The Leontief Input-Output Economic Model

1. The Inter-Industry Economic Network

Wassily Leontief (1973 Nobel Laureate in Economics) developed input-output analysis to model the interdependence of industries in an economy. In an economy divided into $n$ sectors (e.g., Agriculture, Manufacturing, Energy, Transportation), the output of any one sector serves a dual role:

• Intermediate Output: Consumed by other industries (and by itself) as inputs required for production.
• Final Demand: Consumed by households, government, capital investment, or export.

2. The Flow Matrix of Transactions

Let $x_{ij}$ denote the dollar value of output from sector $i$ consumed as intermediate input by sector $j$ during a given production period.
Let $d_i$ be the external final consumer demand for sector $i$'s product.
Let $x_i$ be the total gross output produced by sector $i$. Conservation of economic output requires: $$\mathbf{x_i = \sum_{j=1}^n x_{ij} + d_i, \quad i = 1, 2, \dots, n}$$ Total Gross Output = Total Intermediate Inputs Consumed + Final Demand.

1. The Technological Coefficients

Assuming constant returns to scale and fixed production recipes, the technological coefficient $c_{ij}$ is the dollar amount of sector $i$'s goods required to produce one dollar's worth of sector $j$'s output: $$\mathbf{c_{ij} \equiv \frac{x_{ij}}{x_j} \iff x_{ij} = c_{ij} x_j}$$ The $n \times n$ matrix $C = [c_{ij}]$ is called the consumption matrix (or technological matrix).

2. Properties of the Consumption Matrix

• Every entry is non-negative: $c_{ij} \ge 0$.
• Column $j$ represents the complete cost recipe per dollar produced by industry $j$: $$\mathbf{C_{*, j} = \begin{pmatrix} c_{1j} \\ c_{2j} \\ \vdots \\ c_{nj} \end{pmatrix}}$$
• Economic Profitability Condition: In an economy where industries create value rather than destroying resources, the sum of intermediate material costs per dollar of output must be strictly less than one: $$\sum_{i=1}^n c_{ij} < 1, \quad \text{for all } j = 1, \dots, n$$ The remainder $v_j = 1 - \sum_{i=1}^n c_{ij} > 0$ represents the value added (wages, taxes, and operating profit) per dollar of production!

1. Derivation of the Matrix Equation

Substituting $x_{ij} = c_{ij} x_j$ into the economic conservation balance: $$x_i = \sum_{j=1}^n c_{ij} x_j + d_i \iff X = C X + D$$ where: $$X = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} \quad (\text{Gross Output Vector}), \qquad D = \begin{pmatrix} d_1 \\ d_2 \\ \vdots \\ d_n \end{pmatrix} \quad (\text{Final Demand Vector})$$ Rewriting into canonical linear system form: $$\mathbf{(I_n - C) X = D}$$ The matrix $I_n - C$ is called the Leontief Matrix.

2. The Equilibrium Production Solution

If the Leontief matrix $I_n - C$ is invertible, the gross output required across all sectors to satisfy final demand $D$ is uniquely determined by: $$\mathbf{X = (I_n - C)^{-1} D}$$ The inverse matrix $(I - C)^{-1}$ is termed the Leontief Inverse (or Total Requirements Matrix). Its $(i, j)$-th entry represents the total dollar amount that sector $i$ must produce directly and indirectly to supply one dollar of final demand to sector $j$!

1. The Economic Viability Problem

In real-world economics, negative production is impossible ($X \ge 0$). An economy is defined as economically viable if for every non-negative final demand vector $D \ge 0$, there exists a unique non-negative gross production vector $X \ge 0$ satisfying $(I - C)X = D$.

2. The Hawkins–Simon Theorem

Theorem (David Hawkins & Herbert Simon, 1949): An input-output system with consumption matrix $C \ge 0$ is economically viable if and only if all leading principal minors of the Leontief matrix $I - C$ are strictly positive:

• $\Delta_1 > 0 \iff c_{11} < 1$: Sector 1 cannot consume more of its own product than it produces.
• $\Delta_2 > 0 \iff (1 - c_{11})(1 - c_{22}) > c_{12} c_{21}$: The combined direct and indirect feedback loops between sectors 1 and 2 must not consume more than their collective net capacity.

1. The Neumann Power Series Expansion

If the spectral radius $\rho(C) < 1$, the Leontief inverse can be expanded as a convergent geometric matrix series (the Neumann Series): $$\mathbf{(I - C)^{-1} = I + C + C^2 + C^3 + \dots = \sum_{k=0}^\infty C^k}$$ Substituting into the output equation $X = (I - C)^{-1} D$: $$\mathbf{X = D + C D + C^2 D + C^3 D + \dots}$$ Economic Multiplier Breakdown:

• $D$: Direct final consumer demand.
• $CD$: First-round intermediate inputs required by industries to produce $D$.
• $C^2 D$: Second-round inputs required to produce the intermediate inputs $CD$.
• $C^k D$: $k$-th generation indirect supply chain requirements throughout the economy.

2. The Dual Leontief Price Model

Let $P = (p_1, p_2, \dots, p_n)$ be the unit price row vector across sectors, and let $V = (v_1, v_2, \dots, v_n)$ be the value-added row vector (wages + profits per unit). The equilibrium pricing relation states that price equals intermediate material costs plus value added: $$P = P C + V \iff P(I - C) = V$$ Multiplying by the Leontief inverse from the right yields the equilibrium price structure: $$\mathbf{P = V (I - C)^{-1}}$$ This allows governments and central banks to calculate how changes in wages or energy tax ($V$) propagate throughout the entire price level of the macroeconomy!