Blanchard and Kahn are neither sufficient nor necessary to obtain a solution to a perfect foresight model: if there exist multiple stable solutions to the model, the perfect foresight algorithm may sometimes return just one of them.

Even if the BK conditions are satisfied, there is another condition that must be met: the model must such that given past and future value of the variables, there exists a unique solution for the current variables of the model. In your model, equation 2

```
R(+1)*Lambda(+1) = 1
```

is the culprit: there is no current variable in that equation. This equation doesn’t really determine today’s interest rate. Any process for the today’s interest rate such that the expected value of tomorrow interest rate rate satisfies this equation would do. You should revisit the timing of interest rate in the model.

The simulation with perfect foresight fails because we compute the actual value of the variables on a finite horizon. On the other hand, the approach for the stochastic model finds a solution because we are looking for a policy function in infinite horizon adding the asymptotic stationarity constraint.