**[1]** [PS05](zreferences.html#id22) verified this assertion using a different tactic, namely, by constructing\n",
"analytic formulas for an equilibrium under the incomplete\n",
"information structure and confirming that they match the pooling equilibrium formulas derived here.\n",
"\n",
"

**[2]** See [[Sar87](zreferences.html#id197)], especially\n",
"chapters IX and XIV, for principles that guide solving some roots backwards and others forwards.\n",
"\n",
"

**[3]** As noted by [[Sar87](zreferences.html#id197)], this difference equation is the Euler equation for\n",
"a planning problem that maximizes the discounted sum of consumer plus\n",
"producer surplus.\n",
"\n",
"

**[4]** [[PS05](zreferences.html#id22)] verify the same claim by applying machinery of [[PCL86](zreferences.html#id23)].\n",
"\n",
"

**[5]** See [[AHMS96](zreferences.html#id135)] for an account of invariant subspace methods.\n",
"\n",
"

**[6]** See [[AMS02](zreferences.html#id28)] for a discussion\n",
"of information assumptions needed to create a situation\n",
"in which higher order beliefs appear in equilibrium decision rules. A way\n",
"to read our findings in light of [[AMS02](zreferences.html#id28)] is that, relative\n",
"to the number of signals agents observe, Townsend’s\n",
"section 8 model has too few random shocks to get higher order beliefs to\n",
"play a role."
]
}
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