Production function, the law of variable proportions and the laws of returns to scale

1. Production function

A production function is the technical relationship between the physical inputs of a firm and its physical output, for a given state of technology. It shows the maximum output obtainable from each combination of inputs. Stigler: the production function is “the name given to the relationship between rates of input of productive services and the rate of output of product.”

It is written Q = f(L, K, N, E, T), where Q is output, L labour, K capital, N land, E entrepreneurship and T technology. For analysis it is simplified to Q = f(L, K).

Features. It is a physical (not money) relationship; it is defined for a given technology, and a change in technology shifts the whole function; it is a flow per period; and it distinguishes the short run (at least one input fixed) from the long run (all inputs variable).

Cobb–Douglas production function. C.W. Cobb and P.H. Douglas (1928) fitted Q = A L^α K^β to American manufacturing data. A is technology, and α and β are the output elasticities of labour and capital. If α + β = 1, there are constant returns to scale; if greater than 1, increasing; if less than 1, diminishing.

2. Law of variable proportions (short run)

Statement. When more and more units of a variable input are applied to a given quantity of fixed inputs, total product at first increases at an increasing rate, then at a diminishing rate, and finally falls. Marginal product of the variable input eventually declines. The law is also called the law of diminishing returns (in its later stages).

Assumptions: one input variable, others fixed; technology constant; all units of the variable input are homogeneous; it is possible to vary the proportion in which inputs are combined.

(MP = change in TP; AP = TP ÷ L.)

The three stages.

1. Stage I: increasing returns (L = 1 to 4). MP rises, then starts to fall, but stays above AP, so AP keeps rising. The stage ends where AP is at its maximum and equals MP (L = 4, AP = MP = 13). Cause: the fixed input is under-used; adding labour allows division of labour and fuller use of machinery.
2. Stage II: diminishing returns (L = 4 to 7). MP and AP both fall, but MP is positive, so TP still rises, at a diminishing rate. The stage ends where TP is at its maximum and MP is zero (L = 7). Cause: the fixed input becomes scarce relative to labour.
3. Stage III: negative returns (L beyond 7). MP is negative and TP falls. Too many workers crowd the fixed plant and get in each other’s way.

The diagram, in words. Input on the X-axis, output on the Y-axis. TP rises steeply, bends, flattens to a peak and turns down. MP rises to a peak, falls, cuts AP at AP’s highest point, and crosses the X-axis where TP peaks. AP is an inverted U lying below MP while rising and above it while falling.

Stage of operation. A rational producer operates only in Stage II. In Stage I he can raise the average product by adding labour, so he will not stop there; in Stage III he can raise output by employing less labour. The exact point within Stage II depends on input and output prices.

3. Laws of returns to scale (long run)

Returns to scale show how output responds when all inputs are increased in the same proportion, that is, when the scale of the firm changes. There are three phases.

1. Increasing returns to scale: output increases in a greater proportion than inputs. Causes: internal economies — specialisation and division of labour, use of large and more efficient machines (technical indivisibilities), managerial specialisation, bulk buying and selling, cheaper finance.
2. Constant returns to scale: output increases in the same proportion as inputs. The economies of scale are exhausted and are balanced by emerging diseconomies.
3. Diminishing (decreasing) returns to scale: output increases in a smaller proportion than inputs. Causes: diseconomies of scale — difficulties of coordination and control in a very large firm, delays in decision, labour problems, and the limited supply of entrepreneurial ability.

In terms of isoquants. If the isoquants for 100, 200, 300 units lie closer and closer together along a ray from the origin, returns are increasing; equal distance, constant; wider and wider, diminishing.

4. Variable proportions and returns to scale compared

How this is examined

January 2026: “Explain law of returns to scale” (10 marks). February 2025: “What is production function? Explain the law of returns to scale of production” or “Explain the law of variable proportions with the help of a diagram” (10 marks). August 2024: “Explain the law of returns to scale in production” (10 marks). Define the production function in two lines, state the law, give a numerical table, explain each phase with its causes, and describe the diagram.