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Optimum Input Combination

  Optimum Input Combination

Illustration 49

An industry is manufacturing output using labour and capital in such volumes that marginal product of labour is 30 and marginal product of capital is 16. The remuneration rate of labour is $6 and price of capital is $4. Does the industry using effective factor of combination for production or else what must be done to accomplish economic efficacy?

Solution

Effectiveness stipulation for factor use that is optimum combination of factor) needs that the following condition should be fulfilled.

                                    MPA    =          MPB
                                      r                        p

Now,                           MPA    =          30
                                      r                     6

                                    MPB     =          16
                                       p                   4

As                                30        >          16
                                     6                       4

The given factor combination cannot therefore be effective or optimal factor combination for the reason that the industry is getting more output from a dollar expended on labour than in capital.

To accomplish economic efficacy in the use of resources and to optimise profits the firm must surrogate labour for capital and hence

                                    MPA                parities             MPB
                                       r                                               p

Illustration 50

Consumer product manufacturing company hearsays the following data about its manufacture and factors used:

O

B

A

O

B

A

550

30

90

525

25

120

600

30

120

600

30

120

Where O is the output, A is the labour and B is the capital.

If the remuneration rate of labour is $10 and price of capital is $20, does the combination of input of 30B and 120A denotes the least cost factor combination? If not must it use more labour and less capital or less labour and more capital?

Solution

Least-cost combination factor needs that the following situation must be met.

                                    MPA   =          MPB
                                      r                     p

where r is remuneration and p is price of capital.

Now, in the given problem, when factor combination is changed from 30B + 120A to 30B + 90A, the productivity decreases by 50 units that is (600 – 550). Therefore, MPA = 50.

Likewise, when factor combination is changed from 30B + 120A to 25B + 120A, the productivity decreases to 75 units that is (600 – 525). Therefore, MPB = 75. substituting these values of marginal products of labour and capital and the factor prices in the condition for least cost factor combination, we have the following:

                        MPA   =          MPB
                          r                      p

                        50        >          75       
                        10                     20

The company will utilise more capital and less labour as output per unit of a dollar (50 / 10 = 5) spent on it is greater than the output per unit of a dollar (75 / 20 = 3.75) spent on labour.

Illustration 51

Given the production function O = 200B^0.5A^0.5. Ascertain the maximum input combination for producing 2000 units of output if remuneration rate of labour (r) is $60 and price per unit of capital (p) is $80. what is optimum cost of production?

Solution

The given production function is

                      O         =          200B^0.5A^0.5

MPA   =          dO       =          200 * 0.5B ^ 0.5A^-0.5
                       dA

                                   =          100B^0.5 A^-0.5

                        MPB    =          dO       =          200 * 0.5B^-0.5 A^0.5
                                                dB

                                    =          100B^-0.5 A^0.5

                        MPA   =          100B^0.5ª^-0.5           =          B
                        MPB                100B^-0.5ª^0.5                        A

In equilibrium MRSAB          =          MPA   =          r
                                                        MPB                p

Therefore, in optimal input combination, B / A = r / p

Or,                   B         =          r.A
                                                P

To obtain the value of A we are to substitute B = r / p * A in the production function with O = 2000 units. Therefore,

                        2000    =          200B^0.5A^0.5

                                    =          200 (r.A) ^0.5 A^0.5
                                                        p

                        2000    =          200 * A (r) ^ 0.5
                                                              (p)

Substituting the values of r = 60 and p = 80

                        2000    =          200 A * (60) ^0.5
                                                              (80)

                        2000    =          200 A * 0.75^0.5

                                    =          200 A * 0.866

                                    =          173.2A

                        A         =          2000 / 173.2    =          11.55

Now, using the equation for expansion path (B = r / p . A), we can procure the value of B by substituting the values of A, r and p. Therefore,

                        B         =          60        *          11.55
                                                80       

                                    =          8.67

Therefore, maximum combination of inputs comprises of 11.55 units of labour and 8.67 units of capital. This will make sure minimum feasible cost for producing 2000 units of output.

In order to ascertain this minimum cost we substitute the maximum values of A and B procured above and the given prices of labour and capital (i.e. r and p) in the cost function.

Therefore,

                        F          =          rA + pB

                                    =          60 * 11.55 + 80 * 8.67

                                    =          693 + 693

                                    =          1386

Therefore, minimum cost of producing 2000 units of output is $1386.

Illustration 52

Given that: O = 200B^0.5 A^0.5, F = $1600, r = 60, p = 80. Ascertain the volume of labour and capital that the industry must use with respect to optimise output. What is this level of output?

Solution

The problem of restrained optimisation is:

Optimise O = 200 ^ ½ A ^ ½

Subject to cost restraint:

            1000    =          60A + 80B

MPB    =          dO       =          1          200 * B^- ½ A^ ½
                        dB                    2

MPA   =          dO       =          1          200 * B^ ½ A^- ½
                       dA                    2

MPB    =          100B^- ½ A^ ½          =          A         …..Equation (1)
MPA                 100B^ ½ A^ -½                      B

For output optimisation

            MPB    =          p                                                          …..Equation (2)
            MPA                 r

Substituting MPB       =          A         and r = 60 and p = 80 in Equation (2) we have
                    MPA                    B

                                                A         =         4          or 4B   …..Equation (4)
                                                B                     3               3

Putting the value of A = 4 B in the cost restraint equation we get,
                                        3

                                    1600    =          rA + pB

                                                =          60 * 4B + 40B
                                                                     3
                                                =          80B + 40B

                                    B         =          1600 / 120       =          13.33

Substituting B = 40 in the cost restraint equation we get,

                                    1600    =          60 * A + 80 * 13.33

                                                =          60A + 1066.4

                                    60A     =          1600 – 1066.4

                                                =          533.6

                                    A         =          533.6 / 60        =          8.9

Therefore, output maximising amounts of capital and labour are 13.33 and 8.9 correspondingly. To get the level of output produced we substitute these amounts of capital and labour in the given production function.

                                    O         =          200B^ ½ A^ ½

                                    O         =          200 √13.33 * √8.9

                                                =          200 * √118.6

                                                =          200 * 10.89

                                                =          2178

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