\section{Introduction}
In a dynamic stochastic general equilibrium (DSGE) model, satisfaction of time consistency \citep{kydland77} is crucial, especially for a rational expectation equilibrium (REE).

Without stochastic shocks, DSGE models are equivalent to multi-period static models, except for the question of time consistency, which prohibits some of possible equilibria in dual multi-period static models.

This paper intends to demonstrate that under some initial conditions, there is no time-consistent REE in an unexpected way. There is surprise in that conventional dynamic model analysis \citep{slp89} suggests there should be no equilibrium existence issue. The resolution to this `paradox' is easily found: for an economy $E_0$, if we instead take limit $E \rightarrow E_0$, then there exists the limit of model-corresponding equilibria. There is no problem utilizing this limit equilibrium instead. This limit strategy evades the time consistency issue by taking the limit of economies that do not have time consistency issues.

\section{Time consistency issue}
In fashion of a New Keynesian model \citep{woodford03,gali15}, assume that a central bank controls issuance of $B_{it}$, where $i$ refers to each consumer. When $B_{it}>0$, consumer has government bonds, when $B_{it} <0$, consumer is indebted to the central bank. $B_{it}$ is the only way consumers can build up wealth or be indebted. As standard with many DSGE models, an infinite-horizon economy is assumed.

Suppose that the central bank, for whatever reasons, decides to impose $B_{it} = 0$, along with a right interest rate path for the implementation. That is, central bank decides to run an economy as if there is no finance. While the assumption is quite toxic, this is not expected to produce no equilibrium by itself.

Now the specification of consumer $i$ follows.
\begin{equation}
\label{eqn:girutilitymax}
\max_{\{C_{it}, B_{it},..\}}U_i = \max_{\{C_{it}, B_{it},..\}} \sum_{t=0}^{\infty} (\beta_i)^t u_i(C_{it},..)
\end{equation}
where $C_{it}$ refers to consumption good, $\beta_i$ refers to time discount factor and $u_i$ refers to a single-period utility function with $U$ being time-discounted utility function. It is assumed that generally, $\beta_i \neq \beta_j$. Let the budget constraint of each consumer be:
\begin{equation}
\label{eqn:girbudget}
P_t C_{it} + (1+i_t)^{-1} B_{it} +.. \leq  B_{i(t-1)}+..
\end{equation}
where $..$ terms do not contain any of $C_{it}$ and $B_{it}$. $P_t$ refers to price level and $i_t$ is nominal interest rate set by central bank.

The following main equation may be derived from consumer optimization, with the derivation provided in the appendix:
\begin{equation}
\label{eqn:girmain}
\frac{u_j'(C_{jt})}{u_k'(C_{kt})} = \frac{(1-\gamma_{jk})}{\gamma_{jk}}\left(\frac{\beta_k}{\beta_j}\right)^t
\end{equation}
Note that Equation \eqref{eqn:girmain} can be derived even when $B_{it}$ and central bank do not exist - see the appendix.

Suppose that economic conditions remain time-invariant. For example, there is zero technology factor growth, there is no capital accumulation in an economy as standard in a basic New Keynesian model \citep{woodford03,gali15} and so on. As aforementioned, $B_{it} = 0$, and there are infinitely many time periods. Then Equation \eqref{eqn:girmain} creates the time inconsistency issue not found in a dual multi-period static model.

Since economic conditions remain same across periods and there is no `end' in time, the left-hand side of Equation \eqref{eqn:girmain} should remain constant across time. This is impossible when $\beta_k \neq \beta_j$.

\subsection{Limit economy}
Note that the `initial condition' requirement for the time inconsistency result is relatively strong. First, $B_{it} = 0$ was required. Second, other economic conditions also are assumed to remain same across periods. An infinite horizon also plays a role - however, technically, the time inconsistency result itself can be created even in a finite horizon circumstance.

If any of these conditions is not satisfied, then Equation \eqref{eqn:girmain} cannot be asserted to generate the time consistency issue.

We may then choose to approach an economy with the time consistency issue as a limit of neighboring economies that do have time-consistent equilibria. A limit of equilibria then can be asserted to be an equilibrium for a limit economy (the economy with the time consistency issue) as well.

Therefore, in practical sense, an economy with the time consistency issue may safely be treated as if it has an equilibrium. In case of the example economy, even if fiscal authority does not actually exercise its power, its potential power to reallocate $B_{it}$ helps to entirely avoid theoretical issues. 

\section{Conclusion}
The lesson is that there must be significant care that must be taken of when dealing with rational expectation equilibria (REE). It is well-known that not all REE are meaningful or learnable. \citep{evans01,evans18} The direction also goes from non-existence as well - non-existence of REE has to be carefully considered.

For REE non-existence driven by a time consistency issue induced from consumer heterogeneity, it is easy to construct nearby economies that do not have time consistency issues, thereby evading the issue. In such a case, it is not economics-wise meaningful to declare that there exists no REE.

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