Input / Output

290. Input / Output

290.1. Input Output Model (Leontief)

𝐀 = [\begin{matrix} 𝑎_{11} & 𝑎_{12} & \dots & 𝑎_{1 𝑛} \\ 𝑎_{21} & 𝑎_{22} & \dots & 𝑎_{2 𝑛} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 𝑎_{𝑚 1} & 𝑎_{𝑚 2} & \dots & 𝑎_{𝑚 𝑛} \end{matrix}] 𝐱 = [\begin{matrix} 𝑥_{1} \\ 𝑥_{2} \\ ⋮ \\ 𝑥_{𝑛} \end{matrix}] 𝐝 = [\begin{matrix} 𝑑_{1} \\ 𝑑_{2} \\ ⋮ \\ 𝑑_{𝑛} \end{matrix}]

Where:

$𝑎_{𝑖 𝑗}$ : units of good $𝑖$ needed to produce 1 unit of good $𝑗$
$𝐱$ : total outputs for each sector
$𝐝$ : vector of goods used externally by consumers
$𝐀 𝐱$ : vector of goods used internally in production

Technical coefficient: the amount of input required per unit of output

Balance Equation

Total production = production used as inputs for other sectors+final consumption

𝐱 = 𝐀 𝐱 + 𝐝

Rewrite the Equation

Subtract $𝐀 𝐱$ from both sides:

𝐱 - 𝐀 𝐱 = 𝐝

Factor out $𝐱$ :

𝐱 (𝐈 - 𝐀) = 𝐝

Given final demand $𝐝$ and input requirements $𝐀$ , what total production $𝐱$ is needed?

Solving for Production

𝐱 = {(𝐈 - 𝐀)}^{- 1} 𝐝

This gives exactly how much each sector must produce to satisfy both internal (inputs) and external (final demand) needs.

${(𝐼 - 𝐴)}^{- 1}$ : Leontief inverse, each element shows the total output needed from sector $𝑖$ per unit of final demand in sector $𝑗$ , accounting for indirect effects

Example

Steel (S)
Rubber (R)
Chemicals (C)

\begin{matrix} 𝐱 = {(𝐈 - 𝐀)}^{- 1} 𝐝 \\ 𝐱 = {([\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] - [\begin{matrix} 0.1 & 0.2 & 0.1 \\ 0.1 & 0.1 & 0.2 \\ 0.2 & 0.1 & 0.1 \end{matrix}])}^{- 1} [\begin{matrix} 10 \\ 20 \\ 30 \end{matrix}] \\ 𝐱 = [\begin{matrix} 23.42 \\ 34.23 \\ 42.34 \end{matrix}] \end{matrix}

Steel ( $𝒙_{𝟏} = 23.42$ ): total units needed for final demand and as inputs to other sectors
Rubber ( $𝒙_{𝟐} = 34.23$ ): total units for final demand plus inputs to all sectors
Chemicals ( $𝒙_{𝟐} = 42.34$ ): total units including intermediate uses

290.1.1. Labor

Direct Labor

Each sector requires some amount of direct labor to produce a unit of output. Direct labor coefficient:

$𝑙_{𝑘}$ : hours of labor needed to produce one unit of good $𝑗$

Indirect Labor

Producing goods often requires inputs from other sectors, which themselves required labor. To account for this:

𝝀 = 𝐥 {(𝐈 - 𝐀)}^{- 1}

$𝐥$ : vector of direct labor coefficients
$𝐀$ : input-output matrix
$𝝀$ : total (direct and indirect) labor required per unit of each good

Interpretation: $𝜆_{𝑗}$ is the total socially necessary labor time embedded in one unit of good $𝑗$

Labor as a “Currency”

People earn labor vouchers corresponding to the hours they work.
Each good has a price in labor time: $𝜆_{𝑗}$ hours per unit.
If you work $ℎ$ hours, you get $ℎ$ labor vouchers to spend.

Budget constraint for a worker:

\sum_{𝑗} 𝜆_{𝑗} 𝑐_{𝑗} \leq ℎ

Where:

$𝑐_{𝑗}$ : units of good $𝑗$ the worker wants to consume
$𝜆_{𝑗}$ : labor time required to produce one unit of good $𝑗$ (includes direct and indirect) labor
$ℎ$ : total labor hours the worker has contributed (i.e., the number of labor vouchers they possess)

Example

Suppose:

Good 1: $𝜆_{1} = 2 hours/unit$
Good 2: $𝜆_{1} = 5 hours/unit$
Worker labor: $ℎ = 20 hours$

Constraint:

2 𝑐_{1} + 5 𝑐_{2} \leq 20

If the worker wants 4 units of Good 1: $2 \times 4 = 8 hours$
Then they could afford at most $(20 - 8) \div 5 = 2.4 units$ of Good 2

So their consumption plan is limited by their labor contribution

Labor Planning

Labor also acts as a constraint for the whole economy:

𝐈 𝐱 \leq 𝐿

Where:

$𝐱$ : vector of production
$𝐿$ : total labor available in society
This ensures that planned production doesn’t require more labor than society can provide

Example

Stepup

Suppose the economy produces Bread (B) and Clothes (C). Let’s define:

$𝐱 = (𝑥_{𝐵}, 𝑥_{𝐶})$ : total production of each good
$𝐝 = (𝑑_{𝐵}, 𝑑_{𝐶})$ : final demand (how much society wants to consume)
$𝐀 = [\begin{matrix} 0.1 & 0.2 \\ 0.3 & 0.1 \end{matrix}]$ : input-output matrix
$𝐥 = (2, 5)$ : direct labor per unit (hours) for Bread and Clothes
$𝐿 = 100$ : total labor available in society

Constraints

Input-Output Balance: production must satisfy both internal (inputs) and final demand

\begin{matrix} 𝑥_{𝐵} \geq 0.1 𝑥_{𝐵} + 0.2 𝑥_{𝐶} + 𝑑_{𝐵} \\ 𝑥_{𝐶} \geq 0.3 𝑥_{𝐵} + 0.1 𝑥_{𝐶} + 𝑑_{𝐶} \end{matrix}

Labor Constraint: total labor used cannot exceed available labor

2 𝑥_{𝐵} + 5 𝑥_{𝐶} \leq 100

Non-negativity: you can’t produce negative amounts

𝑥_{𝐵} \geq 0 𝑥_{𝐶} \geq 0

Objective: Maximize production

max 𝑤_{𝐵} 𝑑_{𝐵} + 𝑤_{𝐶} 𝑑_{𝐶}

Where:

$𝑤_{𝐵}, 𝑤_{𝐶}$ : societal priority weights (e.g., bread might be more essential than clothes)
$𝑑_{𝐵}, 𝑑_{𝐶}$ : quantities actually supplied to meet demand

Interpretation

The LP decides how much of each product to product ( $𝑥_{𝐵}, 𝑥_{𝐶}$ )
It respects labor limits ( $2 𝑥_{𝐵} + 5 𝑥_{𝐶} \leq 100$ )
It respects interdependencies (inputs for other products) via the matrix $𝐴$
It maximizes fulfillment of product demand, possibly weighted by social priorities

So, no abstract utility function needed — the LP is about physical goods, labor, and inputs