Financial constraints and firm dynamics

Abstract

This study analyzes the effect of financial constraints (FCs) on firm dynamics. We measure FCs with an official credit rating, which captures availability and cost of external resources. We find that FCs undermine average firm growth, induce anti-correlation in growth patterns and reduce the dependence of growth volatility on size. FCs are also associated with higher volatility and asymmetries in growth shock distributions, preventing young fast-growing firms especially from seizing attractive growth opportunities and further deteriorating the growth prospects of already slow-growing firms, particularly if old. The sub-diffusive nature of the growth process of constrained firms is compatible with the distinctive properties of their size distribution.

Appendix

1.1 Cleaning

We removed a few anomalous data from our sample. Cleaning was performed using Total Sales as a reference variable. For each firm, a missing value was inserted, in the place of the original value of Total Sales, when the latter lay outside the interval

$$ [\hbox{Median}(\hbox{TS}_i)/10; \,\hbox{Median}(\hbox{TS}_i)*10], $$

(7)

where the median is computed over the years for which data are available for firm i. Table 5 shows yearly descriptive statistics computed before and after the cleaning. It is apparent that the procedure does not introduce any relevant change to the data.

Table 5 Total sales: descriptive statistics

1.2 Asymptotic behavior of the autoregressive process

Consider the model of firm size evolution described in Eq. (3), where the shocks $\epsilon$ are independent and identically distributed according to a probability density f with mean c. Let us neglect, for the sake of simplicity, the firm and FC-class specific subscripts and the heteroskedasticity correction (i.e. set σ(s _t−1) = 1). Let also s ₀ be the initial size of the firm. By recursive application of Eq. (3), the size after T time steps, s _T , can be written as the weighted sum of T independent random variables

$$ s_{T} = (1+\lambda)^T s_{0} + \sum_{\tau=0}^{T-1} (1+\lambda)^\tau \epsilon_{T-\tau}. $$

(8)

Consider the cumulant generating function of the size at time T, $\tilde{g}_{s_T}$, defined as the logarithm of the Fourier transform of the unconditional distribution

$$ \tilde{g}_{s_T}(k) = \log {E}[e^{i k s_T}]. $$

(9)

Due to the i.i.d. nature of the shocks it is immediate to see that

$$ \tilde{g}_{s_T}(k) = \tilde{g}_{s_0}( (1+\lambda)^T k) + \sum_{\tau=0}^{T-1} \tilde{f}( (1+\lambda)^\tau k) $$

(10)

where $\tilde{g}_{s_0}$ and $\tilde{f}$ are the cumulant generating functions of the initial size distribution and of the shocks distribution, respectively. As a consequence, if the initial size distribution and the shocks distribution possess the cumulant of order n, then the size distribution at time T also possesses it and it reads

$$ C^{n}_{{\rm s}_T} = \left. \frac{\hbox{d}^n}{\hbox{d} k^n} \tilde{g}_{{\rm s}_T}(k) \right|_{k=0} = (1+\lambda)^{n T} C^{n}_{{\rm s}_0} + \frac{(1+\lambda)^{n T}-1}{(1+\lambda)^{n}-1} C^{n}_{\epsilon}. $$

(11)

Let $M_{{\rm s}_{t}}$ and $V_{{\rm s}_{t}}$ be the mean and variance of the size distribution at time t, respectively. Under the hypothesis of a constant λ the evolution of $M_{{\rm s}_{t}}$ and $V_{{\rm s}_{t}}$ from t = 0 to t = T is given by

$$ M_{{\rm s}_T} \equiv C^{1}_{{\rm s}_T} = (1+\lambda)^T M_{{\rm s}_0} + \frac{(1+\lambda)^T-1}{\lambda} M_{\epsilon}, \quad V_{{\rm s}_T} \equiv C^{2}_{{\rm s}_T} = (1+\lambda)^{2 T} V_{{\rm s}_0} + \frac{(1+\lambda)^{2 T}-1}{(1+\lambda)^{2}-1} V_{\epsilon} $$

(12)

where $M_{\epsilon}$ and $V_{\epsilon}$ are the mean and variance of the shocks $\epsilon$. Therefore, for λ = 0, the variance and mean increase proportionally to T and, for the Central Limit Theorem, the FSD converges in distribution to a lognormal. Conversely, when λ < 0 the FSD converges to a stationary distribution with finite variance $V_{\epsilon}/(1-(1+\lambda)^2).$