Input-Output Analysis. (This exercise builds on Exer cises 1.1.20, 1.2.37, 1.2.38, and

Chapter 2, Problem 49

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Input-Output Analysis. (This exercise builds on Exer cises 1.1.20, 1.2.37, 1.2.38, and 1.2.39.) Consider the industries J j, J2, . ., / in an economy. Suppose the consumer demand vector is b, the output vector is i and the demand vector of the 7th industry is vj. (The *th component aij of dj is the demand industry Jj 0r] industry 7/, per unit of output of Jj.) As we have seer in Exercise 1.2.38, the output jc just meets the aggregat, demand ifX[V\ + *2^2 + + xnvn + b .aggregate demand outputThis equation can be written more succinctly asI I V\ V2 Vn*1 X2_xn _+ b = Jc,or Ax 4- b = jc. The matrix A is called the technolof matrix of this economy; its coefficients ay describe tbc interindustry demand, which depends on the technoio;< used in the production process. The equationAx + b = xdescribes a linear system, which we can write in tk customary form:x Ax by lnx Ax = b,Un - A)x = b.If we want to know the output x required to satidf * given consumer demand b (this was our objectiwp the previous exercises), we can solve this linear sysl* preferably via the augmented matrix. In economics, however, we often ask other tions: If b changes, how will jc change in resp If the consumer demand on one industry increa 1 unit and the consumer demand on the other i tries remains unchanged, how will jc change? is like these, we thinkof the output * as a r ^ Tof the consumer demand b. ^ J l h e matrix (/ - A) is invertible,find a vector b in R4 such that the system Ax = b is inconsistent. See Exercise 51.we can express - as a function of b (in fact, as a linear transformation): * x = (In - A ) - ]b.Consider the example of the economy of Israel in 1958 (discussed in Exercise 1.2.39). Find the tech- oology matrix A, the matrix (/ - A), and its inverseb fo the example discussed in part (a), suppose the consumer demand on agriculture (Industry 1) is 1 unit (1 nv11'1 pounds), and the demands on the other two industries are zero. What output x is required in this case? How does your answer relate to the matrixdn-Ar'i c. Explain, in terms of economics, why the diagonal elements of the matrix (/* - A)-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 two other industries remains the same, how will the output have to change? How does your "answer relate to the matrix (In - A)~1 ? e. Using your answers in parts (a) through (d) as a guide, explain in general (not just for this example) what the colomns and the entries of the matrix (In A)~1 tell you* in terms of economics. Those who have studied multivariable calculus may wish to consider the partial derivatives