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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.

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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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