# Integration in Assignment/Match Model

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.

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.