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In macroeconomics, the study of production networks is a recent approach of the literature that focuses on understanding the impact of the firm-to-firm production networks on aggregate quantities. The role of production networks in the economy was already recognized by classical economists, but its most recent development originates from the combination of two young scientific fields: network science and general equilibrium. Using the tools developed in network analysis and the framework of general equilibrium theory, it has been possible to assess how sectorial shocks propagate through the economy, to quantify sectorial co-movements, and to study the relation between macro-fluctuations and micro-disturbances. This level of analysis was made possible by the recent availability of increasingly granular datasets.

Background and history

While classical economics already recognized the role of production networks in the economy, one of the firsts and most important theoretical contributions in this direction is due to Wassily Leontief. In 1936, Leontief already published his work “Quantitative Input and Output Relations in the Economic System of the United States”, where he stressed the importance of inter-industries connections in the economy. In the 1940s he continued to work on the input-output model, making contributions that earned him the Nobel Prize in Economics in 1973.

Some pioneering works in the direction of production networks were also done by the Hungarian mathematician John von Neumann[1] and the former president of the Federal Reserve Bank of Philadelphia Charles Plosser[2].

With the development of Real Business Cycles theory and New-Keynesian economics in the 80s, the role of production networks was initially downplayed in the literature. This was due in particular to the work of the Nobel-prize winner Robert Lucas. In his 1977’s essay[3], Lucas claimed that idiosyncratic shocks hitting the economy at the sectorial level would on average cancel out when looking at aggregate fluctuations.

In the last decade, Lucas’ argument was questioned by several economists, including Daron Acemoglu. Indeed, researchers were able to show that, given that the firms’ size distribution follows a power law, shocks hitting large firms cannot be balanced out by those hitting small firms[4]. This result revived interest in the role of production networks in macroeconomics, making it a rapidly growing interdisciplinary field of economic analysis.

The baseline model of production networks

In the baseline model of production networks, there is a static economy of  competitive industries producing distinct products, which can be either consumed or used as intermediate goods. Firms have Cobb-Douglas production functions with constant returns to scale. The output of an industry i is given by

    \[y_i = z_i \zeta_i l_i^{\alpha_i}\prod^n_{j=1}x_{ij}^{a_{ij}}\]

where l_i is the quantity of labor in industry i and \alpha_i is its share over the total labor in industry i, \zeta_i is a normalization constant, z_i is a sector-specific productivity shock, x_{ij} is the amount of good  used to produce good ,j and a_{ij} is a parameter that signals the importance of good j as an input for good i. Note that constant returns to scale imply that \alpha_i+\sum^n_{j=1}a_{ij}=1 for all i.

In addition to firms, there is a representative agent in the economy with the following utility function


where \beta is the share of good i in the function.

Notice that the input-output matrix \mathbf{A}=\begin{bmatrix}a_{ij}\end{bmatrix} can be interpreted as the weighted adjacency matrix of the production network, which represents a weighted directed graph with n nodes.

Macroeconomic implications

The first implication of the baseline model of production networks is that the equilibrium output of a sector i is dependent on the shocks of sectors j \neq i. Additionally, it can be shown that a productivity shock (supply shock) propagates downstream from one industry to its customers[5]. On the other hand, a demand-side shock propagates upstream from one industry to its suppliers[6].

Using the same model, it can be shown that the aggregate volatility is

    \[\sigma_{\text{aggregate}}=\frac{\sigma/\alpha}{\sqrt{n}}\sqrt{1+n^2\alpha^2\text{var}\left ( \frac{p_1y_1}{\text{GDP}},...,\frac{p_ny_n}{\text{GDP}}\right )}\]

and depends on the variance of the single sectors. Provided that p_iy_i/\text{GDP} is the same for all i, Lucas argument is true and production networks are not important sources of variations at the aggregate level. However, in presence of significant firm-level heterogeneity, micro shocks can affect macro fluctuations.

By calling \nu_i the Bonacich centrality of node i in the network, the previous equation can be rewritten as

    \[\sigma_{\text{aggregate}}=\frac{\sigma/\alpha}{\sqrt{n}}\sqrt{\alpha^{-2}+\text{var}\left ( \nu_1,...,\nu_n\right )}\]

implying that network-originated macroeconomic fluctuations are important when there are sufficient disparities in industry centralities.

In addition to the aggregate volatility, it can be shown that production networks also have an impact on the distribution of aggregate output. In particular, production networks significantly affect macroeconomic tail risks.

Endogenous production networks

In the baseline model, it is assumed that input-output linkages are the transmission channel for shock propagation, but are invariant to the shock itself. In reality, a shock might alter the network structure of the economy as firms change trading partners. A variety of solutions were proposed in the literature to reflect these aspects of real production networks in a model. Some of the proposed models include a preferential attachment model for firms’ link formation[7], an industry-level network formation model[8], and incentives based models of network formation[9].

Empirical analysis of production networks

By analyzing real production networks, it is possible to assess their empirical regularities. The main reference country in terms of data availability is the US; similar datasets are also available for several other countries, although with lower quality. Recently, very granular datasets were also studied for Belgium and Japan. For all these production networks, it was possible to observe some common characteristics[10],[11],[12]:

  • Production networks are highly sparsely connected.
  • A small number of hubs control the network; these are general-purpose industries that serve a variety of different industries.
  • Production networks exhibit small-world properties: while most industries are not linked to each other, they are topologically close because of the presence of hubs.
  • Production networks have highly skewed distributions of Bonacich centralities.

[1] Von Neumann, J. (1971). A model of general economic equilibrium. In Readings in the Theory of Growth (pp. 1-9). Palgrave Macmillan, London.

[2] Long Jr, J. B., & Plosser, C. I. (1983). Real business cycles. Journal of political Economy91(1), 39-69.

[3] Lucas Jr, R. E. (1977). Understanding business cycles. In Carnegie-Rochester conference series on public policy (Vol. 5, pp. 7-29). North-Holland.

[4] Gabaix, X. (2011). The granular origins of aggregate fluctuations. Econometrica79(3), 733-772.

[5] Carvalho, V. M., & Tahbaz-Salehi, A. (2019). Production networks: A primer. Annual Review of Economics11, 635-663.

[6] Acemoglu, D., Ozdaglar, A., & Tahbaz-Salehi, A. (2015). Networks, shocks, and systemic risk (No. w20931). National Bureau of Economic Research.

[7] Atalay, E., Hortacsu, A., Roberts, J., & Syverson, C. (2011). Network structure of production. Proceedings of the National Academy of Sciences, 108(13), 5199-5202.

[8] Carvalho, V. M., & Voigtländer, N. (2014). Input diffusion and the evolution of production networks (No. w20025). National Bureau of Economic Research.

[9] Oberfield, E. (2018). A theory of input–output architecture. Econometrica86(2), 559-589.

[10] Carvalho, V. M. (2007). Aggregate fluctuations and the network structure of intersectoral trade.

[11] Carvalho, V. M. (2014). From micro to macro via production networks. Journal of Economic Perspectives28(4), 23-48.

[12] Acemoglu, D., Carvalho, V. M., Ozdaglar, A., & Tahbaz‐Salehi, A. (2012). The network origins of aggregate fluctuations. Econometrica80(5), 1977-2016.

Leonardo Rizzo

PhD candidate in Network and Data Science. I am a data scientist with a background in economics and finance, and my research focuses on the economic and managerial applications of network science and agent-based simulation models.

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