(Leontief1 input-output model) Suppose that three industries are interrelated so that

Chapter 8, Problem 8.1.45

(choose chapter or problem)

(Leontief \({ }^{1}\) input-output model) Suppose that three industries are interrelated so that their outputs are used as inputs by themselves, according to the 3 X 3 consumption matrix

\(\mathbf{A}=\left[a_{j k}\right]=\left[\begin{array}{ccc} 0.2  0.5  0 \\ 0.6  0  0.3 \\ 0.2  0.5  0.7\end{array}\right]\)

where \(a_{j k}\) is the fraction of the output of industry k consumed (purchased) by industry j. Let \(p_{j}\) be the price charged by industry j for its total output. A problem is to find prices so that for each industry, total expenditures equal total income. Show that this leads to Ap = p, where \(\mathbf{p}=\left[\begin{array}{lll}p_{1}  p_{2}  p_{3}\end{array}\right]^{\top}\), and find a solution p with nonnegative \(p_{1}, p_{2}, p_{3}\).

Text Transcription:

^1

A = [a_jk] = [0.2  0.5  0 \\ 0.6  0  0.3 \\ 0.2  0.5  0.7]

a_jk

p_j

p = p_{1}  p_{2}  p_{3}]^{top}

p_1, p_2, p_3