Lecture 5: Firm Dynamics

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 \frac{p_j}{P} = \left(\frac{\sigma}{\sigma-1}\right) \frac{w}{Z} Finally we can use this to derive an expression for the wage w = \left(\frac{\sigma-1}{\sigma}\right) Z N^{1/(\sigma-1)} We can also derive the labor utilization \ell = \left[\left(\frac{\sigma-1}{\sigma}\right)\frac{Z}{w}\right]^{\sigma-1} \left(\frac{\sigma-1}{\sigma}\right) \frac{C}{w} Additionally, the profits will be \pi = \frac{1}{\sigma} \left[\left(\frac{\sigma-1}{\sigma}\right)\frac{Z}{w}\right]^{\sigma-1} C = \frac{w\ell}{\sigma-1}

To produce a differentiated good, you need a blueprint. Blue prints depreciate in a one-hoss-shay fashion.This terminology is in reference to the poem "The Deacon's Masterpiece; or The Wonderful One-Hoss Shay" by Oliver Wendell Holmes, Sr. Furthermore, blue prints can be replicated from existing blueprints using labor ($\nu$) or created anew by entrepreneurs ($\mu$). Thus the number of blueprints $N$ grows (or shrinks) at rate $\nu+\mu-\lambda$.

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 r q = \max_{i,j} \left\{w \left[\frac{\ell}{\sigma-1}-(i+j)\right] + (f(i)-g(j))q + \dot{q}\right\} This yields the first order condition q f^{\prime}(i) = -q g^{\prime}(j) = w

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 E \align = s_x \left[\frac{(s_x q)^{\sigma}}{(s_y w)^{\sigma}+(s_x q)^{\sigma}}\right]^{\frac{\sigma-1}{\sigma}} \\ L \align = s_y \left[\frac{(s_y w)^{\sigma}}{(s_y w)^{\sigma}+(s_x q)^{\sigma}}\right]^{\frac{\sigma-1}{\sigma}} Thus the ratio of these two values has constant elasticity with respect to $q/w$ and is given by \frac{E}{L} = \left(\frac{s_x}{s_y}\right)^{\sigma} \left(\frac{q}{w}\right)^{\sigma-1} Consistency with the rate of outside blueprint creation implies \nu N = H E(q/w) while on the labor market side we have (\ell+i+j) N = H L(q/w)

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 \eta = \nu + \mu - \lambda Per capita consumption can be expressed as $C/H = N^{1/(\sigma-1)} Z$ meaning it will grow at rate \kappa = \theta + \frac{\eta}{\sigma-1} Rearranging terms in the value function equation and noting that both $q$ and $w$ will grow at the same rate as per capita consumption, we find \frac{q}{w} = \frac{\frac{\ell}{\sigma-1}+(i+j)}{r-\kappa-(\mu-\lambda)} We now have six equations in the six unknowns $(i,j,\mu,\lambda,\ell,q/w)$. There are certain conditions under which an equilibrium is guaranteed to exist, but we will omit them here.

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 N = \sum_{n=1} n M_n We can write down a system of flow equations describing $M_n$ in steady state \text{Inflows} \align = \text{Outflows} \\ \nu N + 2 \lambda M_2 \align = \mu M_1 + \lambda M_1 \\ (n-1) \mu M_{n-1} + (n+1) \lambda M_{n+1} \align = n \mu M_n + n \lambda M_n \qquad\qquad\qquad It is useful to define a normalized distribution as well, namely $P_n \equiv M_n / F$ where $F = \sum_{n=1}^{\infty} M_n$. In addition, we can define the share of product lines owned by size $n$ firms as $Q_n = n M_n / N$. In this case we can write down the flow equations \eta Q_1 \align = \lambda Q_2 + \nu - (\mu+\lambda) Q_1 \\ \frac{1}{n} \eta Q_n \align = \mu Q_{n-1} + \lambda Q_{n+1} - (\mu+\lambda) Q_n It is actually possible to solve in closed form for the resulting distribution. However, it is sufficiently complicated to lack much clear intuition. However, the main result of the theorem establishes the general shape of the distribution.

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 Suppose that $\eta \gt \mu - \lambda \gt 0$. Then the right tail probabilities $R_n = \sum_{k=n}^{\infty} P_k$ of the stationary firm size distribution satisfy \lim_{n\to\infty} n \left[1 - \frac{R_{n+1}}{R_n}\right] = \zeta \equiv \frac{\eta}{\mu-\lambda} For instance, if some generic distribution is Pareto, then the PDF satisfies $F_n = 1 - n^{-\zeta}$, meaning $R_n = n^{-\zeta}$, then it can be shown that $\zeta$ is simply the tail index as derived in the limit above. Meanwhile, doing the same for the Klette-Kortum distribution yields $\zeta = 0$, meaning it is thin-tailed.

The firm entry rate as a fraction of the number of incumbent firms $\varepsilon$ in this economy should satisfy \varepsilon - \lambda P_1 = \eta Given a mass of firms $F$, the average number of blueprints per firm is $\bar{N} = N/F$ and satisfies Q_n = n P_n \cdot \frac{F}{N} = \frac{n}{\bar{N}} \cdot P_n Consistency of the firm entry rate and the blueprint creation rate implies $\varepsilon F = \nu N$, meaning $\bar{N} = \varepsilon/\nu$. We also know that $P_1 = \bar{N} Q_1 = (\varepsilon/\nu) Q_1$. From here we arrive at a closed form expression for the entry rate \varepsilon = \eta \left(\frac{\nu}{\nu-\lambda Q_1}\right)

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 \zeta = \min\left\{\frac{\eta+\delta_H}{[\mu_H-\lambda_H]^+},\frac{\eta+\delta_L}{[\mu_L-\lambda_L]^+}\right\} Notice that we can generate a thick tail for the firm distribution even in the absence of population growth. With this new setup, one can do a pretty good job of matching the above mentioned targets, inclusive of the median age of very large firms.

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.