of steel; 0.45 dollars of steel to produce 1 dollar's worth of automobile, etc., so the values 0.30 and 0.45 are the (1,2)nd and (2,3)rd entries of the matrix:

It is seen from this matrix, that auto is the largest consumer of steel and steel is the largest consumer of coal. Steel is most dependent on auto to survive.

This matrix is an example of an input-output, or a consumption matrix describing the interdependency of the economic sectors. The entries of such a matrix are nonnegative less than 1. In addition, the sum of the entries of each column should be less than 1 if each sector is to produce more than it consumes. Consumption matrices were introduced and studied by the Harvard economist Wassily W. Leontief (1906-1999) in the 1930s. He received the 1973 Nobel Prize in Economics for this work.

Speaking in general terms, suppose n economic sectors are interrelated in a way described by a consumption matrix C = [cij]. Let xi be the total amount of output needed to be produced by the ith sector to satisfy the demands of all sectors. Then cijxj is the amount needed from commodity i to produce xj units of commodity j. Because the total output of sector i equals to the sum of the demands of all sectors, we have:

x1 = c11x1 + ... + c1nxn
. . . . . . .
xn = cn1x1 + ... + cnnxn

or, in matrix form:

x = Cx

This situation when demand for commodities comes only from producing economic sectors is known as a Leontief closed model.

In reality, there is also demand from nonproducing sectors, such as consumers, government, etc. All nonproducing sectors form the open sector. Then xi = ci1x1 + ... + cinxn + di, and the last matrix equation appears in the form:

x = Cx + d

where d is the demand vector with nonnegative components d1, ..., dn and x is the output vector. This matrix equation describes a Leontief open model.

Economists are usually interested in computing the output vector x given the demand vector d. This requires the following:

x = Cx + d => (I - C)x = d => x = (I - C)^(-1)d

provided that the matrix I - C is invertible. A matrix C is called productive if (I - C)^(-1) exists and has nonnegative entries.

Also, economists often assume that the levels of production are known and want to compute the demand that can be placed upon the producing sectors. In such cases x is given and d is unknown; x is then computed as:

d = x - Cd

Input-output matrices are used to analyze the economy of a country or even an entire geographic region. The producing sectors are usually some key industries, such as agricultural goods, steel, chemicals, coal, livestock, etc. For the U.S. national input-output matrix, the open sectors are the federal, state, and local governments.

B. Problems

1. Consumption & Demand

Let C be the consumption matrix and d be the demand vector, in millions of dollars, for an open-sector economy with 3 independent industries. Compute the output demanded by the industries and the open sector if

2. Trade Income

Consider the international trade model consisting of 3 countries, C1, C2, C3. Suppose that:

• the fraction of C1's income spent on imports from C1 is 1/4, from C2 is 1/2, from C3 is 1/4;
• the fraction of C2's income spent on imports from C1 is 2/5, from C2 is 1/5, from C3 is 2/5;
• the fraction of C3's income spent on imports from C1 is 1/5, from C2 is 1/2, from C3 is 0.

C. Helpful Sources

The Course Text, Chapter 8, Section 8.5.
References from Further Readings sub-section, Section 8.5.

D. Further Readings

Wassily Leontief - 1973 Nobel Laureate in Economics, The Nobel Prize Internet Archive.
Nobelist and Economist Wassily Leontief Dies, The Harvard University Gazette, Feb 11, 1999; On-line version.
Input-Output Economics, 2nd Edition, Edited by W. Leontief, Oxford Press, 1986; The Summary