The Shleifer Model

BEHAVIORAL ECONOMICS

Deciphering Shleifer

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Players

Assets

BehaviorEquilibrium

Profitability of Players

Deciphering Shleifer

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LExample: Safe asset versus unsafe asset

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• Imagine an economy with two assets (financial assets)• Safe asset, s• Unsafe asset, u

• Assume a single consumption good• Safe asset, s

• Suppose that s is always convertible (back and forth between the consumption good and itself)

• That means the price of s is always 1 in terms of the consumption good

• It is called the “safe” asset – its price is always 1, regardless of anything

• Unsafe asset, u• Suppose that u is not convertible back and forth into the

consumption good• You buy u on the open market and sell it on the open

market• That means that the price of u is not fixed• It is called the “unsafe” asset because the price of u is not

fixed

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LExample: Safe asset versus unsafe asset (cont’d)

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• Now imagine both s and u pay the same dividend, d

• d is constant, period after period

• d is paid with complete certainty, no uncertainty at all

• This implies that neither s or u have “fundamental” risk

• If someone gave you 10 units of s and you never sold it, your outcome would be the same as if someone gave you 10 units of u

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LThe Players

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Arbitrageurs NoiseTraders

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LUtility Functions

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Wealth

Utility

• “Expected Utility”, not “Expected Value”

• U = -e-(2λ)w

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LOverlapping Generations Structure

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• All agents live two periods

• Born in period 1 and buy a portfolio (s, u)

• Live (and die) in period 2 and consume

• At time t

• The (t-1) generation is in period 2 of their life

• The (t) generation is in period 1 of their life

• So, they “overlap”

t1 t2 t3 t4

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LHow many are arbitrageurs? How many are noise traders?

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• Total number of traders is the same as the number of real numbers between zero and one - an infinite number

• “Measure” means the size of any interval• Examples

• The measure of the interval between 0 and ½ is ½

• The measure of a single point (a single number) is zero

• The measure of the interval between zero and one is 1

• Think of it as a fraction of the entire interval• Measure of noise traders is µ and measure of

arbitrage traders is 1 - µ. That is, the fraction of noise traders is µ and everybody else is an arbitrage traders

0 1

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LWhat is a noise trader?

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• Pt+1 is the price of the risky asset at time t+1

• Ρt+1 is the “mean misperception” of pt+!

Ρt+!

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LWhat is an arbitrage trader?

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• Arbitrage traders “correctly” perceive the true distribution of pt+1

• There is “systematic” error in estimation of future price, pt+1

• But, arbitrageurs face risk unrelated to the “true” distribution of pt+1

• If there were no “noise traders,” then there would be no variance in the price of the risky asset…but, there are noise traders, hence the risky asset is a risky asset

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LArbitrageur versus Traders

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• Arbitrageurs expectations are “correct;” noise traders expectations are “biased”

Correct mean of pt+1

Difference is ρt+1

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LThe Main Issue

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• What happens in equilibrium

• Undetermined

• Some forces make pt > 1, some forces push pt < 1, result is indeterminant

• Who makes more profit, arbitrageurs or noise traders?

• Depends

• But, it is perfectly possible for arbitrageurs to make more!

• Survival?

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LWhen Do Noise Traders Profit More Than Arbitrageurs?

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• Noise traders can earn more than arbitrageurs when ρ* is positive

• Meaning when noise traders are systematically too optimistic

• Why?• Because they have relatively more of the

risky asset than the arbitrageurs• But, if ρ* is too large, noise traders will

not earn more than arbitrageurs• The more risk averse everyone is (higher

λ in the utility function), the wider the range of values of ρ for which noise traders do better than arbitrageurs

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LWhat Does Shleifer Accomplish?

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• Given two assets that are “fundamentally” identical, he shows a logic where the market fails to price them identically

• Assumes “systematic” noise trader activity

• Shows conditions that lead to noise traders actually profiting from their noise trading

• Shows why arbitrageurs could have trouble (even when there is no fundamental risk)