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Macroeconomics Exam Review 246

# Macroeconomics Exam Review 246 - Solutions for Foundations...

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the second and third constraints. The feasible solution 𝑥 𝑏 = 0, 𝑥 𝑐 = 5, 𝑥 𝑑 = 10, where the constraints are linearly dependent, is known as a degenerate solution. Degeneracy is a significant feature of linear programming, allowing the theoretical possibility of a breakdown in the simplex algorithm. Fortunately, such breakdown seems very rare in practice. Degeneracy at the optimal solution indicates multiple optima. One way to proceed in this example is to arbitrarily designate one constraint as redun- dant, assuming the corresponding multiplier is zero. Arbitrarily choosing 𝜆 𝑚 = 0 and proceeding as before, complementary slackness ( 𝑥 𝑑 > 0) requires that 𝐷 𝑥 𝑑 𝐿 = 3 2 𝜆 𝑓 𝜆 𝑙 = 0 or 𝜆 𝑙 = 3 2 𝜆 𝑓 (5.106) Nonnegativity of 𝜆 𝑙 implies that 𝜆 𝑓 3 2 . Substituting (5.106) in the second first-order condition yields 𝐷 𝑥 𝑐 𝐿 = 1 2 𝜆 𝑓 2 𝜆 𝑙 = 1 2 𝜆 𝑓 2(3 2 𝜆 𝑓 ) = 5 + 2 𝜆 𝑓 < 0 for every
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