Chapter 3 Production And Costs

Production Function

The production function is a fundamental concept in economics that describes the relationship between the inputs used by a firm and the maximum possible output it can produce with a given technology.

It represents the technological relationship that specifies, for various quantities of physical inputs, the maximum possible amount of output that can be produced.

For example, a farmer might use land (input 1) and labour (input 2) to produce wheat (output). The production function tells us the largest quantity of wheat the farmer can grow with different combinations of land and labour.

Production functions are assumed to represent technically efficient use of inputs, meaning it's not possible to get more output from the same amount of inputs.

Technology plays a crucial role; an improvement in technology shifts the production function, allowing more output from the same inputs.

A common way to represent a production function with two inputs, Labour (L) and Capital (K), is:

$q = f(L, K)$

where $q$ is the maximum output that can be produced using quantities L and K of the respective inputs.

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Isoquant

Similar to an indifference curve for consumers, an isoquant is a curve that shows all the possible combinations of two inputs (like Labour and Capital) that yield the same level of maximum output.

Each isoquant corresponds to a specific quantity of output. Higher isoquants represent higher levels of output.

If inputs have positive marginal products, isoquants are generally negatively sloped. This is because if you use more of one input, you must use less of the other input to keep the output level constant.

Isoquants also exhibit the property of convexity, implying a diminishing marginal rate of technical substitution between the inputs.

The diagram shows multiple isoquants, each representing a different output level ($q_1 < q_2 < q_3$). Any point on a single isoquant gives the same output level.

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Returns To Scale

Returns to scale refer to how the maximum output changes when all inputs are increased proportionally. This concept is analysed in the long run, where all factors are variable.

There are three types of returns to scale:

1. Increasing Returns to Scale (IRS): When a proportional increase in all inputs results in a more than proportional increase in output. For example, if you double all inputs and output more than doubles.
2. Constant Returns to Scale (CRS): When a proportional increase in all inputs results in an equal proportional increase in output. If you double all inputs and output exactly doubles.
3. Decreasing Returns to Scale (DRS): When a proportional increase in all inputs results in a less than proportional increase in output. If you double all inputs and output less than doubles.

Mathematically, for a production function $q = f(x_1, x_2)$, if we multiply inputs $x_1$ and $x_2$ by a factor $t > 1$:

• IRS: $f(tx_1, tx_2) > t \cdot f(x_1, x_2)$
• CRS: $f(tx_1, tx_2) = t \cdot f(x_1, x_2)$
• DRS: $f(tx_1, tx_2) < t \cdot f(x_1, x_2)$

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Cobb-Douglas Production Function

A specific and widely used form of the production function is the Cobb-Douglas production function, often written as:

$q = x_1^a x_2^b$

where $q$ is output, $x_1$ and $x_2$ are the quantities of two inputs, and $a$ and $b$ are constants that represent the output elasticity of each input.

The sum of the exponents ($a+b$) in a Cobb-Douglas function determines the type of returns to scale:

• If $a + b = 1$, the function exhibits Constant Returns to Scale (CRS). (Multiplying inputs by $t$ results in output multiplied by $t$: $(tx_1)^a (tx_2)^b = t^{a+b} x_1^a x_2^b = t^1 x_1^a x_2^b = tq$)
• If $a + b > 1$, the function exhibits Increasing Returns to Scale (IRS). (Output increases by more than $t$)
• If $a + b < 1$, the function exhibits Decreasing Returns to Scale (DRS). (Output increases by less than $t$)

The Short Run And The Long Run

The concepts of short run and long run in production analysis are defined not by calendar time but by the flexibility of inputs.

The Short Run is a period of time during which at least one factor of production is fixed and cannot be changed. To increase output in the short run, the firm can only increase the use of its variable factors (e.g., labour), while fixed factors (e.g., factory size or capital) remain constant.

The Long Run is a period of time during which all factors of production are variable. A firm can change the scale of its operations, expand or contract its capital stock, and adjust all other inputs freely. In the long run, there are no fixed factors.

The specific duration of the short run and long run varies depending on the industry and the time it takes to alter different inputs. What is a 'long run' for one firm might be a 'short run' for another with different technology or scale.

Total Product, Average Product And Marginal Product

When analyzing production in the short run, where some inputs are fixed, we examine how output changes as we vary a single input (typically labour). This leads to the concepts of Total Product, Average Product, and Marginal Product.

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Total Product

Total Product (TP) refers to the total quantity of output produced by a firm using a given amount of a variable input, while all other inputs are held constant.

If we fix Capital (K) at a certain level and vary Labour (L), the relationship between the quantity of Labour employed and the resulting output is the Total Product of Labour.

As the quantity of the variable input increases, the Total Product generally increases, at least initially, but the rate of increase may vary.

The figure shows a typical Total Product curve, which rises as labour input increases, but may eventually flatten or even decline.

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Average Product

Average Product (AP) measures the output produced per unit of the variable input. It tells us, on average, how much output each unit of the variable factor contributes.

It is calculated by dividing the Total Product (TP) by the quantity of the variable input used (e.g., Labour, L):

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

Average Product is also known as average physical product or average return to the variable input.

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Marginal Product

Marginal Product (MP) is the additional output produced when one more unit of the variable input is employed, holding all other inputs constant.

It measures the change in Total Product resulting from a one-unit change in the variable input.

For labour, the Marginal Product of Labour ($MP_L$) is calculated as:

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

where $\Delta$ denotes the change in the respective variable.

When output changes by discrete units, the Marginal Product of the $n^{th}$ unit of labour can be calculated as:

$MP_{L_n} = TP_L(n) - TP_L(n-1)$

The Total Product at any given level of the variable input is the sum of the Marginal Products of all units of that input up to that level (assuming $TP=0$ at input=0).

Answer:

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

This fundamental principle describes the pattern of Marginal Product as the variable input is increased while other inputs are fixed.

The Law of Variable Proportions states that as the amount of a variable input is increased, while holding other inputs fixed, the Marginal Product of the variable input will eventually decline.

It is sometimes called the Law of Diminishing Marginal Product because the declining phase is the most significant aspect. The law typically describes three phases:

1. Phase 1: Increasing Marginal Product: Initially, as more units of the variable input are added, the Marginal Product may increase. This often happens because the fixed input is underutilized, and adding more of the variable input allows for specialization and better use of the fixed resource.
2. Phase 2: Decreasing (but still positive) Marginal Product: Beyond a certain point, adding more units of the variable input leads to smaller and smaller increases in total output. The Marginal Product is still positive, but it is falling. This occurs because the fixed input becomes increasingly crowded or limiting relative to the variable input.
3. Phase 3: Negative Marginal Product: If the variable input continues to be increased, the Marginal Product may eventually become negative. This means that adding more units of the variable input actually causes total output to fall (e.g., too many workers on a fixed machine causing interference).

The law is explained by the changing factor proportions. As the variable input increases relative to the fixed input, the efficiency of the variable input first improves (up to an optimal ratio) and then deteriorates.

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Shapes Of Total Product, Marginal Product And Average Product Curves

Based on the Law of Variable Proportions, the shapes of the TP, AP, and MP curves exhibit characteristic patterns:

• Total Product (TP) Curve: Starts from zero, initially increases at an increasing rate (convex shape), then increases at a decreasing rate (concave shape), reaches a maximum, and may eventually decline.
• Marginal Product (MP) Curve: Is typically inverse 'U'-shaped. It rises initially, reaches a maximum (corresponding to the point where TP changes from increasing at an increasing rate to increasing at a decreasing rate), and then falls. It becomes zero when TP is at its maximum and negative when TP is declining.
• Average Product (AP) Curve: Is also typically inverse 'U'-shaped. It rises initially, reaches a maximum, and then falls.

The relationship between AP and MP is crucial:

• When MP is greater than AP ($MP > AP$), AP is rising. Adding a unit that produces more than the current average pulls the average up.
• When MP is less than AP ($MP < AP$), AP is falling. Adding a unit that produces less than the current average pulls the average down.
• When MP is equal to AP ($MP = AP$), AP is at its maximum point. The MP curve intersects the AP curve from above at the peak of the AP curve.

Average Product lags behind Marginal Product. MP rises and falls faster than AP.

Here is a summary of the data from the text's example (Table 3.2) showing these relationships:

Labour (L)	TP	MPL	APL
0	0	-	-
1	10	10	10.00
2	24	14	12.00
3	40	16	13.33
4	50	10	12.50
5	56	6	11.20
6	57	1	9.50

Notice MP increases up to L=3, then falls. AP increases up to L=3, then falls. MP > AP for L=1, 2; MP < AP for L=4, 5, 6. At L=3, MP is 16, AP is 13.33, AP is still rising. The point where MP=AP (approximately where AP is maximum) is somewhere between L=3 and L=4 in a continuous case.

Costs

To produce output, firms incur costs by employing inputs. A specific level of output can often be produced using different combinations of inputs.

A rational firm aims to produce its desired output level at the lowest possible cost. The cost function describes this minimum cost for 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, leading to two categories of costs:

1. Total Fixed Cost (TFC): Costs associated with fixed inputs. These costs do not change with the level of output. Even if the firm produces zero output, it still incurs fixed costs (e.g., rent for factory building). TFC remains constant in the short run.
2. Total Variable Cost (TVC): Costs associated with variable inputs. These costs change directly with the level of output. As output increases, the firm needs more variable inputs (like labour, raw materials), so TVC increases. TVC is zero when output is zero.

Total Cost (TC) is the sum of Total Fixed Cost and Total Variable Cost at any given output level:

$TC = TFC + TVC$

As output increases, TC increases because TVC increases, while TFC remains constant.

In addition to total costs, we also analyse average and marginal costs:

• Average Fixed Cost (AFC): Total fixed cost per unit of output. $AFC = \frac{TFC}{q}$. As output ($q$) increases, AFC continuously decreases because a fixed cost is spread over a larger number of units.
• Average Variable Cost (AVC): Total variable cost per unit of output. $AVC = \frac{TVC}{q}$.
• Short Run Average Cost (SAC): Total cost per unit of output. $SAC = \frac{TC}{q}$. It is also the sum of AFC and AVC: $SAC = AFC + AVC$.
• Short Run Marginal Cost (SMC): The change in total cost (or total variable cost, as TFC is constant) resulting from producing one additional unit of output. $SMC = \frac{\Delta TC}{\Delta q} = \frac{\Delta TVC}{\Delta q}$. SMC represents the cost of the last unit produced.

The Total Variable Cost at a specific output level can be found by summing the Marginal Costs of all units produced up to that level (assuming TVC=0 at q=0).

Here is a table showing various short-run costs based on the example in the text (Table 3.3). Note the costs are given in $\textsf{₹}$):

Output (q)	TFC ($\textsf{₹}$)	TVC ($\textsf{₹}$)	TC ($\textsf{₹}$)	AFC ($\textsf{₹}$)	AVC ($\textsf{₹}$)	SAC ($\textsf{₹}$)	SMC ($\textsf{₹}$)
0	20	0	20	-	-	-	-
1	20	10	30	20.00	10.00	30.00	10
2	20	18	38	10.00	9.00	19.00	8
3	20	24	44	6.67	8.00	14.67	6
4	20	29	49	5.00	7.25	12.25	5
5	20	33	53	4.00	6.60	10.60	4
6	20	39	59	3.33	6.50	9.83	6
7	20	47	67	2.86	6.71	9.57	8
8	20	60	80	2.50	7.50	10.00	13
9	20	75	95	2.22	8.33	10.55	15
10	20	95	115	2.00	9.50	11.50	20

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Shapes Of The Short Run Cost Curves

The shapes of short run cost curves reflect the underlying production relationships (specifically the Law of Variable Proportions).

• Total Fixed Cost (TFC) Curve: A horizontal straight line because TFC is constant regardless of output level.
• Total Variable Cost (TVC) Curve: Starts at the origin (0,0) and increases as output increases. Its shape is related to the TP curve; initially, it increases at a decreasing rate (due to increasing MP), then increases at an increasing rate (due to diminishing MP).
• Total Cost (TC) Curve: Starts at the level of TFC on the y-axis (since $TC = TFC$ when $q=0$). Its shape is parallel to the TVC curve, shifted upwards by the constant amount of TFC.

Now, let's look at the average and marginal cost curves:

• Average Fixed Cost (AFC) Curve: Continuously slopes downwards as output increases, approaching the x-axis but never reaching it. It is a rectangular hyperbola (see separate section). As output increases, the fixed cost is spread over more units, decreasing the cost per unit.
• Short Run Marginal Cost (SMC) Curve: Is typically 'U'-shaped. Initially, it falls as output increases (due to increasing MP, requiring fewer variable inputs for each extra unit), reaches a minimum, and then rises sharply (due to diminishing MP, requiring more variable inputs for each extra unit).
• Average Variable Cost (AVC) Curve: Is also typically 'U'-shaped. It falls initially as output increases, reaches a minimum, and then rises.
• Short Run Average Cost (SAC) Curve: Is also typically 'U'-shaped. It falls initially, reaches a minimum, and then rises. The SAC curve is the vertical sum of the AFC and AVC curves. Since AFC is always decreasing and AVC is U-shaped, SAC first decreases (fall in AFC dominates rise in AVC), reaches a minimum, and then increases (rise in AVC dominates fall in AFC).

The relationship between the Marginal Cost and Average Cost curves is crucial:

• The SMC curve intersects both the AVC curve and the SAC curve from below at their respective minimum points.
• When SMC is below AVC, AVC is falling. When SMC is above AVC, AVC is rising.
• When SMC is below SAC, SAC is falling. When SMC is above SAC, SAC is rising.
• The minimum point of the SAC curve occurs at a higher output level than the minimum point of the AVC curve. This is because SAC includes AFC, which is always falling.

The area under the SMC curve up to a certain output level represents the Total Variable Cost (TVC) at that level.

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Long Run Costs

In the long run, all factors of production are variable. Consequently, there are no fixed costs in the long run. Total Cost (TC) and Total Variable Cost (TVC) are identical in the long run.

We focus on two main cost concepts in the long run:

• Long Run Average Cost (LRAC): The total cost per unit of output when all inputs are variable. $LRAC = \frac{TC}{q}$. The LRAC for any output level represents the minimum cost of producing that output when the firm can choose the optimal scale of operation (i.e., the optimal combination of all variable inputs).
• Long Run Marginal Cost (LRMC): The change in total cost resulting from producing one additional unit of output when all inputs are adjusted optimally in the long run. $LRMC = \frac{\Delta TC}{\Delta q}$.

Similar to the short run, the sum of LRMCs up to a certain output level gives the total cost at that level in the long run.

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Shapes Of The Long Run Cost Curves

The shapes of the long run average and marginal cost curves are primarily determined by the returns to scale experienced by the firm.

• When the firm experiences Increasing Returns to Scale (IRS), LRAC falls as output increases. Doubling output requires less than doubling inputs, so average cost decreases.
• When the firm experiences Constant Returns to Scale (CRS), LRAC remains constant as output increases. Doubling output requires exactly doubling inputs, keeping average cost the same. This typically occurs at the minimum point of the LRAC curve.
• When the firm experiences Decreasing Returns to Scale (DRS), LRAC rises as output increases. Doubling output requires more than doubling inputs, leading to higher average costs.

Assuming a typical firm experiences IRS at low output levels, followed by CRS, and finally DRS at high output levels, the Long Run Average Cost (LRAC) curve is typically 'U'-shaped.

The Long Run Marginal Cost (LRMC) curve is also typically 'U'-shaped.

The relationship between LRAC and LRMC is similar to their short run counterparts:

• When LRAC is falling, LRMC is below LRAC.
• When LRAC is rising, LRMC is above LRAC.
• The LRMC curve intersects the LRAC curve from below at the minimum point of the LRAC curve. This minimum point represents the output level where the firm achieves the lowest possible average cost in the long run.

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Rectangular Hyperbola

The Average Fixed Cost (AFC) curve in the short run is a graphical representation of the relationship $AFC = \frac{TFC}{q}$, where TFC is a constant value.

The AFC curve is a rectangular hyperbola. This means that for any point on the curve, the product of the corresponding output level (q) on the x-axis and the average fixed cost (AFC) on the y-axis is always equal to the constant Total Fixed Cost (TFC).

$q \times AFC = TFC$ (Constant)

Graphically, if you pick any point on the AFC curve and draw lines perpendicular to the x-axis and y-axis, the area of the rectangle formed by these lines and the axes will always be the same, equal to TFC.

As output increases towards infinity, AFC approaches zero. As output approaches zero, AFC approaches infinity.