Very likely thisll lead to new values for the optimal

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Unformatted text preview: ge slightly. Very likely, this’ll lead to new values for the optimal inputs and therefore a new value for the optimal cost. Without solving the problem again, we can use the envelope theorem to compute the (approximate) change in the optimal cost due to a small change in a parameter. Following the three steps of the envelope theorem we would first write down what we are trying to maximize in terms of parameters. In the case of a 2 inputs CMP this would be (here, both inputs have a lower limit of zero but could in general have non-zero minimum limits): [ [ [ ( ( ) ) ⏟ ⏟ { ⏟ ⏟ } { ⏟ } Next, we would differentiate the parameterized Lagrangian with respect to the parameter that is changing and then evaluate that derivative at the optimal solution to get the change in the optimal Lagrangian due to the change in the parameter. For example, here are the expressions for the change in the optimal Lagrangian due to, all else constant, a changes in the parameters , respectively8: But since these derivatives are in fact: ( ) That is, if the price of input 1 rises, total cost must increase or stay the same (why...
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