Mathematical Derivation of Cost Function
S.Zakir Abbas ZaidiSaturday, November 20, 2021
Driving Short-run Cost Function from Cobb-Douglas Production Function
Normally, every firm face two problems while producing their product,
1. How much quantity of goods to produce?
2. And how much labor and capital is required to produce output most efficiently?
The production Function state relationship between output and input such as,
Q = ƒ (K, L)
Cobb-Douglas Production Function
Suppose cost components of a firm are as follow,
Price of a new machine — K = $36 million
The wage of each worker — W = $144 per day
Rental rate of capital or price of capital — r = $200 per annum
“r” = (interest rate + Rate of Depreciation) x Price of capital
Lets determine the value of L
Q = L⁰.5 36⁰.5 = L⁰.5 x 6
We can calculate the Marginal Cost by taking first derivative of TC
TC = 4Q² + 7,200
⸫ MC = dTC/dQ = (4*2) Q²-1 + 0 = 8Q
Minimum Cost condition where,
MC = ATC
8Q = 4Q + 7,200/Q
8Q — 4Q = 7,200/Q
4Q = 7,200/Q
Multiply both sides of equation by Q
Q(4Q) = (7,200/Q) Q
4Q2 =7,200
⸫ Q = (7,200/4)¹/2 = 42.43
Identical value MC and ATC obtained,
MC = 8 * 42.43 = 339.4
ATC= 4(42.43) + 7,200/42.43