0
$\begingroup$

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

$\endgroup$
0
$\begingroup$

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:

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.