Production Function

Production means creating goods and services to satisfy human wants — or, more precisely, creating utility by transforming inputs into output. The things used in production are the inputs (factors of production): land, labour, capital and enterprise. What comes out is the output.

The production function shows the technical relationship between inputs and output — the maximum output that can be produced from a given combination of inputs, with a given technology. In symbols, output Q depends on the factors used:

Q = f (L, K)

meaning the quantity of output Q is a function of labour (L) and capital (K) used. The production function only tells us the physical relation between inputs and output, not costs or prices.

Production is studied over two time periods:

• Short run — a period in which at least one factor is fixed (usually capital, like the factory and machines) and only the other factors (like labour) can be varied. Output is changed only by changing the variable factor.
• Long run — a period long enough for all factors to be varied; there are no fixed factors. The firm can change even its plant size.

The short run gives us the Law of Variable Proportions, and the long run gives us the laws of returns to scale. In Class 11 we focus mainly on the short-run behaviour of output.

To study how output changes as we use more of a variable factor (say labour), we use three measures of product:

• Total Product (TP) — the total quantity of output produced by all the units of the variable factor.
• Average Product (AP) — output per unit of the variable factor: AP = TP ÷ L (where L is the units of the variable factor).
• Marginal Product (MP) — the addition to total product when one more unit of the variable factor is used: MP = change in TP (TPₙ − TPₙ₋₁).

The table shows these for units of labour applied to a fixed amount of land:

Important relationships: TP rises as long as MP is positive, is maximum when MP = 0 (here at L = 6), and falls if MP becomes negative. MP first rises then falls. The MP curve cuts the AP curve at the AP's maximum point; when MP > AP, AP rises, and when MP < AP, AP falls.

The behaviour of marginal product in the short run is summed up by the Law of Variable Proportions (also called the law of diminishing returns). It states: as we use more and more units of a variable factor with a fixed factor, the marginal product first rises, then falls, and finally becomes negative. This happens because the proportion between the fixed and variable factors keeps changing.

The law works in three stages (look back at the table):

• Stage I — Increasing returns: TP rises at an increasing rate and MP rises. This happens because the fixed factor is under-used, and adding variable units improves efficiency (better division of labour).
• Stage II — Diminishing returns: TP rises but at a decreasing rate; MP falls but is still positive (until TP is maximum). This is the most important stage — a rational producer always operates here, because output is still rising. The fixed factor is now being used closer to its best capacity.
• Stage III — Negative returns: TP actually falls and MP becomes negative. There are now too many variable units crowding the fixed factor, so output drops. No sensible producer operates here.

Why do diminishing returns set in? Because the fixed factor (land, machinery) cannot be increased in the short run; as more variable units are added, each has less of the fixed factor to work with, so the extra output (MP) eventually falls. The Law of Variable Proportions is one of the most fundamental laws of production and explains why a firm cannot keep raising output indefinitely just by adding more workers to a fixed plant.