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Unformatted text preview: Solution to Problem Set 1 Econ 210 Core Macro Monika Piazzesi Stanford University Wednesday, September 29 1. Provide a proof for Proposition 6 on page 14 of Krueger (2002). 1 An allocation f ( c 1 t ;c 2 t ) g 1 t =0 is Pareto e cient if and only if it solves the social planners problem for some & 2 [0 ; 1] : Proof: * If: Suppose ( c 1 ;c 2 ) : f ( c 1 t ;c 2 t ) g 1 t =0 solves the social planner problem for some & 2 [0 ; 1] , but is not Pareto e cient. If that is true, then there must exist another feasible allocation (~ c 1 ; ~ c 2 ) : f (~ c 1 t ; ~ c 2 t ) g 1 t =0 such that either (1) u (~ c 1 ) > u ( c 1 ) and u (~ c 2 ) & u ( c 2 ) or (2) u (~ c 1 ) & u ( c 1 ) and u (~ c 2 ) > u ( c 2 ) . Eithe case implies that &u (~ c 1 ) + (1 & ) u (~ c 2 ) > &u ( c 1 ) + (1 & ) u ( c 2 ) which contradicts that ( c 1 ;c 2 ) : f ( c 1 t ;c 2 t ) g 1 t =0 is a solution to the Social Planner problem. Therefore, ( c 1 ;c 2 ) is Pareto e cient. * Only if: Suppose ( c 1 ;c 2 ) : f ( c 1 t ;c 2 t ) g 1 t =0 is Pareto e cient but it does not solve the Social Planner problem. Claim: The ratio of marginal utility over both agents in each period remains constant, i.e. for 8 t u ( c 1 t ) u ( c 2 t ) = & c 1 The solution for question 1 is provided by Rui Xu, Xing Li, Markus Baldauf, and Trevor Davis.Thanks! Some parts have been edited. 1 If not, wlog, suppose u ( c 1 t ) u ( c 2 t ) > u ( c 1 t +1 ) u ( c 2 t +1 ) or u ( c 1 t ) u ( c 1 t +1 ) > u ( c 2 t ) u ( c 2 t +1 ) & That is, agent 1 consumes less in period t but more in t+1 2 , while agent 2 consumes more in t and less in t+1. Then, there exists an r such that: u ( c 1 t ) u ( c 1 t +1 ) > r > u ( c 2 t ) u ( c 2 t +1 ) and u ( c 1 t ) > ru ( c 1 t +1 ) u ( c 2 t ) < ru ( c 2 t +1 ) & Now, we will show that by changing that allocation both agents will be better o/ and thereby we contradict that the current allocation has been Pareto e cient. Start with the proposed allocation ( c 1 ;c 2 ) : f ( c 1 t ;c 2 t ) g 1 t =0 that is Pareto e cient but does not solve the Social Planner problem. Compared to that allocation, we now let agent 1 consume & units more and let agent 2 consume & less in period 1, while in period 2 agent 1 con- sumes more in period 1 and agent 2 consumes units less 3 . Then we can evaluate the e/ect of this allocation change on the utility as: u ( c 1 t + & ) & u ( c 1 t ) & > r ( u ( c 1 t +1 ) & u ( c 1 t +1 & )) u ( c 2 t ) & u ( c 2 t & & ) & < r ( u ( c 2 t +1 + ) & u ( c 2 t +1 )) Next, pick & and s.t. the discount factor = r & . Then rearrang- ing the above two inequalities results in u ( c 1 t + & ) + u ( c 1 t +1 ) > u ( c 1 t ) + u ( c 1 t +1 ) u ( c 2 t & ) + u ( c 2 t +1 + ) > u ( c 2 t ) + u ( c 2 t +1 ) : Hence, we &nd a feasible allocation which pareto dominates the original 2 Thus, agent 1 marginal utility of an additional unit of consumption is greater in t but smaller in t+1 compared to the allocation before.smaller in t+1 compared to the allocation before....
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