= Solution
{c}
Let $f(X,Y)=X[r/(1+bY^2)-\delta]$. At the positive <steady state>, $1+b(N^*)^2=r/s$, where $s=\delta+1$. The two <partial derivatives> are
$$
f_X(N^*,N^*)=1,\qquad f_Y(N^*,N^*)=-\frac{2rb(N^*)^2}{(1+b(N^*)^2)^2}=-\frac{2s(r-s)}r.
$$
The first-order <Taylor expansion> with $N_t=N^*+n_t$ therefore gives
$$
\boxed{n_{t+1}-n_t+a n_{t-1}=0,\qquad a=\frac{2s(r-s)}r=\frac{2(r-(\delta+1))(\delta+1)}r.}
$$
Equivalently, the <linearization> on $(n_t,n_{t-1})$ has matrix $\begin{pmatrix}1&-a\\1&0\end{pmatrix}$ and <characteristic polynomial> $\mu^2-\mu+a$.
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