# Lecture 5: Firm Dynamics

This lecture will be a hybrid of both theoretical and empirical elements. We've already seen a number of growth models featuring basic theories of firm dynamics. Now we'll look at this problem in more detail from two angles.

First, we'll investigate of model of firm dynamics proposed by . This paper starts with the following motivational trends:

- The firm employment size distribution is Pareto with tail index $\zeta \approx 1.05$. This is close to the case of
**Zipf's Law**where $\zeta = 1$. - Average firm growth rates satisfy Gibrat's law approximately, in that they are invariant to firm size for all but the smallest firms. However, the variance of firm growth falls with firm size.
- The largest firms become so very fast. The median age of firms with more than 10,000 employees is 75 years.

## Model of Firm Dynamics

I will follow the notation used in throughout, rather than that used in previous lectures. There is a mass of consumers $H$ that is growing at rate $\eta$. Outcomes are evaluated according to the "dynastic" CRRA utility function with parameter $\gamma$

### Firms

As we have often assumed, differentiated goods producers can use $\ell$ units of labor to produce $Z \ell$ units of the commodity. Furthermore, $Z$ is growing over time at rate $\theta$. Thus the marginal cost is $w/Z$.
Firms will charge a fixed markup over cost

To produce a differentiated good, you need a blueprint. Blue prints depreciate in a **one-hoss-shay** fashion.

Employing $i$ labor on blueprint replication yields a rate $\mu = f(i)$ of success. Similarly, using $j$ labor on maintaining a blueprint yields a loss rate of $\lambda = g(j)$. Thus the value of a blueprint $q$ will satisfy

### Entrepreneurs

Each agent is endowed with a two-dimensional skill vector $(x,y)$, which represent their ability to develop blueprints and do labor respectively. You can think of this as analogous to brains and brawn. There is some fixed distribution over these $T(x,y)$. Thus agents will choose entrepreneurship or wage work according to the relative values of $q x$ and $w y$. So in terms of worker decisions, we only really care about the distribution of $x/y$ as it relates to the ratio $q/w$.

Here we can use a standard trick to simplify the outcome. We assume that $x$ and $y$ are independent and Frechet distributed. They have respective mean parameters $s_x$ and $s_y$ and a common shape parameter $\alpha$. One can show that under these assumptions

### Equilibrium

Restricting attention to balanced growth paths, we can see that for constant $q/w$, the number of products $N$ will grow with the population growth rate $\eta$. Therefore we will have

### Firm Size Distribution

For this, we can proceed with the derivation in a manner similar to how we addressed the model from . Let the mass of firms with $n$ products be $M_n$. Now the consistency equation is

First note that this reduces to the Klette-Kortum distribution when there is no population growth, i.e. $\eta = 0$. We know that in this case, the distribution follows $P_n \propto (\mu/\lambda)^n/n$. Thus the distribution is thin-tailed in that case, contrary to what we see in the data. In the general setting, we have the following proposition to characterize the **tail index** of the distribution

The firm entry rate as a fraction of the number of incumbent firms $\varepsilon$ in this economy should satisfy

Given that we have pretty good information on the entry rate ($\approx 10\%$), the population growth rate, and the tail index of firms, we can get fairly tight constraints on the parameters here. However, even with the best fit, the median firm is still older than the US. This shortcoming is what motivates the introduction of firms types.

### Firm Types

Included is an extension of the basic model where there are two types of firms, high and low. The firms differ only in their level of productivity $Z_H$ and $Z_L$. Productivity growth for both is still $\theta$. Entering firms are of high quality with some probability $\alpha$, while the remainder are low quality. Over time, high quality firms degrade into low quality firms at rate $\delta_H$. One can show that in this case, under certain regularity conditions the tail index is given by

The only remaining question is how this affects Gibrat's Law. That is, will it be the case that high type firms will also be larger on average, producing the result that large firms grow faster? This would certainly problematic, as if Gibrat's Law is only approximately true, it breaks in the other direction, with smaller firms growing faster. It turns out that with a sufficiently large depreciation rate of high type firms (the rate at which they turn into low type), we can alleviate this problem.

## Mapping to Data

There has been a considerable amount of effort put into understanding the dynamics of firm size and productivity in the data. The major source of information in this realm is the Longitudinal Business Database (LBD) put out by the US Census Bureau. For an overview of this literature, see . Also consult for a slightly more concise summary.

One of the major conclusions of these analyses is that there is very large amount of idiosyncratic variation in plant-level outcomes, both in terms of levels and growth. Furthermore, reallocation of inputs (primarily labor) is a particularly salient force. Approximately 10% of jobs are created or destroyed each year, and the majority of these constitute movements within narrowly defined sectors. Of these movements, around 20% of the total can be accounted for by firm entry and exit.

There are also interesting trends at the cyclical level. The general finding is that reallocation is more intense during downturns. This results in lower variability in productivity than what would otherwise be implied by within-firm variations. See -- for a detailed description.

It is important to get a handle on what the source of changes in sectoral productivity are. To this end, there are various productivity growth decompositions one can utilize. At the highest level, we can decompose productivity into contributions from the various constituent firms or plants, which we can just call establishments

In these decompositions, the shares are taken from variables such as output or employment. The productivities are calculated as either the value added per unit labor or something akin to Solow residual under the assumption of Cobb-Douglas production. That is

The results of these decomposition generally come down along the following lines. About half of productivity growth comes from changes within existing firms. Around one quarter comes from reallocation between plants, which here is the sum of the between-plant and covariance terms. Finally, the remaining quarter results from net entry, that is, the sum of contributions from entry and exit. Due to the variability from study to study, sometimes people just think of this as one third from each.