<p>I am trying to replicate the Search and Matching model Walsh (2003) <a href="https://escholarship.org/content/qt6tg550dv/qt6tg550dv.pdf" rel="nofollow noreferrer">https://escholarship.org/content/qt6tg550dv/qt6tg550dv.pdf</a></p>
<p>I am struggling with the aggregate output of the wholesale sector (Equation 21):
<span class="math-container">$$Q_t=\mathbb{E}[a_t|a\geq\tilde{a_t}]z_t\phi_tN_t \\
= \Bigg[\int_{\tilde{a_t}}^{\infty}a_t(\frac{f(a_t)}{1-F(\tilde{a_t})})da\Bigg]z_t\phi_tN_t $$</span></p>
<p>Now to solve the integral analytically, I assumed a logistic distribution and integrating by parts I get:
<span class="math-container">$$\int_{\tilde{a_t}}^{\infty}a_tf(a_t)da=\Bigg[a_tF(a_t)\Bigg]_{\tilde{a_t}}^{\infty}-\int_{\tilde{a_t}}^{\infty}F(a_t)da$$</span></p>
<p>The problem that arises is that it diverges to infinity
<span class="math-container">$$\Bigg[a_tF(a_t)\Bigg]_{\tilde{a_t}}^{\infty}=\Bigg[\infty\frac{1}{1+\exp(\frac{-\infty-a}{b})}\Bigg]-\Bigg[\tilde{a_t}\frac{1}{1+\exp(\frac{-\tilde{a_t}-a}{b})}\Bigg] \\
= \infty*1-\Bigg[\tilde{a_t}\frac{1}{1+\exp(\frac{-\tilde{a_t}-a}{b})}\Bigg]$$</span></p>
<p>Now I do not know how to proceed, as I cannot write the Model in Dynare because of <span class="math-container">$\infty$</span>.</p>
<p>I would really apreciate any help.
Best</p>

<search-and-matching><dynare><simultaneous-equation-model>
Search and Matching model Endogenous separation Walsh (2003)
Paolo Bontempo
# Search and Matching model Endogenous separation Walsh (2003)

(Asked by: username_0 on 2023-12-11)

I am trying to replicate the Search and Matching model Walsh (2003) <https://escholarship.org/content/qt6tg550dv/qt6tg550dv.pdf>

I am struggling with the aggregate output of the wholesale sector (Equation 21):
$$Q\_t=\mathbb{E}[a\_t|a\geq\tilde{a\_t}]z\_t\phi\_tN\_t \\
= \Bigg[\int\_{\tilde{a\_t}}^{\infty}a\_t(\frac{f(a\_t)}{1-F(\tilde{a\_t})})da\Bigg]z\_t\phi\_tN\_t $$

Now to solve the integral analytically, I assumed a logistic distribution and integrating by parts I get:
$$\int\_{\tilde{a\_t}}^{\infty}a\_tf(a\_t)da=\Bigg[a\_tF(a\_t)\Bigg]\_{\tilde{a\_t}}^{\infty}-\int\_{\tilde{a\_t}}^{\infty}F(a\_t)da$$

The problem that arises is that it diverges to infinity
$$\Bigg[a\_tF(a\_t)\Bigg]\_{\tilde{a\_t}}^{\infty}=\Bigg[\infty\frac{1}{1+\exp(\frac{-\infty-a}{b})}\Bigg]-\Bigg[\tilde{a\_t}\frac{1}{1+\exp(\frac{-\tilde{a\_t}-a}{b})}\Bigg] \\
= \infty\*1-\Bigg[\tilde{a\_t}\frac{1}{1+\exp(\frac{-\tilde{a\_t}-a}{b})}\Bigg]$$

Now I do not know how to proceed, as I cannot write the Model in Dynare because of $\infty$.

I would really apreciate any help.
Best

**Tags:** search-and-matching, dynare, simultaneous-equation-model

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## 0 answers

