# 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:

- The industry’s short run aggregate and average costs.

- 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 = O = 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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**Other topics under Theory of Production and Cost analysis:**

- Break Even and Leverage Analysis
- Concept of Cost
- Cost Volume Profit Analysis for Accomplishing Target Profits
- Elasticity of Supply and Its Function
- Establishment of Cost Function Analysis
- Estimation of Returns To Scale
- Isoquants, Equal Product Curves
- Linearity Assumptions
- Linearity Assumptions and Choice of Product and Process
- Long Run average Cost Curve
- Optimum Input Combination
- Production Function with Two Variable Inputs
- Short Run Cost Function
- Survival Technique
- Theory of Production - Returns to One Variable Factor