Definition
Cobb-Douglas (C-D) function:
α and 
are (strictly) positive constants.
α: total factor productivity, or shift parameter, or efficiency parameter
β: output elasticity, or function exponents, or value share.
Note: C-D function may not have 
!
If the exponents sum to 1, this function has constant return to scale and is concave but not strictly concave.
If the exponents sum less than 1, the function has decreasing return to scale and is strictly concave:
If the exponents sum greater than 1, the function is neither convex nor concave
At optimum, all arguments of a C-D function will be strictly positive.
Cobb-Douglas production function
Consider a Cobb-Douglas production with two inputs L and K and has constant return to scale technology.
Solve cost minimization function (CMP):
Note: advantage of solving is that the solution does not include price of output (we can link price of output back from zero-profit condition and unit cost function).
Problem is:
Write Lagrange:
FOC conditions
(1)
(2)
(3)
From (1) / (2), we have:
(4)
Substitute (4) into (3):
(5)
Substitute (5) into (4):
(6)
(5) and (6) are conditional demand of inputs.
From (5) and (6), we have the cost function to be:
Unit cost function is
From the cost function: we can further check Shephard's lemma:
Where
- c is cost function
- q is output level
- wi is the price of input i
- zi is the conditional demand for input i
To show that, take partial derivative of cost function with respect to pL, we have:
in (5)
Note
For more general solution (not CTRS case), see http://coin.wne.uw.edu.pl/jhagemejer/wp-content/uploads/cobb_douglas.pdf