production function...between one variable factor (keeping all other factors fixed) and the output....
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Production Function
To understand production and costs it is important to grasp the
concept of the production function and understand the basics in
mathematical terms. We break down the short run and long run
production functions based on variable and fixed factors. Let us get
started!
What is the Production Function?
The functional relationship between physical inputs (or factors of
production) and output is called production function. It assumed
inputs as the explanatory or independent variable and output as the
dependent variable. Mathematically, we may write this as follows:
Q = f (L,K)
Here, ‘Q’ represents the output, whereas ‘L’ and ‘K’ are the inputs,
representing labour and capital (such as machinery) respectively. Note
that there may be many other factors as well but we have assumed
two-factor inputs here.
Time Period and Production Functions
The production function is differently defined in the short run and in
the long run. This distinction is extremely relevant in
microeconomics. The distinction is based on the nature of factor
inputs.
Those inputs that vary directly with the output are called variable
factors. These are the factors that can be changed. Variable factors
exist in both, the short run and the long run. Examples of variable
factors include daily-wage labour, raw materials, etc.
On the other hand, those factors that cannot be varied or changed as
the output changes are called fixed factors. These factors are normally
characteristic of the short run or short period of time only. Fixed
factors do not exist in the long run.
Consequently, we can define two production functions: short-run and
long-run. The short-run production function defines the relationship
between one variable factor (keeping all other factors fixed) and the
output. The law of returns to a factor explains such a production
function.
For example, consider that a firm has 20 units of labour and 6 acres of
land and it initially uses one unit of labour only (variable factor) on its
land (fixed factor). So, the land-labour ratio is 6:1. Now, if the firm
chooses to employ 2 units of labour, then the land-labour ratio
becomes 3:1 (6:2).
The long-run production function is different in concept from the
short run production function. Here, all factors are varied in the same
proportion. The law that is used to explain this is called the law of
returns to scale. It measures by how much proportion the output
changes when inputs are changed proportionately.
Solved Example for You
Question: What is meant by returns to a factor?
Answer: Returns to a factor is used to explain the short run production
function. It explains what happens to the output when the variable
factor changes, keeping the fixed factors constant. Thus, it can be said
that ‘returns to a factor’ is a short run phenomenon.
Question: Production function is a _______.
a. Catalogue of Output possibilities
b. Catalogue of input possibilities
c. Catalouge of price
d. None of the above
Ans: The correct option is A. Production function is a catalogue of
output possibilities
Total Product, Average Product and Marginal Product
What is the production function in economics? Let us study the
definitions of Total Product, Average Product and Marginal Product in
simple economic terms along with the methods of calculation for each.
We will also look at the law of variable proportions and the
relationship between Marginal product and Total Product.
Production Function
The function that explains the relationship between physical inputs
and physical output (final output) is called the production function.
We normally denote the production function in the form:
Q = f(X1, X2)
where Q represents the final output and X1 and X2 are inputs or factors
of production.
Browse more Topics under Production And Costs
● Production Function
● Shapes of Total Product, Average Product and Marginal
Product
● Return to scale and Cobb Douglas Function
● Behaviour of Cost in the Short Run
● Long-Run Cost Curves
Learn more about Production Function here in more detail.
Total Product
In simple terms, we can define Total Product as the total volume or
amount of final output produced by a firm using given inputs in a
given period of time.
Marginal Product
The additional output produced as a result of employing an additional
unit of the variable factor input is called the Marginal Product. Thus,
we can say that marginal product is the addition to Total Product when
an extra factor input is used.
Marginal Product = Change in Output/ Change in Input
Thus, it can also be said that Total Product is the summation of
Marginal products at different input levels.
Total Product = Ʃ Marginal Product
Average Product
It is defined as the output per unit of factor inputs or the average of the
total product per unit of input and can be calculated by dividing the
Total Product by the inputs (variable factors).
Average Product = Total Product/ Units of Variable Factor Input
Source: FreeEconHelp
Relationship between Marginal Product and Total Product
The law of variable proportions is used to explain the relationship
between Total Product and Marginal Product. It states that when only
one variable factor input is allowed to increase and all other inputs are
kept constant, the following can be observed:
● When the Marginal Product (MP) increases, the Total Product
is also increasing at an increasing rate. This gives the Total
product curve a convex shape in the beginning as variable
factor inputs increase. This continues to the point where the
MP curve reaches its maximum.
● When the MP declines but remains positive, the Total Product
is increasing but at a decreasing rate. Thisgiveends the Total
product curve a concave shape after the point of inflexion. This
continues until the Total product curve reaches its maximum.
● When the MP is declining and negative, the Total Product
declines.
● When the MP becomes zero, Total Product reaches its
maximum.
Relationship between Average Product and Marginal Product
There exists an interesting relationship between Average Product and
Marginal Product. We can summarize it as under:
● When Average Product is rising, Marginal Product lies above
Average Product.
● When Average Product is declining, Marginal Product lies
below Average Product.
● At the maximum of Average Product, Marginal and Average
Product equal each other.
Learn more about the Shapes of Total Product, Average Product, and
Marginal Product.
Solved Example for You
Question: What are Returns to a Factor? What do you mean by the
Law of Diminishing Returns?
Answer: Returns to a Factor is used to explain the behaviour of
physical output as only one factor is allowed to vary and all other
factors are kept constant. This is a short-run concept.
The law of diminishing returns to a factor states that as the variable
factor is allowed to vary (increase), keeping all other factors constant,
the Marginal Product first increases, reaches its maximum and then
declines and even becomes negative.
Shapes of Total Product, Average Product and Marginal Product
What shapes do Total Product, Marginal product and Average Product
take in the short-run? Let us understand the three stages of production
and the significance of each stage. Let us take a detailed look.
Total Product
The total product refers to the total amount (or volume) of output
produced with a given amount of input during a period of time.
Therefore, a firm wanting to increase its Total Product in the short run
will have to increase its variable factors as the fixed factors remain
unchanged (that is why they are ‘fixed’ in the short run).
In the long run, as we know that all factors become variable, the firm
can increase its total product by increasing any of its factors as all
factors become variable. The concept of Total Product helps us
understand what is called the Marginal Product.
Marginal Product
The total product can be calculated by adding subsequent marginal
returns to an input (also known as the marginal product). The increase
in output per unit increase in input is called Marginal Product. Thus, if
we were to assume Labour as the input used in the production process
(say), then Marginal Product can be calculated as-
MP = Change in output/ Change in input (here, labour)
TP = ƩMP
Average Product
Average product, as the name suggests, refers to the per unit total
product of the variable factor (here, labour). Hence, the calculation of
Average Product is also very simple.
AP = Total Product/ units of variable factor input = TP/L
Note that Total Product can also, therefore, be calculated as TP = AP x
L
TP, MP, AP: Shape of the Curves
When we take a look at the figure below, the following can be noted
about the shapes of the TP, MP and AP curves.
Source: TeX StackExchange
● The TP curve first increases at an increasing rate, after which it
continues to increase but at a decreasing rate, giving the curve
an S-shape. This trend continues till TP reaches its maximum.
Here, MP =0. After the maximum, TP starts to fall or it
declines.
● The MP curve also initially increases, reaches its maximum and
then declines. Note that the maximum of MP is reached at the
point where TP starts to increase at a diminishing rate. An
interesting fact is that MP can also be negative, whereas TP is
always positive even when it declines.
● The AP curve also shows a similar trend as the MP. It rises,
reaches its maximum and then falls. At the point where AP
reaches its maximum, AP = MP.
● All – TP, MP and AP curves, are inverted U-shaped.
Law of Variable Proportion
The law of variable proportions explains the peculiar shape of the TP
curve. It is based on the following assumptions:
● Only one input is variable and all other inputs are held
constant.
● The proportion in which factor units are used may be changed.
● The state of technology and factor prices are assumed to be
constant.
● The time period is the short-run.
It states that if we increase one variable factor, keeping all other
factors constant, the TP curve first increases at an increasing rate
(convex shape) and then at a diminishing rate (concave shape) after
which it starts to fall. This lends it an S-shape till the point where TP
reaches its maximum.
Stages of Production
Based on the shapes of the TP, MP and AP curves, we can identify
different stages of the production process faced by a firm.
Source: JBDON
Stage I
Called the stage of increasing returns to a factor, his stage refers to
that phase in the production process where MP is increasing and
reaches its maximum point. It is the phase where TP is increasing at
an increasing rate. The stage starts from the origin and extends till the
point of inflexion – the point on the TP curve after which TP increases
at a diminishing rate
Since TP is increasing at an increasing rate in this phase, it is
profitable for the firm to continue employing more units of the
variable factor to increase its production. Hence, the firm never
operates in Stage I.
Stage II
This stage is called the stage of diminishing returns to a factor. It
refers to the phase where TP increases at a diminishing rate and
reaches its maximum. In this phase, MP is declining but note that it
still remains positive. The stage ends where MP = 0. Since this implies
efficient utilization of the fixed factor, a firm always operates in the
second stage of production.
Stage III
This is the final phase, called the stage of negative returns to a factor,
where the TP curve starts to decline. MP in this phase becomes
negative. This stage is not at all feasible for operation for any firm as
the TP starts to decline, which means that production has surpassed
the optimum level of specialization.
Solved Example for You
Question: What might be possible reasons for negative returns to a
factor in Stage III of production?
Answer: There can be various reasons for negative returns to a factor:
● The fixed factor is limited in the short run. If we go on
increasing the variable factor beyond a certain point, it will
mean inefficient usage of the fixed factor, acted upon by the
variable factor. This is why MP becomes negative.
● The efficiency of variable factor may also be a reason for
negative returns. If more and more labour is added to fixed
capital (say, machinery), the marginal contribution of each
variable factor becomes less, leading to overcrowding.
● The efficiency of the fixed factor is also affected in case of
overcrowding of variable factor. Too many labourers may
cause chaos and wear and tear of machinery, which ultimately
causes TP to fall.
Returns to Scale and Cobb Douglas Function
What are returns to scale and what are its three types? Let us
understand each case with a diagram for the production function. We
will also learn about the famous Cobb-Douglas production function.
Let us get started!
Returns to Scale
The long run refers to a time period where the production function is
defined on the basis of variable factors only. No fixed factors exist in
the long run and all factors become variable. Thus, the scale of
production can be changed as inputs are changed proportionately.
Thus, returns to scale are defined as the change in output as factor
inputs change in the same proportion. It is a long run concept.
Browse more Topics under Production And Costs
● Production Function
● Total Product, Average Product and Marginal Product
● Shapes of Total Product, Average Product and Marginal
Product
● Behaviour of Cost in the Short Run
● Long-Run Cost Curves
Types of Returns to Scale
There are three defined types of returns to scales, which include:
Increasing Returns to Scale
When the output increases more than proportionately when all the
inputs increase proportionately, it is known as increasing returns to
scale. This represents a kind of decreasing the cost to the firm.
External economies of scale might be one of the reasons behind such
increase in output in increasing returns to scale. Thus, when inputs
double, output more than doubles in this case.
Decreasing Returns to Scale
When the output increases less than proportionately as all the inputs
increase proportionately, we call it decreasing returns to scale or
diminishing returns to scale. In this case, internal or external
economies are normally overpowered by internal or external
diseconomies. Thus, if we double the inputs, the output will increase
but by less than double.
Constant Returns to Scale
When the output increases exactly in proportion to an increase in all
the inputs or factors of production, it is called constant returns to
scale. For example, if twice the inputs are used in production, the
output also doubles. Thus, constant returns to scale are reached when
internal and external economies and diseconomies balance each other
out.
A regular example of constant returns to scale is the commonly used
Cobb-Douglas Production Function (CDPF). The figure given below
captures how the production function looks like in case of
increasing/decreasing and constant returns to scale.
Source: FAO
Cobb-Douglas Production Function
As we know, a production function explains the functional
relationship between inputs (or factors of production) and the final
physical output. Let us begin with a simple form a production function
first –
Q = f(L, K)
The above mathematical equation tells us that Q (output) is a function
of two inputs (assumption). These inputs are L (amount of labour) and
K (hours of capital). Basing our understanding of the function above,
we can now define a more specific production function – the Cobb
Douglas Production Function.
Q = A Lβ Kα
Here, Q is the output and L and K represent units of labour and capital
respectively. A is a positive constant (also called the technology
coefficient). α and β are constants lying between 0 and 1.
We can calculate the Marginal Product for the CDPF and derive
interesting results. Marginal Product captures the change in output due
to an infinitesimal change in an input. It is calculated by first-order
differentiation of the CDPF. Hence,
MPL = A β Lβ-1 Kα , and MPK = A α Lβ Kα-1
Let us now find out the implications of returns to scale on the
Cobb-Douglas production function: If we are to increase all inputs by
‘c’ amount (c is a constant), we can judge the impact on output as
under.
Q (cL, cK) = A (cL)β (cK)α = Acβ cα Lβ Kα = Acα+β Lβ Kα
Note that if α+β > 1 there will be increasing returns to scale. If α+β <
1 there will be decreasing returns to scales. And, if α+β = 1 there will
be constant returns to scale (case of linear homogenous CDPF). Thus,
depending on the nature of the CDPF, there will be increasing,
decreasing or constant returns to scale.
Solved Example for You
Question: What is the shape of the production in case of constant
returns to scale?
Answer: When the output increases exactly in proportion to an
increase in all the inputs or factors of production, it is called constant
returns to scales. This means if inputs are increased ‘x’ times, output
also increases by ‘x’ times.
This means that the shape of the production function is a linear
straight line passing through the origin, where the x-axis measures
inputs and y-axis measures output. The line is at an angle of 45 to the
origin.
Behavior of Cost in the Short Run
Short-run costs are important to understanding costs in economics.
The distinction between short-run and long-run based on fixed and
variable factors of production makes the concept of understanding
short run costs simpler. Let us understand the concepts by way of
examples, diagrams for graphical representation.
The Concept of Short Run
It is key to understand the concept of the short run in order to
understand short run costs. In economics, we distinguish between
short run and long run through the application of fixed or variable
inputs.
Fixed inputs (plant, machinery, etc.) are those factors of production
that cannot be changed or altered in a short span of time because the
time period is ‘too small’. This makes the short run. Here, the inputs
are of two types: fixed and variable.
In the long-run, all the inputs become variable (eg. raw materials). By
this, we mean that all inputs can be changed with a change in the
volume of output. Thus, the concept of fixed inputs applies only to the
short-run. It is to short-run costs that we now turn.
Browse more Topics under Production And Costs
● Production Function
● Total Product, Average Product and Marginal Product
● Shapes of Total Product, Average Product and Marginal
Product
● Return to scale and Cobb Douglas Function
● Long-Run Cost Curves
Short Run Cost Function
The cost function is a functional relationship between cost and output.
It explains that the cost of production varies with the level of output,
given other things remain the same (ceteris paribus). This can be
mathematically written as:
C = f(X)
where C is the cost of production and X represents the level of output.
Total Fixed Cost
Fixed cost refers to the cost of fixed inputs. It does not change with
the level of output (thus, fixed). Fixed inputs include building,
machinery etc. Hence the cost of such inputs such as rent or cost of
machinery constitutes fixed costs. Also referred to as overhead costs,
supplementary costs or indirect costs, these costs remain the same
irrespective of the level of output.
Hence, if we plot the Total Fixed Cost (TFC) curve against the level
of output on the horizontal axis, we get a straight line parallel to the
horizontal axis. This indicates that these costs remain the same and
that they have to be incurred even if the level of output is zero.
Total Variable Cost
The cost incurred on variable factors of production is called Total
Variable Cost (TVC). These costs vary with the level of output or
production. Thus, when production level is zero, TVC is also zero.
Thus, the TVC curve begins from the origin.
The shape of the TVC is peculiar. It is said to have an inverted-S
shape. This is because, in the initial stages of production, there is
scope for efficient utilization of fixed factor by using more of the
variable factor (eg. Workers employing machinery).
Hence, as the variable input employed increases, the productive
efficiency of variable inputs ensures that the TVC increases but at a
diminishing rate. This makes the first part of the TVC curve that is
concave.
As the production continues to increase, more and more variable
factor is employed for a given amount of fixed input. The productive
efficiency of each variable factor falls and it adds more to the cost of
production. So the TVC increases but now at an increasing rate. This
is where the TVC curve is convex in shape. And so the TVC curve
gets an inverted-S shape.
Total Cost
Total cost (TC) refers to the sum of fixed and variable costs incurred
in the short-run. Thus, the short-run cost can be expressed as
TC = TFC + TVC
Note that in the long run, since TFC = 0, TC =TVC. Thus, we can get
the shape of the TC curve by summing over TFC and TVC curves.
Fig.1
(Source: economicsdiscussion)
The following can be noted about the TC curve:
● The TC curve is inverted-S shaped. This is because of the TVC
curve. Since the TFC curve is horizontal, the difference
between the TC and TVC curve is the same at each level of
output and equals TFC. This is explained as follows: TC –
TVC = TFC
● The TFC curve is parallel to the horizontal axis while the TVC
curve is inverted-S shaped.
● Thus, the TC curve is the same shape as TVC but begins from
the point of TFC rather than the origin.
● The law that explains the shape of TVC and subsequently TC is
called the law of variable proportions.
Solved Example for You
Question: Comment on the shape of the TC, TVC and TFC curves
based on the following table:
Output Fixed Cost Variable Cost Total Cost
0 40 0 40
1 40 20 60
2 40 30 70
3 40 32 72
4 40 34 74
5 40 36 76
6 40 38 78
7 40 40 80
8 40 46 86
Answer:
1. We see that the Fixed Cost remains the same even as
production increases from 0 to 8 units. Thus the value of FC =
40
2. It can be noted that the Variable Cost increases as the output
increases. The VC increases at a diminishing rate till 7 units of
output, after which it starts increasing at an increasing rate.
3. The final column shows the Total Cost which is the sum of FC
and VC and increases as the output increases.
Long Run Cost Curves
The long run is different from the short run in the variability of factor
inputs. Accordingly, long-run cost curves are different from short-run
cost curves. This lesson introduces you to Long run Total, Marginal
and Average costs. You will learn the concepts, derivation of cost
curves and graphical representation by way of diagrams and solved
examples.
The Concept of the Long Run
The long run refers to that time period for a firm where it can vary all
the factors of production. Thus, the long run consists of variable inputs
only, and the concept of fixed inputs does not arise. The firm can
increase the size of the plant in the long run. Thus, you can well
imagine no difference between long-run variable cost and long-run
total cost, since fixed costs do not exist in the long run.
Long Run Total Costs
Long run total cost refers to the minimum cost of production. It is the
least cost of producing a given level of output. Thus, it can be less
than or equal to the short run average costs at different levels of output
but never greater.
In graphically deriving the LTC curve, the minimum points of the
STC curves at different levels of output are joined. The locus of all
these points gives us the LTC curve.
Long Run Average Cost Curve
Long run average cost (LAC) can be defined as the average of the
LTC curve or the cost per unit of output in the long run. It can be
calculated by the division of LTC by the quantity of output.
Graphically, LAC can be derived from the Short run Average Cost
(SAC) curves.
While the SAC curves correspond to a particular plant since the plant
is fixed in the short-run, the LAC curve depicts the scope for
expansion of plant by minimizing cost.
Derivation of the LAC Curve
Note in the figure, that each SAC curve corresponds to a particular
plant size. This size is fixed but what can vary is the variable input in
the short-run. In the long run, the firm will select that plant size which
can minimize costs for a given level of output.
You can see that till the OM1 level of output it is logical for the firm
to operate at the plat size represented by SAC2. If the firm operates at
the cost represented by SAC2 when producing an output level OM2,
the cost would be more.
So in the long run, the firm will produce till OM1 on SAC2. However,
till an output level represented by OM3, the firm can produce at SAC2,
after which it is profitable to produce at SAC3 if the firm wishes to
minimize costs.
(Source: test.blogspot)
Thus, the choice, in the long run, is to produce at that plant size that
can minimize costs. Graphically, this gives us a LAC curve that joins
the minimum points of all possible SAC curves, as shown in the
figure. Thus, the LAC curve is also called an envelope curve or
planning curve. The curve first falls, reaches a minimum and then
rises, giving it a U-shape.
We can use returns to scale to explain the shape of the LAC curve.
Returns to scale depict the change in output with respect to a change
in inputs. During Increasing Returns to Scale (IRS), the output
doubles by using less than double inputs. As a result, LTC increases
less than the rise in output and LAC will fall.
● In Constant Returns to Scale (CRS), the output doubles by
doubling the inputs and the LTC increases proportionately with
the rise in output. Thus, LAC remains constant.
● In Decreasing Returns to Scale (DRS), the output doubles by
using more than double the inputs so the LTC increases more
than proportionately to the rise in output. Thus, LAC also rises.
This gives LAC its U-shape.
Long Run Marginal Cost
Long run marginal cost is defined at the additional cost of producing
an extra unit of the output in the long-run i.e. when all inputs are
variable. The LMC curve is derived by the points of tangency between
LAC and SAC.
Note an important relation between LMC and SAC here. When LMC
lies below LAC, LAC is falling, while when LMC is above LAC,
LAC is rising. At the point where LMC = LAC, LAC is constant and
minimum.
Solved Example for You
Question: Why is the LAC also called the envelope curve?
Answer: The LAC curve suggests the long run optimization problem
of the firm. The firm can choose a plant size to operate at in the
long-run where all inputs are variable. Thus, the firm shall choose that
plant at which it can minimize costs.
So, the LAC is derived by joining the minimum most points of all
possible SAC curves of the firm at different output levels. Since the
LAC thus obtained almost ‘envelopes’ the SAC curves faced by the
firm, it is called the envelope curve.