Information, Trade and Common Knowledge

1. Paul Milgrom and Nancy Stokey Journal of Economic Thoery,1982 Information, Trade and Common Knowledge

2. 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.

3. 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 ωϵΩ = 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).

4. 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.

5. Model III • Trades • A trade, t = (t1, …, tn) is a function from Ωto 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 concordantif:

6. 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.

7. 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.

8. 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.

9. 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.

10. 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…

11. 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.

12. 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.

13. 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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