Production Function

Production is the process of using different factor inputs like land, labour, capital, and entrepreneurship, along with raw materials, to produce goods and services.

A production function shows the functional (cause‑and‑effect) relationship between physical inputs and physical output of a firm.

In simple words: Production function tells us how much output a firm can produce with given quantities of inputs in a given period, using the best available technique, or how much input is needed to produce a given level of output.

Example (shoe factory):
To produce shoes, a firm uses workers, machines, leather, glue, etc. The production function shows either:

• the maximum number of pairs of shoes per day that can be produced with given workers and machines, or
• the minimum number of workers, machines, and raw materials needed to produce, say, 100 pairs of shoes per day.

A production function can be shown as:

• A table (showing different combinations of inputs and output),
• A graph, or
• An equation (algebraic form).

General form:

\[Q_x=f(f_1,f_2,\ldots,f_n)\]

Where:

• Q_x = quantity of output of commodity X,
• f₁, f₂,…, f_n = quantities of different factor inputs,
• Q_x is the dependent variable (depends on inputs) ,
• f₁, f₂,…, f_n are independent variables.

If we assume only two inputs: labour (L) and capital (K):

\[Q_x=f(L,K)\]

1) Time‑based (flow concept)

• It is always related to a specific time period – per day, per month, per year.
• Both inputs and output are measured as flows per period (e.g., workers per day, units per day).

2) Physical relationship

• Inputs and outputs are measured in physical units, not in money.
• Example: “5 workers and 2 machines produce 200 units” (physical quantities).

3) Technological relation

• It reflects the state of technology.
• With better technology, the same inputs can produce more output; the production function shifts.

Economists do not fix short run and long run in terms of exact months or years.
Instead, they define them by how easily inputs can be changed.

1) Short run

• In the short run, at least one factor is fixed (usually plant, building, major machines).
• Other factors are variable (labour, raw materials, electricity, etc.).

To change output in the short run, the firm changes only variable factors while fixed factors remain the same.

Example 1 (garment factory):

• Factory building and big stitching machines are fixed for the next 6 months.
• The firm can hire more workers and buy more cloth to increase production during that time.

Example 2 (small bakery):

• The oven is fixed in the short run.
• The owner can buy more flour and hire more helpers today but cannot immediately install a new oven.

2) Long run

• In the long run, all factors of production are variable.
• The firm can change plant size, buy new machines, shift to a bigger building, or even open a new plant.

Examples:

• A steel company may need at least 3 years to build a new plant. For it, anything less than that may still be a short run.
• A coaching centre can rent a new classroom within a few weeks, so its short run is much shorter.

Economists classify production functions according to which inputs can be changed.

1) Short‑run production function

• Only one input is variable; the others are fixed.
• Often, labour is variable and capital is fixed.

Short‑run production function:
\[Q_x=f(L)\]

Here, output changes when labour (L) changes, but capital and other factors remain fixed.

The study of how output changes when only one factor varies is called returns to a factor.
This is the basis of the Law of Variable Proportions (studied later with TP, AP, MP).

Example:
A firm has 5 identical machines (fixed). It increases the number of workers from 1 to 2 to 3, etc., and observes changes in output. This is a short‑run production function.

2) Long‑run production function

• All inputs are variable.
• The firm can change both labour and capital together.

Long‑run production function:
\[Q_x=f(K,L)\]

Here, output depends on both capital and labour, and both can change.

When all inputs are changed in the same proportion (e.g., both doubled), the resulting change in output is called returns to scale (increasing, constant, or decreasing returns to scale).

Example:
If a firm doubles labour and capital (from 10 workers and 5 machines to 20 workers and 10 machines) and output more than doubles, it faces increasing returns to scale.

Tjalling Charles Koopmans (1910–1985) was a Dutch‑American economist and mathematician.
He shared the 1975 Nobel Prize in Economics with Leonid Kantorovich for work on optimal allocation of resources and the study of how inputs and outputs are related in efficient production systems.

His research helped develop a general theory of how to allocate scarce resources among different productive activities in a way that maximises efficiency.

• A production function shows the technical relationship between physical inputs and maximum possible output in a given time.
• Short run: At least one factor is fixed; the firm changes output by changing only variable factors.
• Long run: All factors are variable; the firm can change the scale of production and plant size.
• Short‑run production function Q = f (L) → study of returns to a factor and Law of Variable Proportions.
• Long‑run production function Q = f (K, L) → study of returns to scale.