The Dynamic Distribution in the Fixed Cost Model: An Analytical Solution

I derive an analytical solution to the Kolmogorov forward equation for fixed cost models. This is a challenging PDE because the dynamic distribution depends on the flow of resetting agents, which is endogenously determined by the distribution itself. I show that there is a shortcut that allows the reset flow to be derived without first finding the entire distribution of agents. This shortcut is also valuable because many aggregate variables can be written in terms of the reset flow alone. Steady-state conditional adjustment behavior recovers the entire marginal reset-flow response to a common state shift. As an example, I solve an investment model with fixed costs of adjustment and study the effects of increased productivity growth. Aggregate dynamics can be derived from a one-dimensional weighted distribution of capital gaps. Because of adjustment frictions, the shock causes a boom-lull investment pattern whose shape depends on the shock size.