Building Leontief Input-Output Models

From Production Data to a Matrix Model

An input-output model records how the output of one sector is used by other sectors. For example, a manufacturing sector may purchase energy and materials from other sectors to produce its own output. Some output is used in this intermediate production; the rest goes to final users such as households, government, or export markets. The Leontief balance equation puts this accounting into a matrix equation. Its meaning depends on using a consistent convention for the rows, columns, and vectors.

Suppose an economy has \(n\) sectors. Use the same ordering of sectors throughout the model. Let \(x_j\) be the gross output of sector \(j\) over a specified period. Let \(z_{ij}\) be the amount of output from sector \(i\) used as an intermediate input by sector \(j\) during that period. Thus the first index identifies the supplying sector, and the second identifies the sector using the input. Put these transactions in the matrix \(\mathbf{Z}\), with \(z_{ij}\) in row \(i\), column \(j\). The vector \(\mathbf{x}\) of gross outputs is a column vector.

Each column of \(\mathbf{Z}\) describes the intermediate inputs used by one sector. For example, column \(j\) lists the inputs from all sectors that are used to produce sector \(j\)'s output. A transaction and its corresponding output must use compatible units and refer to the same period.

The condition \(x_j>0\) matters: if sector \(j\) has zero output, the ratio \(z_{ij}/x_j\) is undefined, so its technical coefficients cannot be calculated from that period's transaction data by this formula. When positive output is observed, these ratios give the period's measured input proportions. Using them to model other output levels assumes those proportions remain fixed.

Final demand is output delivered to users outside the modeled intermediate production process. Let \(d_i\) be total final demand for sector \(i\)'s output, and collect these amounts in the column vector \(\mathbf{d}\). Final demand may combine several categories, such as household consumption, government purchases, and exports. These categories are uses of output, not additional columns of intermediate inputs in \(\mathbf{Z}\).

Deriving the Balance Equation

For each sector \(i\), its gross output must account for all intermediate use of that sector's output and its final demand. The intermediate use of sector \(i\)'s output by sector \(j\) is \(z_{ij}\); summing across the using sectors gives \(\sum_{j=1}^{n}z_{ij}\). Thus sector \(i\)'s accounting identity is \(x_i=\sum_{j=1}^{n}z_{ij}+d_i\). In vector form, the sum of intermediate use across each row is \(\mathbf{Z}\mathbf{1}\), where \(\mathbf{1}\) is the column vector whose entries are all \(1\).

By the definition \(a_{ij}=z_{ij}/x_j\), each transaction satisfies \(z_{ij}=a_{ij}x_j\). Consequently, the intermediate use of sector \(i\)'s output is \(\sum_{j=1}^{n}a_{ij}x_j\), which is entry \(i\) of \(\mathbf{A}\mathbf{x}\). This gives the standard column-vector form of the balance equation.

Proof. For each pair \(i,j\), the coefficient definition and \(x_j>0\) give \(z_{ij}=a_{ij}x_j\). Substituting these equalities into the sector \(i\) accounting identity gives \(x_i=\sum_{j=1}^{n}a_{ij}x_j+d_i\). By matrix multiplication, \(\sum_{j=1}^{n}a_{ij}x_j\) is entry \(i\) of \(\mathbf{A}\mathbf{x}\). Since this holds for every \(i\), the component identities combine to \(\mathbf{x}=\mathbf{A}\mathbf{x}+\mathbf{d}\). Subtracting \(\mathbf{A}\mathbf{x}\) from both sides yields \((\mathbf{I}_n-\mathbf{A})\mathbf{x}=\mathbf{d}\). \(\square\)

The equation is an accounting model, not by itself a guarantee that a production plan exists or is unique. Its purpose here is to express the balance consistently: gross output equals intermediate requirements plus final demand. Determining a production plan from a given final-demand vector is a separate question.

Worked Examples: Constructing and Checking the Model

Keeping Sector Order Consistent

The formulas rely on one shared sector ordering. If the order changes, the rows and columns of the coefficient matrix and the entries of both vectors must change together. A useful way to state this precisely is with a permutation matrix. Such a matrix \(\mathbf{P}\) reorders coordinates: each column is a standard basis vector, and each standard basis vector appears exactly once.

Proof. A permutation matrix has \(\mathbf{P}^{\top}\mathbf{P}=\mathbf{I}_n\), because its columns are distinct standard basis vectors, which have dot product \(1\) with themselves and \(0\) with one another. Using \(\mathbf{x}'=\mathbf{P}\mathbf{x}\), we obtain \(\mathbf{A}'\mathbf{x}'+\mathbf{d}'=\mathbf{P}\mathbf{A}\mathbf{P}^{\top}\mathbf{P}\mathbf{x}+\mathbf{P}\mathbf{d}=\mathbf{P}(\mathbf{A}\mathbf{x}+\mathbf{d})=\mathbf{P}\mathbf{x}=\mathbf{x}'\). For the converse, multiply \(\mathbf{x}'=\mathbf{A}'\mathbf{x}'+\mathbf{d}'\) by \(\mathbf{P}^{\top}\). Since \(\mathbf{P}^{\top}\mathbf{P}=\mathbf{I}_n\), substitution of the definitions gives \(\mathbf{x}=\mathbf{A}\mathbf{x}+\mathbf{d}\). \(\square\)

This result explains why simply reordering the entries of \(\mathbf{x}\) and \(\mathbf{d}\), while leaving \(\mathbf{A}\) unchanged, generally breaks the model. Both the supplying-sector rows and using-sector columns of \(\mathbf{A}\) must be reordered.

Common Pitfall: Transposing the Coefficient Matrix

Why the Formulation Matters

The technical-coefficient matrix separates input proportions from the scale of production. Each coefficient is a per-unit requirement, while \(\mathbf{x}\) records the output levels to which those proportions are applied. Final demand then accounts for output not used as an intermediate input. This separation makes the bookkeeping explicit and allows the same model structure to represent different sector sizes, provided the fixed-proportion assumption is appropriate.

Before using a model, check three things: rows and columns follow the stated supplier/user convention; each coefficient divides a transaction by the output of its column's sector; and the final-demand vector is indexed in the same sector order as \(\mathbf{x}\). Finally, verify the accounting componentwise by comparing \(\mathbf{x}\) with \(\mathbf{A}\mathbf{x}+\mathbf{d}\). A mismatch may indicate an arithmetic error, inconsistent units, omitted final-demand categories, or a convention that has been applied inconsistently.