First read about Leontief Economic Models in Section 1.6 of the text. Now consider an exchange model economy which has f

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First Read About Leontief Economic Models In Section 1 6 Of The Text Now Consider An Exchange Model Economy Which Has F 1 (1.44 MiB) Viewed 28 times

only question (7)and (8)
First read about Leontief Economic Models in Section 1.6 of the text. Now consider an exchange model economy which has five sectors, Chemicals, Metals, Fuels, Power and Agriculture; and assume the matrix T below gives an exchange table for this economy: T= C M F P A C .20 .17 .25 .20 .10 .20 .10 .30 0 M .25 F .05 .20 .10 .15 .10 P .10 .28 .40 .20 0 A .40 .15 .15 .15 .80 Notice that each column of T sums to one, indicating that all output of each sector is distributed among the five sectors, as should be the case in an exchange economy. The system of equations Tx = x must be satisfied for the economy to be in equilibrium. As you saw above, this is equivalent to the system Bx = 0. (1) Write out the five equations in the equation Tx= x. (2) Obtain a homogeneous linear system Br = 0 equivalent to Tx= x. What is B? Hint: Collect the like terms after moving all non-zero terms to LHS. (3) (Optional) Solve Bx = 0 directly using any kind linear system solver provided by any computing tools. Specify what calculator or computing language you used. (4) Reduce augmented matrix [B10] to RREF form step by step. You may use ei- ther hand-computation or programming. Attach codes at the end of the report if you used programming. A reference to python coding for Gaussian elimination is HERE, and video is HERE (5) Write the general solution of Br = 0. (6) Suppose that the economy described above is in equilibrium and TA = 100 million dollars. Calculate the values of the outputs of the other sectors. (7) As already observed, each column of T sums to one. Consider how you obtained B from T and explain why each column of B must sum to zero. (8) (Bonus 1 pt) Let B be any square matrix with the property that each column of B sums to zero. Explain why the reduced echelon form of B must have a row of zeros. Hint: The proof has two steps: ● In step 1, we can prove (how?) that Br = 0 must have infinitely many solutions due to its each column sum to zero. In step 2, to the contrary, if we assume the last row of REF for B is non- zero, we can show (how?) the uniqueness of Br = 0. But this leads to a contradiction and completes the desired proof.