Suppose the central bank sets interest rates according to the following rule.

i_t=\sigma \pi_t +\theta,

where \pi is inflation, i_t is the nominal interest rate, and \theta is a AR(1) error term with mean zero. \sigma is a parameter that governs how strongly the central bank responds to inflation.

The Euler equation gives us the following expression

Y_t=E(Y_t)-(i_t-E(\pi_{t+1})),

which gives us i_t=E(\pi_{t+1}) if Y_t is constant. Combining the two expressions we get

E(\pi_{t+1})=\sigma \pi_t +\theta.

We also have the boundary condition \lim_{t \to +\infty} E(\pi_{t+1})=0

The above two equations and boundary condition give us an infinite number of finite solutions in the case where \sigma is less than 1. Any value of inflation today is a solution to the model. If, on the other hand, \sigma is greater than 1 we only have one solution, where \pi_t=(1-\sigma) \theta. By turning solutions that exist if \sigma is less than 1 into solutions where inflation goes to infinity the central bank somehow ensures they do not happen in reality, and thus inflation is made determinate.

To extend the above logic we simply need to realize that the model is indeterminate in every variable that we have not explicitly excluded elsewhere in the model. The standard model says absolutely nothing about the birth rate, for example. Using the same approach it used to eliminate inflation indeterminacy the central bank in the model can eliminate these other indeterminacies, and set the birth rate to whatever it wants.

To do so we simply modify the central bank’s reaction function to add a term that only exists if the birth rate is a particular value. If the birth rate is not equal to the birth rate target then the central bank reacts to inflation as follows
i_t=\sigma\pi_t+\theta_t+ 10), while if the birth rate is on target it uses the normal reaction function There are no finite solutions to the model using the modified reaction function unless the birth rate is equal to the birth rate target, so, using the same logic as above, we can conclude that the birth rate must now always equal the birth rate target.