Abstract
In this paper we study the relevance of links among firms in explaining the mean and the auto-correlation property of the aggregate total factor productivity rate of growth (Solow residual). Our approach relies on the interaction between idiosyncratic shocks of firms and the network structure of firms. We analytically study this phenomenon using the adjacency matrix of a complete network and we present a simulation with more general random adjacency matrices. We also check empirically, using Italian data, the relationship between the network structure and the Solow residual. In particular, in the empirical part, we find two main results: firstly, the relationship between the Solow residual and the measure of connectivity of firms is positive, in accordance with the analytical results. Secondly, we find that the measure of connectivity is pro-cyclical with the annual growth rate of industrial production.
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Notes
This is a standard assumption for adjacency matrices in graph theory, meaning that firms have no self-loops. The general case with self-loops and different kinds of network is analyzed in Palestrini (2013).
The specification of the link structure implicitly assumes a Markovian property regarding the \(z_t\) vector. The analysis can be easily generalized adding more lags to the \(z_t\) specification.
Note that the linear algebra operator \(\frac{u^T}{N}\) simply obtains the mean of a vector.
In the empirical part the size is measured by employees.
The case with different sizes will be analyzed in the Sect. 2.2
Since the greatest eigenvalue of the complete network is \(N-1\), \(\beta \) has to be less than \(\frac{1}{N-1}\). See Cvetković et al. (1980).
Random adjacency matrices are matrices in which the links are generated independently with equal probability.
Note that in the case of the probability of a link between two firms being one (\(p=1\)) then we come back to the complete network case; i.e., \(A(1)=A\).
For a complete description of the dataset, see Sect. 3.
Therefore, according to our definition, links are bidirectional. Indeed if firm \(i\) is connected with firm \(i+1\), the latter is necessarily connected with the former.
Matrix \(B\) is a block diagonal adjacency matrix.
In general, by denoting with \(k\) the number of production chains in the economy and by \(n_{i}\) the number of firms in chain \(i\), the total number of links among firms is \(\sum _{i=1}^k 2(n_{i}-1)\), as proven in Appendix 2. Also in Appendix 2 we show that in analyzing production chains with equal length (i.e. \(n\) is divisible by \(k\)) the above formula simplifies to \(2(n-k)\).
We assume, for simplicity, that firm(s) belonging to this first stage are totally integrated, i.e. do not acquire any goods, material or machinery. This assumption do not alter the subsequent analysis.
Therefore, in this first situation we observe two firms extracting raw materials and producing the capital goods and two firms producing the semi finished product and the finished one
This hypothesis implicitly assumes that firms have equal size. Therefore in the empirical analysis (Sect. 4) we control also for firm size.
The fact that the Adelman index is sensitive to the stage of production in which the firm is specialized does not depend on the assumptions we made in our numerical example. It is a general characteristic of the Adelman index. This characteristic was noted and discussed, among others, by Holmes (1999). Although this fact is considered a limitation of the Adelman index, we do not think that it would affect our results. Indeed, since we consider a balanced panel, it is unlikely that changes in Adelman index are the result of changes of firms’ ‘position’ in the supply chain. It is more likely that changes in Adelman index reflects other factors such as, ‘make or buy’ decisions, etc.
Even if we allow, following Levine (2012), that with longer production chains the total output increases, by assuming that each firm adds the same value added in the production process, the situation remains unchanged.
More precisely, we compute the Adelman index for each year and for each firm. Then we drop ‘pathological’ observations with a negative Adelman index or with a value of this index greater than one. These observations represent only a small part of the sample. In particular, for the years from 2000 to 2006 there are respectively 48, 24, 28, 24, 44, 31 and 53 observations. Finally we compute the complement of the Adelman index.
As it is standard, we obtain this deflator by computing the ratio between the manufacturing investment at current prices and real manufacturing investment at constant prices.
See, e.g. Romer (2006).
The Hausman test rejects the null hypothesis of the random effect consistency, with a significant level below 0.1 %.
The analysis of this part is conducted by using robust SEs to cope with the problem of heteroskedasticity.
Indeed as Table 3 shows the average values of \(LIA\) are about 0.7 and the SDs are about 0.12. In conclusion, the relevant part of the distribution seems to be in the second half of the [0–1] support.
Indeed the equation \(-7.4056 LIA + 7.8668 LIA^{2}\) is a parable with upper concavity and with minimum at about 0.47. This implies that in the upper range of the LIA distribution there exists a significant positive relationship.
However the statistical independence seems to hold only in mean. Indeed many works, starting from Stanley et al. (1996), found a scaling of variance; i.e., the variance of firm growth seems to be negatively related to firm’s size according to a power law relationship.
The Hausman test rejects the null hypothesis of the random effect consistency, with a significant level below of 0.1 %.
The second line of the tables does not reject the hypothesis that the distribution of the second year is greater than the distribution of the year before.
Note that diff \(<0\) means that the average value of 2004 is above the average of 2003 since the estimation procedure computes the difference between the average of 2003 and the average of 2004.
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For helpful comments and suggestions we would like to thank the anonymous referees and the Editors of the journal.
Appendices
Appendix 1
The following pseudo-code shows the implemented Monte Carlo simulation:
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1.
Set parameters \(N\), \(\beta \), \(\mu _\varepsilon \), \(\sigma _\varepsilon \), \(M\), \(J\) and set starting value \(z_0\).
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2.
Initialize \(p=0.1\).
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3.
Generate \(M\) random matrices \(A(p)\). For each of them simulate \(J\) iterations of the system \(z_t = \varepsilon _t + \beta A(p) z_{t-1}\) starting from \(z_0\) and sampling \(\varepsilon _t\) from a normal distribution with mean \(\mu _\varepsilon \) and standard deviation \(\sigma _\varepsilon \).
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4.
Save the mean of the last iterated random vector \(z_{J}\) and the average auto-correlation of the random vectors \(z_1,...,z_{J}\) of the \(J\) iterations.
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5.
Increment probability, \(p \leftarrow p+0.1\).
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6.
If \(p \le 1 \) then go to point 3. Else stop.
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7.
Save results.
Appendix 2
Given \(n\) firms, \(k\) production chains and \(n_{i}\) firms in each production chain we will show that the total number of links among firms is \(\sum _{i=1}^k 2(n_{i}-1)\).
First of all, note that in the presence of \(k\) different production chains the adjacency matrix \(A\) can be decomposed in a \(k\)-block diagonal matrix. This means that each block of \(A\) has one chain with \(n_i\) firms and thus with \(2(n_i - 1)\) links. Therefore the total number of links is \(\sum _{i=1}^k 2(n_{i}-1)\).
In the case of chains of equal length, each chain has \(n/k\) firms. The above formula becomes \(2k(\frac{n}{k}-1)\) which simplifies to \(2(n-k)\).
Appendix 3
See Table 11.
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Guzzini, E., Palestrini, A. Growth in total factor productivity and links among firms. J Econ Interact Coord 11, 35–55 (2016). https://doi.org/10.1007/s11403-014-0138-0
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DOI: https://doi.org/10.1007/s11403-014-0138-0