Saving increasing in interest rate Romer equation 2.55 on the Diamond model, says:
  
  Let $s(r)$ being the fraction of income saved, $r$ the interest rate,$\rho$ the discount rate, and $\theta$ the household's willingness to shift consumption between periods, with
  
  $$ s(r) = \frac{(1+r)^{(1-\theta)/\theta}}{ (1+\rho)^{1/\theta} + (1+r)^{(1-\theta)/\theta} } $$
  implies that saving is increasing in $r \iff (1+r)^{(1 - \theta)/\theta)}$ is increasing in r; equivalently $ds / dr &gt; 0$ if $\theta &lt; 1$


Is the first statement related to monoticity?  Also, how to show this graphically, that is to show, $s(r)$ a function of $\theta$?  Considering this, plotting $(\theta, s(\theta, r=.1))$ I do not see a change at $\theta=1$, but when I graphed $(r, s(r))$ with values $\theta=0.5,$ and $\theta=1.5$ I saw the desired change in orientation.  Which way is right?
$$y = \frac {a(x)}{b+a(x)} \implies \frac{\partial y}{\partial x} = \frac{\partial y}{\partial a}\cdot \frac{\partial a}{\partial x}$$

Since

$$\frac{\partial y}{\partial a} = \frac {b+a -a}{(b+a)^2} = \frac {b}{(b+a)^2} &gt;0$$

because in your case "$b$"$&gt;0$,

the sign of $\frac{\partial y}{\partial x}$ will be the same as the sign of $\frac{\partial a}{\partial x}$, which is what Romer is asserting.

$\theta &gt;1$ would make $\frac{\partial a}{\partial x} &lt;0$, so 

$$\theta &lt;1 \implies \frac{\partial a}{\partial x}&gt;0\implies \frac{\partial y}{\partial x}&gt;0$$