Any ideas on the following would be much appreciated. I’m interested in how the shadow price of non-renewable resource changes when new constraints are introduced to the problem. So, assume that there is optimal extraction path and a corresponding shadow price path for the resource. Next, an additional constraint is introduced, as a consequence of which the original optimal path is no longer attainable. Is there some general result or intuition based on which it is possible to say that on the new optimal path the shadow price is always lower?
A simple example follows. Assume two-periods, and that the resource stock is equal to $100$, consumption is $q$. The problem is the following.
$\max_{q_1,q_2} -(100-q_1)^2 -(100-q_2)^2$
subject to
$q_1 + q_2 \leq 100$ (corresponding KKT multiplier is $\lambda$)
The additional constraint is
$q_1 \leq 40$ (corresponding KKT multiplier is $\varphi$)
Without the additional constraint the shadow price of the stock is $\lambda = 100$. With the additional constraint, the shadow price of the stock is less, $\lambda = 80$.
Is it possible to generalise this result, somehow in terms of the original optimal path not being attainable in the presence of the additional constraint, and thus the shadow price being lower?