problem1sol

# problem1sol - Solution to Problem Set 1 Econ 210 Core Macro...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

### Page1 / 14

problem1sol - Solution to Problem Set 1 Econ 210 Core Macro...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online