2.6 Exercises

20 points · 4 extra credit

The Leontief Input–Output Model

An economy has two sectors with consumption matrix

$C = \begin{bmatrix} 0.2 & 0.4 \\ 0.2 & 0.4 \end{bmatrix},$

where column $j$ lists the inputs (from each sector) needed to produce one unit of sector $j$'s output. Production $\mathbf{x}$ and final demand $\mathbf{d}$ satisfy $\mathbf{x} = C\mathbf{x} + \mathbf{d}$, i.e. $(I - C)\mathbf{x} = \mathbf{d}$.

1. 1. (3 pts) For final demand $\mathbf{d} = \begin{bmatrix} 20 \\ 40 \end{bmatrix}$, solve $(I - C)\mathbf{x} = \mathbf{d}$ for the production vector $\mathbf{x}$.

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2. 2. (3 pts) Compute the Leontief inverse $(I - C)^{-1}$.

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3. 3. (2 pts) Determine whether each column sum of $C$ is less than $1$ (the condition that makes $C$ productive), and state what productivity guarantees about $(I - C)^{-1}$.

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4. 4. (3 pts) Compute the partial sum $I + C + C^2$ and compare it entrywise with the exact inverse from Problem 2. What does the comparison illustrate?

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5. 5. (3 pts) State whether each is true or false, and justify.

   (a) In the Leontief model, increasing the final demand for one good can never decrease the required production of any good (all entries of $(I-C)^{-1}$ are nonnegative).

   (b) The equation $\mathbf{x} = C\mathbf{x} + \mathbf{d}$ says that total output equals intermediate consumption plus final demand.

   (c) If some column sum of $C$ exceeds $1$, the economy can still always meet any positive demand.

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6. 6. (3 pts) Interpret the meaning of column $1$ of $(I - C)^{-1}$ in economic terms. What does its entry in row $2$ represent?

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7. 7. (3 pts) Starting from the demand in Problem 1, suppose final demand for sector $1$ rises by $10$ units while sector $2$'s demand is unchanged, so $\Delta\mathbf{d} = \begin{bmatrix} 10 \\ 0 \end{bmatrix}$. Find the resulting change $\Delta\mathbf{x}$ in production.

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8. 8. (4 pts ••) Let $C$ be a consumption matrix. Prove that for every $m \ge 0$,

   $(I - C)\big(I + C + C^2 + \cdots + C^{m}\big) = I - C^{m+1},$

   and explain why, when $C^{m+1} \to 0$ as $m \to \infty$, the series $I + C + C^2 + \cdots$ converges to $(I - C)^{-1}$.

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