# Search and Matching model Endogenous separation Walsh (2003)

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

Id leave this as a comment but cannot. I don't think you want to(or cannot) integrate by parts here. Also look at $$\int^\infty F(a)da$$ that will also be $$\infty$$. $$F(a)$$ is increasing function so $$\int_{\tilde{a}}^\infty F(a)da\geq F(\tilde{a})\int_{\tilde{a}}da=\infty$$
• Sorry to intrude, but I think that number1AWL can’t answer in a comment because of reputation points. I think that the number1AWL’s answer says that you can’t use integration by parts here because this way you get an integral that is not convergent, the integral of the cumulative distribution function $F(a)$. The necessary condition for an improper integral to be convergent is that, if the function to be integrated has a limit for $\infty$, this limit must be zero. This is not our case, as the cumulative function has not zero limit. Commented Dec 12, 2023 at 14:39