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Establishment Of Short Run Cost Function

  Establishment of Short Run Cost Function

Illustration 55

An industry manufacturing javelin sticks contains a production function provided by O = 2√BA. In the short run the industry’s volume of capital equipment is set at B = 400. The rental cost for B is $2 and the remuneration rate is $8.

Compute the following:

  1. The industry’s short run aggregate and average costs.
  1. What are STC, SAC and SMC for manufacturing 50 javelin sticks?

Solution

The provided production function of the industry is

                        O         =          2√BA

With B = 200, in the short run the short run manufacturing function is as follows:

                        O         =          2√400*A

                                    =          2*20*√A         =          40√A

Cost Function                         F          =          rA + pB

Where r and p are remuneration rate and price;

Provided that              r = 8 and w = 2

                                    F       =          8*A + 2*B

With the given            B        =          400

                                    F       =          8*A + 2*400               …..Equation (1)

The short run manufacturing function when B = 400 as procured above is:

                                    O         =          40√A

Considering square of both the sides we obtain,

                                    O^2     =          1600A

Or                                O^2     =          A                                 …..Equation (2)
                                    1600

Substituting (2) in (1) we have the following:

                                    F          =          800 + 8*O^2
                                                                          1600

                                                =          800 + O^2                   …..Equation (3)
                                                                        200

The above equation (3) depicts the short run aggregate cost function, and to obtain the short run average cost function, we divide the short run aggregate cost function in (3) by productivity O.

Therefore,

                                    SAC    =          800 + O^2
                                                                        200
                                                                   O

                                                =          800 +    O       
                                                                O       200

Short run Marginal Cost Function

Short run marginal cost function can be procured by considering the first derivative of the short run cost function.

Short run cost function as obtained above are:

                                    F          =          800 + O^2
                                                                        200

            SMC    =          dF        =          2*O     =         O       
                                    dO                    200                100

In case if the Javelin Sticks are equal to 50,

Then,

            STC     =          800 + 50^2      =          800 + 2500
                                                200                                 200

                        =          800 +   12.5

            STC    =          812.5

            SAC    =          STC    =          812.5   =          16.25
                                      O                     50

            SMC   =                =          50       =          0.5
                                  100                  100

Illustration 56

If O = X(BA)^0.5, what would be the short run cost function when B = 900, Also determine the Marginal Cost Function (MC).

Solution

With B = 900, the short run manufacturing function can be rewritten as follows:

                        O         =          X . (900.A)^0.5

Squaring both the sides we have the following:

                        O^2     =          900*X^2*A                            …..Equation (1)

Now, the short run cost function is,

                        F          =          Total Fixed Cost + Total Variable Cost

                        TFC     =          Bp       =          900p

                        TVC    =          rA

Where p is the rental price of capital and r is the remuneration rate and provided B = 900.

                        F          =          900p    +          rA                    …..Equation (2)

From Equation (1) we have

                        A         =            O^2 
                                                900X^2

Substituting the value of A in Equation (2) we obtain the following short run cost function:

                        F          =          900p    +          r. O^2              …..Equation (3)
                                                                         900X^2

Note that total variable cost function is       r. O^2
                                                                                900X^2

Deviating the total variable cost function with regards to productivity or output O we procure the following Marginal Cost Function MC:

                        MC      =          dTVC
                                                   dO

                                    =            2.r.O
                                                900X^2

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