The Law of Variable Proportions

The Law of Variable Proportions (also called the Law of Non-Proportional Returns, and known in its older form as the Law of Diminishing Returns) describes what happens to output in the short run, when one factor (say capital) is held fixed and another (say labour) is increased unit by unit, i.e., the proportion between the fixed and variable factor keeps changing.

Three related output measures

• Total Product (TP) — the total output produced by a given quantity of the variable factor, all other factors held constant.
• Average Product (AP) — output per unit of the variable factor: $AP = \dfrac{TP}{L}$, where $L$ is the number of units of the variable factor (labour).
• Marginal Product (MP) — the addition to total product from employing one more unit of the variable factor: $MP_n = TP_n - TP_{n-1}$.

An illustrative schedule

The three stages

• Stage I — Increasing Returns: TP rises at an increasing rate, MP rises and stays above AP, pulling AP up (Labour units 1–3 above). A rational firm never stops production here because the fixed factor is still under-utilised relative to the variable factor.
• Stage II — Diminishing Returns: TP continues to rise but at a diminishing rate; MP falls but stays positive, and AP also falls once MP < AP (Labour units 4–5 above). MP equals AP at the point AP is maximum. A rational producer always operates in Stage II — it is the only stage where both TP is still rising and MP is positive but falling, giving the firm a genuine trade-off to optimise.
• Stage III — Negative Returns: the variable factor becomes so large relative to the fixed factor that TP itself starts falling and MP turns negative (Labour unit 6 above). No rational producer operates here, since adding more of the variable factor actually reduces output.

Why diminishing returns set in …