Chapter 3 Production And Costs

Production Function

This chapter examines the behaviour of a producer, also known as a firm. A firm is an entity that transforms inputs into output through a process called production.

Inputs are resources acquired by the firm, such as labour, machines, land, raw materials, etc. Output is the goods or services produced by using these inputs. Output can be consumed by individuals or used by other firms in further production processes.

Examples of production include a tailor making shirts from cloth and thread, a farmer producing wheat using land, labor, and equipment, or a manufacturer producing cars from steel, rubber, and labor.

For simplicity, we often make assumptions such as production being instantaneous (inputs are immediately converted to output) and using the terms production and supply interchangeably.

Firms incur costs of production to acquire inputs. They earn revenue by selling the output. The difference between revenue and cost is the firm's profit. The primary objective of a firm is assumed to be profit maximisation.

The production function of a firm describes the technical relationship between the inputs used and the maximum output that can be produced from those inputs. For any given combination of inputs, it specifies the highest possible quantity of output obtainable.

If a firm uses two inputs, say Labour (L) and Capital (K), to produce output (q), the production function can be written as $q = f(L, K)$. This function represents the maximum output q for any given amounts of L and K.

A production function represents efficient production, meaning it's not possible to produce more output with the same inputs, given the current technology.

The production function is defined for a specific level of technology. Improvements in technology allow higher maximum output levels to be produced from the same input combinations, resulting in a new production function.

Table 3.1 provides a numerical example of a production function showing the maximum output for different combinations of Labour and Capital. It illustrates that increasing inputs generally leads to increased output, and in this example, both inputs are necessary for production.

Factor	Capital
Labour	0	1	2	3	4	5	6
0	0	0	0	0	0	0	0
1	0	1	3	7	10	12	13
2	0	3	10	18	24	29	33
3	0	7	18	30	40	46	50
4	0	10	24	40	50	56	57
5	0	12	29	46	56	58	59
6	0	13	33	50	57	59	60

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The Short Run And The Long Run

In the analysis of production, economists distinguish between two time periods: the short run and the long run. This distinction is based on the variability of factors of production.

In the short run, at least one factor of production is fixed and cannot be changed. This fixed factor (e.g., factory size, amount of land) restricts the firm's ability to adjust output freely. To change the output level in the short run, the firm can only vary the amount of other factors, called variable factors (e.g., labour, raw materials).

In the long run, all factors of production are variable. The firm has enough time to adjust the quantity of any input, including those that were fixed in the short run (e.g., building a new factory, acquiring more land). In the long run, there are no fixed factors.

The terms short run and long run are relative to the specific production process and do not correspond to fixed calendar periods like days, months, or years. The key is the variability of inputs.

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Isoquant

An isoquant is a concept similar to an indifference curve, used to represent the production function graphically when there are two inputs (Labour and Capital).

An isoquant is a curve that shows all the different combinations of two inputs (e.g., L and K) that can produce the same specific maximum level of output.

Each isoquant corresponds to a particular output level and is labeled with that quantity.

Referring to Table 3.1, the output level of 10 units can be produced with combinations like (4 units of L, 1 unit of K), (2 units of L, 2 units of K), or (1 unit of L, 4 units of K). All these combinations lie on the same isoquant for output q=10.

Isoquants are typically negatively sloped. This is because if marginal products are positive, producing the same output with more of one input requires using less of the other input.

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Total Product, Average Product And Marginal Product

These concepts are used to describe the output achieved when varying only one input while keeping others constant (which happens in the short run).

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Total Product

The Total Product (TP) of a variable input is the relationship between the quantity of that input used and the total output produced, assuming all other inputs are held constant.

For example, if Capital is fixed at 4 units in Table 3.1, the column for K=4 shows the total output (TP) for different amounts of Labour (L). This is the TP schedule of Labour when K=4.

Table 3.2 provides a numerical example of TP for Labour.

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Average Product

The Average Product (AP) of a variable input is the output produced per unit of that input. It is calculated by dividing the Total Product (TP) by the quantity of the variable input (L).

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

The values in the last column of Table 3.2 are the AP of Labour, calculated by dividing TP by L.

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Marginal Product

The Marginal Product (MP) of a variable input is the change in total output resulting from using one additional unit of that input, holding all other inputs constant.

$MP_L = \frac{\Delta TP_L}{\Delta L}$ (where $\Delta$ means change)

For discrete units of input, MP can be calculated as the difference in TP when one more unit of the variable input is used: $MP_L = TP_L - TP_{L-1}$.

Table 3.2 shows the MP of Labour. MP is typically undefined at zero input level.

Total product ($TP_n$) at a certain level of input employment (n) is the sum of the marginal products of each unit of that input up to level n: $TP_n = \sum_{i=1}^n MP_i$.

Average product at any level is the average of the marginal products of all preceding units up to that level.

Labour (L)	TP	MPL	APL
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

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The Law Of Diminishing Marginal Product And The Law Of Variable Proportions

The Law of Variable Proportions (also known as the Law of Diminishing Marginal Product or Marginal Returns) describes the pattern of change in the marginal product of a variable input when other inputs are fixed.

This law states that as employment of one factor (variable input) is increased while holding other factors (fixed inputs) constant, the marginal product of the variable input will eventually decline.

Typically, the process involves three phases:

1. Initially, MP rises: As the variable input is increased from a low level, the factor proportions (ratio of variable to fixed input) become more favourable, allowing for better specialisation and utilisation of fixed inputs. Each additional unit of the variable input adds more to total output than the previous unit.
2. MP reaches a maximum.
3. Eventually, MP falls: After a certain point, as more variable input is added to the fixed factor, the fixed factor becomes relatively scarce. Adding more variable input leads to 'crowding' or less efficient use of the fixed input, causing the marginal product of additional units to decrease.
4. MP may become zero or negative: If the variable input continues to increase, MP can eventually fall to zero (total output maximum) or become negative (total output decreases).

This law operates because the factor proportions change when one input is varied and others are fixed. When both inputs can change, this law does not necessarily apply (see Returns to Scale).

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Shapes Of Total Product, Marginal Product And Average Product Curves

Based on the Law of Variable Proportions, the Total Product (TP), Marginal Product (MP), and Average Product (AP) curves have characteristic shapes when one input is varied while others are fixed.

• Total Product (TP) Curve:
  • Starts from the origin (0 output with 0 variable input).
  • Initially increases at an increasing rate (corresponding to rising MP).
  • Increases at a decreasing rate after the point where MP is maximum (corresponding to falling but positive MP).
  • Reaches a maximum level (where MP is zero).
  • May eventually fall if MP becomes negative.
  • The TP curve is positively sloped initially.
• Marginal Product (MP) Curve:
  • Undefined at zero input.
  • Initially rises, reaches a maximum, and then falls.
  • It is typically an inverse 'U'-shaped curve.
• Average Product (AP) Curve:
  • Undefined at zero input.
  • Initially rises (as long as MP > AP).
  • Reaches a maximum.
  • Then falls (when MP < AP).
  • It is also typically an inverse 'U'-shaped curve.

Relationship between AP and MP Curves:

• When AP is rising, MP is greater than AP.
• When AP is falling, MP is less than AP.
• The MP curve intersects the AP curve from above at the maximum point of the AP curve.

Figure 3.2 illustrates the typical shapes and relationship between the AP and MP curves.

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Returns To Scale

The Law of Variable Proportions applies in the short run when at least one factor is fixed. In the long run, all factors are variable. Returns to Scale describes the relationship between a proportional change in *all* inputs and the resulting proportional change in output, which is a long-run concept.

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Returns To Scale

Consider increasing the employment level of all inputs by the same proportion, say by a factor 't' (where $t > 1$). Returns to Scale are classified based on how output changes:

• Constant Returns to Scale (CRS): Output increases by exactly the same proportion 't' as all inputs. If $q = f(L, K)$, then $f(tL, tK) = t \cdot f(L, K)$.
• Increasing Returns to Scale (IRS): Output increases by a larger proportion than 't'. $f(tL, tK) > t \cdot f(L, K)$.
• Decreasing Returns to Scale (DRS): Output increases by a smaller proportion than 't'. $f(tL, tK) < t \cdot f(L, K)$.

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Cobb-Douglas Production Function

A common example of a production function is the Cobb-Douglas form: $q = L^a K^b$, where L and K are inputs, and 'a' and 'b' are positive constants.

For a Cobb-Douglas production function, the type of returns to scale is determined by the sum of the exponents (a+b):

• If $a + b = 1$, the production function exhibits Constant Returns to Scale (CRS).
• If $a + b > 1$, it exhibits Increasing Returns to Scale (IRS).
• If $a + b < 1$, it exhibits Decreasing Returns to Scale (DRS).

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Costs

Firms incur costs to acquire inputs for production. For any given level of output, a firm aims to produce it using the least costly combination of inputs.

The cost function describes the minimum cost of producing each level of output, given input prices and the production technology.

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Short Run Costs

In the short run, some inputs are fixed. The costs associated with these fixed inputs are called Total Fixed Cost (TFC). TFC does not change with the level of output produced.

Costs associated with variable inputs are called Total Variable Cost (TVC). TVC changes as the output level changes (since variable inputs must be adjusted).

Total Cost (TC) in the short run is the sum of Total Fixed Cost and Total Variable Cost: $TC = TFC + TVC$.

As output increases, variable input usage increases, leading to increases in both TVC and TC. TFC remains constant.

Table 3.3 provides a numerical example of short run costs.

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

Other short run cost concepts derived from Total Costs:

• Short Run Average Cost (SAC): Total cost per unit of output: $SAC = \frac{TC}{q}$. Also, $SAC = AFC + AVC$.
• Average Variable Cost (AVC): Total variable cost per unit of output: $AVC = \frac{TVC}{q}$.
• Average Fixed Cost (AFC): Total fixed cost per unit of output: $AFC = \frac{TFC}{q}$.
• Short Run Marginal Cost (SMC): Change in total cost resulting from producing one additional unit of output: $SMC = \frac{\Delta TC}{\Delta q}$. In the short run, $\Delta TC = \Delta TVC$, so $SMC = \frac{\Delta TVC}{\Delta q}$. The sum of SMCs up to a certain output level equals TVC at that level. TVC is the area under the SMC curve.

Shapes of Short Run Cost Curves:

• TFC Curve: A horizontal straight line because TFC is constant regardless of output.
• TVC Curve: Starts from the origin (0 cost at 0 output) and increases as output increases. Its shape reflects the Law of Variable Proportions (initially increasing at a decreasing rate due to rising MP, then increasing at an increasing rate due to falling MP).
• TC Curve: Starts at the TFC level for 0 output and increases with output, parallel to the TVC curve (it is TVC shifted upward by TFC).
• AFC Curve: A downward sloping curve that is a rectangular hyperbola. As output (q) increases, TFC is spread over more units, so AFC ($=TFC/q$) decreases. As q approaches zero, AFC becomes very large; as q approaches infinity, AFC approaches zero. The area under the AFC curve up to any output level equals TFC.
• SMC Curve: Typically 'U'-shaped. Initially falls (due to increasing marginal returns of the variable input), reaches a minimum, and then rises (due to diminishing marginal returns). TVC at any output is the area under the SMC curve up to that output.
• AVC Curve: Also typically 'U'-shaped. Initially falls (as long as SMC is below AVC), reaches a minimum, and then rises (when SMC rises above AVC). The SMC curve cuts the AVC curve from below at the minimum point of the AVC curve.
• SAC Curve: Also typically 'U'-shaped. It is the sum of AVC and AFC. Initially falls (both AVC and AFC fall). It rises after a point when the increase in AVC outweighs the decrease in AFC. The SAC curve lies above the AVC curve, with the vertical distance between them being AFC. The minimum point of the SAC curve is to the right of the minimum point of the AVC curve. The SMC curve cuts the SAC curve from below at the minimum point of the SAC curve.

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Long Run Costs

In the long run, all inputs are variable. Consequently, there are no fixed costs in the long run; Total Cost (TC) is equal to Total Variable Cost (TVC).

• Long Run Average Cost (LRAC): Total cost per unit of output in the long run: $LRAC = \frac{TC}{q}$.
• Long Run Marginal Cost (LRMC): Change in total cost from producing one extra unit of output in the long run. The sum of LRMCs up to a certain output level gives the total cost at that level.

Shapes of Long Run Cost Curves:

The shapes of LRAC and LRMC curves are related to Returns to Scale.

• LRAC Curve: Typically 'U'-shaped.
  • It falls when the firm experiences Increasing Returns to Scale (IRS). Increasing inputs by a proportion increases output by a larger proportion, so average cost falls.
  • It is constant when the firm experiences Constant Returns to Scale (CRS). Increasing inputs by a proportion increases output by the same proportion, so average cost remains constant.
  • It rises when the firm experiences Decreasing Returns to Scale (DRS). Increasing inputs by a proportion increases output by a smaller proportion, so average cost rises.
  • The minimum point of the LRAC curve corresponds to the output level where CRS is observed.
• LRMC Curve: Typically 'U'-shaped.
  • For the first unit, LRMC and LRAC are the same.
  • As output increases, LRMC initially falls, then rises.
  • When LRAC is falling, LRMC is less than LRAC.
  • When LRAC is rising, LRMC is greater than LRAC.
  • The LRMC curve cuts the LRAC curve from below at the minimum point of the LRAC curve.

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Summary

• The production function shows the maximum output from input combinations, given technology.

• The short run has fixed inputs, while the long run has all variable inputs.

• Total product (TP) relates output to a variable input, holding others constant.

• Marginal product (MP) is the change in output per unit change in variable input; Average product (AP) is output per unit of variable input.

• The Law of Variable Proportions states that MP of a variable input eventually falls as its employment increases (with fixed inputs).

• TP, MP, and AP curves have characteristic shapes (TP initially increasing at increasing rate, then decreasing rate; MP and AP are inverse 'U'-shaped; MP intersects AP at AP's maximum).

• Returns to Scale (CRS, IRS, DRS) describe the relationship between proportional changes in all inputs and output in the long run.

• The cost function gives the minimum cost to produce each output level.

• Short run costs include Total Fixed Cost (TFC), Total Variable Cost (TVC), and Total Cost (TC=TFC+TVC).

• Average costs are AFC, AVC, and SAC (SAC=AFC+AVC).

• Short run marginal cost (SMC) is the change in TC/TVC per unit output.

• AFC is downward sloping. SMC, AVC, and SAC are 'U'-shaped.

• SMC intersects AVC and SAC from below at their minimum points.

• Long run costs have no fixed costs; TC=TVC. Long run average cost (LRAC) and marginal cost (LRMC) are typically 'U'-shaped, with LRMC intersecting LRAC from below at LRAC's minimum, reflecting returns to scale.

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Key Concepts

Production function

Short run

Long run

Total product

Marginal product

Average product

Law of diminishing marginal product

Law of variable proportions

Returns to scale

Cost function

Marginal cost

Average cost

Exercises

Exercises are excluded as per user instructions.