287. Input / Output
287.1. Input Output Model (Leontief)
Where:
- : units of good needed to produce 1 unit of good
- : total outputs for each sector
- : vector of goods used externally by consumers
- : 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
Rewrite the Equation
- Subtract from both sides:
- Factor out :
Given final demand and input requirements , what total production is needed?
- Solving for Production
This gives exactly how much each sector must produce to satisfy both internal (inputs) and external (final demand) needs.
: Leontief inverse, each element shows the total output needed from sector per unit of final demand in sector , accounting for indirect effects
Example
- Steel (S)
- Rubber (R)
- Chemicals (C)
- Steel (): total units needed for final demand and as inputs to other sectors
- Rubber (): total units for final demand plus inputs to all sectors
- Chemicals (): total units including intermediate uses
287.1.1. Labor
- Direct Labor
Each sector requires some amount of direct labor to produce a unit of output. Direct labor coefficient:
- : hours of labor needed to produce one unit of good
- Indirect Labor
Producing goods often requires inputs from other sectors, which themselves required labor. To account for this:
- : vector of direct labor coefficients
- : input-output matrix
- : total (direct and indirect) labor required per unit of each good
Interpretation: is the total socially necessary labor time embedded in one unit of good
- Labor as a “Currency”
- People earn labor vouchers corresponding to the hours they work.
- Each good has a price in labor time: hours per unit.
- If you work hours, you get labor vouchers to spend.
Budget constraint for a worker:
Where:
- : units of good the worker wants to consume
- : labor time required to produce one unit of good (includes direct and indirect) labor
- : total labor hours the worker has contributed (i.e., the number of labor vouchers they possess)
Example
Suppose:
- Good 1:
- Good 2:
- Worker labor:
Constraint:
- If the worker wants 4 units of Good 1:
- Then they could afford at most 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:
Where:
- : vector of production
- : 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:
-
: total production of each good
-
: final demand (how much society wants to consume)
-
: input-output matrix
-
: direct labor per unit (hours) for Bread and Clothes
-
: total labor available in society
- Constraints
Input-Output Balance: production must satisfy both internal (inputs) and final demand
Labor Constraint: total labor used cannot exceed available labor
Non-negativity: you can’t produce negative amounts
- Objective: Maximize production
Where:
- : societal priority weights (e.g., bread might be more essential than clothes)
- : quantities actually supplied to meet demand
Interpretation
- The LP decides how much of each product to product ()
- It respects labor limits ()
- It respects interdependencies (inputs for other products) via the matrix
- 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