[Calculus] Cobb-Douglas Production Function
By: Tao Steven Zheng (郑涛)
【Problem】
The Cobb-Douglas production function is an econometric model formulated by American economist Paul Douglas (1892 - 1976) and mathematician Charles Cobb (1875 - 1949) in 1927. They modelled total production as a function of labour and capital investment
:
where .
If the budget constraint is , where
are the unit cost of labour and capital. Determine the labour and capital values that maximizes production in terms of
.
【Solution】
This problem requires the use of Lagrange multipliers. Consider the following functions:
Take the following partial derivatives in order to generate the equations for .
Therefore, we get the equations
Isolate for for both of the above equations and isolate for either
or
(here I chose
).
Subsequently,
Substitute this value of into the budget constraint to solve for
. Then use the newfound value of
to solve for
.
Therefore, the solution set that maximizes production is
