Heterogeneous Distributions of Firms Sustained by Innovation Dynamics—A Model with Empirical Illustrations and Analysis

Abstract

This paper develops a framework of innovation dynamics to appreciate observed heterogeneity of firm size distributions, in which dynamics refer to exit and entry of product varieties and variety markets of individual firms. The analysis is based on a model of variety-triplets where every such triplet in the economy is identified by a unique combination of a variety, destination and firm. New variety triplets are introduced by innovating firms in a quasi-temporal setting of monopolistic competition. Ideas for variety-triplets arrive to firms according to a firm-specific and state dependent Poisson process, whereas variety triplets exit according to a destination-specific Poisson process. The empirical analysis employs a detailed firm-level data base which provides information about all variety triplets. Firm size is measured by a firm’s number of variety triplets. The empirical results are compatible with the model predictions of (i) a persistent distribution of firm sizes, (ii) frequent events of exit and entry, and (iii) state dependent entry, where a state may be given by each firm’s composition of triplets and/or other firm attributes.

Appendices

Appendix 1: Transforming poisson processes into a Markov process

Consider that at a given point in time, t, there is a vector $f(t) = [{f_0}(t),{f_1}(t),...]$ of probabilities or shares such that $\sum\nolimits_k {{f_k}(t) = 1}$, where f _n (t) denotes the share of firms with n distinct (i, j)-combinations. Following Clegg (2008) we shall study the Poisson arrival of innovation ideas and exit of (i, j)-combinations with the help of a continuous-time Markov chain, where σ _kl denotes the flow rate between state k and l. With small discrete time steps, δt, the transition pattern can be depicted by

$$ P(\delta t) = \left[ {\begin{array}{*{20}{c}} {1 - {\sigma_{{00}}}\delta t} \hfill & {{\sigma_{{{01}}}}\delta t{ }...} \hfill & {} \hfill \\{{\sigma_{{{10}}}}\delta t} \hfill & {{1 - }{\sigma_{{{11}}}}\delta t{ }...} \hfill & {} \hfill \\{{\sigma_{{{20}}}}\delta t} \hfill & {{\sigma_{{{21}}}}\delta t} \hfill & {{1 - }{\sigma_{{{22}}}}\delta t{ }...} \hfill \\{ - } \hfill & {} \hfill & {} \hfill \\\end{array} } \right] $$

(A1.1)

It is not meaningful to take the limit for δt → 0. Instead we introduce ${q_{{kl}}}(t,t + \tau )$ to denote the probability of a transition from state k to state l. Since there is no temporal memory (homogenous time), we may construct the following transition rates:

$$ {q_{{kl}}} = \mathop{{\lim }}\limits_{{\tau \to 0}} {q_{{kl}}}(\tau )/\tau $$

(A1.2)

The transitions depicted are like a Poisson process, and then it is possible to write

$$ {f_k}(t + \delta t) = {f_k}(t) - \sum\nolimits_{{l \ne k}} {{f_k}(t){\sigma_{{kl}}}\delta t + } \sum\nolimits_{{l \ne k}} {{f_l}(t){\sigma_{{lk}}}\delta t + {\rm O}(\delta t)} $$

(A1.3)

Differentiating with respect to time yields the following result

$$ \partial {f_k}(t)/\partial t = - \sum\nolimits_{{l \ne k}} {{f_k}(t){\sigma_{{kl}}} + } \sum\nolimits_{{l \ne k}} {{f_l}} (t){\sigma_{{lk}}} $$

(A1.4)

Now, formula (A1.3) can be replicated in terms of q _kl (t), which yields

$$ {f_k}(t + \delta t) = {f_k}(t) - \sum\nolimits_{{l \ne k}} {{f_k}(t){q_{{kl}}}(\delta t) + } \sum\nolimits_{{l \ne k}} {{f_l}(t){q_{{lk}}}(\delta t)} $$

(A1.5)

From this we can replicate (A1.4) to obtain $\partial {f_k}(t)/\partial t = - \sum\nolimits_{{l \ne k}} {{f_k}(t){q_{{kl}}} + } \sum\nolimits_{{l \ne k}} {{f_l}} (t){q_{{lk}}}$. Then, to make the two processes consistent, we assume that q-coefficients and σ-coefficients correspond so that q _kl = σ _kl for k ≠ l and that ${q_{{kk}}} = - \sum\nolimits_{{l \ne k}} {{q_{{kl}}}}$. These q-coefficients can be arranged in a Q-matrix, Q = {q _kl }, for which we have that $df(t)/dt = f(t)Q$.

Given that the Q-matrix is irreducible to satisfy ergodicity properties, the equilibrium distribution of firms exists and can be obtained as $f* = \mathop{{\lim }}\limits_{{t \to \infty }} f(t)$, satisfying f* Q = 0. The introduced transition rate q _kl reflects the rate at which a firm with k variety-destination pairs experiences the change l − k = a _k − e _k , where k is the firm’s current number of variety-destination pairs, a _k is the number of variety-destination pair ideas that arrive to the firm, and where e _k is the number of variety-destination pairs that exit from the current stock of such varieties. All these transition rates can be derived from Poisson probabilities related to each individual firm.

Appendix 2: Limit values of fixed costs for scope-firms

We shall consider three categories of scope firms and make precise the limit of possible effects from scope economies. An M1-firm has a strong scope effect, compared with an L1-firm. The former reduces the fixed cost (F +G + H) to half the value for the L1-firm. M-firms have always scope effects by reducing the importance of F for each destination delivery. On the other hand, as one extreme an M-firm may have many destination links for few varieties, and as the other extreme many varieties sold to few destination markets.

Consider now that we apply an average flow size $\tilde{x}$. When all destination markets are in the neighborhood of an L1-equilibrium, we have that $\tilde{x} \approx {x^o}$. We denote a firm’s gross profit per average delivery by $\tilde{V} = ({p^o} - v)\tilde{x}$ Table 5 presents the break-even values for three categories of firms: M1, MH (with only one variety and many destinations), and MG (with many varieties and only one destination besides the domestic market.

Table 5 Break-even values of gross profit per average delivery flow

The limit values in the table are obtained by simply assigning large values to H* and G*. In this way F/(H* +1) and G/(H* +1) approach zero for MH-firms, while F/2G* and H/2G* approach zero for MG-firms. Given that innovation ideas arrive according to a stochastic process, the two extreme outcomes, represented by MH and MG firms, are unlikely events.

Now, in case that F ≈ H ≈ G, then we can conclude that ${\tilde{V}_M}$ ≈ ${\tilde{V}_{{M1}}}$, which implies that an M1-equilibrium will be robust. This equilibrium is defined by the supply quantity x ^oo such that $({p^o} - v){x^{{oo}}} = (F + G + H)/2$.