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Unformatted text preview: Alejandro Badel Homework 1, Macro II Jonathan Heathcote : Answer 1) a) The set of Pareto E¢ cient allocations coincides with the solutions to the planning problem. By the Inada conditions, solutions to this problem are always interior. Also, by strict concavity of the planners objective, the FOC are necessary and su¢ cient. u & c 1 t ¡ u ( c 2 t ) = 1 & & & c 1 t + c 2 t = e 1 t + e 2 t := e t Using the functional form for u ( : ) , we get: ¢ c 1 t c 2 t £ & & = 1 & & & Imposing the resource constraint: Pareto = f ¤ c p 2 t ; c p 1 t ¥ 2 R 2 + such that c p 1 t = & 1 & & 1 & + (1 & & ) 1 & e t c p 2 t = (1 & & ) 1 & & 1 & + (1 & & ) 1 & e t & ¡ [0 ; 1] g Note that consumption varies across periods but the ratio of consumption remains constant. b) Denoting the Lagrange multiplier for the resource constraint of the Plan- ner&s problem above by ¡ t = 2 , and recalling one of the FOC, ¢ t & c 1 t ¡ & & = ¡ t 2 & The FOC of the decentralized sequential markets problems are & c 1 t ¡ & & =...
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- pareto, lagrange multipliers, Joseph Louis Lagrange