Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 2: Q2.6-9Q (page 93)

Solve the Leontief production equation for an economy with three sectors, given that

\(C = \left[ {\begin{array}{*{20}{c}}{.2}&{.2}&{.0}\\{.3}&{.1}&{.3}\\{.1}&{.0}&{.2}\end{array}} \right]\)and \({\mathop{\rm d}\nolimits} = \left[ {\begin{array}{*{20}{c}}{40}\\{60}\\{80}\end{array}} \right]\).

Short Answer

Expert verified

\({\mathop{\rm x}\nolimits} = \left( {82.8,131.0,110.3} \right)\)

Step by step solution

Achieve better grades quicker with Premium

  • Unlimited AI interaction
  • Study offline
  • Say goodbye to ads
  • Export flashcards
Start your free trial Skip

Over 22 million students worldwide already upgrade their learning with Vaia!

01

Determine \(I - C\)

The Leontief input-output model or production equationis \(x = Cx + {\mathop{\rm d}\nolimits} \). The equation \(x = Cx + {\mathop{\rm d}\nolimits} \) can be written as \(I{\mathop{\rm x}\nolimits} - C{\mathop{\rm x}\nolimits} = {\mathop{\rm d}\nolimits} \) or \(\left( {I - C} \right){\mathop{\rm x}\nolimits} = {\mathop{\rm d}\nolimits} \).

\(\begin{array}{c}I - C = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}{.2}&{.2}&{.0}\\{.3}&{.1}&{.3}\\{.1}&{.0}&{.2}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{.8}&{ - .2}&{.0}\\{ - .3}&{.9}&{ - .3}\\{ - .1}&{.0}&{.8}\end{array}} \right]\end{array}\)

02

Write the augmented matrix \(\left[ {\begin{array}{*{20}{c}}{I - C}&{\mathop{\rm d}\nolimits} \end{array}} \right]\)

It is given that \({\mathop{\rm d}\nolimits} = \left[ {\begin{array}{*{20}{c}}{40}\\{60}\\{80}\end{array}} \right]\).

The augmented matrix \(\left[ {\begin{array}{*{20}{c}}{I - C}&{\mathop{\rm d}\nolimits} \end{array}} \right]\) is shown below:

\(\left[ {\begin{array}{*{20}{c}}{.8}&{ - .2}&{.0}&{40.0}\\{ - .3}&{.9}&{ - .3}&{60.0}\\{ - .1}&{.0}&{.8}&{80.0}\end{array}} \right]\)

03

Apply the row operation

At row one, multiply row one by \(\frac{1}{{0.8}}\).

\( \sim \left[ {\begin{array}{*{20}{c}}1&{ - 0.25}&0&{50}\\{ - .3}&{.9}&{ - .3}&{60.0}\\{ - .1}&{.0}&{.8}&{80.0}\end{array}} \right]\)

At row two, multiply row one by 0.3 and add it to row two. At row three, multiply row one by 0.1 and add it to row three.

\( \sim \left[ {\begin{array}{*{20}{c}}1&{ - 0.25}&0&{50}\\0&{0.825}&{ - .3}&{75}\\0&{ - .025}&{.8}&{85}\end{array}} \right]\)

At row two, multiply row two by \(\frac{1}{{0.825}}\).

\( \sim \left[ {\begin{array}{*{20}{c}}1&{ - 0.25}&0&{50}\\0&1&{ - .3636}&{90.90}\\0&{ - .025}&{.8}&{85}\end{array}} \right]\)

At row one, multiply row two by 0.25 and add it to row one. At row three, multiply row two by 0.025 and add it to row three.

\( \sim \left[ {\begin{array}{*{20}{c}}1&0&{ - 0.0909}&{72.727}\\0&1&{ - 0.3636}&{90.90}\\0&0&{0.7909}&{87.272}\end{array}} \right]\)

At row three, multiply row three by 0.79090.

\( \sim \left[ {\begin{array}{*{20}{c}}1&0&{ - 0.0909}&{72.727}\\0&1&{ - 0.3636}&{90.90}\\0&0&1&{110.3448}\end{array}} \right]\)

At row one, multiply row three by 0.09090 and add it to row one.

\( \sim \left[ {\begin{array}{*{20}{c}}1&0&0&{82.7586}\\0&1&{ - 0.3636}&{90.90}\\0&0&1&{110.3448}\end{array}} \right]\)

At row two, multiply row three by 0.3636 and add it to row two.

\( \sim \left[ {\begin{array}{*{20}{c}}1&0&0&{82.7586}\\0&1&0&{131.03}\\0&0&1&{110.3448}\end{array}} \right]\)

Thus, \({\mathop{\rm x}\nolimits} = \left( {82.8,131.0,110.3} \right)\).

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

a. Verify that \({A^2} = I\) when \(A = \left[ {\begin{array}{*{20}{c}}1&0\\3&{ - 1}\end{array}} \right]\).

b. Use partitioned matrices to show that \({M^2} = I\) when\(M = \left[ {\begin{array}{*{20}{c}}1&0&0&0\\3&{ - 1}&0&0\\1&0&{ - 1}&0\\0&1&{ - 3}&1\end{array}} \right]\).

Let \(A = \left( {\begin{aligned}{*{20}{c}}1&1&1\\1&2&3\\1&4&5\end{aligned}} \right)\), and \(D = \left( {\begin{aligned}{*{20}{c}}2&0&0\\0&3&0\\0&0&5\end{aligned}} \right)\). Compute \(AD\) and \(DA\). Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Find a \(3 \times 3\) matrix B, not the identity matrix or the zero matrix, such that \(AB = BA\).

Suppose \({A_{{\bf{11}}}}\) is an invertible matrix. Find matrices Xand Ysuch that the product below has the form indicated. Also,compute \({B_{{\bf{22}}}}\). [Hint:Compute the product on the left, and setit equal to the right side.]

\[\left[ {\begin{array}{*{20}{c}}I&{\bf{0}}&{\bf{0}}\\X&I&{\bf{0}}\\Y&{\bf{0}}&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{A_{{\bf{1}}1}}}&{{A_{{\bf{1}}2}}}\\{{A_{{\bf{2}}1}}}&{{A_{{\bf{2}}2}}}\\{{A_{{\bf{3}}1}}}&{{A_{{\bf{3}}2}}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{B_{11}}}&{{B_{12}}}\\{\bf{0}}&{{B_{22}}}\\{\bf{0}}&{{B_{32}}}\end{array}} \right]\]

Suppose \(CA = {I_n}\)(the \(n \times n\) identity matrix). Show that the equation \(Ax = 0\) has only the trivial solution. Explain why Acannot have more columns than rows.

In Exercises 27 and 28, view vectors in \({\mathbb{R}^n}\) as \(n \times 1\) matrices. For \({\mathop{\rm u}\nolimits} \) and \({\mathop{\rm v}\nolimits} \) in \({\mathbb{R}^n}\), the matrix product \({{\mathop{\rm u}\nolimits} ^T}v\) is a \(1 \times 1\) matrix, called the scalar product, or inner product, of u and v. It is usually written as a single real number without brackets. The matrix product \({{\mathop{\rm uv}\nolimits} ^T}\) is an \(n \times n\) matrix, called the outer product of u and v. The products \({{\mathop{\rm u}\nolimits} ^T}{\mathop{\rm v}\nolimits} \) and \({{\mathop{\rm uv}\nolimits} ^T}\) will appear later in the text.

27. Let \({\mathop{\rm u}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{ - 2}\\3\\{ - 4}\end{aligned}} \right)\) and \({\mathop{\rm v}\nolimits} = \left( {\begin{aligned}{*{20}{c}}a\\b\\c\end{aligned}} \right)\). Compute \({{\mathop{\rm u}\nolimits} ^T}{\mathop{\rm v}\nolimits} \), \({{\mathop{\rm v}\nolimits} ^T}{\mathop{\rm u}\nolimits} \),\({{\mathop{\rm uv}\nolimits} ^T}\), and \({\mathop{\rm v}\nolimits} {{\mathop{\rm u}\nolimits} ^T}\).
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free