Showing that the elasticity of arrival rate for workers wrt $\theta$ is between $0$ and $1$ Let $L=E+U$ where $L$ is labour force, $E$ is number of employed, and $U$ is unemployed people. 

Let $u = \frac{U}{L}$ and $v = \frac{V}{L}$.

Given $m(u,v)$ as a matching function that determines the flow of matches between workers and firms, and assume $m$ has constant return to scale (CRS) in the two arguments $u$ and $v$. 

The elasticity of the arrival rate for workers is given by $$\theta q(\theta),$$ where $\theta = \frac{v}{u}$ represents "tightness" in the labour market. 

Elasticity ($\epsilon$) = $$\frac{\mathrm{d} f(x)}{\mathrm{d} x} \frac{x}{f(x)} = \frac{\partial m}{\partial \theta} \frac{\theta}{m(1,\theta)},$$

where $\theta q(\theta) = m(\theta^{-1},1) = m(1,\theta)$ (by using CRS property of $m$ in the two arguments $u$,$v$).

The rate at which unemployed workers find job: $$\theta q(\theta) = m(\theta^{-1},1)=m(1,\theta),$$ since the matching function, $m$, is constant returns to scale in the each argument. 

Since $\theta q(\theta) = m(1,\theta)$, $$\frac{\partial m(1,\theta)}{\partial \theta} = q(\theta) + \theta q'(\theta).$$

So the elasticity of the arrival rate for workers, $$\epsilon = \frac{\partial m}{\partial \theta} \frac{\theta}{m} = \left [q(\theta) + \theta q'(\theta)  \right ] \frac{\theta}{\theta q(\theta)} = 1 + \frac{\theta q'(\theta)}{q(\theta)}$$.

Since $q'(\theta)&lt;0$, $\epsilon&lt;1$, but how can I formally show that it 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.