I see, I’ll try to sketch the key part of the model and hope it highlights what you would want to know. Turns out that in equilibrium, it should be ok to not keep track of the full distribution but in response to a shock, I am not sure why it should hold in the way Den Haan et al. (2021) use it.

There is a constant mass of entrepreneurs, *Gamma*, that can, upon deciding to enter the labor market and matching with a worker, form a one-worker firm. Every period with probability delta, this worker-firm match is exogenously severed, the worker becomes unemployed while the entrepreneur dies and is replaced by a new-born entrepreneur. There is no capital in this model (yet), labor is the sole input.

While an entrepreneur is idle i.e. not matched with a worker, he/she draws individual productivity *z_i* from a Uniform distribution with bounds *(z_min,z_max)* at the beginning of each period. The individual productivity that an entrepreneur enters into a match with will be constant until exogenous separation & death.

The productivity with which a one-worker firm produces is given by the sum of aggregate productivity *Z_t* & individual productivity z_i, such that *(z_i + Z_t)*, (Note that I omit the subscript t here because once entered into production, *z_i* remains constant.)

In equilibrium, there is a value z_bar for which an idle entrepreneur is indifferent between entering today or waiting for a new draw of individual productivity tomorrow. (The standard free-entry condition in these types of matching models is replaced by one where entrepreneurs weight entering today vs entering in the future)

Thus, average productivity of producing one-worker firms is given by z_star = z_bar + 1/2 *(z_max - z_bar) & the mass of active worker-firm matches can be inferred given the parameters like Gamma, delta, & equilibrium vacancy postings etc.

The value of an idle entrepreneur with individual productivity z is given by:

```
J_t(z) = p_t *(Z_t + z) - w_t + E_t [beta (1-delta) J_{t+1}(z)]
```

Then, keeping track of this equation at z_bar and z_star is enough (according to Den Haan et al. (2021)).

When labor is the sole input, the fact that individual productivity is additive to aggregate productivity allows to split the expectations term into a part for *Z_t* and another for *z* => using the infinite sum, the part for *z* can be reformulated analytically such that it only depends on values today. When adding capital to the model, this simplification breaks down and that’s where I am struggling to implement it in Dynare.

For me, it makes sense that in equilibrium one does not have to keep track of the full distribution but only *{z_bar,z_star}*.

Something that I realized while writing this outline and that I honestly do not completely understand is that given a shock to *Z_t*, *z_bar* will fall to *z_barPrime* because the rise in productivity will motivate entrepreneurs with lower individual draws to enter than before.

However, it is not immediately clear to me that the amount of firms entering with individual productivity right above z_bar is necessarily the exact amount such that the mass of firms that enter with *z_i* *\in* *(z_barPrime, z_bar)* are exactly enough to maintain the generally existing uniform distribution of active firms.

The paper and the appendix do not touch on this specifically so I’ll might have to contact the authors and ask.