Production and Costs

A production function is the relationship between the inputs used and the output produced by a firm. It shows the maximum quantity of output that can be produced from different combinations of inputs, given the technology. In short, it tells us how much a firm can produce with a given set of factors of production such as labour and capital.

The total product (TP) of an input is the relationship between a variable input and output, when all other inputs are held constant. It shows the total amount of output produced at different levels of that input.

Average product (AP) is defined as the output per unit of variable input. It is calculated as $AP_L = \frac{TP_L}{L}$ for labour.

Marginal product (MP) of an input is the change in output per unit of change in the input, when all other inputs are held constant. For labour, $MP_L = \frac{\Delta TP_L}{\Delta L}$.

The total product is the sum of marginal products of all units of the variable input up to that level of employment. So, if we keep adding units of labour, each unit contributes its marginal product to total output.

- When MP rises, TP rises at an increasing rate.
- When MP is positive but falling, TP still rises, but at a decreasing rate.
- When MP becomes zero, TP reaches its maximum.
- If MP becomes negative, TP falls.

Thus, the shape of TP is determined by MP: TP increases as long as MP is positive, and the rate of increase depends on whether MP is rising or falling.

In the short run, at least one factor of production, such as labour or capital, cannot be varied and remains fixed. The other factor can be changed, so it is called the variable factor.

In the long run, all factors of production can be varied. There is no fixed factor in the long run. The distinction depends not on days or months, but on whether all inputs can be changed.

The law of diminishing marginal product says that the marginal product of a factor input initially rises as employment increases, but after reaching a certain level, it starts falling.

The law of variable proportions says that when one factor is held constant and the other is increased, the marginal product of the variable input initially rises with employment and then, after a certain point, falls.

A production function satisfies constant returns to scale (CRS) when a proportional increase in all inputs leads to an equal proportional increase in output. If all inputs are increased by a factor $t>1$, output also increases by the same factor: $f(tx_1, tx_2)=t\,f(x_1,x_2)$.

A production function satisfies increasing returns to scale (IRS) when a proportional increase in all inputs results in an increase in output by a larger proportion. Mathematically, if inputs are scaled up by $t>1$, then $f(tx_1,tx_2)>t\,f(x_1,x_2)$.

A production function satisfies decreasing returns to scale (DRS) when a proportional increase in all inputs leads to an increase in output by a smaller proportion. That is, if inputs are increased by a factor $t>1$, then $f(tx_1,tx_2)<t\,f(x_1,x_2)$.

The cost function describes the least cost of producing each level of output, given the prices of factors of production and the technology. Since a given output can usually be produced in more than one way, the firm chooses the least expensive input combination for each output level.

- Total fixed cost (TFC) is the cost of employing fixed inputs in the short run. It does not change with output.
- Total variable cost (TVC) is the cost of employing variable inputs. It changes with output.
- Total cost (TC) is the sum of the two:

$$TC = TVC + TFC$$

So, for any level of output, total cost = fixed cost + variable cost.

- Average fixed cost (AFC) is fixed cost per unit of output: $AFC = \frac{TFC}{q}$.
- Average variable cost (AVC) is variable cost per unit of output: $AVC = \frac{TVC}{q}$.
- Average cost (AC or SAC in short run) is total cost per unit of output: $SAC = \frac{TC}{q}$.

They are related by:

$$SAC = AVC + AFC$$

No, there cannot be any fixed cost in the long run. In the long run, all inputs are variable, so there are no fixed factors. Therefore, the cost of employing fixed inputs does not arise; total cost and total variable cost become the same in the long run.