Leontief Input-Output

Leontief Input-Output Model §

Suppose we have an economy, divided into sectors $A,B,C$, and each sector produces a product. For a sector to produce its product, it could consume some of its own product as well as some of the products produced by the other sectors.

For example, let $A,B,C$ be our products. Suppose that

• To produce 1 unit of product $A$, we need 0.2 units of $A$, 0.15 units of $B$, and 0.1 units of product $C$.
• To produce 1 unit of product $B$, we need 0.25 units of $A$, and 0.05 units of $B$.
• To produce 1 unit of product $C$, we need 0.2 units of $B$, and 0.1 units of $C$.

Assume that all units are on the same scale, presented in monetary value (making the unit standardized across all products).

For each product, we can create a consumption vector, representing what it takes to produce that product.

$$A:\left[\begin{array}{c}0.2\\ 0.15\\ 0.1\end{array}\right]B:\left[\begin{array}{c}0.25\\ 0.05\\ 0\end{array}\right]C:\left[\begin{array}{c}0\\ 0.2\\ 0.1\end{array}\right]$$

We can use these vectors to determine the cost of producing each product! For example, if we wanted to produce $p_A,p_B,p_C$ units of $A,B,C$ (respectively), we can find our cost as

$$p_A\left[\begin{array}{c}0.2\\ 0.15\\ 0.1\end{array}\right]+p_B\left[\begin{array}{c}0.25\\ 0.05\\ 0\end{array}\right]+p_C\left[\begin{array}{c}0\\ 0.2\\ 0.1\end{array}\right]=\left[\begin{array}{ccc}0.2& 0.25& 0\\ 0.15& 0.05& 0.2\\ 0.1& 0& 0.1\end{array}\right]\left[\begin{array}{c}{p}_A\\ p_B\\ p_C\end{array}\right]=M\vec{p}$$

This gives us a matrix vector product, where $\vec{p}$ contains the amounts produced, $M$ is called the consumption matrix, and $M\vec{p}$ is called the internal demand (amounts of products that are consumed).

Now say we have some demand outside of our economy for these products, say $d_A=50$ units of $A$, $d_B=80$ units of $B$, and $d_C=100$ units of $C$. How much of each product do we need to produce?

Because each product uses the other products, we can’t just supply exactly $d_A,d_B,d_C$. We need to supply more to be used internally as input elsewhere in the economy! This motivates the Leontief Input-Output Model, stating that for any product, it’s amount produced is equal to the sum of its internal and external demand.

$$\text{Amount Produced}=\text{Internal Demand}+\text{External Demand}$$

In mathematical terms,

$$\vec{p}=M\vec{p}+\vec{d}$$

So given our consumption matrix $M$ and external demand $\vec{d}$, we ask: what must we set $\vec{p}$ to so that we can satisfy the external demand?

We can solve this equation for $\vec{p}$ as so.

$$\begin{array}{rl}\vec{p}& =M\vec{p}+\vec{d}\\ \vec{p}-M\vec{p}& =\vec{d}\\ (I-M)\vec{p}& =\vec{d}\end{array}$$

This gives us a linear system that we can solve!

So, in our example, we can solve to find our production amounts

$$\vec{p}={\left(I-\left[\begin{array}{ccc}0.2& 0.25& 0\\ 0.15& 0.05& 0.2\\ 0.1& 0& 0.1\end{array}\right]\right)}^{-1}\left[\begin{array}{c}50\\ 80\\ 100\end{array}\right]\approx \left[\begin{array}{c}101.89\\ 126.07\\ 122.43\end{array}\right]$$

101.89 of $A$, 126.07 of $B$, and 122.43 of $C$.

Note that our solution requires that $(I-M)$ is invertible. While it may not always be, “reasonable” conditions in economies typically imply that $I-M$ would be invertible.

Conceptually Interpreting $(I-M{)}^{-1}$

Note that each column of $(I-M{)}^{-1}$ gives us the increase tells us the amount of extra production needed for each product! For example, if we wanted to produce $(1,0,0)$ of products, we can find it as $\vec{p}=(I-M{)}^{-1}(1,0,0{)}^T$, which is just the first column!

In more mathematical terms, the $j^{th}$ colum of $(I-M{)}^{-1}$ contains the additional production required to satisfy an additional unit of external demand for product $j$.

Given this interpretation, it should make sense that:

• The diagonal entries of $(I-M{)}^{-1}$ are $\ge 1$.
• The entries of $(I-M{)}^{-1}$ should be non-negative.

But is there a mathematical explanation for this? Under certain conditions of $M$, it is true that

$$(I-M{)}^{-1}=I+M+M^2+M^3+\dots$$

First, we observe that all terms on the right-hand side should have non-negative entries (as $M$ has all non-negative entries- we cannot consume negative products)! Furthermore, as $I$ has 1’s down the diagonal, this means that our inverse must have values $\ge 1$, as we are only adding positive numbers to 1!

These conditions are $M$ are as follows. We define a column sum of a matrix to be the sum of the entires in any given column of the matrix. Any matrix $M$ with column sums $<1$ satisfy the above condition, and in fact, any “reasonable” economy is one where each column sum is $<1$.

Otherwise, it would take more than one unit to produce a unit of a product, which is not economically feasible!

Theorem: Economically Feasible Consumption Matrices

Suppose that $M$ is $n\times n$ with non-negative entries and suppose every column sum of $M$ is less than 1.

Then,

• $(I-M{)}^{-1}=1+M+M^2+M^3+\dots$
• In particular, $(I-M{)}^{-1}$ exists and has non-negative entries.
• Given an external demand $\vec{d}$ with non-negative entries, then we have a unique solution $\vec{p}=(I-M{)}^{-1}\vec{d}$ and furthermore, $\vec{p}$ has non-negative entries too.

Example: A Nonsensical Economy

Suppose we have consumption matrix

$$M=\left[\begin{array}{cc}0.6& 0.7\\ 0.5& 0.5\end{array}\right]$$

and we want to satisfy some external demand $\vec{d}$. In this example,

$$(I-M{)}^{-1}=\left[\begin{array}{cc}-3.33& -4.66\\ -3.33& -2.66\end{array}\right]$$

It’s not feasible to satisfy $\vec{d}$, and this is because of the column sums!