Production Function

1) What is a production function?

A production function describes the technological relationship between inputs used in production and the maximum output that can be produced from those inputs in a given period, given the state of technology.

General form:

Q = f(X_1, X_2, \ldots, X_n)

For two inputs, typically Labour (L) and Capital (K):

Q = f(L, K)

Key phrase: “maximum output” implies firms are technically efficient—production function represents the production frontier.

2) Production function vs production process vs productivity

Production function: mathematical/relationship mapping inputs → maximum feasible output.
Production process: actual method/sequence of operations; may be inefficient.
Productivity: output per unit input (e.g., labour productivity $Q/L$ , total factor productivity).

CUET trap: A production function is not “cost function.” Cost depends on input prices; production depends on technology.

3) Time horizon: short run vs long run

3.1 Short run

At least one input is fixed (usually capital K fixed), others variable (labour L variable).

Q = f(L, \bar{K})

3.2 Long run

All inputs are variable.

Q = f(L, K)

This matters because:

In the short run, we study Law of Variable Proportions (next module).
In the long run, we study Returns to Scale.

4) Key concepts: total, average, and marginal product

Assume in the short run K fixed, output depends on L.

4.1 Total Product (TP)

TP(L) = Q

Total output produced with L units of labour.

4.2 Average Product of Labour (AP)

AP_L = \frac{Q}{L}

Output per unit of labour.

4.3 Marginal Product of Labour (MP)

MP_L = \frac{\Delta Q}{\Delta L}

(or $dQ/dL$ in continuous case)

Important relationship:

If $MP > AP$ , then AP is rising.
If $MP < AP$ , then AP is falling.
When $MP = AP$ , AP is at its maximum.

This relationship is frequently tested.