Say "production function" and most people picture a dry accounting formula. It's actually one of the most foundational concepts in production economics — and understanding it explains why EVERY line, EVERY factory has a natural output ceiling, no matter how many people you hire.
What is a production function
A production function is the quantitative relationship between input factors (materials, labor, machinery...) and the maximum output achievable with that combination of inputs. General notation: M = f(r₁, r₂, ..., r_H) — M is output, each r_h is the quantity of one input factor.
Three things worth keeping straight when working with production functions:
- It always represents maximum achievable output — the technically efficient outcome, not just any arbitrary combination.
- It describes a purely technical input-output relationship — no prices or costs attached yet (that's the next layer, cost theory).
- It assumes a fixed technology level — changing technology means switching to a different production function, not moving along the same one.
Two types: substitutional and limitational
- Substitutional — inputs can be traded off against each other to some degree while output stays constant (add labor, use less machinery, same output). This is the case where the law of diminishing returns applies, and it's the focus of this post.
- Limitational (fixed-proportions) — the ratio between inputs and output is always fixed, no trade-off possible. The simplest linear version is the Leontief production function: one tabletop always needs exactly 4 legs — more legs doesn't create more tabletops. A more complex version (the Gutenberg function) applies to industrial machinery, where material consumption depends on how intensively the machine runs.
Partial factor variation — the condition for the law to appear
The law of diminishing returns only shows up under one specific condition: partial factor variation — only ONE input is allowed to change (typically labor), while the other input(s) stay fixed (machine count, floor space). This is the classic "short run" problem in economics: in the short run, at least one production factor can't be adjusted in time, so a business can only change output by adjusting the remaining factor.
Seeing the whole surface — a 3D simulation
With 2 inputs (labor L, capital/machines K), the production function M(L,K) can be pictured as a 3D surface — every combination of (L,K) produces a height M. The simulation below draws exactly that surface for a concrete example: a bakery with K fixed ovens and L bakers on shift. Drag to rotate, and try the sliders yourself.
The gold line on the surface is a slice — holding K fixed at one value and letting only L vary. That's exactly the "partial factor variation" condition above, and exactly what the law of diminishing returns describes.
The four phases of the law of diminishing returns
Three concepts to keep separate when reading this curve:
- Total product — total units produced.
- Marginal product — the output added by the single newest unit.
- Average product — total output divided evenly across all variable-input units.
The curve always moves through exactly 4 phases, regardless of industry:
- Phase I: total output climbs at an accelerating rate, marginal product is rising.
- Phase II: total output still climbs but more slowly; marginal product has started declining but is still above average product.
- Phase III: the point where marginal product = average product happens at the start of this phase (average product peaks exactly here); marginal product keeps declining toward zero.
- Phase IV: total output has peaked (marginal product = 0), then starts falling — marginal product goes negative.
One detail that's easy to miss: the relationship between marginal and average product always follows the same rule — while marginal product is above average product, average product is rising; once marginal product drops below it, average product is falling; the two lines cross exactly at average product's peak. This isn't a coincidence specific to the bakery example — it holds for every substitutional production function.
With both inputs allowed to vary (not just one), the equivalent concept is the isoquant — a curve connecting every (L,K) combination that yields the same output — and the marginal rate of technical substitution (MRTS) measures that curve's slope. That's a broader topic, outside the scope of this post.
Where this applies
This isn't an academic exercise — it answers a question every factory owner has asked: "should I hire more people?" The post Why adding more workers doesn't always increase output applies exactly this theory to a real example — a 3-CNC-machine line — with its own simulation to test against your own factory's numbers.
Nguyễn Hải Minh
I build custom software and data solutions for manufacturing ERP systems, including INFOR, for clients in Germany. As a Staatlich geprüfter IT-Techniker (Fachrichtung Informatik) and Informationselektroniker, I combine deep technical skill with business-systems thinking to help manufacturers automate operations and optimize cross-border import and export.
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