Application of Cost Function
The estimated cost function is useful in determination of optimum output level and optimum scale. It is also a useful concept to measure optimal lot size and derive the supply schedule.
Optimum output: The average cost is minimum at the optimum output level or where the average cost is equal to marginal cost (AC = MC). For example optimum output level is to be estimated for the following cost function.
TC = 128 + 6Q + 2Q2
Then Ac= TQ/Q = 128/Q + 6 + 2Q
And MC = dc/dQ = 6+4Q
Therefore d(AC)/dQ = [-128/Q2 +2] = 0 Hence Q = 8
Alternately setting AC = MC
128/Q + 6 + 2Q = 6 + 4Q
Or 2Q2 = 128
Therefore Q = 8 which is the optimum output level for the short run. Note that it is a short run case as TFC = 128.
Optimum scale: The optimum scale is given by the value of plant size (K) at which the total cost is least. The two conditions for optimum scale are:
Necessary condition: ∂C/∂K = 0
Sufficient condition: ∂C /∂K2 > 0
For example, the optimum scale has to be found for the cost function;
C = 0.04Q3 - 0.9Q2 +(11 - K) + 5K2
Or K = 0.1Q
∂2C /∂K2 = 10 >0
Thus at K = 0.1Q total cost is the least. If the firm plans to produce 10 units of output its optimum scale equals 1, if it plans to produce 50 units, the optimum plant size is 5, and so on.
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