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
Unformatted text preview: Solutions to Problem Set 7 EC720.01  Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Thursday, October 29 Consider an economy populated by a large number of identical consumers, each of whom takes s as given, and chooses sequences { c t } t =0 and { s t } t =1 to maximize the utility function X t =0 t u ( c t ) subject to the budget constraint d t s t c t p t s t +1 s t for all t = 0 , 1 , 2 ,... . 1. The KuhnTucker Formulation With the Lagrangian for this problem written as L = X t =0 t u ( c t ) + X t =0 t +1 s t + d t s t c t p t s t +1 the firstorder condition for c t is t u ( c t ) t +1 p t = 0 and the firstorder condition for s t is t +1 + t +1 d t p t t = 0 . The first of these two conditions must hold for all t = 0 , 1 , 2 ,... and the second must hold for all t = 1 , 2 , 3 ,... . Together with the binding constraint d t s t c t p t = s t +1 s t for all t = 0 , 1 , 2 ,... , these conditions form a system of three equations in the three unknowns c t , s t , and t +1 . 1 2. The Maximum Principle The Hamiltonian for the consumers problem can be defined as H ( s t , t +1 ; t ) = max c t t u ( c...
View Full
Document
 Fall '09
 IRELAND
 Economics

Click to edit the document details