Production function

The essentials

A production function shows the maximum output a firm can produce from given quantities of inputs, with a given technology: q = f(L, K). In the short run at least one factor (usually capital) is fixed, so output changes only by varying labour. In the long run all factors can be varied.

Key points

• Total product (TP): total output from a given amount of the variable input.
• Average product (AP) = TP ÷ L.
• Marginal product (MP) = ΔTP ÷ ΔL, the extra output from one more unit of labour.
• Law of variable proportions (diminishing marginal product): as more units of a variable factor are combined with a fixed factor, MP first rises and eventually falls. TP first rises at an increasing rate, then at a diminishing rate, and falls when MP becomes negative.
• When MP > AP, AP rises; when MP < AP, AP falls; MP = AP at the maximum of AP.
• TP is maximum when MP = 0.
• Returns to scale (long run): if all inputs are increased in the same proportion and output rises by a greater proportion, it is increasing returns to scale (IRS); by the same proportion, constant (CRS); by a smaller proportion, decreasing (DRS).
• An isoquant joins input combinations that produce the same output.

Exam tips

• Returns to a factor belong to the short run; returns to scale belong to the long run. Check which one a question is about.
• MP can be computed from a TP table by subtraction; AP by division.

Common mistakes

• Thinking AP falls as soon as MP starts falling. AP keeps rising while MP is still above AP.
• Calling a 150% rise in output from doubling inputs "constant returns". Doubling inputs is a 100% rise; output rising by more than 100% shows IRS.