Macroeconomics Exam Review 152

Macroeconomics Exam Review 152 - Solutions for Foundations...

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and Γ( 𝑥 0 ) = { x 𝑋 : 𝑥 𝑡 +1 𝐺 ( 𝑥 𝑡 ) , 𝑡 = 0 , 1 , 2 , . . . } Since an optimal policy exists (Exercise 2.125), the maximum is attained and 𝑣 ( 𝑥 0 ) = max x Γ( 𝑥 0 ) 𝑈 ( x ) (3.57) It is straightforward to show that 𝑈 ( x ) is strictly concave and Γ( 𝑥 0 ) is convex Applying the Concave Maximum Theorem (Theorem 3.1) to (3.57), we conclude that the value function 𝑣 is strictly concave. 2. Assume to the contrary that x and x ′′ are distinct optimal plans, so that 𝑣 ( 𝑥 0 ) = 𝑈 ( x ) = 𝑈 ( x ′′ ) Let ¯ x = 𝛼 x + (1 𝛼 ) x ′′ . Since Γ( 𝑥 0 ) is convex, ¯ x is feasible and 𝑈 x ) > 𝛼𝑈 ( x ) + (1 𝛼 ) 𝑈 ( x ′′ ) = 𝑈 ( x ) which contradicts the optimality of x . We conclude that the optimal plan is unique. 3.159 We observe that 𝑢 ( 𝐹 ( 𝑘 ) 𝑦 ) is supermodular in 𝑦 (Exercise 2.51) 𝑢 ( 𝐹 ( 𝑘 ) 𝑦 ) displays strictly increasing differences in ( 𝑘, 𝑦 ) (Exercise 3.129) 𝐺 ( 𝑘 ) = [0 , 𝐹 ( 𝑘 )] is increasing. Applying Exercise 2.126, we can conclude that the optimal policy (
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