Production Function with two variable inputs

Rachita Gulati

Production in the Long-Run All inputs are now considered to be variable (both L and K

in our case) How to determine the optimal combination of inputs?

To illustrate this case we will use production isoquants.

An isoquant is a curve showing all possible combinations of inputs physically capable of producing a same level of output. Isoquant are also known as equal-product curves.

Isoquants

0

20

40

60

80

100

120

140

160

0 10 20 30 40Labour

Ca

pit

al

An isoquant shows the

combinations of capital and labour that produce the

samesame level of output

240

358

Factor combination Labour Capital

A 10 100

B 20 60

C 30 50

These Are Called IsoquantsThey slope down and to the right

Due to Marginal Rate of Technical Substitution

Higher isoquant represent higher output levelThey are convex to the origin They do not cross, although they are not necessarily

parallel

MRTSLK would be the amount of capital that the firm would be willing to give up for an additional unit of labour.

MRTSLK = -K/L = MPL/MPK

MRTSKL would be the amount of labour that the firm would be willing to give up for an additional unit of capital.

MRTSKL = -L/K = MPK/MPL

Marginal Rate of Technical Substitution (MRTS)

Law of Diminishing Marginal Rate of Technical Substitution:

for Isoquant Q = 52Combination K L

A 12 1B 8 2C 5 3D 3 4E 2 5

K L MRTSKL

-4 1 4 -3 1 3 -2 1 2 -1 1 1

L = 1

K= -2

L = 1

K = -3

L= 1

K=- 4

A

B

C

DE

Downward sloping isoquant

MRTS = K/L = - MPL/MPK

Units of KEmployed Output Quantity (Q)

18 37 60 83 96 107 117 127 12812 52 64 78 90 101 110 119 1208 37 52 64 73 82 90 97 1045 31 47 52 67 75 82 89 953 24 39 47 52 67 73 79 852 17 29 41 47 52 64 69 731 8 18 29 39 47 52 56 52

1 2 3 4 5 6 7 8Units of L Employed

52

52

52

5252

52

Higher isoquant represents higher output

Units of KEmployed Output Quantity (Q)

8 37 60 83 96 107 117 127 1287 42 64 78 90 101 110 119 1206 37 52 64 73 82 90 97 1045 31 47 58 67 75 82 89 954 24 39 52 60 67 73 79 853 17 29 41 52 58 64 69 732 8 18 29 39 47 52 56 521 4 8 14 29 27 24 21 17

1 2 3 4 5 6 7 8Units of L Employed

Isoquant

Economic region of production

Isoquants

Economic Region of Production

Ridge Line

Substituting inputsThere exists some degree of substitutability between inputs.

Different degrees of substitution:

Sugar

a) Perfect substitution b) Perfect complementarity

All other ingredients

Natural flavoring

Q

Q

Capital

Labor L1 L2 L3 L4

K1 K

2 K

3

K4

Cornsyrup

c) Imperfect substitution

Isocost linesIsocost line is the locus of points of all the different

combinations of labour and capital that a firm can employ, given the total cost and prices.

Isocost lines represent all combinations of two inputs that a firm can purchase with the same total cost.

If the price of labour is wage (w), the price of capital is interest (r), then total cost incurred by the firm is summarized as

C wL rK (M=Px. X+ Py. Y)

Isocost line

Budget line

Isocost curvesIsocost Lines

AB C = $100, w = r = $10

A’B’ C = $140, w = r = $10

A’’B’’ C = $80, w = r = $10

AB* C = $100, w = $5, r = $10

intercept slope

C wK L

r r

The intercept of isocost line on the capital axis is the maximum amount of capital employed, when labour is not used in production process and is given by C/r.

The intercept on the labour axis is the maximum amount of labour used in the production process and is given by C/w.

Slope of the isocost lineSlope= Price of labour/Price of capital= w/r

Changes in the Isocost Line

Changes in total costDecrease lead to a parallel,

inward shift in the isocost line.Increases lead to a parallel,

outward shift.

Changes in Price of labourA decreases in the price of

labour L rotates the isocost line counter-clockwise.

An increases rotates the budget line clockwise.

L

L

L

K

Rs. 500

Rs. 400

Rs. 300

10050

60 If price of labour falls from Rs. 4 to Rs. 3, isocost line shifts outwards.

Producer’s Equilibrium: Optimal Combination of Inputs

MRTSLK = -K/L=w/r (slope of isoquant=slope of isocost line)

Producer’s equilibriumProducer’s equilibrium shows lowest cost producing a given

level of output, where the isoquant corresponding to this output is tangent to the isocost line.

Thus, optimal combination of factor inputs depends on the relative prices of factor inputs and on the degree to which they can be substituted for one another.

This relationship can be stated as follows:MRTS=-K/L=MPL/MPK = PL/PK = w/r

(or MPL/PL= MPK/PK)

MPL/w= MPK/r

Optimal level of inputsThe optimality conditions given in the previous slides ensure

that a firm will be producing in the least costly way, regardless of the level of output.

But how much output should the firm be producing?Answer to this depends on the demand for the product (like

in the one input case as well).

Expansion pathThe locus of equilibrium points of isoquant with the

lowest possible isocost lineIt shows all the cost minimising input combinations for

various levels of output the firm could produce in the long run.

Un

its

of

cap

ita

l (K

)

O Units of labor (L)

100

200

300

Expansion path

TC =Rs. 20 000

TC =Rs. 40 000

TC =Rs. 60 000

The long-run situation:both factors are variable

Producer’s equilibrium vs. Consumer’s equilibrium

Producer’s equilibrium Producer’s theory Maximization of output Isoquant and isocost line Expansion path Equilibrium condition

MRTS= -K/L=w/r

Consumer’s equilibrium Consumer’s theory Maximization of satisfaction Indifference curve and

budget line Income consumption curve

or price consumption curve Equilibrium condition

MRS= -Y/X=Px/Py

Returns to ScaleLet us now consider the effect of proportional increase in

all inputs on the level of output produced.To explain how much the output will increase, we will use

the concept of returns to scaleReturns to scale refers to the degree by which output changes

as a result of a given change in the quantity of all the factor inputs used in production.

Returns to Scale

What happens to output when allall inputs are increased by a given percentage?

Three will be three situations:Increasing Returns

output increases by a largerlarger percentage Increase in factors by 10%, output increase by 20%

Constant Returnsoutput increases by the samesame percentage

Increase in factors by 10%, output increase by 10%Decreasing Returns

output increases by a smallersmaller percentage Increase in factors by 10%, output increase by 5%

Returns to Scale

Units of LEmployed Output Quantity (Q)

8 37 60 83 96 107 117 127 1287 42 64 78 90 101 110 119 1206 37 52 64 73 82 90 97 1045 31 47 58 67 75 82 89 954 24 39 52 60 67 73 79 853 17 29 41 52 58 64 69 732 8 18 29 39 47 52 56 521 4 8 14 20 27 24 21 17

1 2 3 4 5 6 7 8Units of K Employed

In the previous table we are experiencing increasing returns to scale

Similarly, constant returns to scale and decreasing returns to scale are possible.

Measurement of Returns to Scale

Coefficient of output elasticity:

EQ=

So if,EQ>1, increasing returns (proportionate increase in output is more

than proportionate increase in inputs)

EQ=1, constant returns (proportionate increase in output is in same

proportion to that of increase in inputs)

EQ<1, decreasing returns (proportionate increase in output is less

than proportionate increase in inputs)

percentage change in output

percentage change in all inputs

Constant Returns to Scale

fig

Un

its o

f ca

pita

l (K

)

Units of labor (L)

100

200

300

400

500

a

b

cR

1K+1L (100)2K+2L (200)3K+3L (300)

Increasing Returns to Scale (beyond point b)

fig

Un

its

of

cap

ita

l (K

)

Units of labor (L)

100

200

300

400

500

a

b

cR

600

1K+1L (100)1.7K+1.7L (200)2.2K+2.2L (300)

Decreasing Returns to Scale (beyond point b)

fig

Un

its o

f ca

pita

l (K

)

Units of labor (L)

200

300

400

500

a

b

cR

1K+1L (100)2.5K+2.5L (200)4.5K+4.5L (300)

Reasons for Increasing Returns to Scale:Division of labor (specialization)

Indivisibility of machinery or more sophisticated machinery justified

Economies of scale

Decreasing returns to scale can result from certain managerial inefficiencies:problems in communicationincreased bureaucracyIncreased use of fixed factor

Production function as power function

Power function is the most frequently used type of production function in empirical work, even though it cannot exhibit two directions for marginal product on the same function.One reason for its popularity is that it can be readily transformed into a function with two or more independent variables:

mn

d3

c2

b1 VVVaVQ ...

Production Function as Power FunctionProduction function with one variable input

Q = aLb

ifb > 1, Q increasing at increasing rate: MPL increasing

b = 1, Q increasing at constant rate: MPL constant

b < 1, Q increasing at decreasing rate: MPL decreasing

Major advantage of the power function is the fact that it can be transformed in a log-linear function

log Q = log a + b log L

The Cobb-Douglas Production Function

A special case of power functions is Cobb-Douglas Production Function

Q = AKaLb

Estimated using natural logarithms

ln Q = ln A + a ln K + b ln L

Original version with constant returns to scale ( b + 1 - b = 1) introduced by Cobb in 1928.

Q = aLbK1-b,

Reformulation by Cobb and Douglas:Q = aLbKc

b + c = 1, constant returns

b + c > 1, increasing returns

b + c < 1, decreasing returns

Properties of the Cobb-Douglas function that have kept it so popular for 80 years

1. Both inputs have to be used simultaneously to get an output

2. Can exhibit different returns to scale

3. Allows to investigate MP for any factor while holding all others constant. So it is useful both in short-run and long-run analysis.

4. Elasticities are equal to the exponents b and c (constant in this formulation)

K

Qc

K

QMP

L

Qb

L

QMP

K

L

Assignment problems1. If an estimated Cobb-Douglas production function is

Q = 10L0.8K0.6 (a) what are the output elasticities of capital and labour? If

the firm increases only the quantity of capital or only the labour used by 10%, by how much would output increase?

(c) What type of returns to scale does this production function indicate? If the firm increases at the same time both the quantity of capital and labour used by 10%, by how much would output increase?

2. Suppose that the production function for a commodity is given by Q = 10L0.5K0.5 , where Q is the quantity of output, L is the labour units, and K is the capital units.

(a)Indicate whether this production function exhibits constant, increasing or decreasing returns to scale?

(b)Does the production function exhibits diminishing returns? If so, when does the law of diminishing returns begin to operate?

3. The Royal furnishing manufacturers office furniture with the following production function:

Q = 20L0.1K0.9

the firm currently is producing with maximum efficiency and using 20 units of capital and 50 units of labour.

(a)What is the rate of output?

(b)What are the relative prices of capital and labour? What will be the actual price of labour and capital? Explain