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Input-Output Analysis. (This exercise builds on Exercises 1.1.24, 1.2.39. 1.2.40, and 1.2.41). Consider the industries
This equation can be written more succinctly as
describes a linear system, which we can write in the customary form:
If we want to know the output
In economics, however, we often ask oilier questions: If
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11The relevance of questions like these became particularly clear during World War II, when the demand on certain industries suddenly changed dramatically. When U.S. President F.D. Roosevelt asked for 50,000 airplanes to be built, it was easy enough to predict that the country would have to produce more aluminum. Unexpectedly, the demand for copper dramatically increased (why?). A copper shortage then occurred, which was solved by borrowing silver from Fort Knox. People realized that input—output analysis can be effective in modeling and predicting chains of increased demand like this. After World War II, this technique rapidly gained acceptance and was soon used to model the economies of more than 50 countries.
ask questions like these, we think of the output
If the matrix
a. Consider the example of the economy of Israel in 1958 (discussed in Exercise 1.2.4 1). Find the technology matrix A, the matrix
b. In the example discussed in part (a), suppose the consumer demand on agriculture (Industry 1) is 1 unit (1 million pounds), arid the demands on the other two industries are zero. What output
c. Explain, in terms of economics, why the diagonal elements of the matrix
d. If the consumer demand on manufacturing increases by 1 (from whatever it was), and the consumer demand on the other two industries remains the same, how will the output have to change? How does your answer relate to the matrix
e. Using your answers in parts (a) through (d) as a guide, explain in general (not just for this example) what the columns and the entries of the matrix
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Linear Algebra With Applications
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