FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (1)
Markus K. Brunnermeier LECTURE 05: ONE PERIOD MODEL GENERAL EQUILIBRIUM, EFFICIENCY & AGGREGATION
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A FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (2)
specify observe/specify Preferences & existing Technology Asset Prices
• evolution of states NAC/LOOP NAC/LOOP • risk preferences • aggregation
State Prices q absolute (or stochastic discount relative asset pricing factor/Martingale measure) asset pricing
LOOP
derive derive Asset Prices Price for (new) asset
Only works as long as market completeness doesn’t change 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 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.
′ ′ ′ 푐0, 푐1, … , 푐푆 ≻ 푐0, 푐1, … , 푐푆 ⇔ ′ ′ ′ 푈 푐0, 푐1, … , 푐푆 > 푈 푐0, 푐1, … , 푐푆
FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (5) Agent’s Optimization
푆 • Consumption vector 푐0, 푐1 ∈ ℝ+ × ℝ+ 푖 푆 • Agent 푖 has 푈 : ℝ+ × ℝ+ → ℝ 푆 endowments 푒0, 푒1 ∈ ℝ+ × ℝ+ • 푈푖 is quasiconcave 푐: 푈푖 푐 ≥ 푣 is convex for each real 푣 푈푖 is concave: for each 0 ≤ 훼 ≤ 1, 푈푖 훼푐 + 1 − 훼 푐′ ≥ 훼푈푖 푐 + 1 − 훼 푈푖 푐′ 휕푈푖 휕푈푖 • > 0, ≫ 0 휕푐0 휕푐1
FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (6) Agent’s Optimization
• Portfolio consumption problem 푖 max 푈 푐0, 푐1 푐0,푐1,ℎ subject to (i) 0 ≤ 푐0 ≤ 푒0 − 푝 ⋅ ℎ ′ and (ii) 0 ≤ 푐1 ≤ 푒1 + 푋 ℎ
푖 ′ ℒ = 푈 푐0, 푐 1 − 휆 푐0 − 푒0 + 푝ℎ − 휇 푐 1 − 푒 1 − ℎ 푋 • FOC 푖 푖 휕푈 ∗ 휕푈 ∗ 푐0: 푐 = 휆, 푐푠: 푐 = 휇푠 휕푐0 휕푐푠 ℎ: 휆푝 = 푋휇 휇 ⇔ 푝푗 = 푠 푥푗 푠 휆 푠 FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (7) Agent’s Optimization
푖 푗 휕푈 휕푐푠 푗 푝 = 푖 푥푠 휕푈 휕푐0 푠 • For time separable utility function 푖 푈 푐0, 푐 1 = 푢 푐0 + 훿푢 푐 1 • And vNM expected utility function 푖 푈 푐0, 푐 1 = 푢 푐0 + 훿퐸 푢 푐
푖 푗 휕푢 휕푐푠 푗 푝 = 휋푠훿 푖 푥푠 휕푢 휕푐0 푠 FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (8) Stochastic Discount Factor
푖 푗 휕푢 휕푐푠 푗 푝 = 휋푠 훿 푖 푥푠 휕푢 휕푐0 푠 푚푠 푞푠 푞푠 • That is, stochastic discount factor 푚푠 = for 휋푠 all 푠 푗 푗 푗 푝 = 휋푠푚푠푥푠 = 퐸 푚푥 푠 FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (9) Agent’s Optimization
To sum up • Proposition 3: Suppose 푐∗ ≫ 0 solves problem. Then there exists positive real numbers 휆, 푚1, … 푚푆, such that 휕푈푖 = 휆 휕푐0 휕푈푖 = 휇1, … , 휇푆 휕푐1 푗 푗 휆푝 = 휇푠푥푠 , ∀푗 = 1, … , 퐽 푠 (The converse is also true.) • The vector of marginal rate of substitutions 푀푅푆푠,0 is a (positive) state price vector. 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 FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (11)
Suppose 푐 ′ is fixed 퐴 푠 푐푠 since it can’t be traded
퐴 퐴 퐴 퐴 푐0 , 푐1 : 푈 푐0 , 푐1 = 푈 푒 퐴 퐴 퐴 퐴 휕푈 푐0 , 푐1 퐴 휕푈 푐0 , 푐1 퐴 푑푐0 + 퐴 푑푐푠 = 0 휕푐0 휕푐푠
Slope A’s indifference curve 휕푈퐴 휕푐퐴 퐴 0 푒 −푀푅푆0,푠 = − 퐴 퐴 휕푈 휕푐푠
Mr. A 퐴 푐0 FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (12)
퐵 푐0 Ms. B
푒 퐵 푐푠 FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (13)
퐵 푐0 Ms. B 퐴 푐푠
퐴 퐵 푒 , 푒 퐵 푐푠
Mr. A 퐴 푐0 FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (14)
퐵 푐0 Ms. B 퐴 푐푠
1 푞 퐴 퐵 푒 , 푒 퐵 푐푠 Mr. A 푐퐴 퐴 퐴 0 1 휕푈 휕푐푠 푞 = = 푀푅푆푠,0 = − 퐴 퐴 푀푅푆0,푠 휕푈 휕푐0 FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (15) Set of PO allocations (contract curve)
퐵 푐0 Ms. B 퐴 푐푠
퐴 퐵 푒 , 푒 퐵 푐푠
Mr. A 퐴 푐0 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
FIN501 Asset Pricing Lecture 05 GE, Efficiency, Aggregation (17) Set of PO allocations (contract curve) MRSA = MRSB B c 0 Ms. B A c s
eA, eB B c s
Mr. A A c 0 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 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 푞. All 푀푅푆푖 푐∗ coincide with unique state price 푞. – Despite (pecuniary) externalities • Second Welfare Theorem. Any Pareto efficient allocation can be decentralized as a competitive equilibrium.
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 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 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 푘 푈 푐 = 훼푘푢 푐 푘 where 훼푘 is the welfare weight of agent 푘. and the single agent consumes the aggregate endowment. 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 Quasi- complete 1 푅퐴 푐 = ⇒ linear risk sharing rule 퐴푖+퐵푐
Then asset prices in economy with many agents are identical to a single agent economy with HARA preferences with 1 푅퐴 푐 = 푖 퐴푖 + 퐵푐
– Recall fund separation theorem in Lecture 04.