<p>Given <span class="math-container">$m(u,v)$</span> is CRTS we have the property that <span class="math-container">$m(u,v)=\frac{\delta{m}}{\delta{v}}v+\frac{\delta{m}}{\delta{u}}u$</span>. Divide by <span class="math-container">$v$</span> to get in terms of <span class="math-container">$\theta$</span>: <span class="math-container">$m(\theta)=\frac{\delta{m}}{\delta{v}}+\frac{\delta{m}}{\delta{u}}\theta$</span>.</p>

<p>Thus, it clearly follows that:</p>

<p><span class="math-container">$\frac{df}{d\theta}=\mu\frac{dm(\theta)}{d\theta}=\mu\frac{\delta{m}}{\delta{u}}&gt;0$</span> given that we have positive marginal products</p>

CC BY-SA 4.0
2019-04-22T14:42:07.860Z
2019-04-22T14:42:07.860Z
Robin Liao
<p>In a simplified model in the search theory of unemployment, let <span class="math-container">$M=\mu{m(u,v)}$</span> be the matching function where <span class="math-container">$u$</span> and <span class="math-container">$v$</span> is unemployment and vacancies respectively. Given <span class="math-container">$m(u,v)$</span> has constant returns to scale and positive marginal products, how do we show that the job finding rate <span class="math-container">$f = M/u$</span> increases with labour market tightness <span class="math-container">$\theta=\frac{v}{u}$</span>? </p>

<p>Given CRTS, I can rewrite <span class="math-container">$f$</span> as follows: <span class="math-container">$f=\mu{m(1,\theta)}=\mu{}m(\theta)$</span>. How can you then show that given positive marginal products, we have <span class="math-container">$\frac{df}{d\theta}&gt;0?$</span></p>

<search-and-matching>
Constant Returns to Scale and Positive Marginal Products
Robin Liao
# Constant Returns to Scale and Positive Marginal Products

(Asked by: username_0 on 2019-04-16)

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

**Tags:** search-and-matching

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## 1 answers

### Answer by username_0 on 2019-04-22. Score: 0 | Accepted answer

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


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