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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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