Markus K. Brunnermeier LECTURE 05: ONE PERIOD MODEL ... · Lecture 05 GE, Efficiency, Aggregation...
Transcript of Markus K. Brunnermeier LECTURE 05: ONE PERIOD MODEL ... · Lecture 05 GE, Efficiency, Aggregation...
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (1)
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LECTURE 05: ONE PERIOD MODEL GENERAL EQUILIBRIUM, EFFICIENCY & AGGREGATION
Markus K. Brunnermeier
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (2)
specify Preferences &
Technology
observe/specify existing
Asset Prices
State Prices q (or stochastic discount
factor/Martingale measure)
derive Price for (new) asset
• evolution of states • risk preferences • aggregation
absolute asset pricing
relative asset pricing
NAC/LOOP
LOOP
NAC/LOOP
Only works as long as market completeness doesn’t change
derive Asset Prices
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (3)
Overview
1. Marginal Rate of Substitution (MRS)
2. Pareto Efficiency
3. Welfare Theorems
4. Representative Agent Economy
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (4)
Representation of Preferences
A (i) complete, (ii) transitive, (iii) continuous [and (iv) relatively stable] preference ordering can be represented by a utility function, i.e. 𝑐0, 𝑐1, … , 𝑐𝑆 ≻ 𝑐0
′ , 𝑐1′ , … , 𝑐𝑆
′ ⇔ 𝑈 𝑐0, 𝑐1, … , 𝑐𝑆 > 𝑈 𝑐0
′ , 𝑐1′ , … , 𝑐𝑆
′
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (5)
Agent’s Optimization
• Consumption vector 𝑐0, 𝑐1 ∈ ℝ+ × ℝ+𝑆
• Agent 𝑖 has 𝑈𝑖: ℝ+ × ℝ+𝑆 → ℝ
endowments 𝑒0, 𝑒1 ∈ ℝ+ × ℝ+𝑆
• 𝑈𝑖 is quasiconcave 𝑐: 𝑈𝑖 𝑐 ≥ 𝑣 is convex for
each real 𝑣
𝑈𝑖 is concave: for each 0 ≤ 𝛼 ≤ 1, 𝑈𝑖 𝛼𝑐 + 1 − 𝛼 𝑐′ ≥ 𝛼𝑈𝑖 𝑐 + 1 − 𝛼 𝑈𝑖 𝑐′
•𝜕𝑈𝑖
𝜕𝑐0> 0,
𝜕𝑈𝑖
𝜕𝑐1≫ 0
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (6)
Agent’s Optimization
• Portfolio consumption problem
max𝑐0,𝑐1,ℎ𝑈𝑖 𝑐0, 𝑐1
subject to (i) 0 ≤ 𝑐0 ≤ 𝑒0 − 𝑝 ⋅ ℎ and (ii) 0 ≤ 𝑐1 ≤ 𝑒1 + 𝑋
′ℎ
ℒ = 𝑈𝑖 𝑐0, 𝑐 1 − 𝜆 𝑐0 − 𝑒0 + 𝑝ℎ − 𝜇 𝑐 1 − 𝑒 1 − ℎ′𝑋
• FOC
𝑐0:𝜕𝑈𝑖
𝜕𝑐0𝑐∗ = 𝜆, 𝑐𝑠:
𝜕𝑈𝑖
𝜕𝑐𝑠𝑐∗ = 𝜇𝑠
ℎ: 𝜆𝑝 = 𝑋𝜇
⇔ 𝑝𝑗 = 𝜇𝑠
𝜆𝑥𝑠𝑗
𝑠
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (7)
Agent’s Optimization
𝑝𝑗 = 𝜕𝑈𝑖 𝜕𝑐𝑠
𝜕𝑈𝑖 𝜕𝑐0 𝑠
𝑥𝑠𝑗
• For time separable utility function 𝑈𝑖 𝑐0, 𝑐 1 = 𝑢 𝑐0 + 𝛿𝑢 𝑐 1
• And vNM expected utility function 𝑈𝑖 𝑐0, 𝑐 1 = 𝑢 𝑐0 + 𝛿𝐸 𝑢 𝑐
𝑝𝑗 = 𝜋𝑠𝛿𝜕𝑢𝑖 𝜕𝑐𝑠
𝜕𝑢𝑖 𝜕𝑐0 𝑥𝑠𝑗
𝑠
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (8)
Stochastic Discount Factor
𝑝𝑗 = 𝜋𝑠 𝛿𝜕𝑢𝑖 𝜕𝑐𝑠
𝜕𝑢𝑖 𝜕𝑐0 𝑚𝑠
𝑞𝑠
𝑥𝑠𝑗
𝑠
• That is, stochastic discount factor 𝑚𝑠 =𝑞𝑠
𝜋𝑠 for
all 𝑠
𝑝𝑗 = 𝜋𝑠𝑚𝑠𝑥𝑠𝑗
𝑠
= 𝐸 𝑚𝑥𝑗
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (9)
To sum up • Proposition 3: Suppose 𝑐∗ ≫ 0 solves problem. Then there exists
positive real numbers 𝜆,𝑚1, …𝑚𝑆, such that
𝜕𝑈𝑖
𝜕𝑐0= 𝜆
𝜕𝑈𝑖
𝜕𝑐1= 𝜇1, … , 𝜇𝑆
𝜆𝑝𝑗 = 𝜇𝑠𝑥𝑠𝑗
𝑠
, ∀𝑗 = 1,… , 𝐽
(The converse is also true.)
• The vector of marginal rate of substitutions 𝑀𝑅𝑆𝑠,0 is a (positive) state price vector.
Agent’s Optimization
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (10)
Overview
1. Marginal Rate of Substitution (MRS)
2. Pareto Efficiency
3. Welfare Theorems
4. Representative Agent Economy
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (11)
Mr. A 𝑐0𝐴
𝑐𝑠𝐴
Suppose 𝑐𝑠′ is fixed since it can’t be traded
𝑐0𝐴, 𝑐1𝐴: 𝑈 𝑐0
𝐴, 𝑐1𝐴 = 𝑈 𝑒
𝜕𝑈 𝑐0𝐴, 𝑐1𝐴
𝜕𝑐0𝐴 𝑑𝑐0
𝐴 +𝜕𝑈 𝑐0
𝐴, 𝑐1𝐴
𝜕𝑐𝑠𝐴 𝑑𝑐𝑠 = 0
𝑒
Slope
−𝑀𝑅𝑆0,𝑠𝐴 = −
𝜕𝑈𝐴 𝜕𝑐0𝐴
𝜕𝑈𝐴 𝜕𝑐𝑠𝐴
A’s indifference curve
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (12)
Ms. B
𝑐𝑠𝐵
𝑐0𝐵
𝑒
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (13)
Ms. B
Mr. A
𝑐𝑠𝐵
𝑐0𝐵
𝑐𝑠𝐴
𝑐0𝐴
𝑒𝐴, 𝑒𝐵
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (14)
Ms. B
Mr. A
1
𝑞
𝑐𝑠𝐵
𝑐0𝐵
𝑐𝑠𝐴
𝑐0𝐴
𝑒𝐴, 𝑒𝐵
𝑞 =1
𝑀𝑅𝑆0,𝑠= 𝑀𝑅𝑆𝑠,0 = −
𝜕𝑈𝐴 𝜕𝑐𝑠𝐴
𝜕𝑈𝐴 𝜕𝑐0𝐴
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (15)
Ms. B
𝑐𝑠𝐵
𝑐0𝐵
Mr. A
𝑐𝑠𝐴
𝑐0𝐴
𝑒𝐴, 𝑒𝐵
Set of PO allocations (contract curve)
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (16)
Pareto Efficiency
• Allocation of resources such that
– there is no possible redistribution such that
• at least one person can be made better off
• without making somebody else worse off
• Note
– Allocative efficiency Informational efficiency
– Allocative efficiency fairness
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (17)
Ms. B
cBs
cB0
Mr. A
cAs
cA0
eA, eB
Set of PO allocations (contract curve) MRSA = MRSB
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (18)
Overview
1. Marginal Rate of Substitution (MRS)
2. Pareto Efficiency
3. Welfare Theorems
4. Representative Agent Economy
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (19)
Welfare Theorems
• First Welfare Theorem. If markets are complete, then the equilibrium allocation is Pareto optimal. – State price is unique 𝑞. All 𝑀𝑅𝑆𝑖 𝑐∗ coincide
with unique state price 𝑞.
– Despite (pecuniary) externalities
• Second Welfare Theorem. Any Pareto efficient allocation can be decentralized as a competitive equilibrium.
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (20)
Knife-edginess of Welfare Theorem
• In multi-period (or multiple good) setting if markets are incomplete, then equilibrium allocation is generically not only Pareto inefficient but also constrained Pareto inefficient. – i.e. a social planner can do better even if restricted to
the same trading space
– Pecuniary externalities can lead to wealth shifts • With incomplete markets not all MRS are equalized,
hence pecuniary externalities generically lead to inefficiencies.
– … more when we study multi-period settings
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (21)
Overview
1. Marginal Rate of Substitution (MRS)
2. Pareto Efficiency
3. Welfare Theorems
4. Aggregation/Representative Agent Economy
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (22)
Representative Agent & Complete Markets
• Aggregation Theorem 1: Suppose – markets are complete
Then asset prices in economy with many agents are identical to an economy with a single agent/planner whose utility is
𝑈 𝑐 = 𝛼𝑘𝑢𝑘 𝑐
𝑘
where 𝛼𝑘 is the welfare weight of agent 𝑘. and the single agent consumes the aggregate endowment.
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FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (23)
Representative Agent & HARA utility world
• Aggregation Theorem 2: Suppose – riskless annuity and endowments are tradable. – agents have common beliefs – agents have a common rate of time preference – agents have LRT (HARA) preferences with
𝑅𝐴 𝑐 =1
𝐴𝑖+𝐵𝑐⇒ linear risk sharing rule
Then asset prices in economy with many agents are identical to a single agent economy with HARA preferences with
𝑅𝐴 𝑐 =1
𝐴𝑖𝑖 + 𝐵𝑐
– Recall fund separation theorem in Lecture 04.
Quasi- complete