Bak et al. (1993) was an early attempt to understand aggregate economic fluctuations in terms of idiosyncratic shocks to individual agents. This also asks the question that should we consider idiosyncratic shocks as a source of volatility? There were several irrelevance theorems proved which essentially had shown that idiosyncratic shocks
tend to cancel each other in a multi-sector set-up, but the debate was not settled (see e.g. Dupor1999; Horvath1998). Newer results, both theoretical and empirical, came up later which showed that the answer could well be positive which we describe below.
5.3.1 Input-Output Structures
One of the first network structures studied in great details is the input-output network (Leontief1936,1947,1986). The essential postulate is that an economy is made up of a number of sectors that are distinct in their inputs and outputs. Each sector buys inputs from every other sector (some may buy zero inputs from some particular sectors) and sells output to other sectors. The workers provide labor, earn wage and consume the output net of input supplies.
Thus if one sector receives a negative shock, it can potentially transfer the effects to all downstream sectors. The question is whether that will so dispersed that none of it would be seen in the aggregate fluctuations or not. Acemoglu et al. (2012) studied it directly in the context of an input-output structure and showed that the degree dis- tribution is sufficiently skewed (fat-tailed; there are some sectors disproportionately more important than the rest) so that idiosyncratic shocks to those sectors do not die completely. This provides a theoretical solution to the debate (it also shows that this channel is empirically relevant).
A simple exposition of the model is as follows (for details see Acemoglu et al.
2012). There is an unit mass of households with utility function defined over a consumption bundle{ci}i∈N as
u =ξ.
i∈N
(ci)1/N (5.3)
whereξ is a parameter. The production function for each sector is such that it uses some inputs from other sectors (or at least zero),
xi =(zili)α(
j∈N
xi jωi j)1−α (5.4)
whereziis an idiosyncratic shock to thei-th sector and{ωi}j∈Ncaptures the indegrees of the production network. To understand how it captures the network structure of production, take logs on both sides to get
log(xi)=αlog(zi)+αlog(li)+(1−α)
j∈N
ωi jlogxi j. (5.5)
Since all sectors are profit maximizing, their optimal choice of inputs (how much would they buy from other sectors) will depend in turn on the prices and how much
they themselves are producing. The markets clear at the aggregate level. After sub- stitution, we can rewrite the equation above as (x∗ being the solution to the above equation)
log(x∗)=F(log(z),log(x∗)) (5.6) which means that the aggregate output of one sector is a function of all productivity shocks and optimal outputs of all sectors. Hence, this becomes a recursive system.
Acemoglu et al. (2012) shows that the final GDP can be expressed as
GDP=wlog(z) (5.7)
wherewis a weight vector. Thus the output of all sectors are functions of the vector of all idiosyncratic shocks.
Foerster et al. (2011) considered the same question and provided a (neoclassical multi-sector) model to interpret data. They showed that idiosyncratic shocks explain about half of the variation in industrial production during the great moderation (which basically refers to two-decades long calm period before the recession in 2007).
5.3.2 Trade Networks
A very simple trade model can be provided on the basis of the input-output model presented above. Suppose there are N countries and for the households of thei-th country, the utility function is given by the same simple form:
ui=ξi.
(ci)1/N (5.8)
Each country gets an endowment of country-specific goods (e.g. Italian wine, German cars):
yi =zi (5.9)
whereziis a positive random variable. Now we can assume that there is a Walrasian market (perfectly competitive) across all countries. Each country is small enough so that it takes the world price as given. The we can solve the trading problem and can generate a trade flow matrix where weight of each edge is a function of the preference and productivity parameters,
ei j = f(ξ,z,N) (5.10)
whereei j is the weight of the directed edge from countryi to j. This is of course, a very simple model and apart from generating a network, it does not do much.
Many important details are missing. For example, it is well known that trade volume is directly proportional to the product of GDPs and inversely proportional to the distance between a pair of countries. This feature is referred to as the gravity equation
of trade (Barigozzi et al.2010; Fagiolo2010). Explaining the first part (trade being proportional to the product of GDPs) is not that difficult. Even in the model stated above a similar feature is embedded. The more difficult part is to understand why trade exactly inversely proportional (in the actual gravity equation, force of attraction is inversely proportional to the distance squared). Chaney (2014) presents a framework to understand that type of findings.
5.3.3 Migration Networks
Another important type of network is the migration network. People are moving across the world form country to country. An important motivation comes from productivity reasons which is related to wages or job-related reasons. The gravity equation kind of behavior also seen in migration as well. The network perspective in less prevalent in this literature even though there are instances of its usage (e.g.
Stark and Jakubek2013). Fagiolo and Mastrorillo (2014) connects the literature on trade and migration establishing that the trade network and the migration network are very correlated.
5.3.4 Financial Networks
A big part of the literature has focused on financial networks which broadly includes bank-to-bank transfers (Bech et al.2010), firm-credit network (Bigio and Lao2013), asset networks (Allen and Gale2000; Babus and Allen2009) etc. Jackson et al. (2014) proposes an extremely simple and abstract way to model interrelations between such entities. Suppose there are N primitive assets each with valuern. There are organizations with cross-holding of claims. Thus the value of the i-th organization is
Vi =
k
wi krk+
k
˜
wi kVk (5.11)
wherewis a matrix containing relative weights of primitive asset holdings andw˜ is a matrix containing relative weights of cross-holding of claims. One can rewrite it in vector notations as
V =wr+ ˜wV (5.12)
which can be rearranged to obtain
V =(I− ˜w)−1wr. (5.13)
One very interesting feature of this model is that
j
Vj ≥
j
rj. (5.14)
The reason is that each unit of valuation held by one such organization contributes exactly 1 unit to its equity value, but at the same time through cross-holding of claims, it also increases value of other organizations. The net value is defined as
vi =(1−
j
˜
wj i)Vi (5.15)
which is simplified as
v=(I−w)(I− ˜w)−1wr (5.16) withwcapturing the net valuation terms. Thus eventually this system also reduces to a recursive structure which we have already encountered in the input-output models.
Finally they also introduce non-linearities and show how failures propagate through such a network. It is easily seen that any shock (even without any nonlinearities) to the primitive assets will affect all valuations through the cross-holding of claims channel. See Jackson et al. (2014) for details.
5.3.5 Dispersion on Networks: Kinetic Exchange Models
A descriptive model of inequality was forwarded by Dr˘agulescu and Yakovenko (2000) and Chakraborti and Chakrabarti (2000) (see Chakrabarti et al.2013for a general introduction and description of this class of models). The essential idea is that just like particles colliding against each other, people meet randomly in market places where they exchange money (the parallel being energy for particles). Thus following classical statistical mechanics, the the steady state distribution of money (energy) has an exponential feature which is also there in real world income and wealth distribution. Lack of rigorous micro-foundation is still a problem for such models, but the upside is that this basic model along with some variants of it can very quickly generate distributions of tradable commodities (money here) that resemble the actual distribution to a great extent including the power law tail. The equation describing evolution of assets (w) due to binary collision (between thei-th and the
j-th agents) as
wi(t+1)= f(wi(t),wj(t), ε)
wj(t+1)= f(wj(t),wi(t),1−ε) (5.17) ε is a shock and f(.) usually denotes a linear combination of its arguments Chakrabarti et al. (2013).
In principle, one can think of it as essentially a network model where agents are the nodes of the network and links randomly form among those agents and then destroyed. When the links are forms, they trade with each other and there is a microscopic transaction. After a sufficient number of transactions, the system reaches a steady state where the distribution of money does not change. Hence, inequality appears as a robust feature. An intriguing point is that the basic kinetic exchange models are presented in such a way that this underlying network structure is immaterial and does not have any effect on the final distribution. An open question is under what type of trading rule the topology of the network will have significant effect on the distribution (Chatterjee2009). So far this issue has not been studied much.
5.3.6 Networks and Growth
So far all of the topics discussed are related to the idea of distribution and dispersion.
Traditionally, economists have employed representative agent models to understand growth. In recent times, one interesting approach to understanding growth prospects has been proposed through applications of network theory. Hidalgo and Hausman (2009), suggested that existing complexity of an economy contains information about possible economic growth and development and they provide a measure of complex- ity of the production process using the trade network.
One unanswered point of the above approach is that it does not explain how such complexity is acquired and accumulated. A potential candidate is technology flow network. Theoretical attempts have mostly been concentrated on symmetric and linear technology flow networks (see e.g. Acemoglu2008). An open modeling question is to incorporate non-trivial asymmetry in the technology network and model its impact on macroeconomic volatility.
5.3.7 Correlation Networks
This stream of literature is fundamentally different in its approach compared to the above. It is almost exclusively empirical with little theoretical (economic/finance) foundation. During the last two decades, physicists have shown interests in modeling stock market movements. The tools are obviously very different from the ones that economists use. In general, economists include stock market (if at all) in a macro model by assuming existence of one risky asset which typically corresponds to a mar- ket index. Sometimes multiple assets are considered but usually their interrelations are not explicitly modeled as the basic goal often is to study the risk-return trade off (hence one risky asset and one risk-free asset suffices in most cases). However, physi- cists took up exactly this problem: how to model joint evolution and interdependence of a large number of stocks? Plerou et al. (2002) introduced some important tools.
Later extensions can be found in Bonanno et al. (2003), Tumminello et al. (2010) and references therein. These studies were done in the context of developed coun- tries with little attention to less developed financial markets. Pan and Sinha (2007) extended the study on that frontier.
The basic object of study here is the correlation network betweenN number of stocks. Each stock generates a return{rnt}of lengthT. Given any two stocksiand
j, one can compute the correlation matrix with thei,j-th element ci j = E(riãrj)−E(ri)ãE(rj)
σiãσj
. (5.18)
Clearly the correlation matrix is symmetric. Then the question is if there is any way one can divide the correlation matrix into separate modes defining the market (m), group (g) and idiosyncratic (r) effects,
C=Cm+Cg+Cr. (5.19)
A very important tool is provided by the random matrix theory which allows us to pin down the idiosyncratic effects. Through eigenvalue decomposition, one can construct the random mode by the so-called Wishart matrix which is essentially a random correlation matrix (Pan and Sinha 2007). This helps to filter the random component of the correlation matrix (Crin Eq.5.19). The market mode is the global factor that affects all stocks simultaneously in the same direction. The mode in between these two, is described as the group mode.
Empirically, the group effects in certain cases seems to be very prominent and cor- respond to actual sector-specific stocks (Plerou et al.2002), but not always (Kuyya- mudi et al.2015; Pan and Sinha2007). Onnela et al. (2003) also studied contraction and expansion along with other properties of the correlation network. However, this field is still not very matured and the applications of such techniques are not wide- spread.
One can also construct networks based on the correlation matrix. A standard way to construct a network would be to consider only the group correlation matrix and apply a threshold to determine if stocks are connected or not. A complementary approach without using a threshold would be to construct a metric based on the correlation coefficients. This is very useful to construct a minimum spanning tree and study its evolution over time, (see Bouchaud and Potters (2009) for related discussions).