Objective; Find the input levels that minimize the production cost for a given level of output. (Cost minimizer assumption.)
This is an optimization problem, in particular choose input levels to minimize costs. But subject to a constraint: the inputs must produce a given level of output, y.
Mathematical technique for solution of constrained optimization: Lagrange multipliers.
It turns out that, assuming the Cobb Douglas production function, then
the optimal level of inputs produce a COST FUNCTION of the form
It looks a mess, but notice that it is multiplicative, so taking logs will achieve a linear expression ready for regression.
Further, using the fact that
the logged
version can be rewritten as
From this lot we can get at what's of interest,
.