Chapter 7

Production Theory

Chapter Objectives

• To examine the economic analysis of a firm’s technology, different types of inputs and the process of production.

• To help develop an understanding of the distinction between short run and long run production functions.

• To build up a critical appraisal of the law of variable proportions and returns to scale.

• To introduce the concepts of isoquant, isocost line, marginal rate of technical substitution, elasticity of substitution and expansion path.

• To develop an understanding of technical progress and its nuances.

Production• is the process of transformation of resources (like land,

labour, capital and entrepreneurship) into goods and services of utility to consumers and/or producers.

• is the process of creation of value or wealth through the production of goods and services that have economic value to either consumers or other producers.

• process of adding value may occur – by change in form (input to output, say steel into car), or – by change in place (supply chain, say from factory to

dealers/retailer), or– by changing hands (exchange, say from retailer to consumer).

Production of goods includes all tangible items such as furniture, house, machine, food, car, television etc services include all intangible items, like banking, education, management, consultancy, transportation.

Types of InputsTechnology• determines the type, quantity and proportion of inputs. • also determines the maximum limit of total output from a

given combination of inputs. • at any point of time, technology will be given; impact of

technology can be seen only over a period of time. Fixed and Variable Inputs:• Production analysis of a firm uses two distinct time frames:

– the short run: refers to a period of time when the firm cannot vary some of its inputs.

– the long run., refers to a time period sufficient to vary all of its inputs, including technology.

• Variable input : that can be made to vary in the short run, e.g. raw material, unskilled/semi skilled labour, etc.

• Fixed input: that cannot be varied in the short run, e.g. land, machine, technology, skill set, etc.

Factors of Production 5 factors of productioni. Land: anything which is gift of nature and not the result of

human effort. ii. Labour: physical or mental effort of human being that

undertakes the production process.iii. Capital: wealth which is used for further production; it is

not a gift of nature, but is produced by human beings. Capital is the output of one production process that generally goes as input in another.

iv. Enterprise: The ability and action to collect, coordinate, and utilize all the factors of production for the purpose of economic gains. is also defined as the ability to take risk.

v. Organization: acknowledges a special kind of function, which is a combination of highly skilled labour and specialized human capital. It is organizational efficiency that differentiates one company from another in terms of success.

Production Function• is a technological relationship between physical inputs

and physical outputs over a given period of time.• shows the maximum quantity of the commodity that can

be produced per unit of time for each set of alternative inputs, and with a given level of production technology.

Hence it can be said that production function is:• Always related to a given time period• Always related to a certain level of technology• Depends upon relation between inputs. • Normally a production function is written as:

Q = f (L,K,I,R,E) where Q is the maximum quantity of output of a good being

produced, and L=labour; K=capital; l=land; R=raw material; E= efficiency parameter.

Production Function with One Variable Input

• also termed as variable proportion production function• is essentially a short term production function in which

production is planned with one variable input (say L). • shows the maximum output a firm can produce when only

one of its inputs can be varied, other inputs remaining fixed.

• where Q = output, L = labour and K = fixed amount of capital.

Total product is a function of labour :Average Product (AP) is total product per unit of variable input :

Marginal Product (MP) is defined as addition in total output per unit change in variable input. Thus marginal product of labour (MPL) would be:

),( KLfQ

),( LKfTPL

L

TPAPL

L

TPMPL

Labour (’00 units) Total Product (’000 tonnes)

Marginal

Product

Average Product

Stages

1 20 - 20 Increasing returns

2 50 30 25

3 90 40 30

4 120 30 30 Diminishing returns

5 140 20 28

6 150 10 25

7 150 0 21.5

8 130 -20 16.25 Negative returns

9 100 -30 11.1

-50

0

50

100

150

200

1 2 3 4 5 6 7 8 9

Labour

Ou

tpu

t

Total Product(’000 tonnes)

MarginalProduct

AverageProduct

Law of Variable Proportions

states that as the quantity of the variable factor is increased with other fixed factors, the marginal product and average product of the variable factor will eventually decline.

Labour

Total Output

OMPL

APL

A*B*

TPL

Labour

Total Output

O

A

B

C

C*

Panel a

Panel b

Stage I Stage II Stage III

Increasing Returns to the Variable Factor: very first stage. MP>0 and MP>APDiminishing Returns to a Variable Factor: the second stage. MP>0 and MP<APNegative Returns: technically inefficient stage of production and a rational firm will never operate in this stage. MP<0 while AP is falling but positive.

Law of Variable Proportions

Elasticity of Substitution• measures the percentage change in factor proportions due

to a change in marginal rate of technical substitution.

б =

• б is effectively a measure of the curvature of an isoquant. • It is a pure number and independent of the units of labour

and capital, since both the numerator and denominator are measured in the same units.

• more curved or convex is the isoquant, the lower is б. • In Leontief (zero substitution) technology, with L shaped

isoquants, there is no substitutability between the inputs and thus б = 0.

• In perfect substitution or linear production technology, the MRTS does not change at all along the isoquant; б is infinite

MRTSMRTSd

LKLKd

)(/

)/(

Expansion Path

is the line formed by joining the tangency points between various isocost lines and the corresponding highest attainable isoquants.O

E

E1

E2

Expansion Path

Labour

Capital

When the production function is homogeneous and factor prices (and hence factor ratio) are given, the expansion path is a straight line through the origin.

If the production function is not homogeneous, the optimal expansion path will not be linear.

Cobb-Douglas Production Function

was proposed by Wicksell and tested against statistical evidence by Charles W. Cobb and Paul H. Douglas in 1928.

where α, β are constants. A is the technological parameter, α is the elasticity of output with respect to capital and β is the elasticity of output with respect to labour.

Properties: i. is a homogeneous of degree (+).If (+) = 1 then Production function exhibits CRS.If (+) > 1 the production function exhibits IRS.If (+) < 1 the production function exhibits DRS.

ii. Isoquants are negatively sloped and convex to the origin iii. MRTSLK is a function of input ratio.iv. Elasticity of substitution is equal to 1.

LAKQ

Leontief Production Function• represents the extreme case of perfect complements • such production functions have ‘L’ shaped or right angled

isoquants.• is also known as fixed coefficient production function.

• production technology always involves inputs labour (L) and capital (K) in fixed proportions to produce a unit of output and αand в are the fixed coefficients.

• a certain amount of each input is required technologically to produce one unit to output. So, the demand for the inputs is uniquely determined, because inputs are required in exact quantities per unit of output.

• Any change in MRTS will not lead to any change in the factor proportions and б = 0.

),min(KL

Q

CES Production Function• Constant Elasticity of Substitution (CES) production

function, was introduced by Arrow, Chenery, Minhas and Solow (also known as ACMS)

Q=A[αK -ρ +(1-α)L -ρ ] -r/ρ where A(>0) is the efficiency parameter which represents

the "size" of the production function; α is the distribution parameter which will help us explain

relative factor shares (so 0<α<1); ρ is the substitution parameter, which will help us derive the

elasticity of substitution and r is the scale parameter which determines the degree of

homogeneity. • It is homogeneous of degree r.

Technical Progress(TP) and its Implications

refers to research and development and investments made to manage technical know how.– may be embodied or investment specific, – may also be disembodied or investment neutral

Q1

Labour

Capital

O

Q

Q

L

K

L1

K1

OLabour

Capital

Q

QQL

QL

O

QK

Labour

Capital

Q

Q

QK

Panel a Panel b Panel c

Neutral TP: if changes in the marginal product of labour and capital are same. Labour augmenting TP: if MPL increases faster than the MPK. Capital augmenting TP :if MPK increases faster than MPL.

Summary

• A production function shows the relationship between inputs and outputs given the state of technology.

• Technical efficiency is a situation when use of more of one input and either the same amount or more of the other input increases output.

• According to the law of variable proportions, as the quantity of the variable factor is increased with other fixed factors, the marginal product and the average product will eventually decline.

• Isoquants are locus of points of different combinations of labour and capital inputs that produce the same level of output. MRTS is the absolute slope of an isoquant; it is equal to the ratio of the marginal products of factor inputs.

Summary• The isocost line represents the locus of points of all the

different combinations of labour and capital that a firm can purchase, given the total cost and prices of the inputs. The (absolute) slope of the isocost line is equal to the ratio of the input prices; i.e. the relative price of labour to capital.

• The optimum level of inputs needed for objective of either minimizing cost of producing a given level of output, or maximizing output at a given cost would be at the point of tangency of an isoquant and an isocost line.

• Expansion path is defined as the line formed by joining such tangency points between the isocost lines and the highest attainable isoquants.

Summary• If a proportional increase in all inputs yields a more than

proportional increase in output, the phenomenon is known as increasing returns to scale; if a proportional increase in all inputs yields an equal proportional increase in output, then there is constant returns to scale. If a proportional increase in all inputs yields a less than proportional increase in output, then there is decreasing returns to scale.

• A homogeneous function one in which if each input is multiplied by , then is factored out of the function. The power of is known as the degree of homogeneity and is a measure of returns to scale.

• Technical progress refers to improvements made in research and development and various investments made to manage technical know how. Technical progress is neutral if changes in marginal products of labour and capital are same; labour augmenting if MPL increases faster than MPK and capital augmenting if MPK increases faster than MPL.