Chapter 2.4, Problem 49E

Input-Output Analysis. (This exercise builds on Exercises 1.1.24, 1.2.39. 1.2.40, and 1.2.41). Consider the industries ${\text{J}}_{\text{1}},{\text{J}}_{\text{2}},\mathrm{...},{\text{ J}}_n$ in an economy. Suppose the consumer demand vector is $\stackrel{\to }{b}$ , the output vector is $\stackrel{\to }{x}$ , and the demand vector of the jth Industry is ${\stackrel{\to }{v}}_j$ . (The ith component $a_{ij}$ of ${\stackrel{\to }{v}}_j$ is the demand industry $J_j$ puts on industry $J_i$ , per unit of output of $J_j$ .) As we have seen in Exercise 1.2.40, the output $\stackrel{\to }{x}$ just meets the aggregate demand if $\underset{\text{aggregate}\text{}\text{demand}}{\underset{︸}{x_1{\stackrel{\to }{v}}_1+x_2{\stackrel{\to }{v}}_2+\cdots +x_n{\stackrel{\to }{v}}_n+\stackrel{\to }{b}}}=\underset{\text{output}}{\underset{︸}{\stackrel{\to }{x}}}$ .

This equation can be written more succinctly as $\left[\begin{array}{cccc}|& |& & |\\ {\stackrel{\to }{v}}_1& {\stackrel{\to }{v}}_2& \cdots & {\stackrel{\to }{v}}_n\\ |& |& & |\end{array}\right]\left[\begin{array}{l}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]+\stackrel{\to }{b}=\stackrel{\to }{x}$ , or $A\stackrel{\to }{x}+\stackrel{\to }{b}=\stackrel{\to }{x}$ . The matrix A is called the technology matrix of this economy; its coefficients $a_{ij}$ describe the interindustry demand, which depends on the technology used in the production process. The equation $A\stackrel{\to }{x}+\stackrel{\to }{b}=\stackrel{\to }{x}$
describes a linear system, which we can write in the customary form:
$\begin{array}{l}\stackrel{\to }{x}-A\stackrel{\to }{x}=\stackrel{\to }{b}\\ I_n\stackrel{\to }{b}-A\stackrel{\to }{x}=\stackrel{\to }{b}\\ \left(I_n-A\right)\stackrel{\to }{x}=\stackrel{\to }{b}.\end{array}$
If we want to know the output $\stackrel{\to }{x}$ required to satisfy a given consumer demand $\stackrel{\to }{b}$ (this was our objective in the previous exercises), we can solve this linear system, preferably via the augmented matrix.
In economics, however, we often ask oilier questions: If $\stackrel{\to }{b}$ changes, how will $\stackrel{\to }{x}$ change in response? If the consumer demand on one industry increases by 1 unit and the consumer demand on the other industries remains unchanged, how will $\stackrel{\to }{x}$ change?11 If we
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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 $\stackrel{\to }{x}$ as a function of the consumer demand $\stackrel{\to }{b}$ .
If the matrix $\left(I_n-A\right)$ is invertible,12 we can express $\stackrel{\to }{x}$ as a function of $\stackrel{\to }{b}$ (in fact, as a linear transformation): $\stackrel{\to }{x}={\left(I_n-A\right)}^{-1}\stackrel{\to }{b}$ .
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 $\left(I_n-A\right)$ , and its inverse ${\left(I_n-A\right)}^{-1}$ .
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 $\stackrel{\to }{x}$ is required in this case? How does your answer relate to the matrix ${\left(I_n-A\right)}^{-1}$ ?
c. Explain, in terms of economics, why the diagonal elements of the matrix ${\left(I_n-A\right)}^{-1}$ you found in part (a) must be at least 1.
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 ${\left(I_n-A\right)}^{-1}$ ?
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 ${\left(I_n-A\right)}^{-1}$ tell you, in terms of economics. Those who have studied multivariable calculus may wish to consider the partial derivatives $\frac{\partial x_i}{\partial b_j}$ .