Linear Models in Economics and Engineering
It was late summer in 1949. Harvard Professor Wassily Leontief was carefully feeding the last of his punched cards into the university’s Mark II computer. The cards contained economic information about the U.S. economy and represented a summary of more than 250,000 pieces of information produced by the U.S. Bureau of Labor Statistics after two years of intensive work. Leontief had divided the U.S. economy into 500 “sectors,” such as the coal industry, the automotive industry, communications, and so on. For each sector, he had written a linear equation that described how the sector distributed its output to the other sectors of the economy. Because the Mark II, one of the largest computers of its day, could not handle the resulting system of 500 equations in 500 unknowns, Leontief had distilled the problem into a system of 42 equations in 42 unknowns.
Mark II computer: Bettmann/Corbis; Wassily Leontief: Hulton Archive/Getty Images. Programming the Mark II computer for Leontief’s 42 equations had required several months of effort, and he was
anxious to see how long the computer would take to solve the problem. The Mark II hummed and blinked for 56 hours before finally producing a solution. We will discuss the nature of this solution in Section 6.
Leontief, who was awarded the 1973 Nobel Prize in Economic Science, opened the door to a new era
in mathematical modeling in economics. His efforts at Harvard in 1949 marked one of the first significant uses of computers to analyze what was then a largescale mathematical model. Since that time, researchers in many other fields have employed computers to analyze mathematical models. Because of the massive amounts of data involved, the models are usually linear; that is, they are described by systems of linear equations.
The importance of linear algebra for applications has risen in direct proportion to the increase in computing power, with each new generation of hardware and software triggering a demand for even greater capabilities.
Computer science is thus intricately linked with linear algebra through the explosive growth of parallel processing and large-scale computations. Scientists and engineers now work on problems far
more complex than even dreamed possible a few decades ago. Today, linear algebra has more potential value for students in many scientific and business fields than any other undergraduate mathematics subject! The material in this text provides the foundation for further work in many interesting areas. Here are a few possibilities; others will be described later.