Transcript Slide 1
Paul Milgrom and Nancy Stokey
Journal of Economic Thoery,1982
Motivation
How should traders respond to new, private information?
Ex. If a farmer receives a mid-year, private report on the state of
the crop, should he:
Use that information to speculate
Assume futures prices already impound so much information that his own
information is valueless
Examines the case of risk-averse, fully-rational traders.
Ex ante and ex post markets.
Model I
Agents are rational
Know that other agents are rational.
Know that other agents know that they are rational.
Know that other agents know that they know that they… etc…
In this case, it is common knowledge that any agreed-upon trade is
“feasible” and “mutually acceptable”.
There are n agents, i = 1,…, n.
Let the state of the world be described by ω ϵ Ω = ϴ x X.
ϴ: the set of payoff-relevant events; effects endowments, utilities.
X: the set of payoff-irrelevant events; may be statistically related to ϴ.
Let each agent, i’s, information be represented by a partition, Pi on Ω.
For any ω, Pi (ω) represents the element of Pi which contains ω.
This represents imperfect information for agent i: When the state of the world is ω,
agent i only knows that the state is Pi (ω).
Model II
There are L commodities in each state of the world.
Assume consumption set is RL+.
Each trader i is described by:
his endowment, ei: ϴ RL+
his utility function, Ui: ϴ x RL + R
his (subjective) prior beliefs about ω, pi (.)
and his (prior) informational partition, Pi
Utility
Assumed that Ui(ϴ, .): RL + R is increasing for all i, ϴ.
If Ui(ϴ, .) is concave for all ϴ, trader i is said to be weakly risk-
averse.
If strictly concave, then strictly risk-averse.
Model III
Trades
A trade, t = (t1, …, tn) is a function from Ω to RnL, where ti(ω)
describes trader i’s net trade of physical commodities in state
ω.
If a trade can be described as a function from ϴ to RnL, it is
called a ϴ-contingent trade.
E.g. a bet: I’ll bet you $100 it doesn’t rain tomorrow.
A trade is feasible if:
Beliefs
Assume pi (ω) > 0 for all ω and every i.
Let Ei [.] denote i’s expectation under pi.
Say that beliefs are concordant if:
Theorem I
The idea of the proof: traders are at an ex ante pareto-optimal
allocation. Thus, in order for a trader to wish to trade, he must
become strictly more well-off leads to the other traders to be
less-well off.
This information is conveyed by the first trader’s willingness to
trade no trade occurs.
This is made strict if traders prefer less risk to more.
Example
Suppose that two agents hold the following beliefs on (ϴ, x):
And that their information
structures are described by the
following partition on X:
And the following bet is proposed:
if ϴ = 1 agent 2 pays one dollar to agent 1,
and if ϴ = 2 agent 1 pays one dollar to agent 2.
Suppose x = 3 occurs.
Example II
x = 3. Consider the following
types of behavior:
Naïve Behavior
Since at x = 3, p(ϴ = 1 | P1) = 2/3 > ½, agent 1 accepts the bet.
Since at x = 3, p(ϴ = 2 | P2) = 2/3 > ½, agent 2 accepts the bet.
First-order sophistication
Agent 1 reasons: “I know that either x = 3 or x = 4…
○ If x =3, p(ϴ = 2 | P2) = 2/3 > ½, so I would expect agent 2 to accept the bet.
○ If x =4, p(ϴ = 2 | P2) = 5/9 > ½, so I would expect agent 2 to accept the bet.
○ Therefore, if agent 2 accepts the bet, it tells me nothing new.
…and since p(ϴ = 1 | P1) = 2/3 > ½, I will accept the bet.”
Agent 2 reasons similarly and also accepts the bet.
Example III
Rational Expectations
Agent 1 reasons:
○ If x = 5, I will refuse.
○ If x = 1, agent 2 knows that x = 1 and will refuse the bet. Hence if agent 2
accepts, x ≠ 1.
○ Thus, if I observe {1, 2} I know that agent 2 will only accept if x = 2, so I
should always refuse.
○ Agent 2 will use the same line of reasoning to exclude {4, 5}.
○ If I see {3, 4}, I know that agent 2 will refuse to bet if x = 4, and since trade
must be mutually agreed upon, I can condition my decision on x = 3.
○ By the same logic, agent two will condition his decision on x =3.
○ Both thus have an expected payoff of zero if x = 3 and trade occurs only if
both agents are risk neutral or risk seeking.
○ If agents are risk risk-averse, then both agent prefers less risk to more,
and will decide to not trade at all.
Model III
Suppose that before any information is revealed, a round of
trading is conducted using a market mechanism.
Let e denote the competitive equilibrium allocation
and let q(ϴ) denote the prices supporting e.
Let Q be the coarsest common refinement of P1,…Pn
Thus, Q contains all of the information contained in P1,…Pn but
no more.
If markets are reopened after private information is
revealed, we know from Theorem 1, that e is still a
competitive equilibrium allocation (there are no
consequence trades).
This leads us to…
Theorem II
(3): concordant beliefs.
Notice that it is the change in prices that reveals all of the
information about ϴavailable to all traders (i.e.
Theorem III
Idea of the proof:
Since e is a competitive equilibrium ex-ante AND ex-post we have two
sets of equilibrium equations involving e that must be satisfied.
Using this fact, it is possible to show algebraically that i’s posterior on ϴ
depends on his private signal only through the new equilibrium prices.
Conclusion
No trade: If both ex ante and ex post markets are available, new
private information provides no private benefit and no social benefit.
If traders’ expectations are rational, any attempt to speculate on the
basis of new information must result in that information being
impounded in prices, so that profitable speculation is impossible.
When markets are available both before and after information is
released, it is the change of prices that reveals information.
Information conveyed by ex-post prices “swamps” the private signal
received by any agent.
Why do traders bother to gather information if they cannot profit
from it?
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