<p>The answer is extremely simple.</p>
<p>Because the definition of a positive assortative matching with continuous variable is that <span class="math-container">$1-G(g)=1-K(k(g)) \forall g$</span>, where <span class="math-container">$G$</span> and <span class="math-container">$K$</span> are the cdf of <span class="math-container">$g$</span> and <span class="math-container">$\mathrm{k}$</span>.</p>
<p>Then from the definition of the cdf of lognormal distribution we can solve this matching function analytically <span class="math-container">$k(g)=g^{\frac{\sigma_{k}}{\sigma g}}$</span>. And the integration is then trivial.</p>

CC BY-SA 4.0
2022-02-07T09:11:17.693Z
2022-02-07T09:11:17.693Z
Alalalalaki
<p>Say a match model where worker skill <span class="math-container">$g$</span> matches with machine size <span class="math-container">$k$</span>. Suppose the production function takes CD form <span class="math-container">$g^{\alpha} k^{\beta}$</span> and the workers and machine sizes are lognormally distributed with variances of logarithms <span class="math-container">$\sigma_{g}^{2}$</span> and <span class="math-container">$\sigma_{k}^{2}$</span>. Because the production function is complementary, the assignment will be a positive assortative matching <span class="math-container">$k(g)$</span>.</p>
<p>From FOC we have <span class="math-container">$w^{\prime}\left(g_{0}\right)=\left[\frac{\partial f\left(g_{0}, k^{*}\right)}{\partial g_{0}}\right]_{k^{*}=k\left(g_{0}\right)} = \alpha g^{\alpha-1}k(g)^\beta$</span>.</p>
<p>Integrate this FOC and we can get the wage function <span class="math-container">$w(g)=A g^{\left(\alpha \sigma_{g}+\beta \sigma_{k}\right) / \sigma_{g}}+C_{w}$</span>, where <span class="math-container">$A$</span> is a constant and <span class="math-container">$C_{w}$</span> is the constant from the integration. I wonder how to do this integration.</p>

<mathematical-economics><matching>
Integration in Assignment/Match Model
Alalalalaki
# Integration in Assignment/Match Model

(Asked by: username_0 on 2022-02-01)

Say a match model where worker skill $g$ matches with machine size $k$. Suppose the production function takes CD form $g^{\alpha} k^{\beta}$ and the workers and machine sizes are lognormally distributed with variances of logarithms $\sigma\_{g}^{2}$ and $\sigma\_{k}^{2}$. Because the production function is complementary, the assignment will be a positive assortative matching $k(g)$.

From FOC we have $w^{\prime}\left(g\_{0}\right)=\left[\frac{\partial f\left(g\_{0}, k^{\*}\right)}{\partial g\_{0}}\right]\_{k^{\*}=k\left(g\_{0}\right)} = \alpha g^{\alpha-1}k(g)^\beta$.

Integrate this FOC and we can get the wage function $w(g)=A g^{\left(\alpha \sigma\_{g}+\beta \sigma\_{k}\right) / \sigma\_{g}}+C\_{w}$, where $A$ is a constant and $C\_{w}$ is the constant from the integration. I wonder how to do this integration.

**Tags:** mathematical-economics, matching

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## 1 answers

### Answer by username_0 on 2022-02-07. Score: 1 | Accepted answer

The answer is extremely simple.

Because the definition of a positive assortative matching with continuous variable is that $1-G(g)=1-K(k(g)) \forall g$, where $G$ and $K$ are the cdf of $g$ and $\mathrm{k}$.

Then from the definition of the cdf of lognormal distribution we can solve this matching function analytically $k(g)=g^{\frac{\sigma\_{k}}{\sigma g}}$. And the integration is then trivial.


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