In (environmental) econ class we had to solve the following problem:
min
costs = clean*2 + 1*dirty (where 2 and 1 are the respective input prices)
s.t.
production: 8 = (clean^0.4)*(dirty^0.6)
climate constraint: clean > 2*dirty
Using the Lagrange-multiplier of production as the price for the output good, all costs are exactly covered. However, if I change the climate constaint to:
climate constraint: clean > 14 (just to give an example)
the costs are not covered anymore if I take the the Lagrange-multiplier as price for the output good. Interestingly, there is also no Problem if I set the climate constraint to:
climate constraint: clean > 2 (which is more than would be optimal without the climate constraint)
Can anyone explain to me why this price-interpretation of the Lagrange-multiplier sometimes works and sometimes doesn't? Could it be connected to whether the solution is interior or not?
Best and thanks in advance,
F.