consider the same function (0.4), and suppose now that the choice of the optimal x is constrained by x x, where x is an arbitrary number, satisfying 0 < x < b.

consider the same function (0.4), and suppose now that the choice of the optimal x is constrained by x x, where x is an arbitrary number, satisfying 0 < x < b. (0.5) 1. Graphically display the nature of the new constraint relative to the optimization problem and the function f. 2 2. Solve via the Lagrangian method for the optimal solution. 3. Lagrange multiplier. Suppose that we increase the bound x by dx. What is the marginal e§ect this has on the value of the function to be optimized in terms of a and b. How does it compare to the value of the Lagrangian multiplier which you just computed. 4. Interpretation of Lagrangian multiplier. Imagine now that we cannot insist that the decision maker respect the constraint but that we can ask a penalty, say , for every unit of x over and above x. What is the price that we would have to charge so that the decision maker would just be happy to exactly choose x = x and thus in fact respect the constraint even so did not face the constraint directly. In the process we have replaced the strict constraint with a price for the constraint (equal to the Lagrange multiplier). At the optimum the price is equal to the marginal value of the objective function. For this reason we refer to it as the shadow price of the constraint. It is the smallest price we can associate to the constraint so that the decision maker, facing this price would respect the constraint.

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