Asset Pricing Chapter VIII. Arrow-Debreu Pricing June 22, 2006 - PowerPoint PPT Presentation

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Asset Pricing Chapter VIII. Arrow-Debreu Pricing June 22, 2006 Asset Pricing

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Setting: An Arrow Debreu Economy General Equilibrium Supply = Demand With production (if desired) Static but Multi-period No restrictions on preferences The most general of the theories we shall consider Asset Pricing

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks 1. Two dates: 0, 1 . This set-up, however, is fully generalizable to multiple periods. 2. N possible states of nature at date 1, which we index by θ = 1 , 2 , ..., N with probabilities π θ 3. One perishable (non storable) consumption good 4. K agents, indexed k = 1,...,K, with preferences: N � � π θ U k � � U k c k + δ k � c k ; 0 0 θ θ = 1 5. In addition, agent k’s endowment is described by the vector { e k e k � � 0 , θ = 1 , 2 ,..., N } Possibly: θ u k ( c k 0 , c k θ 1 , c k θ 2 , ..., c k θ N ) . Asset Pricing

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Traded Securities Exclusively Arrow-Debreu securities (contingent claims): Security θ priced q θ promises delivery of one unit of commodity tomorrow if state θ is realized and nothing otherwise How to secure one unit of consumption tomorrow? Asset Pricing

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks N U k 0 ( c k 0 ) + δ k π θ U k ( c k � max θ ) ( c k 0 , c k 1 ,...., c k N ) θ = 1 s.t. (P) N N c k q θ c k θ ≤ e k q θ e k � � 0 + 0 + θ θ = 1 θ = 1 c k 0 , c k 1 , ...., c k N ≥ 0 Equilibrium is a set of prices ( q 1 , .... q N ) such that: � c k 0 , ..., c k � 1. at those prices solve problem (P), for all k, N and K K K K c k e k C k e k 2. � 0 = � 0 , � θ = � θ , for every θ k = 1 k = 1 k = 1 k = 1 Asset Pricing

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated Checking Pareto Optimality 8.3 Pareto Optimality and risk Sharing Interior vs. Corner Solution 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Table 8.1. Endowments and Preferences in Our Reference Example Agents Endowments Preferences t = 0 t = 1 θ 1 θ 2 � 1 2 c 1 1 c 1 + 2 c 1 � � � �� Agent 1 10 1 2 0 + 0 . 9 3 ln 3 ln 1 2 � 1 1 2 c 2 c 2 + 2 c 2 � � � �� Agent 2 5 4 6 0 + 0 . 9 3 ln 3 ln 1 2 Asset Pricing

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated Checking Pareto Optimality 8.3 Pareto Optimality and risk Sharing Interior vs. Corner Solution 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks The respective agent problems are: Agent 1: � 1 max 1 10 + 1 q 1 + 2 q 2 − c 1 1 q 1 − c 1 c 1 + 2 c 1 � � � � � �� 2 q 2 + 0 . 9 3 ln 3 ln 2 1 2 S.t. c 1 1 q 1 + c 1 2 q 2 ≤ 10 + q 1 + 2 q 2 , and c 1 1 , c 1 2 ≥ 0 Agent 2: � 1 max 1 5 + 4 q 1 + 6 q 2 − c 2 1 q 1 − c 2 c 2 + 2 c 2 � � � � � �� 2 q 2 + 0 . 9 3 ln 3 ln 2 1 2 S.t. c 2 1 q 1 + c 2 2 q 2 ≤ 5 + 4 q 1 + 6 q 2 and c 2 1 , c 2 2 ≥ 0 Note: q 1 , q 2 solve «key finance problem»! 1 : q 1 1 : q 1 c 1 “ ” 8 1 8 c 2 1 1 = 0 . 9 (Π θ ) = 0 . 9 c 1 > 2 > 2 3 c 2 < < c 1 2 + c 2 1 1 Agent 1 : Agent 2 : 2 = 8 2 : q 2 q 2 c 1 “ 2 ” 1 “ ” c 2 2 1 = 0 . 9 2 : = 0 . 9 > 2 3 c 1 > 2 3 c 2 : : 2 2 Asset Pricing

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated Checking Pareto Optimality 8.3 Pareto Optimality and risk Sharing Interior vs. Corner Solution 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks δπ θ ∂ U k ∂ c k θ q θ = , k , θ = 1 , 2 (1) ∂ U k 0 ∂ c k 0 = MU k price of gold if state θ is realized θ , MU k price of gold today 0 c 1 1 = c 2 1 = 2 . 5 c 1 2 = c 2 2 = 4 � 1 � 1 � � 1 � 1 � � 4 � � � 1 � � q 1 = 2 ( 0 . 9 ) = 2 ( 0 . 9 ) = ( 0 . 9 ) = 0 . 24 3 c 1 3 2 . 5 3 5 1 � 2 � 2 � � 1 � 2 � � 4 � � � 1 � � q 2 = 2 ( 0 . 9 ) = 2 ( 0 . 9 ) = ( 0 . 9 ) = 0 . 3 3 c 1 3 4 3 8 2 Asset Pricing

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated Checking Pareto Optimality 8.3 Pareto Optimality and risk Sharing Interior vs. Corner Solution 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Table 8.2: Post-Trade Equilibrium Consumptions t = 0 t = 1 θ 1 θ 2 Agent 1: 9.04 2.5 4 Agent 2: 5.96 2.5 4 Total 15.00 5.0 8 It is Pareto optimal? H1 and H2 are satisfied! Asset Pricing

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated Checking Pareto Optimality 8.3 Pareto Optimality and risk Sharing Interior vs. Corner Solution 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Checking Pareto Optimality u 1 ( c 1 0 , c 1 1 , c 1 2 ) + λ u 2 ( c 2 0 , c 2 1 , c 2 max 2 ) { c 1 2 } 0 , c 1 1 , c 1 s.t. c 1 0 + c 2 0 = 15 ; c 1 1 + c 2 1 = 5 ; c 1 2 + c 2 2 = 8 , c 1 0 , c 1 1 , c 1 2 , c 2 0 , c 2 1 , c 2 2 ≥ 0 u 1 = u 1 = u 1 0 1 2 = λ (2) u 2 u 2 u 2 0 1 2 ( 0 . 9 ) 1 1 ( 0 . 9 ) 2 1 1 / 2 3 c 1 3 c 1 1 2 1 / 2 = = ( 0 . 9 ) 1 1 ( 0 . 9 ) 2 1 c 2 c 2 3 3 1 2 Asset Pricing

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated Checking Pareto Optimality 8.3 Pareto Optimality and risk Sharing Interior vs. Corner Solution 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Box 8-1 Interior vs. Corner Solution ∂ U k ∂ U k 0 , if c k q 0 ≤ δπ θ 0 > 0 , and k , θ = 1 , 2 (3) ∂ c k ∂ c k 0 θ Asset Pricing

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated Checking Pareto Optimality 8.3 Pareto Optimality and risk Sharing Interior vs. Corner Solution 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks If one security ( q 1 ) only (incomplete markets): Table 8.3 : The Post-Trade Allocation t = 0 t = 1 θ 1 θ 2 Agent 1: 9.64 2.5 2 Agent 2: 5.36 2.5 6 Total 15.00 5.0 8 Agent 1: 5.51 instead of 5.62 Agent 2: 4.03 instead of 4.09 Asset Pricing

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Table 8.4: The New Endowment Matrix t = 0 t = 1 θ 1 θ 2 Agent 1 4 1 5 Agent 2 4 5 1 Asset Pricing

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Table 8.5: Agents’ Utility In The Absence of Trade State-Contingent Utility Expected Utility in Period 1 θ 1 θ 2 Agent 1 ln(1) = 0 ln(5) = 1.609 1/2 ln(1) + 1/2 ln(5) = 8047 Agent 2 ln(5) = 1.609 ln(1) = 0 1/2 ln(1) + 1/2 ln(5) = 8047 Table 8.6: The Desirable Trades And Post-Trade Consumptions Date 1 Endowments Pre-Trade Consumption Post-Trade θ 1 θ 2 θ 1 θ 2 Agent 1 1 5 [ ⇓ 2] 3 3 Agent 2 5 [ ⇑ 2] 1 3 3 Post-trade allocation is Pareto Optimal After trade, EU in period 1 is approx. 1.1 for both agents. Asset Pricing