Constant Returns to Scale and Positive Marginal Products In a simplified model in the search theory of unemployment, let $M=\mu{m(u,v)}$ be the matching function where $u$ and $v$ is unemployment and vacancies respectively. Given $m(u,v)$ has constant returns to scale and positive marginal products, how do we show that the job finding rate $f = M/u$ increases with labour market tightness $\theta=\frac{v}{u}$? 

Given CRTS, I can rewrite $f$ as follows: $f=\mu{m(1,\theta)}=\mu{}m(\theta)$. How can you then show that given positive marginal products, we have $\frac{df}{d\theta}&gt;0?$
Given $m(u,v)$ is CRTS we have the property that $m(u,v)=\frac{\delta{m}}{\delta{v}}v+\frac{\delta{m}}{\delta{u}}u$. Divide by $v$ to get in terms of $\theta$: $m(\theta)=\frac{\delta{m}}{\delta{v}}+\frac{\delta{m}}{\delta{u}}\theta$.

Thus, it clearly follows that:

$\frac{df}{d\theta}=\mu\frac{dm(\theta)}{d\theta}=\mu\frac{\delta{m}}{\delta{u}}&gt;0$ given that we have positive marginal products