Showing that the elasticity of arrival rate for workers wrt $\theta$ is between $0$ and $1$
Let $m$ cobb-douglas. This means $m(u,v) = A u^\alpha v^{1-\alpha}$ for $\alpha &lt; 1$ and $q(\theta) = A \theta^{-\alpha}$.

Then 

$$\frac{\theta q'(\theta)}{q(\theta)} = -\alpha &gt; -1$$

To what extent this holds for other functional forms I'm not sure, but I haven't seen anything but Cobb-Douglas being used as the matching function.