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.