UNIT 4 THEORY OF PRODUCTION

Structure 4.0 4.1 4.2 4.3 Objectives Introduction Short Period Analysis
4.2.1 Returns to a Factor

Long Period Analysis

4.3.1 Iso-quant 4.3.2 Elasticity of Substitution

4.4 4.5 4.6 4.7 4.8 4.9

Returns to Scale Homogeneous Production Function Let Us Sum Up Key Words Some Useful Books Answers to Check Your Progress

4.0

OBJECTIVES

After studying this unit, you will be able to: appreciate the relationship between input and output in production; measure productivities of inputs; analyse laws of production with single input variation and multiple inputs variation; assess of production in short-and long-run; and measure to degree of substitution between factors of production.

4.1

INTRODUCTION

Production activities related to goods and services require inputs. Typically, the set of inputs includes labour, capital equipments and raw materials. The producing unit usually has to solve the choice problem as a given amount of output can be produced from various combinations of inputs. Firms, therefore, look for production possibilities that are technologically feasible. A production function describes the relation between input and output with a given the technology. More formally, it shows the maximum amount of output that can be produced from any specified set of inputs, given the existing technology. If we assume that there are only two factors of production labour (L), capital (K) and a single output q, mathematically a production function can be written as, q = F(L, K)

Producer Behaviour

4.2

SHORT PERIOD ANALYSIS

Short period in production refers to a time when some inputs remain fixed. A fixed input is one, whose quantity cannot be changed readily, whereas, a variable input varies with production. Inputs like land, building and major pieces of machinery cannot be varied easily and, therefore, can be called fixed inputs. On the other hand, inputs like labour (labour hours), raw materials, and processed materials can be easily increased or decreased. Therefore, these are categorised as variable inputs. Depending on whether inputs can be kept fixed or not, we have a short period or a long period. To put this more precisely, if inputs being used in the production process have just enough time such that they cannot be varied, then the analysis pertains to the short-run. On the other hand, if the inputs employed have enough time such that they are amenable to variation, then the analysis is based on the frame work of long-run. Generally, the firms do not readily change their capital, which could be land, machinery, managerial and technical personnel. Therefore, these are fixed input in the short-run. When we treat these under , the production function can be written as
q = F(L, K)

where L = Labour, a variable factor

K = Capital, a fixed factor

Marginal Product (MP) of a Factor

From the above mentioned production function, immediately we can study the effect on total output when there is a variation in labour utlilisation, keeping the other factor K , fixed. Thus, we have the marginal physical product, which shows the change in output quantity for a unit change in the quantity of an input, (L), when all other inputs (K) are held constant. Mathematically, it is given by the first partial derivative of a production function with respect to labour. Thus, q = F(L, K) = Total Product (TP). q = fL MP of Labour = MPL = L

If L = L , then marginal product of capital = MPK =

q = K

fK

It is reasonable to expect that the marginal product of an input depends on the quantity used of that input. In the above example, use of labour is made keeping the amount of other factors (such as equipments and land) fixed. Continued use of labour would eventually exhibit deterioration in its productivity. Thus, the relationship between labour input and total output can be recorded to show the declining marginal physical productivity. Mathematically, the diminishing marginal physical productivity is assessed through the second-order partial derivative of the production function. Thus, change in labour productivity can be presented as:

MPl 2 q = 2 = f ll < 0 L L
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Similarly, change in productivity of capital is denoted by MPk 2 q = = f kk < 0 K K 2

Average Product (AP) of a Factor

Theory of Production

The productivity of a factor is often seen in terms of its average contribution. Although not very important in the theoretical discussions, where analytical insight is tried to be drawn from marginal productivities, average productivity finds a platform in empirical evaluations. Deriving it from the total product is relatively easy. It is the output per unit of a factor. So, AP of labour = APL = TP q = . Similarly, L L TP q = . K K

AP of capital = APK =

Relation between TP and MP

Graphically, given the total product curve, MP is the slope of the tangent at any point on the TP curve. This is shown in Figure 4.1.
TP, MP, AP

TP = q

O
Fig. 4.1: Total Product Curve and Marginal Product

See that MPL = tan =

TP L

Relation between TP and AP

Given the TP curve, AP is the slope of a ray from the origin to any point on the TP curve. APL = tan = QL q = OL L

Producer Behaviour

TP, AP, MP

TP Q

L
Fig. 4.2: Derivation of AP from TP Curve

Relation between AP and MP

This relation between AP and MP is true for all average and marginal productivity conditions. When AP is rising, MP > AP; when AP is maximum, MP = AP and when AP is falling, MP < AP.
Proof:

Since APL =

q , differentiating APL partially with respect to L, we get, L

L q L q - q L - q L L = L L2 L2

q (APL ) = = L L L or,

q 1 q (APL ) = . L L L L2
1 q q 1 (APL ) = - 2 = ( MPL - APL ) L L L L L (1)

or,

From equation 1, (APL ) > 0 when MPL > APL L (APL ) = 0 when MPL = APL; L (APL ) < 0 when MPL > APL L (since L > 0) (since L > 0);

It is an empirically observed feature that all inputs have positive but diminishing marginal products. If L and K are the only factors of production, then FL > 0, FK > 0, FLL < 0, FKK < 0.

Thus, for any factor, initially, MP is positive. Then a situation arises when MP is zero, and further on it falls to negative as more of the factor is employed in production.

Theory of Production

Check Your Progress 1

1) What is a production function? .. .. .. .. .. .. 2) In short period why you cannot increase the managerial inputs? .. .. .. .. .. 3) Define marginal productivity of labour .. .. .. .. .. ..

4.2.1 Returns to a Factor

This is a short-run concept which deals with the variability of only one factor keeping the others constant. There are three kinds of returns: i) ii) iii) Increasing returns: when the AP of a factor rises and MP > AP Constant returns: when AP is constant and MP = AP Diminishing returns: when AP is falling MP < AP

The concept of returns to a factor can also be expressed in terms of the partial input elasticity of output.
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Producer Behaviour

Partial Input Elasticity of Output

Partial input elasticity of output is also called elasticity of output with respect to a factor. It is the percentage change in output quantity for one per cent change in the quantity of a factor when all other factors remain constant. Elasticity of q with respect to L is given by, q q L L q L MPL . = L q APL q K MPK . = APK K q

E q,L =

Similarly, E q,K =

Let us now relate this to returns to scale. q Since APL = , L q L q L - q. L - q q L L = L (APL ) = = L L L L2 L2 = = 1 q q L L L2

q q L q (2) . - 1 = 2 ( E q,L - 1) 2 L L q L q As 2 > 0 from equation (2), L increasing returns = ( APL ) > 0 = Eq,L > 1 L constant returns = ( APL ) = 0 = Eq, L = 1 L diminishing returns = ( APL ) < 0 = E q,L < 1 L Besides the above mentioned three returns, there can be another type known as the non-proportional returns to a variable factor. Under it, initially, there is increasing returns to a factor up to a certain level beyond which there is diminishing returns.
Graphical Representation of Various Returns Diminishing Returns: If the TP curve is as shown in the adjacent Figure 4.3, then the MPL given by tan is throughout less than the APL given by tan .

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MP L, AP L

Theory of Production

MP L

AP L

O
Fig. 4.3: Derivation of MPL from TP

As APL is falling from the relation between MP and AP, MP < AP we have the adjoining Figure 4. 4.
TP, MP, AP

TP

O
Fig. 4.4: Diminishing MP and AP

Increasing Returns

Here APL rises and tan < tan , for all L. Therefore, MP > AP. This is shown in the adjacent Figure 4.5.

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Producer Behaviour

TP

(a) MP L, AP L MP L AP L

O (b)
Fig. 4.5: Increasing Returns as given by TP, MP and AP

Constant Returns

Here, APL is constant and tan = tan , therefore, MPL = APL as is shown by a horizontal straight line in the next Figure 4.6(a,b)

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Theory of Production

TP

= O

(a)

MP L, AP L

MP L = AP L

(b)
Fig. 4.6: Constant Returns as given by TP, MP and AP

Non-Proportional Returns

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Producer Behaviour

stage I B

II

III TP

A MP L AP L

O AP P L, M L

MP L O
Fig. 4.7: Derivation of Stages of Production

AP L L

The TP curve is such that upto point A, MP is rising and so is AP and MP > AP, as shown in the diagram below. Beyond point A, MP falls but AP rises, till the two are equated at point B. At B, AP is maximised. AP falls beyond the point B. At point C, the TP curve flattens out and therefore, MP = 0. Beyond C, MP is negative and AP is falling. Therefore, in the case of nonproportional return, both MP and AP rise, initially. MP reaches a maximum earlier than AP. When they both are equated, AP is maximised. Finally, there is a situation where both are falling. Depending on the nature of MP and AP, the production process can be divided into three stages I, II, and III, as shown in Figure 4.7. Characteristics of the three stages are : Stage I: MP > 0, AP rising, thus MP > AP Stage II: MP > 0, AP falling, thus MP < AP Stage III: MP < 0, AP falling

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In stage I, by adding one more unit of L, the producer can increase the average productivity of all the units. Thus, it would be unwise for the producer to stop production in this stage. In stage III, MP < 0, so that by reducing the L input, the producer can increase the total output and save the cost of a unit of L. Therefore, it is impractical for a producer to produce in this stage. Hence, stage II represents the economically meaningful range. This is so because here MP > 0 and AP > MP. So that an additional L input would raise the total production. Besides, it is in this stage that the TP reaches a maximum.
Check Your Progress 2

Theory of Production

1)

Why marginal productivity of labour declines?

2)

The producer must choose the second stage from the total production curve. Why?

3)

What is partial input elasticity of output?

4.3

LONG PERIOD ANALYSIS

Long period refers to a time when all the factors are variable. Earlier in the short period analysis, we had considered capital (K) to be fixed factor. Here in this part, we assume both L and K to be variable factors. Therefore, the production function would be: q = F(L, K) The producer can now employ L and K units at her will to produce output q as per the production technology. Therefore, in the K L input space the producer can choose any combination of K and L to produce output.
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Producer Behaviour

O
Fig. 4.8: K-L Input Space Available to Producer

4.3.1 Iso-quant
The dots in the above Figure 4.8 denotes the various combinations of (L, K) that the producer can pick up from form to produce. Among these combinations, there can be those, which produce the same level of output. Herein comes the concept of an iso-quant. An iso-quant is a locus of combinations of (L, K) that produce the same level of output q (see Figure 4.9).
K

K1 K2 K3

A B C

q L

L 1

L2

L3
Fig. 4.9: Shape of Iso-quant

In the above Figure 4.9, q amount of output can be produced by the input combinations (L1, K1), (L2, K2), (L3, K3). Joining these, we get an isoquant, which is also denoted by q . Therefore, we see that the same level of output q can be produced using different techniques either more K, less L (e.g., technique A), or more L and less K (e.g., technique C).

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Slope of an Iso-quant

Theory of Production

Since along an iso-quant the level of output remains the same, if L units of L are substituted for K units of K, the increase in output due to L units L (namely, L.MPL) should match the decrease in output due to a decrease of K units of K (namely K.MPK). In other words,
L.MPL = K.MPK

K MPL = L MPK K MPL =L MPK is a very small amount we can write K dK . In the K L input as L dL

or,

when space,

dK represents the slope of the iso-quant at any point on it. dL

Slope of the iso-quant =

Check Your Progress 3

MPL dK =. dL MPK

1)

Define an iso-quant.

2)

Derive the slope of an iso-quant mathematically, if the production function is q = F(L,K).

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Producer Behaviour

The absolute value of

dK dK , denoted by is known as the marginal rate of dL dL technical substitution of L for K. (MRTSLK). By definition, it measures the reduction in one input per unit increase in the other that is just sufficient to maintain a constant level of output. It is equal to the ratio of the marginal product of L to the marginal product of K.

Curvature of the Iso-quant

An iso-qunat is convex to the origin. This is so because as more and more units labour are employed, the producer would prefer to give up less and less of the other input to produce the same amount of output. This is shown in the following Figure 4.10:
K

K 1

S |T
K2

q L3 L4 L

L 1

L2

Fig. 4.10: Curvature of an Iso-quant

For a rise in L from L1 to L2, the producer gives up K1 amount of K. For the same output level q , as L increases from L3 to L4, she gives up K2 amount of K. As K2 < K1, it implies that for more units of L, the producer is willing to give up less of K. MPL ,a MPK diminishing MRTS means as L increases, MPL decreases and MPK increases. A convex iso-quant implies a diminishing MRTSLK. As MRTSLK =
Economic Region of Iso-quants

With help of production function, we generate iso-quant maps as shown in the following Figure 4.11. The only difference these have with the previous isoquants is that presently we have positively sloped segments.

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K q3 q2 q1

Theory of Production

O
Fig. 4.11: Economic Region of Production and Iso-quants

Let us examine the characteristic of one such iso-quant with the help of the following Figure 4.12.
K

O
Fig. 4.12: Economic Region of Production

We know that the slope of an iso-quant is given by,

dK MPL . = dL MPK

From the figure 4.12, at point A, the slope is infinite or undefined. This dK implies that at A, MPK = 0. Beyond A, > 0 , implying MPK < 0. As we dL had seen in the short-run analysis, no producer has an incentive to undertake production at this portion (this zone is similar to stage III of short run analysis). At point B, dK = 0 , implying MPL = 0. Beyond point B, MPL falls. By dL similar logic, no producer would want to employ L beyond point B.

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Producer Behaviour

Therefore, production beyond points A and B are irrelevant. Hence, segment AB of the iso-quant is the economically feasible region. This is true for all iso-quants belonging to a family with positively sloped portions. For these types of iso-quants we can obtain an economic region of production comprising the economically feasible portions of the iso-quants. This is obtained by constructing ridge lines, which are loci of points where MPK = 0. The ridge lines provide a boundary to the economic region of production. This is shown in the following Figure 4.13.
K R 1 R 2 q1 < q2 < q3

q3 q2 q1 O
Fig. 4.13: Selection Economic Region of Production

In Figure 4.13, R1 represents the ridgeline which is a locus of points where MPK = 0. R2 represents the ridge line where MPL = 0.
Properties of Iso-quants

1) Iso-quants are non-intersecting, so that one and only one iso-quant will pass through a given point. 2) If both the inputs have positive marginal products, then the iso-quants are negatively sloped. 3) Iso-quants are convex to the origin. 4) An iso-quant lying away from the origin represents higher quantity of output.
Types of Iso-quants

Besides the smooth convex iso-quants assuming continuous substitutability, we get other types as follows: 1) Linear: This type of iso-quant assumes perfect substitutability between the factors of production. A given output quantity may be produced by using either only L or only K or by a combination of both (see Figure 4.14).

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Theory of Production

q1 O

q2

q3 L

Fig. 4.14: Linear Iso-quant

2) L-shaped: Assuming strict complementarity between the inputs i.e., no substitutability of factors of production, we get L-shaped iso-quant (see Figure 4.15). It implies there is only one method of production.

q2 q1 O
Fig. 4.15: L-shaped Iso-quants

4.3.2 Elasticity of Substitution

MRTS has a drawback as a measure of the degree of substitutability because it depends on the units of measurement of factors. Therefore, we need to consider the concept of elasticity. The elasticity of substitution denoted by , is a pure number that measures the rate at which substitution takes place. It is defined as the percentage change in K L ratio for one percentage change in the MPL MPK ratio (or MRTSLK). Thus, = % change in K/L 1% change in MRTS
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Producer Behaviour

In terms of differential calculus,

d(K/L) d ( K L ) fL fK (K/L) = . = d( f L f K ) d ( fL fK ) K L ( fL fK )
K d log L = f d log L fK Generally, varies from point to point on an iso-quant. But in some special . cases = 0 or
Check Your Progress 4

1) What is MRTS ? 2) What is elasticity of substitution? 3) Find for which iso-quants = 0 and for which .

4) What do you mean by returns to scale? 5) What is economic region of production?

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Theory of Production

4.4

RETURNS TO SCALE

It is a typical long-run concept and involves the effect of change in inputs on the quantity of output produced. There are three types of returns to scale. For 1% change in all the factors, if correspondingly, output changes by a) b) c) 1%, then there is said to be Constant Returns to Scale (CRS) Less than 1%, then there is decreasing Returns to Scale (DRS) More than 1%, then there is Increasing Returns to Scale (IRS).

Diagrammatic Representation Constant Returns to Scale

As there is CRS, along a ray through the origin, the distance between the consecutive iso-quants remains the same (see Figure 4.16).
K

3K 2K K q O L 2L 3L L 2q 3q

Fig. 4.16: Derivation of Line of Scale

Decreasing Returns to Scale

As there is DRS, along a ray through the origin, the distance between the consecutive iso-quants increases (see Figure 4.17).

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Producer Behaviour

3q 2q q O
Fig. 4.17: DRS and Distance between Iso-quants

Increasing Returns to Scale

As there is IRS, along a ray through the origin, the distance between the consecutive iso-quants is less (see Figure 4.18).
K

4q 3q 2q q O
Fig. 4.18: IRS and Distance between Iso-quants

4.5

HOMOGENEOUS PRODUCTION FUNCTION

The concept of returns to scale can be captured asily using the concept of homogeneity of production functions. A production function say, z = f(x, y) is said to be homogeneous of degree n in x and y, if and only if, f(tx, ty) = tnz for any t > 0. If n = 1, then f(tx, ty) = tz, then the function is said to exhibit CRS. It is also known as a linearly homogeneous production function.
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If n > 1, then the function exhibits IRS and when n< 1, it exhibits DRS.
Example:

Theory of Production

Suppose we have a production function, q = ALK If L and K are increased by a factor t(>0) then, we have, A(tL) (tK) = AtLtK = t(+)ALK = t(+)q The function is homogeneous of degree (+). It is said to exhibit CRS, if + = 1; DRS if + < 1; IRS if + > 1 When + = 1, the function is also called a linearly homogeneous function. Properties of a homogeneous production function. 1) If q = F(K, L) is a homogeneous production function, then we can write it as, F(tK, tL) = tnq for any t > 0. Let t = 1 for any L > 0. L q 1 = n L

Then the function can be written as, K F(tK, tL) = F , L Let K = K and L

K F = , L
q Ln

1 = f(k) .

we have f(k) =
or, 2) q = Lnf(k) MPL = MPK =

q = Lnf(k)

q = nLn- 1f(k) - KLn- 2 f (k) L q = Ln- 1f (k) . K

K Both MPL and MPK are functions of k = . L

It may be useful to remember that homogenous production function is a special case of homothetic production functions. To take note of the concept, you must look for the ratio MPL/MPK, which does not change with any
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Producer Behaviour

proportionate change in L and K in case of homothetic production function. The difference this formulation has with that of homogenous production function can be seen from the iso-quants given in Figure 4.19. The production function is homothetic if slope of Q1 at A1 = slope of Q2 at A2 and slope of Q1 at B1 = slope of Q2 at B2 In contrast to this feature, the homogenous production function would have OA2/OA1 = OB2/OB1. K

A2 B2 A1 Q2 B1 O L
Fig. 4.19: Homothetic Production Function

Q1

Check Your Progress 5

1) What is a homogeneous production function? 2) Prove for q = Lnf(k) MPL = nLn1f(k) KLn2f (k); MPK = Ln1f (k). 3) When MPL and MPK are functions of (K/L)?
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Theory of Production

4.6

LET US SUM UP

This unit covers theoretical insights on production process. It starts with production function and points out that it a technical relation between inputs and output. Production decisions are based on short-and long-run considerations. While the short-run offers enough time to a producer to change the variable inputs like labour and raw materials, the long-run allows changing all inputs, including machinery and building etc. To carry on production in both the periods, it helps understand the productivity concepts of average and marginal products. The average product is the output produced per unit of an input. Marginal product, on the other hand, gives change in the total product due to a unit change in one of the inputs. It is convenient to speak in terms of average produce of labour (AR) and marginal product of labour ( MP2 ) . The production process brings out the relationship between AP and MP. It is seen that when AP is rising MP>AP, when AP is maximum, MP = AP and when AP is falling MP<AP. The total production passes through three distinct stages (I, II and III). In stage I, the total product curve shows an increasing trend, while in stage III it declines. A producer usually chooses to operation in stage-II of the product curve when AP reaches the maximum point. A single level of output can be produced by different combinations of inputs. The curve that captures this feature is called an iso-quant. The slope of an isoquant allows one to analyse the process of input substitution for a given level of output. When there are two inputs, labour and capital, the slope of the isoMPL dK dK is called marginal rate of quant is given by , where = dL dL MPK technical substitution of L for K

( MRTS )
LK

and MPL and MPK are marginal productivities of labour and

capital. The rate at which the substitution between the factors takes place is given by the elasticity of substitution. It is usually devoted by . Thus

% change in K/L % change in MRTS

Simultaneous change of all inputs results in a change in output in proportion to greater, equal or lower scale. Such a feature is studied through returns to scale. The last part of the discussion is focussed on returns to scale through homogenous production function. The degree of homogeneity is related to an assessment of returns to scale and the properties of the homogenous production function is highlighted to analyse the partial productivities of capital and labour.

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Producer Behaviour

4.7

KEY WORDS

Average Product: Total product per unit of an input Cobb-Douglas Production Function: A production function of the form Q= f(aKbLc) where a, b and c are constants, Q is output, and L and K are inputs Constant Returns to Scale: The case where a proportionate change in all inputs changes output by the same proportion. Decreasing Returns to Scale: The case where a proportionate increase in all inputs leads output to increase by a small proportion Diminishing Marginal Rate of Substitution: The declining marginal rate of substitution as one input is substituted for another. Economic Region of Production: The downward sloping segment of an isoquant Elasticity of Substitution: A measure of the responsiveness of the input ratio to a change in the input-price ratio. Homogeneous Production Function: A special case of homothetic production function in which a proportionate change in inputs causes output to change by a proportion which does not vary changes in the inputs. Homothetic Production Function: A production function where the ratio of marginal product is unaffected by a proportionate change in inputs. Increasing Returns to Scale: A situation where proportionate increase in all inputs causes output to increase by a large proportion. Isoquant Line: The locus of points representing various combinations of inputs yielding a specified and of output. Long Period Production: A period of time sufficient for altering the quantities of all inputs into the production process. Marginal Rate of Technical Substitution: The rate at which one input can be substituted for another without affecting the level of output. Production Function: The functional relationship between inputs and output.

4.8

SOME USEFUL BOOKS

Ferguson and Gould (1989), Microeconomic Theory, Irwin Publications in Economics; Homewood, IL: Irwin. Koutsoyiannis, A. (1979), Modern Microeconomics, Second edition, London: Macmillian. Ferguson, C. E. (1969), The Neoclassical Theory of Production and Distribution, Cambridge: Cambridge University Press.

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4.9

ANSWER OR HINTS TO CHECK YOUR PROGRESS

Theory of Production

Check Your Progress 1

1) A technical relation between inputs and output. 2) It is likely to conflict with the scale of production and may contribute to the increased cost. 3) Change in output due to a unit change in labour with level of other factors kept constant.
Check Your Progress 2

1) Because contribution of other factors to production cannot be perfectly substituted by labour. 2) At this state AP reaches the maximum. 3) It is elasticity of output with respect to a factor (show the derivation).
Check Your Progress 3

1) Different combinations of inputs producing a fixed level of output. 2) See section on slope of an iso-quant
Check Your Progress 4

1) Rate of substitution between inputs allowed by the production function. 2) Define as given in the text and explain the meaning. 3) Consider the iso-quants which are not of regular shape, e.g., complementary inputs. 4) See the change in output by changing all inputs simultaneously.
Check Your Progress 5

1) Change in inputs changes the value of function to a certain degree. 2) See the derivation given the text in section on Homogenous Production Function. 3) In Homogenous Production Function (see, the derivation in the text on homogenous production function).

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