# 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 $$\frac{df}{d\theta}>0?$$

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 $$\theta$$: $$m(\theta)=\frac{\delta{m}}{\delta{v}}+\frac{\delta{m}}{\delta{u}}\theta$$.
$$\frac{df}{d\theta}=\mu\frac{dm(\theta)}{d\theta}=\mu\frac{\delta{m}}{\delta{u}}>0$$ given that we have positive marginal products