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The disappearance of higher order beliefs means that
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decision makers in this model do not really face a problem of
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problem of extracting a signal from equilibrium prices that decision
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makers face in Townsendâs model.
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<p><aid=footnote0href=#footnote0-link><strong>[1]</strong></a> [[PS05](https://python-advanced.quantecon.org/zreferences.html#id22)] verified this assertion using a different tactic, namely, by constructing
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[^footnote0]: [PS05](zreferences.html#id22) verified this assertion using a different tactic, namely, by constructing
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analytic formulas for an equilibrium under the incomplete
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information structure and confirming that they match the pooling equilibrium formulas derived here.
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<p><aid=footnote3href=#footnote3-link><strong>[2]</strong></a> See [[Sar87](https://python-advanced.quantecon.org/zreferences.html#id197)], especially
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[^footnote3]: See [[Sar87](zreferences.html#id197)], especially
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chapters IX and XIV, for principles that guide solving some roots backwards and others forwards.
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<p><aid=footnote4href=#footnote4-link><strong>[3]</strong></a> As noted by [[Sar87](https://python-advanced.quantecon.org/zreferences.html#id197)], this difference equation is the Euler equation for
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[^footnote4]: As noted by [[Sar87](zreferences.html#id197)], this difference equation is the Euler equation for
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a planning problem that maximizes the discounted sum of consumer plus
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producer surplus.
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<p><aid=footnote5href=#footnote5-link><strong>[4]</strong></a> [[PS05](https://python-advanced.quantecon.org/zreferences.html#id22)] verify the same claim by applying machinery of [[PCL86](https://python-advanced.quantecon.org/zreferences.html#id23)].
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[^footnote5]:[[PS05](zreferences.html#id22)] verify the same claim by applying machinery of [[PCL86](zreferences.html#id23)].
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<p><aid=footnote1href=#footnote1-link><strong>[5]</strong></a> See [[AHMS96](https://python-advanced.quantecon.org/zreferences.html#id135)] for an account of invariant subspace methods.
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[^footnote1]: See [[AHMS96](zreferences.html#id135)] for an account of invariant subspace methods.
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<p><aid=footnote2href=#footnote2-link><strong>[6]</strong></a> See [[AMS02](https://python-advanced.quantecon.org/zreferences.html#id28)] for a discussion
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[^footnote2]: See [[AMS02](zreferences.html#id28)] for a discussion
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of information assumptions needed to create a situation
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in which higher order beliefs appear in equilibrium decision rules. A way
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to read our findings in light of [[AMS02](https://python-advanced.quantecon.org/zreferences.html#id28)] is that, relative
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to read our findings in light of [[AMS02](zreferences.html#id28)] is that, relative
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to the number of signals agents observe, Townsendâs
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section 8 model has too few random shocks to get higher order beliefs to
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