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A reader of [[PS05](https://python-advanced.quantecon.org/zreferences.html#id22)] will notice that their representation of the equilibrium of
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A reader of {cite}`Pearlman_Sargent2005` will notice that their representation of the equilibrium of
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Townsend’s model exactly matches that of the **pooling equilibrium** presented here.
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We have structured our notation in this lecture to faciliate comparison of the **pooling equilibrium**
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constructed here with the equilibrium of Townsend’s model reported in [[PS05](https://python-advanced.quantecon.org/zreferences.html#id22)].
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constructed here with the equilibrium of Townsend’s model reported in {cite}`Pearlman_Sargent2005`.
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The computational method of [[PS05](https://python-advanced.quantecon.org/zreferences.html#id22)] is recursive:
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The computational method of {cite}`Pearlman_Sargent2005` is recursive:
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it enlists the Kalman filter and invariant subspace methods for
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solving systems of Euler
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equations [^footnote1] .
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As {cite}`singleton87`,
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[[Kas00](https://python-advanced.quantecon.org/zreferences.html#id24)], and {cite}`sargent91_equilibrium` also
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As {cite}`singleton87`, {cite}`kasa`, and {cite}`sargent91_equilibrium` also
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found, the equilibrium is fully revealing: observed prices tell
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participants in industry $ i $ all of the information held by
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participants in market $ -i $ ($ -i $ means not $ i $).
@@ -1598,7 +1596,7 @@ those forecasts are the same as their own, they know them.
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Sargent {cite}`sargent91_equilibrium` proposed a way to compute an equilibrium
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without making Townsend’s approximation.
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Extending the reasoning of [[Mut60](https://python-advanced.quantecon.org/zreferences.html#id110)], Sargent noticed that it is possible to
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Extending the reasoning of {cite}`Muth1960`, Sargent noticed that it is possible to
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summarize the relevant history with a low dimensional object, namely, a
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small number of current and lagged forecasting errors.
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@@ -1614,16 +1612,16 @@ appropriate orders of the autoregressive and moving average pieces of
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the equilibrium representation.
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By working in the frequency
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domain [[Kas00](https://python-advanced.quantecon.org/zreferences.html#id24)] showed how to discover the appropriate
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domain {cite}`kasa` showed how to discover the appropriate
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orders of the autoregressive and moving average parts, and also how to
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compute an equilibrium.
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The [[PS05](https://python-advanced.quantecon.org/zreferences.html#id22)] recursive computational method, which stays in the time domain, also
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The {cite}`Pearlman_Sargent2005` recursive computational method, which stays in the time domain, also
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discovered appropriate orders of the autoregressive and moving
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average pieces.
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In addition, by displaying equilibrium representations
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in the form of [[PCL86](https://python-advanced.quantecon.org/zreferences.html#id23)], [[PS05](https://python-advanced.quantecon.org/zreferences.html#id22)]
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in the form of {cite}`PCL`, {cite}`Pearlman_Sargent2005`
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showed how the moving average piece is linked to the innovation process
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of the hidden persistent component of the demand shock.
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@@ -1632,25 +1630,25 @@ innovation process is the additional state variable contributed by the
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problem of extracting a signal from equilibrium prices that decision
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makers face in Townsend’s model.
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[^footnote0]: [PS05](zreferences.html#id22) verified this assertion using a different tactic, namely, by constructing
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[^footnote0]: {cite}`Pearlman_Sargent2005` 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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[^footnote3]: See [[Sar87](zreferences.html#id197)], especially
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[^footnote3]: See {cite}`Sargent1987`, especially
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chapters IX and XIV, for principles that guide solving some roots backwards and others forwards.
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[^footnote4]: As noted by [[Sar87](zreferences.html#id197)], this difference equation is the Euler equation for
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[^footnote4]: As noted by {cite}`Sargent1987`, 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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[^footnote5]: [[PS05](zreferences.html#id22)] verify the same claim by applying machinery of [[PCL86](zreferences.html#id23)].
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[^footnote5]: [PS05](Pearlman_Sargent2005) verify the same claim by applying machinery of {cite}`PCL`.
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[^footnote1]: See [[AHMS96](zreferences.html#id135)] for an account of invariant subspace methods.
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[^footnote1]: See [AHMS96](ahms) for an account of invariant subspace methods.
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[^footnote2]: See [[AMS02](zreferences.html#id28)] for a discussion
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[^footnote2]: See [AMS02](ams) 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](zreferences.html#id28)] is that, relative
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to read our findings in light of [AMS02](ams) is that, relative
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to the number of signals agents observe, Townsend’s
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section 8 model has too few random shocks to get higher order beliefs to
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