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Topic:4 Attributes of Production function

Homogeneity of production function:

There are occasions throughout this book when we use the rules of indices and definitions

ofbn. For the moment, we concentrate on one specific application where we see theseideas

in action. The output, Q, of any production process depends on a variety of inputs,

known as factors of production. These comprise land, capital, labour and enterprise. For

simplicity we restrict our attention to capital and labour. Capital, K, denotes all man-

made aids to production such as buildings, tools and plant machinery. Labour, L, denotes

all paid work in the production process. The dependence of Q on K and L may be

written

Q = f K, L! which is called a production function. "nce this relationship is made

e#plicit, in the form of a formula, it is straightforward to calculate the level of production

from any given combination of inputs.

For e#ample, if 21

3

1

100 LKQ = , then the inputs K = $% and L = &'' lead to an output

2

1

3

1

)100()27(100=Q

= &''(!&'!

= ('''

"f particular interest is the effect on output when inputs are scaled in some way. )fcapital and labour both double, does the production level also double, does it go up bymore than double or does it go up by less than double* For the particular productionfunction,

2

1

3

1

100 LKQ =

we see that, when K and L are replaced by $K and $L, respectively,

2

1

3

1

)2()2(100 LKQ =

3

1

3

1

3

1

.2)2( KK = and 21

2

1

2

1

.2)2( LL =

The second term, 21

3

1

100 LK , is +ust the original value of Q, so we see that the output is

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multiplied by 65

2.

oreover, because / is less than &, the scale factor is smaller than $. )n fact, my

calculator gives =65

2 &.%0 to $ decimal places! so output goes up by +ust less thandouble.

)t is important to notice that the above argument does not depend on the particular value,$, that is taken as the scale factor. 1#actly the same procedure can be applied if the

inputs,K and L, are scaled by a general number 2reek letter pronounced 3lambda4!.

5eplacing K and L by K and L respectively in the formula Q = ( ) ( ) 21

3

1

.100 LK

2ives Q = 21

3

1

6

5

.100. LK

6e see that the output gets scaled by 65

, which is smaller than since the power, /,

is less than &. 6e describe this by saying that the production function e#hibits decreasingreturns to scale.

)n general, a function Q = f K, L! is said to be homogeneous if

f K, L! =n

f K, L! for some number, n. This means that when both variables

K and L are multiplied by we can pull out all of the s s as a common factor,n

.

The power, n, is called the degree of homogeneity.

)n the previous e#ample we showed that

),(),( 65

LKfLKf = and so it is homogeneous of degree /. )n general, if thedegree of homogeneity, n, satisfies:

n 7 &, the function is said to display decreasing returns to scale

n = &, the function is said to display constant returns to scale

n 8 &, the function is said to display increasing returns to scale.