Consider a deterministic stationary overlapping generations 1-good exchange economy with a representative generation consisting of a number *H* of 2-period-lived agents with utility function *u*^{h} and endowments ${e}^{h}=({e}_{1}^{h},{e}_{2}^{h})$, for all $h=1,\dots ,H$.

Agents have access to historical records or a memory of length *m* (maybe infinity), so that they know the price of the good in the last *m* periods. I will assume more-over, without loss of generality, that the agents *believe* that prices follow a *k*-state Markov chain over *k* prices.

In what follows, histories of prices ${\left\{{p}_{t}\right\}}_{t\in T}$ (with *T* either $\mathbb{N}$ or $\mathbb{Z}$) taking at any period any of a finite number *k* of possible values ${p}^{1},\dots ,{p}^{k}$, are denoted by means of a function ${\mathrm{\delta}}_{t}^{i}$ indicating whether the price *p*^{i} has been realized at period *t* or not. Thus ${\mathrm{\delta}}_{t}^{i}=1$ whenever ${p}_{t}={p}^{i}$, and equals 0 otherwise. Since only one price can prevail at any period *t*, it must hold that ${\sum}_{i=1}^{k}{\mathrm{\delta}}_{t}^{i}=1$ for all $t\in T$. Therefore, a history of realizations is a sequence $\mathrm{\delta}={\left\{{\mathrm{\delta}}_{t}\right\}}_{t\in T}$ of *k*-tuples of $k-1$ zeros and one 1 at the position of the price realized at that period, that is to say, for all $t\in T$, ${\mathrm{\delta}}_{t}\in {\left\{0,1\right\}}^{k}$ and ${\sum}_{i=1}^{k}{\mathrm{\delta}}_{t}^{i}=1$. Let Δ denote the set of such sequences.

A specific instance of a *rationally formed expectations equilibrium* is defined next.

**Definition:** *A (**k*-state Markovian) **Rationally Formed Expectations Equilibrium** of the deterministic stationary overlapping generations exchange economy with representative generation $({u}^{h},{e}^{h}{)}_{h=1}^{H}$ with memory *m* consists of

*(i)*

*a finite number of positive prices for consumption ${p}^{i}>0$, $i=1,\dots ,k$
*

*(ii)*

*nonnegative first-period consumptions and contingent plans of second-period consumptions $\left({c}_{1}^{hi},{\left\{{c}_{2}^{hj}\right\}}_{j=1}^{k}\right)$ for each agent $h=1,\dots ,H$ at each possible price when young, i.e. for all $i=1,\dots ,k$*

*(iii)*

*beliefs about the probabilities of transition between prices, i.e. a Markov matrix $({\mathrm{\pi}}_{t\mathrm{\delta}}^{hij}{)}_{i,j=1}^{k}$, for each agent $h=1,\dots ,H$ and any history of prices $\mathrm{\delta}\in \mathrm{\Delta}$ up to his date of birth $t\in T$,
*

such that

(c.1) the allocation is feasible, i.e. for all *i*= 1, …, *k*
$\sum _{h=1}^{H}({c}_{1}^{hi}+{c}_{2}^{hi})=\sum _{h=1}^{H}({e}_{1}^{h}+{e}_{2}^{h})$[9]

(c.2) for any history $\mathrm{\delta}\in \mathrm{\Delta}$ and every agent *h*= 1, …, *H* born at any date $t\in T$, his first-period consumption and contingent plan of second-period consumptions $\left({c}_{1}^{hi},{\left\{{c}_{2}^{hj}\right\}}_{j=1}^{k}\right)$ are optimal, given his beliefs, whenever at *t* the price is *p*^{i}, for any *i*= 1, …, *k*, so that it solves
$\begin{array}{rl}& max\phantom{\rule{1em}{0ex}}\sum _{j=1}^{k}{\mathrm{\pi}}_{t\mathrm{\delta}}^{hij}{u}^{h}({c}_{1}^{i},{c}_{2}^{j})\\ & \mathrm{s}.\mathrm{t}.\phantom{\rule{1em}{0ex}}{p}^{i}({c}_{1}^{i}-{e}_{1}^{h})+{p}^{j}({c}_{2}^{j}-{e}_{2}^{h})=0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}j\end{array}$[10]

(c.3) for any history $\mathrm{\delta}\in \mathrm{\Delta}$ and every agent *h*= 1, …, *H* born at any date $t\in T$, no other beliefs for which $\left({c}_{1}^{hi},{\left\{{c}_{2}^{hj}\right\}}_{j=1}^{k}\right)$ is optimal when at *t* the price is *p*^{i}, for any *i*= 1, …, *k*, provide a higher likelihood to the history of prices he remembers, i.e. if ${\mathrm{\pi}}^{i\cdot}\in {S}^{k-1}$ (the *k* − 1-dimensional unit simplex) is such that $\left({c}_{1}^{hi},{\left\{{c}_{2}^{hj}\right\}}_{j=1}^{k}\right)$ solves
$\begin{array}{rl}& max\phantom{\rule{1em}{0ex}}\sum _{j=1}^{k}{\mathrm{\pi}}^{ij}{u}^{h}({c}_{1}^{i},{c}_{2}^{j})\\ & \mathrm{s}.\mathrm{t}.\phantom{\rule{1em}{0ex}}{p}^{i}({c}_{1}^{i}-{e}_{1}^{h})+{p}^{j}({c}_{2}^{j}-{e}_{2}^{h})=0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}j\end{array}$[11]

then
$\prod _{j=1}^{k}{({\mathrm{\pi}}^{ij})}^{{\sum}_{\mathrm{\tau}=1}^{m}{\mathrm{\delta}}_{t-\mathrm{\tau}}^{i}{\mathrm{\delta}}_{t-\mathrm{\tau}+1}^{j}}\le \prod _{j=1}^{k}{({\mathrm{\pi}}_{t\mathrm{\delta}}^{hij})}^{{\sum}_{\mathrm{\tau}=1}^{t}{\mathrm{\delta}}_{t-\mathrm{\tau}}^{i}{\mathrm{\delta}}_{t-\mathrm{\tau}+1}^{j}}$[12]

where ${\mathrm{\delta}}_{t-\mathrm{\tau}}^{i}=0$ for $\mathrm{\tau}\ge t$ if $T=\mathbb{N}$ – i.e. the likelihood of the observed transitions from *p*^{i} in history *δ* up to period *t* if prices follow the Markov chain $({\mathrm{\pi}}_{t\mathrm{\delta}}^{hij}{)}_{i,j=1}^{k}$ (the RHS in eq. [12]), is at least as high as for any other Markov chain $({\mathrm{\pi}}^{ij}{)}_{i,j=1}^{k}$ (the LHS in eq. [12]) – and

(c.4)when $T=\mathbb{Z}$ and $m=\mathrm{\infty}$, for any history $\mathrm{\delta}\in \mathrm{\Delta}$ and every agent *h*= 1, …, *H* born at any $t\in T$, his beliefs are not falsified by the information available then, i.e. for all *i*, *j*= 1, …, *k*,
${\mathrm{\pi}}_{\mathrm{\delta}t}^{hij}=\underset{{t}^{{\prime}^{}}\to -\mathrm{\infty}}{lim}\frac{{\sum}_{\mathrm{\tau}={t}^{{\prime}^{}}}^{t-1}{\mathrm{\delta}}_{\mathrm{\tau}}^{i}{\mathrm{\delta}}_{\mathrm{\tau}+1}^{j}}{{\sum}_{\mathrm{\tau}={t}^{{\prime}^{}}}^{t-1}{\mathrm{\delta}}_{\mathrm{\tau}}^{i}}$[13]

whenever the limit exists.

Some remarks on the definition above are in order. Note first that if the beliefs are constrained to be history and agent independent (so that ${\mathrm{\pi}}_{\mathrm{\delta}t}^{hij}$ becomes ${\mathrm{\pi}}^{ij}$) and the last conditions (c.3) and (c.4) are dropped, then the definition above becomes that of a stationary rational expectations (sunspot) equilibrium following a *k*-state Markov chain, or *k*-SSE.
Note that condition (c.4) is trivially satisfied by such a *k*-SSE but, crucially, (c.3) is not. As a consequence, in a rational expectations equilibrium there exist typically, for every agent, beliefs about the probabilities of transition that are consistent with his consumption choice but that make the history he observes likelier than the equilibrium beliefs do. Of course the discrepancy between the agents’ beliefs and those maximizing the likelihood of history while rationalizing the choices vanishes in the limit if, as in the sunspot equilibrium interpretation, the prices are supposed to actually follow a given Markov chain. But the determination of prices by a specific stochastic process is difficult to justify in the absence of shocks to the fundamentals.

Secondly, condition (c.3) is not superfluous. If instead of condition (c.3) only the existence of subjective beliefs rationalizing the agents’ choices was required (regardless of history), that would imply a set of equilibrium allocations and prices that is a strict superset of the set of rationally-formed expectations equilibria. In effect, while any rationally-formed expectations equilibrium clearly satisfies the existence of subjective beliefs rationalizing the agents’ choices, there are consumption plans, prices and arbitrary, history-independent subjective beliefs rationalizing the agents’ choices that are not rationally-formed expectations equilibria, since history-independent beliefs cannot solve the problem [23] in the proof of Proposition 2 below – equivalent to condition (c.3) – *for all realizations of history*.

Finally, note also that, as previously claimed, the restriction to beliefs in Markovian prices is not constraining for finite memory or $T=\mathbb{N}$. In effect, such an assumption cannot be refuted by the agents unless the data available to them is able to falsify it, but for that to be the case it must at least allow to establish that the empirical frequencies are not Cauchy (if the sequence of empirical frequencies of transitions from any *p*^{i} to *p*^{j} was Cauchy, then completeness would imply its convergence, which would support the Markovian assumption). That is to say, it must allow to conclude that the distance between any two empirical frequencies at dates $t<{t}^{{\prime}^{}}$ from any *p*^{i} to *p*^{j} does not become arbitrarily small, for *t*, ${t}^{{\prime}^{}}$ sufficiently far away down the sequence. In other words, for the agents to be able to discard the assumption of Markovian prices they would need to have infinite histories and memories, so that no data can falsify that assumption if $T=\mathbb{N}$ or *m* is finite (on the contrary, when $T=\mathbb{Z}$ and *m* is infinite, the agents can compute the empirical frequency at any given date *t* of the transitions from any price *p*^{i} to *p*^{j} as the limit
$\underset{{t}^{{\prime}^{}}\to -\mathrm{\infty}}{lim}\frac{{\sum}_{\mathrm{\tau}={t}^{{\prime}^{}}}^{t-1}{\mathrm{\delta}}_{\mathrm{\tau}}^{i}{\mathrm{\delta}}_{\mathrm{\tau}+1}^{j}}{{\sum}_{\mathrm{\tau}={t}^{{\prime}^{}}}^{t-1}{\mathrm{\delta}}_{\mathrm{\tau}}^{i}}.$

Should this limit not exist, the Markovian assumption would then be falsified by the data in this case).

As a matter of fact, for any two consecutive terms the distance between the empirical frequencies of transitions converges to zero along the sequence, since
$\left|\frac{{\sum}_{\mathrm{\tau}=1}^{t+1}{\mathrm{\delta}}_{\mathrm{\tau}}^{i}{\mathrm{\delta}}_{\mathrm{\tau}+1}^{j}}{{\sum}_{\mathrm{\tau}=1}^{t+1}{\mathrm{\delta}}_{\mathrm{\tau}}^{i}}-\frac{{\sum}_{\mathrm{\tau}=1}^{t}{\mathrm{\delta}}_{\mathrm{\tau}}^{i}{\mathrm{\delta}}_{\mathrm{\tau}+1}^{j}}{{\sum}_{\mathrm{\tau}=1}^{t}{\mathrm{\delta}}_{\mathrm{\tau}}^{i}}\right|=\frac{{\mathrm{\delta}}_{t+1}^{i}}{{\sum}_{\mathrm{\tau}=1}^{t+1}{\mathrm{\delta}}_{\mathrm{\tau}}^{i}}\cdot \left|{\mathrm{\delta}}_{t+2}^{j}-\frac{{\sum}_{\mathrm{\tau}=1}^{t}{\mathrm{\delta}}_{\mathrm{\tau}}^{i}{\mathrm{\delta}}_{\mathrm{\tau}+1}^{j}}{{\sum}_{\mathrm{\tau}=1}^{t}{\mathrm{\delta}}_{\mathrm{\tau}}^{i}}\right|$[14]

and

(1)

either *p*^{i} is visited finitely many times and then for some *t* onwards ${\mathrm{\delta}}_{t}^{i}=0$, so that the distance between the empirical frequencies of transition becomes 0 from that term on, and the empirical frequency of transition from *p*^{i} to *p*^{j} becomes constant and therefore convergent,

(2)

or *p*^{i} is visited countably many times and then the first factor in the right-hand side converges to zero (the numerator is bounded and the denominator is both non-decreasing and not non-increasing), while the second factor between brackets is bounded in [0, 1] (the first term is in {0, 1} and the second is in [0, 1]), so that the distance between empirical frequencies of transition from *p*^{i} to *p*^{j} converges to zero.

Thus, when $T=\mathbb{N}$ and agents have unrestricted memory, not only the agents do not have enough information to falsify the Markovian prices assumption, but also they will see vanish progressively any dependence of the probabilities of transition on earlier prices (as differences between subsequent empirical frequencies converge to zero), i.e. Markovian prices tend to be confirmed (although not proved), rather than falsified.

Of course, agents can all believe in Markovian prices while not necessarily agreeing on the specific probabilities of transition governing that process, since they have access to different bits of history when $T=\mathbb{N}$ or memory is finite. On the contrary, if memory is infinite and $T=\mathbb{Z}$, they all have to agree on the probabilities of transitions as well if the limit in eq. [13] above exists for every *t*; while, if memory is infinite and $T=\mathbb{N}$, they all “eventually agree”, meaning that discrepancies of subsequent generations tend to vanish. In the last two cases, in which agents agree (maybe asymptotically) on the probabilities of transition, the limit of the empirical frequencies would necessarily have to be in the intersection on the unit simplex of the linear subspaces determined by the agents’ first-order conditions, as proclaimed in Proposition 1 below (the proof is straightforward). In other words, if memory is infinite, the only rationally formed expectations equilibria are those for which such an intersection exists, but these equilibria are allocationally equivalent to the rational expectations (sunspot) equilibrium associated with such an intersection.

**Proposition 1:** *If the agents’ memory m is infinite, any rationally formed expectations equilibrium of the stationary deterministic overlapping generations exchange economy* $({u}^{h},{e}^{h}{)}_{h=1}^{H}$ *is allocationally equivalent to a k-state sunspot equilibrium*.

Rationally formed expectations equilibria distinct from a rational expectations equilibrium exist in this setup, therefore, only if memory is finite. There can be many reasons why *m* finite is the relevant case. People tend to make forecasts based on their recent experiences, with memories of variable lengths, but certainly of finite length if only because of their actual limited recording and computing abilities. Thus the limited memory case seems to be the relevant one, while the equivalence of rationally formed expectations equilibria and rational expectations sunspot equilibria in the infinite memory case rather highlights the role played by limited knowledge in making possible rationally formed expectations equilibria distinct from rational expectations equilibria.

The next proposition establishes the main result of the paper, namely that any deterministic stationary overlapping generations economy with sunspot equilibria can be perturbed robustly in order to produce rationally formed expectations equilibria that no sunspot equilibrium can match.

**Proposition 2:** *Arbitrarily close*
*to every deterministic stationary overlapping generations economy (with at least two agents of a given type) with a k-state stationary sunspot equilibrium, there exists an economy with finite-memory rationally formed expectations equilibria distinct from any rational expectations equilibrium*.

**Proof** Let $({u}^{h},{e}^{h}{)}_{h\in H}$ be the utility and endowments *types* of the members of the representative generation of a stationary overlapping generations economy, with at least one type of agents *h*_{0} with two agents or more. Let ${\left\{{p}^{i},{({\stackrel{\u02c9}{c}}_{1}^{hi},{\stackrel{\u02c9}{c}}_{2}^{hi})}_{h\in H}\right\}}_{i=1}^{k}$ be the contingent prices and consumptions of a *k*-state stationary sunspot equilibrium of the economy driven by a Markov chain with probabilities of transition $({\mathrm{\pi}}^{ij}{)}_{i,j=1}^{k}$.

Consider a new economy with a representative generation consisting of replacing in *H* just one agent of type *h*_{0} by an agent *h*_{1} with the same endowments and consumptions as agent *h*_{0} – the new allocation of the new economy is therefore feasible – and a utility function ${u}^{{h}_{1}}$ with gradients at the consumption bundles $({\stackrel{\u02c9}{c}}_{1}^{{h}_{1}i},{\stackrel{\u02c9}{c}}_{2}^{{h}_{1}j}{)}_{i,j=1}^{k}$ such that, for some *i*= 1, …, *k*, the vector of products ${A}_{{u}^{h}}^{ij}\equiv {D}_{1}{u}^{h}({\stackrel{\u02c9}{c}}_{1}^{hi},{\stackrel{\u02c9}{c}}_{2}^{hj})({\stackrel{\u02c9}{c}}_{1}^{hi}-{e}_{1}^{h})+{D}_{2}{u}^{h}({\stackrel{\u02c9}{c}}_{1}^{hi},{\stackrel{\u02c9}{c}}_{2}^{hj})({\stackrel{\u02c9}{c}}_{2}^{hj}-{e}_{2}^{h})$ – where ${D}_{i}{u}^{h}$ stands for the partial derivative of *u*^{h} with respect to *c*_{i} – of the gradient of utility and the trade implied by the planned consumptions for each possible transition starting from *i* for agent *h*_{1}
${A}_{{u}^{{h}_{1}}}^{i}=({A}_{{u}^{{h}_{1}}}^{i1},\dots ,{A}_{{u}^{{h}_{1}}}^{ik})$[15]

is distinct from, but in the span of, ${A}_{{u}^{{h}_{0}}}^{i}$ and **1**=(1, …, 1), i.e.
${A}_{{u}^{{h}_{1}}}^{i}=\mathrm{\alpha}{A}_{{u}^{{h}_{0}}}^{i}+\mathrm{\beta}\mathbf{1}$[16]

for some *α* and some $\mathrm{\beta}\ne 0$.
Then the system
$\begin{array}{rl}& {\mathrm{\pi}}^{i1}{A}_{{u}^{{h}_{0}}}^{i1}+\cdots +{\mathrm{\pi}}^{ik}{A}_{{u}^{{h}_{0}}}^{ik}=0\\ & {\mathrm{\pi}}^{i1}{A}_{{u}^{{h}_{1}}}^{i1}+\cdots +{\mathrm{\pi}}^{ik}{A}_{{u}^{{h}_{1}}}^{ik}=0\end{array}$[17]

has no solution in the probabilities ${\mathrm{\pi}}^{i1},\dots ,{\mathrm{\pi}}^{ik}$. Indeed, should there be one, using eq. [16] above, the second equation in [17] can be written equivalently as
$\mathrm{\alpha}({\mathrm{\pi}}^{i1}{A}_{{u}^{{h}_{0}}}^{i1}+\cdots +{\mathrm{\pi}}^{ik}{A}_{{u}^{{h}_{0}}}^{ik})+\mathrm{\beta}({\mathrm{\pi}}^{i1}+\cdots +{\mathrm{\pi}}^{ik})=0$[18]

but from the first equation in [17] and $\mathrm{\beta}\ne 0$, then it should hold
${\mathrm{\pi}}^{i1}+\cdots +{\mathrm{\pi}}^{ik}=0$[19]

which cannot be since these probabilities should add up to 1. Since a solution to eq. [17] is needed for the given allocation to be that of a sunspot equilibrium, this establishes that the prices and consumptions ${\left\{{p}^{i},{({\stackrel{\u02c9}{c}}_{1}^{hi},{\stackrel{\u02c9}{c}}_{2}^{hi})}_{h\in H}\right\}}_{i=1}^{k}$, with $({\stackrel{\u02c9}{c}}_{1}^{{h}_{1}i},{\stackrel{\u02c9}{c}}_{2}^{{h}_{1}j}{)}_{i,j=1}^{k}=({\stackrel{\u02c9}{c}}_{1}^{{h}_{0}i},{\stackrel{\u02c9}{c}}_{2}^{{h}_{0}j}{)}_{i,j=1}^{k}$, are *not* those of a sunspot equilibrium of the new, arbitrarily close economy.

They are, nevertheless, the allocation and prices of a rationally formed expectations equilibrium of an arbitrarily close economy. In effect, for all $h\in H$, all $t\in T$, and all $\mathrm{\delta}\in \mathrm{\Delta}$, there exists $({\mathrm{\pi}}_{t\mathrm{\delta}}^{hij}{)}_{i,j=1}^{k}$ solution to
$\begin{array}{rl}& \underset{{\mathrm{\pi}}^{ij}}{max}\prod _{i,j=1}^{k}{({\mathrm{\pi}}^{ij})}^{{\sum}_{\mathrm{\tau}=1}^{m}{\mathrm{\delta}}_{t-\mathrm{\tau}}^{i}{\mathrm{\delta}}_{t-\mathrm{\tau}+1}^{j}}\\ & \mathrm{s}.\mathrm{t}.\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall}i,\phantom{\rule{thickmathspace}{0ex}}{\mathrm{\pi}}^{i\cdot}\in {S}^{k-1}\\ & \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{\forall}i,({\stackrel{\u02c9}{c}}_{1}^{hi},{\left\{{\stackrel{\u02c9}{c}}_{2}^{hj}\right\}}_{j})=arg\phantom{\rule{thinmathspace}{0ex}}max{\sum}_{j}{\mathrm{\pi}}^{ij}{u}^{h}({c}_{1}^{i},{c}_{2}^{j})\\ & \mathrm{s}.\mathrm{t}.\phantom{\rule{thickmathspace}{0ex}}{p}^{i}({c}_{1}^{i}-{e}_{1}^{h})+{p}^{j}({c}_{2}^{j}-{e}_{2}^{h})=0,\mathrm{\forall}j\end{array}$[20]

– where ${\mathrm{\delta}}_{t-\mathrm{\tau}}^{i}=0$ for $\mathrm{\tau}\ge t$ if $T=\mathrm{N}$ – since the objective function is continuous, and the constrained set is non-empty and compact. The same is true for any ${\tilde{u}}^{{h}_{1}}$ close enough to ${u}^{{h}_{1}}$ in the topology of *C*^{1}-convergence over compacta, i.e. with ${A}_{{\tilde{u}}^{{h}_{1}}}^{i}$ not necessarily in the span of ${A}_{{u}^{{h}_{0}}}^{i}$ and **1**.

Finally, since *m* is finite, the remembered empirical frequencies of the transitions do not falsify the agents’ beliefs.

Q.E.D.

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