I'm following Atkeson, Burstein (2019 JPE), and cannot understand the aggregation result.

There is measure $M(z)$ of firms with productivity $z$, with production function $$ y(z) = z k(z)^\alpha l(z)^{1-\alpha} $$ Aggregate output is $$ Y = \left[ \sum_z y(z)^{\frac{\rho-1}{\rho}} M(z) \right]^{\frac{\rho}{\rho-1}} $$ Assuming that markup is constant across firms and factor markets are competitive, they get the result $$ Y = Z K^\alpha L^{1-\alpha} $$ where $Z = \left[ \sum_z z^{\rho-1} M(z) \right]^{\frac{1}{\rho-1}}$, $K = \sum_z k(z) M(z)$ and $L = \sum_z l(z) M(z)$.

I don't quite understand this jump. Following the usual steps for intermediate goods, I can derive the marginal cost and prices for each firm. They are dependent on $z$, so I can't simply apply the trick that all firms behave in the same way. What I do get is $PZ = p(z) z$ and $y(z)=(z/Z)^\rho Y$. Now I'm lost which way to proceed.


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