11

Optimal Combination of Inputs Now we are ready to answer the

question stated earlier, namely, how to determine the optimal combination of inputs

As was said this optimal combination depends on the relative prices of inputs and on the degree to which they can be substituted for one another

This relationship can be stated as follows:

MPL/MPK = PL/PK

(or MPL/PL= MPK/PK)

22

Optimal Combination of Inputs: An Example

Let’s consider a company that wants to minimize the cost of producing a given output per hour Q

Number of workers used per hour is L and number of machines used per hour is K and

Q = 10(LK)0.5.

Wage rate is $8 per hour, and the price of a machine is $2 per hour

How many workers and machines the company should use, if the company wants to produce 80 units of the output per hour?

33

Optimal Combination of Inputs Example Continues

Q = 10(LK)0.5

Calculate the marginal products:

MPL = 0.5(10)K0.5L-0.5 = 5(K/L)0.5

MPK = 0.5(10)L0.5K-0.5 = 5(L/K)0.5

Thus, if MPL/PL = MPK/PK

2)(5

8)(5 5.0

KL5.0

LK

44

Optimal Combination of Inputs Example Continues

Multiplying both sides of the equation by (K/L)0.5 we get

5K/8L = 5/2,

which means that K = 4L. And since Q = 80,

10(LK)0.5 = 80

10[L(4L)0.5 = 80 L = 4 and K = 16

55

Isocost Curves

Example 3

Table 7.11 Input Combinationsfor $1000 BudgetCombination L K

A 0 5B 2 4C 4 3D 6 2E 8 1G 10 0

Assume PL =$100 and PK =$200

66

Isocost Curve and Optimal Combination of L and K

Isocost and isoquant curve for inputs L and K

5

10 L

K

“Q52”

100L + 200K = 1000

77

Optimal Levels of Inputs

The 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)

88

Optimal Levels of Inputs continued

The earlier rule, MRP = MLC, can be generalized: a firm in competitive markets should

use each input up to the point wherePi = MRPi

where Pi = price of input iMRPi = marginal revenue product of input i

99

So in the two input case, firm’s optimality condition is

PX = MRPX and PY = MRPY

A profit maximizing firm will always try to operate at the point where the extra revenue received from the sale of the last unit of output produced is just equal to the additional cost of producing this output.This is same as

MR = MC

1010

Uni

ts o

f cap

ital (

K)

O Units of labor (L)

100

200

300

Expansion path

TC =£20 000

TC =£40 000

TC =£60 000

The long-run situation:both factors variable

Expansion Path: the locus of points which presents the optimal input combinations for different isocost curves

1111

Returns to Scale

Let 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 scale

1212

Returns to Scale continued

If all inputs into the production process are doubled, three things can happen: output can more than double

increasing returns to scale (IRTS) output can exactly double

constant returns to scale (CRTS) output can less than double

decreasing returns to scale (DRTS)

1313

Returns to Scale An Example:

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 20 27 24 21 17

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

In this production process we are experiencing increasing returns to scale

1414

Reasons for Increasing or Decreasing Returns to Scale:

Often we can assume that firms experience constant returns to scale:

for example doubling the size of a factory along with a doubling of workforce and machinery should lead to a doubling of output

why could a greater (or smaller) than proportional increase occur?

1515

Reasons for Increasing Returns to Scale:

Division of labor (specialization) increased labor productivity

Indivisibility of machinery or more sophisticated machinery justified increased productivity

Geometrical reasons

Decreasing returns to scale can result from certain managerial inefficiencies: problems in communication increased bureaucracy

1616

Measurement of Returns to Scale

Coefficient of output elasticity

EQ=

So if, EQ > 1, increasing returns

EQ = 1, constant returns

EQ < 1, decreasing returns

percentage change in Qpercentage change in all inputs

1717

Measurement of Returns to Scale continued

Multiplying the coefficients of the production function:

If original production function isQ = f(X,Y)

and if the resulting equation after the multiplication of inputs by k is

hQ = f(kX, kY)

where h presents the magnitude of increase in production

1818

Then, ifh > k, increasing returnsh = k, constant returnsh < k, decreasing returns

1919

Graphically, the returns to scale concept can be illustrated using the following graphs

Q

X,Y

IRTSQ

X,Y

CRTSQ

X,Y

DRTS

2020

Constant Returns to Scale

Uni

ts o

f cap

ital (

K)

0

1

2

3

4

0 1 2 3Units of labor (L)

200

300

400

500

600

a

b

cR

2121

Increasing Returns to Scale (beyond point b)

0

1

2

3

4

0 1 2 3

Uni

ts o

f cap

ital (

K)

Units of labor (L)

200

300

400

500

600

a

b

cR

700

2222

Decreasing Returns to Scale (beyond point b)

0

1

2

3

4

0 1 2 3

Uni

ts o

f cap

ital (

K)

Units of labor (L)

200

300

400

500

a

b

cR