Rational herds: Economic models of social learning by Christophe P. Chamley
By Christophe P. Chamley
Penguins leaping off a cliff, monetary forecasters and fiscal advisors speculating opposed to a forex, and farmers utilizing conventional tools in India are all practicing social studying. Such studying from the habit of others may possibly and does bring about herds, crashes, and booms. those matters became, over the past ten years, an exhilarating box of analysis in theoretical and utilized economics, finance, and in different social sciences. This 2004 e-book presents either a casual advent and in-depth insights into the topic. each one bankruptcy is dedicated to a separate factor: contributors study from the observations of activities, the results of those activities, and from what others say. they could hold up or make a right away choice; they could compete opposed to others or achieve from cooperation; they make judgements approximately funding, crop offerings, and monetary investments. The publication highlights the similarities and the diversities among many of the circumstances.
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8 n A Heuristic Remark on the Proof of the Martingale Convergence Theorem The main intuition of the proof is important for our understanding of Bayesian learning. It is a formalization13 of the metaphor of the drunkard. In words, the deﬁnition of a martingale states that agents do not anticipate systematic errors. This implies that the updating difference µt+1 − µt is uncorrelated with µt . The same property holds for more distant periods: conditional on the information in period t, the random variables µt+k+1 − µt+k are uncorrelated for k ≥ 0.
These principles hold for a wider class of models, but only the Gaussian model provides such a simple formulation: 1. The normal distribution is summarized by the most intuitive two parameters of a distribution, the mean and the variance (or its reciprocal, the precision). 2. The updating rules for both the mean and the precision are linear. This makes any computation easier. 3. The weight of the private signal s depends on the noise-to-signal ratio in the most intuitive way. When the variance of the noise term, σ 2 , tends to zero, or equivalently its precision tends to inﬁnity, the signal’s weight α tends to one and the weight of the ex ante expected value of θ tends to zero.
An agent has an incentive to change less than he would with no evaluation, because he wants to show that he already knows much (that his ρ is high). Prendergast and Stole (1996) analyze the distortion that is created by the evaluation of the agent’s actions. 5), the expected value of θ at the end of period t is N (µt+1 , ρt+1 ) with µt+1 = ρ ρθ + (t − 1)ρ st + µt . 5 Appendix 41 The weight on the new signal s t increases with ρ. Note, however, that the weight does not tend to 1 as ρ → ∞ (as it would in the ﬁrst period), because a high ρ means that previous estimates of θ have also a high precision (because they are made with signals of high precision).