Fin Econ2-HW2

# Fin Econ2-HW2 - Jaime Frade ECO5282-Dr Garriga Homework#2 1...

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Unformatted text preview: Jaime Frade ECO5282-Dr. Garriga Homework #2 1. A economy consists of two infintely lived consumers named i = 1 , 2. There is one nonstorable consumption good. Consumer i consumers c i at time t . Consumer i ranks consumptions streams by ∑ ∞ t =0 β t u ( c i t ) where β ∈ (0 , 1) and u ( c ) is strictly increasing, concave, and twice continiously differentiable. Comsumer 1 is endowed with a stream of the consumption good y 1 t = 1 , , , 1 , , , 1 ,... . Consumer 2 is endowed with a stream of the consumption good y 1 t = 0 , 1 , 1 , , 1 , 1 , ,... . Assume that markets are complete with time-0 trading. (a) Define a competitive equilibrium. Definition An allocation for agent i is defined as state contingent function c i = { c i t ( s t ) } ∞ t =0 for i = 1 , 2 Definition A allocation is said to be a feasible allocation if it satisfies 2 X i =1 c i t ( s t ) = 2 X i =1 y i t ( s t ) (1) Definition A competitive equilibrium is a feasible allocation, { c i } 2 i =1 = {{ c i t ( s t ) } ∞ t =0 } 2 i =1 , and a price sys- tem, { p t ( s t ) } ∞ t =0 , such that the allocation solves each household problem. (b) Compute a competitive equilibrum. For a given household i solves U ( c i ) = max { c T t ( s t ) } E ∞ X t =0 β t u ( c i t ) U ( c i ) = max { c T t ( s t ) } ∞ X t =0 X s t β t π ( s t | s ) u ( c i t ( s t )) (2) subject to ∞ X t =0 X s t p t ( s t ) c i t ( s t ) = ∞ X t =0 X s t p t ( s t ) y i t ( s t ) , and c 1 t ( s t ) ,c 2 t ( s t ) ≥ (3) First Order Conditions: β t π ( s t | s ) u ( c i t ( s t )) = γ i p t ( s t ) p t ( s t ) = β t π ( s t | s ) u ( c i t ( s t )) γ i (4) at t = 0, we have the following: p ( s ) = u ( c i ( s )) γ i solving for p ( s ) = 1, we have the following: 1 = u ( c i ( s )) γ i ⇒ γ i = u ( c i ( s )) Subsititing into (4), obtain an system of price equations for the Arrow securities p t ( s t ) = β t π ( s t | s ) u ( c i t ( s t )) u ( c i ( s )) (5) For the problem above for i = 1 , 2, there is no aggregate risk or uncertainity (hence c i t = c i ), thus β t π ( s t | s ) u ( c 1 t ( s t )) u ( c 1 ( s )) = p t ( s t ) = β t π ( s t | s ) u ( c 2 t ( s t )) u ( c 2 ( s )) 1 In equilibrium, from MRS 1 ,T = MRS 2 ,T obtains u ( c 1 t ( s t )) u ( c 1 ( s )) = u ( c 2 t ( s t )) u ( c 2 ( s )) (6) Subsituting into the buget constraints (16) to find the feasible allocation for each agent for a competitive equilibrium. ∞ X t =0 X s t β t π ( s t | s ) u ( c i ( s t )) γ i [ c i t ( s t )- y i t ( s t )] = 0 u ( c i ( s t )) γ i ∞ X t =0 X s t β t π ( s t | s )[ c i t ( s t )- y i t ( s t )] = 0 Because u ( c i ( s t )) γ i 6 = 0, then obtain the following to solve for c i c i o ∞ X t =0 β t | {z } = 1 1- β X s t π ( s t | s ) | {z } =1 = ∞ X t =0 X s t β t π ( s t | s ) y i t ( s t ) Since the states are deterministics, the sum of probabilities in all states, ( s t has one element), π ( s t | s ) = 1 ∀ s t , and using a geometric series for for β u...
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Fin Econ2-HW2 - Jaime Frade ECO5282-Dr Garriga Homework#2 1...

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