TUT04Questions

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Tutorial 4 Questions ECO204Y1 (Ajaz) Week 4, Oct 2nd - 6th 2023 1. Consider the inequality-constrained profit maximization problem max x π ( x ) = 10 x − x 2 subject to x ≤ c . Answer the following questions (a) Clearly state the Lagrangian and all the Kuhn-Tucker conditions associated with this problem. (b) Solve the problem. 1
(c) Interpret the Lagrange multiplier of the solution. (d) Differentiate the Lagrangian L with respect to c and find the value of this derivative at the solution x ⋆ and λ ⋆ . What do you notice? (e) The method in part d) can be used to find how your profits change when you change any parameter. Consider the new problem max x π ( x ) = ax − x 2 subject to x ≤ 1 i. Repeat parts a) and b) for this new problem assuming a > 0. 2
ii. Find d L da evaluated at the solution x ⋆ , λ ⋆ . What is its interpretation? 2. Consider the inequality-constrained profit maximization problem max x π ( x ) = 10 x − x 2 subject to x ≥ c . (a) Why do we express the constraint as − x + c ≤ 0? WARNING: ALWAYS DO THIS WHEN FACED WITH A CONSTRAINT x ≥ c . (b) Clearly state the Lagrangian and all the Kuhn-Tucker conditions associated with this problem. Make sure to use your answer in a). 3
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