2.8E: An Application to Input-Output Economic Models Exercises

Exercises for 1

solutions

2

Find the possible equilibrium price structures when the input-output matrices are:

$\left[ \begin{array}{rrr} 0.1 & 0.2 & 0.3 \\ 0.6 & 0.2 & 0.3 \\ 0.3 & 0.6 & 0.4 \end{array} \right]$ $\left[ \begin{array}{rrr} 0.5 & 0 & 0.5 \\ 0.1 & 0.9 & 0.2 \\ 0.4 & 0.1 & 0.3 \end{array} \right]$ $\left[ \begin{array}{rrrr} 0.3 & 0.1 & 0.1 & 0.2 \\ 0.2 & 0.3 & 0.1 & 0 \\ 0.3 & 0.3 & 0.2 & 0.3 \\ 0.2 & 0.3 & 0.6 & 0.5 \end{array} \right]$ $\left[ \begin{array}{rrrr} 0.5 & 0 & 0.1 & 0.1 \\ 0.2 & 0.7 & 0 & 0.1 \\ 0.1 & 0.2 & 0.8 & 0.2 \\ 0.2 & 0.1 & 0.1 & 0.6 \end{array} \right]$

2. $\left[ \begin{array}{r} t \\ 3t \\ t \end{array} \right]$
3. $\left[ \begin{array}{r} 14t \\ 17t \\ 47t \\ 23t \end{array} \right]$

Three industries $A$, $B$, and $C$ are such that all the output of $A$ is used by $B$, all the output of $B$ is used by $C$, and all the output of $C$ is used by $A$. Find the possible equilibrium price structures.

$\left[ \begin{array}{r} t \\ t \\ t \end{array} \right]$

Find the possible equilibrium price structures for three industries where the input-output matrix is $\left[ \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right]$. Discuss why there are two parameters here.

Prove Theorem [thm:007013] for a $2 \times 2$ stochastic matrix $E$ by first writing it in the form $E = \left[ \begin{array}{cc} a & b \\ 1 - a & 1 - b \end{array} \right]$, where $0 \leq a \leq 1$ and $0 \leq b \leq 1$.

$P = \left[ \begin{array}{c} bt \\ (1 - a)t \end{array} \right]$ is nonzero (for some $t$) unless $b = 0$ and $a = 1$. In that case, $\left[ \begin{array}{r} 1 \\ 1 \end{array} \right]$ is a solution. If the entries of $E$ are positive, then $P = \left[ \begin{array}{c} b \\ 1 - a \end{array} \right]$ has positive entries.

If $E$ is an $n \times n$ stochastic matrix and $\mathbf{c}$ is an $n \times 1$ matrix, show that the sum of the entries of $\mathbf{c}$ equals the sum of the entries of the $n \times 1$ matrix $E\mathbf{c}$.

Let $W = \left[ \begin{array}{ccccc} 1 & 1 & 1 & \cdots & 1 \end{array} \right]$. Let $E$ and $F$ denote $n \times n$ matrices with nonnegative entries.

1. Show that $E$ is a stochastic matrix if and only if $WE = W$.
2. Use part (a.) to deduce that, if $E$ and $F$ are both stochastic matrices, then $EF$ is also stochastic.

Find a $2 \times 2$ matrix $E$ with entries between $0$ and $1$ such that:

1. $I - E$ has no inverse.
2. $I - E$ has an inverse but not all entries of $(I - E)^{-1}$ are nonnegative.

2. $\left[ \begin{array}{rr} 0.4 & 0.8 \\ 0.7 & 0.2 \end{array} \right]$

If $E$ is a $2 \times 2$ matrix with entries between $0$ and $1$, show that $I - E$ is invertible and $(I - E)^{-1} \geq 0$ if and only if $tr \;E < 1 + \det E$. Here, if $E = \left[ \begin{array}{rr} a & b \\ c & d \end{array} \right]$, then $tr \;E = a + d$ and $\det E = ad - bc$.

If $E = \left[ \begin{array}{rr} a & b \\ c & d \end{array} \right]$, then $I - E = \left[ \begin{array}{cc} 1 - a & -b \\ -c & 1 - d \end{array} \right]$, so $\det (I - E) = (1 - a)(1 - d) - bc = 1 - tr \;E + \det E$. If $\det (I - E) \neq 0$, then $(I - E)^{-1} = \frac{1}{\det (I - E)} \left[ \begin{array}{cc} 1 - d & b \\ c & 1 - a \end{array} \right]$, so $(I - E)^{-1} \geq 0$ if $\det (I - E) > 0$, that is, $tr \;E < 1 + \det E$. The converse is now clear.

In each case show that $I - E$ is invertible and $(I - E)^{-1} \geq 0$.

$\left[ \begin{array}{rrr} 0.6 & 0.5 & 0.1 \\ 0.1 & 0.3 & 0.3 \\ 0.2 & 0.1 & 0.4 \end{array} \right]$ $\left[ \begin{array}{rrr} 0.7 & 0.1 & 0.3 \\ 0.2 & 0.5 & 0.2 \\ 0.1 & 0.1 & 0.4 \end{array} \right]$ $\left[ \begin{array}{rrr} 0.6 & 0.2 & 0.1 \\ 0.3 & 0.4 & 0.2 \\ 0.2 & 0.5 & 0.1 \end{array} \right]$ $\left[ \begin{array}{rrr} 0.8 & 0.1 & 0.1 \\ 0.3 & 0.1 & 0.2 \\ 0.3 & 0.3 & 0.2 \end{array} \right]$

2. Use $\mathbf{p} = \left[ \begin{array}{r} 3 \\ 2 \\ 1 \end{array} \right]$ in Theorem [thm:007060].
3. $\mathbf{p} = \left[ \begin{array}{r} 3 \\ 2 \\ 2 \end{array} \right]$ in Theorem [thm:007060].

[ex:ex2_8_10] Prove that (1) implies (2) in the Corollary to Theorem [thm:007060].

[ex:ex2_8_11] If $(I - E)^{-1} \geq 0$, find $\mathbf{p} > 0$ such that $\mathbf{p} > E\mathbf{p}$.

[ex:ex2_8_12] If $E\mathbf{p} < \mathbf{p}$ where $E \geq 0$ and $\mathbf{p} > 0$, find a number $\mu$ such that $E\mathbf{p} < \mu\mathbf{p}$ and $0 < \mu < 1$.

[Hint: If $E\mathbf{p} = (q_{1}, \dots, q_{n})^{T}$ and $\mathbf{p} = (p_{1}, \dots, p_{n})^{T}$, take any number $\mu$ where $\func{max}\left\lbrace \frac{q_{1}}{p_{1}}, \dots, \frac{q_{n}}{p_{n}} \right\rbrace < \mu < 1$.]