Economics: Class 12 (Part 2) — Chapter 3: Production and Costs

3.1 Production Function

The production function of a firm is a relationship between inputs used and output produced by the firm. For various quantities of inputs used, it gives the maximum quantity of output that can be produced.

Factors of Production

The inputs that a firm uses in the production process are called factors of production. For simplicity, we consider a firm that produces output using only two factors of production: Labour (L) and Capital (K).

We may write the production function as:

$$q = f(L, K)$$

where $L$ is labour, $K$ is capital, and $q$ is the maximum output that can be produced.

Isoquant

An isoquant is the set of all possible combinations of the two inputs (labour and capital) that yield the same maximum possible level of output. Each isoquant represents a particular level of output and is labelled with that amount of output. Isoquants are typically negatively sloped.

3.2 The Short Run and the Long Run

Before analyzing production further, we distinguish between two time horizons:

Short Run: A time period in which at least one factor of production (e.g., capital) cannot be varied and remains fixed. The factor that remains fixed is called the fixed factor, while the factor that can be varied is called the variable factor.
Long Run: A time period in which all factors of production can be varied. There are no fixed factors in the long run.

3.3 Total, Average, and Marginal Product

Total Product (TP)

Total Product (TP) is the relationship between the variable input and output, keeping all other inputs constant. It is also referred to as the total product of labour ($TP_L$) when capital is held constant.

Average Product (AP)

Average Product (AP) is defined as the output per unit of variable input. It is calculated as:

$$AP_L = \frac{TP_L}{L}$$

Marginal Product (MP)

Marginal Product (MP) of an input is defined as the change in output per unit of change in the input when all other inputs are held constant. When capital is held constant, the marginal product of labour is:

$$MP_L = \frac{\Delta TP_L}{\Delta L}$$

Alternatively, for discrete changes, it can be expressed as:

$$MP_L = TP_L - TP_{L-1}$$

Product Schedule Table

The following table shows the relationship between Labour, Total Product, Marginal Product, and Average Product (with capital fixed at 4 units):

Labour ($L$)	Total Product ($TP$)	Marginal Product ($MP_L$)	Average Product ($AP_L$)
0	0	-	-
1	10	10	10
2	24	14	12
3	40	16	13.33
4	50	10	12.5
5	56	6	11.2
6	57	1	9.5

3.4 The Law of Diminishing Marginal Product and Variable Proportions

The Law of Variable Proportions (or the Law of Diminishing Marginal Product) states that the marginal product of a factor input initially rises with its employment level, but after reaching a certain level of employment, it starts falling.

This occurs because of changing factor proportions. Initially, adding more variable inputs makes the factor proportions more suitable for production. However, beyond a certain point, the production process becomes too crowded with the variable input, causing the marginal product to decline.

3.5 Shapes of Product Curves

Total Product (TP) Curve: Positively sloped, initially increasing at an increasing rate, and then increasing at a decreasing rate.
Marginal Product (MP) Curve: Inverse 'U'-shaped. It rises initially, reaches a maximum, and then starts falling.
Average Product (AP) Curve: Inverse 'U'-shaped. It rises as long as $MP > AP$, reaches its maximum where $MP = AP$, and falls when $MP < AP$. The $MP$ curve always cuts the $AP$ curve from above at its maximum point.

3.6 Returns to Scale

In the long run, when all factors are increased by the same proportion, we examine returns to scale:

Constant Returns to Scale (CRS): A proportional increase in all inputs results in an increase in output by the same proportion. $$f(tx_1, tx_2) = t \cdot f(x_1, x_2)$$
Increasing Returns to Scale (IRS): A proportional increase in all inputs results in an increase in output by a larger proportion. $$f(tx_1, tx_2) > t \cdot f(x_1, x_2)$$
Decreasing Returns to Scale (DRS): A proportional increase in all inputs results in an increase in output by a smaller proportion. $$f(tx_1, tx_2) < t \cdot f(x_1, x_2)$$

Cobb-Douglas Production Function

A popular production function is the Cobb-Douglas production function:

$$q = x_1^{\alpha} x_2^{\beta}$$

where $\alpha$ and $\beta$ are constants. The returns to scale depend on the sum of the exponents:

If $\alpha + \beta = 1$, the function exhibits CRS.
If $\alpha + \beta > 1$, the function exhibits IRS.
If $\alpha + \beta < 1$, the function exhibits DRS.

3.7 Costs

To produce output, a firm incurs costs to employ inputs. The cost function describes the least cost of producing each level of output given input prices and technology.

Short Run Costs

In the short run, costs are divided into fixed and variable components:

Total Fixed Cost (TFC): Costs incurred on fixed inputs. These do not vary with the level of output.
Total Variable Cost (TVC): Costs incurred on variable inputs. These increase as output increases.
Total Cost (TC): The sum of fixed and variable costs. $$TC = TFC + TVC$$
Average Fixed Cost (AFC): Fixed cost per unit of output. It is a rectangular hyperbola. $$AFC = \frac{TFC}{q}$$
Average Variable Cost (AVC): Variable cost per unit of output. It is U-shaped. $$AVC = \frac{TVC}{q}$$
Short Run Average Cost (SAC): Total cost per unit of output. It is U-shaped. $$SAC = \frac{TC}{q} = AFC + AVC$$
Short Run Marginal Cost (SMC): The change in total cost per unit change in output. It is U-shaped and cuts both $AVC$ and $SAC$ at their minimum points. $$SMC = \frac{\Delta TC}{\Delta q}$$

Short Run Cost Schedule

Output ($q$)	TFC (Rs)	TVC (Rs)	TC (Rs)	AFC (Rs)	AVC (Rs)	SAC (Rs)	SMC (Rs)
0	20	0	20	-	-	-	-
1	20	10	30	20	10	30	10
2	20	18	38	10	9	19	8
3	20	24	44	6.67	8	14.67	6
4	20	29	49	5	7.25	12.25	5
5	20	33	53	4	6.6	10.6	4
6	20	39	59	3.33	6.5	9.83	6
7	20	47	67	2.86	6.7	9.57	8
8	20	60	80	2.5	7.5	10	13
9	20	75	95	2.22	8.33	10.55	15
10	20	95	115	2	9.5	11.5	20

Long Run Costs

In the long run, all inputs are variable, meaning there are no fixed costs ($TFC = 0$, so $TC = TVC$).

Long Run Average Cost (LRAC): Cost per unit of output in the long run. $$LRAC = \frac{TC}{q}$$
Long Run Marginal Cost (LRMC): The change in total cost per unit change in output in the long run. $$LRMC = \frac{\Delta TC}{\Delta q}$$
Shapes of Long Run Curves: Both $LRAC$ and $LRMC$ are U-shaped. The declining phase of $LRAC$ corresponds to Increasing Returns to Scale (IRS), the minimum point corresponds to Constant Returns to Scale (CRS), and the rising phase corresponds to Decreasing Returns to Scale (DRS).