Comput Econ DOI 0.007/s06-0-96- Modellng the Evoluton of Natonal Economes Based on Input Output Networks Wen-Q Duan Accepted: 6 February 0 Sprnger Scence+Busness Meda, LLC. 0 Abstract Uncoverng the evolutonary dynamcs of economes s helpful for us to desgn economc polcy. Ths paper develops an economc evoluton model by examnng the coupled dynamcs of growth and nter structure. For each, ts economc propertes (manly characterzed by prce and quantty) are ncorporated nto the model. The nput output relatonshps among dfferent ndustres are descrbed as nput output networks, n whch nodes represent ndustres, and weghts represent technologcal and economc constrants between parng ndustres. By measurng the dynamc mportance of each node, we fnd that all the nodes n the nput output networks have the same dynamc mportance. On the bass of ths emprcal regularty and the ratonal expectaton assumpton, we show that the coupled dynamcs of the economc propertes of nodes and nput output networks can explan the evolutonary dynamcs of natonal economes. Keywords Natonal economy Evoluton model Input output network Dynamc mportance of node Introducton Natonal economes, composed of a large number of heterogeneous ndustres wth adaptvely tradng nteractons, are typcally complex systems. As an effectve method to descrbe the nteractons n complex systems, the complex network theory has already been appled to economc systems (Duan and Sun 009; Ior et al. 008; Song et al. 009). Some emprcal studes have shown that economc networks have some W.-Q. Duan (B) School of Busness Admnstraton, Zhejang Normal Unversty, Jnhua 00, People s Republc of Chna e-mal: wenqduan@6.com
Wen-Q Duan unversal characterstcs smlar to other complex networks, such as scale-free degree dstrbutons n the nterbank network (Ior et al. 008), node strengths obeyng powerlaw regularty n the world nvestment networks (Song et al. 009), and exponental functon relatonshp between degree and strength n the weghted nternatonal trade networks (Duan and Sun 009). However, the topologcal propertes of economc networks mght not be of much help n understandng the underlyng dynamcs of ndvdual ndustres or the entre economc system, because t s stll not clear how to cross the gap between the aggregated economc system dynamcs and the topologcal propertes of the underlyng nteracton network (Schwetzer et al. 009). In ths paper, we focus on the dynamc propertes of economc networks n determnng the economc system dynamcs by connectng the evolutonary dynamcs of the economy and the dynamc mportance of nodes n nput output networks. The evolutonary dynamcs of natonal economes often exhbt synchronzed expanson or contracton across ndustres (Bernd 00; Horvath 998; Laura and Justn 007; Shea 00), whch s one of the most mportant drvng forces leadng to the cycles of boom and slump. Several mechansms were proposed to explan the synchronzaton of the economc cycle. Horvath showed that the synchronzed movements n output across ndustres were manly caused by complementartes that propagate shocks across ndustres (Horvath 998). By assessng the mportance of nput output lnkages, Shea (00) demonstrated that the nter pattern of comovement depends on the nter pattern of lnkages. Unlke prevous explanatons that emphaszed producton complementartes, nformaton complementarty was proposed as another source of economc cycle because smlar nformaton leads to smlar nvestment decsons and co-movement n nputs and outputs (Bernd 00; Laura and Justn 007). From the vewpont of physcsts, an economc cycle mght result from the economy spontaneously evolvng towards a self-organzed crtcalty state due to the presence of sgnfcantly nonlnear, strongly localzed nteractons between dfferent parts of the economy (Chenkman and Woodford 99). Though these studes acknowledged that the nter structure of the economy played an mportant role n the synchronzaton process of the economc cycle, we stll need to study the specfc mechansm and process of how the nteractons among ndustres affect the evoluton of the aggregated natonal economy. To fulfll ths gap, ths paper ntroduces the complex network dynamcs method to examne the ssues of economc cycle. Frst, we propose the general conceptual framework, n whch the economc evoluton s assumed to be determned by the coupled dynamcs of prces, outputs, and nput output network. Next, we measure the dynamc mportance of each node, and fnd that all nodes n nput output networks have the same dynamc mportance. On the bass of the emprcal result on the dynamc mportance of nodes n nput output networks, we then propose an economc evoluton model that manly descrbes the coupled dynamcs between node economc propertes (characterzed by prce and output quantty) and nput output networks. Our results demonstrate that our model can reproduce the evolutonary dynamcs of prce, output quantty, and nput output network smultaneously, whch suggests the possblty of developng an economc cycle model by examnng the evolutonary dynamcs of nput output networks.
Modellng the Evoluton of Natonal Economes Based on Input Output Networks The General Modellng Framework To produce goods (outputs), an needs to consume outputs from other ndustres, whch are nputs to ths. These nter connectons among all ndustres consst of an nput output matrx, whch was developed by Nobel laureate Wassly Leontef n the 90s. From the pont of vew of network theory, we can easly transform the nput output matrx nto a network n whch a node represents an and an edge stands for a connecton between two ndustres. By the defnton of the nput output matrx, the weght of an edge represents the amount of products of one requred by another to produce outputs. To better characterze the economc propertes of a node, we choose two essental measures for the correspondng, the prce and output quantty. Obvously, these two measures are strongly related to the nput output network. For nstance, f the prce of outputs for a gven s ncreased, ts output quantty wll be stmulated, but ts consumpton by other ndustres wll be suppressed. Therefore, the whole nput output network wll be changed. Smlarly, f the output quantty of an s ncreased, the better avalablty of these products mght boost ther usage n the related ndustres. Thus, the correspondng nput output network also needs to be adjusted. In concluson, the fnal prce, output quantty, and local nput output network of an are determned by the collectve dynamcs of all ndustres, and the evolutonary dynamcs of the economy can be descrbed as the prce and output quantty co-evolvng wth the nput output network. To llustrate the coupled dynamcs between node economc propertes and the nput output networks, we depct the evoluton process of a small demonstraton nput output network n Fg.. Data and Emprcal Results of the Dynamc Importance of Nodes The nput output matrx s an economc analyss method used worldwde. Many countres or nternatonal organzatons publsh Input Output tables; for example, the Organzaton for Economc Cooperaton and Development (OECD) comples Input Output tables for the OECD countres. However, most tables are ssued wth ntervals of fve years and at the sector level. Therefore, most data are not sutable for us to model the evolutonary dynamcs of natonal economes year by year. Fortunately, the Bureau of Economc Analyss (BEA) has recently started ssung US tables for benchmark years (years endng n and 7) wth three levels of detal. From 998 to 007, yearly tables are avalable at the level. Ths allows us to test the deas presented n Fg.. All data analysed n ths paper are downloaded from the BEA webste http://www.bea. gov. The dataset nclude the prce and output quantty of each and the Input Output matrx, whch covers the perod from 998 to 007. All ndustres are coded accordng to the 997 North Amercan Industral Classfcaton System (NAICS). The entres of prce and output quantty are recorded n the GDP-by- format wth the 6-dgt NAICS code. Consderng the -dgt NAICS code used n the Input Output matrx, we obtan aggregate statstcs of prces and output quanttes consstent wth the nput output data. We have adjusted for the slght nconsstences between the entres n the GDP-by- table and the Input Output matrx. Though outputs are
Wen-Q Duan p= q=6 p= q=7 p= q=9 5 5 5 5 p= q= p=7 q= p= q=9 p=6 q= 5 p= q= t t +t + Fg. Illustraton of the coupled dynamcs of the prce, the output quantty, and the nput output network. Node,,, and represent four ndustres, respectvely. The edges connectng two nodes and the numbers near them are the weghts of the nput output network. Weght represents the amount of products of one requred by another so as to produce the outputs of that. For example, n the left subfgure, unts of the product of are requred by to produce 6 unts of ts output. Further, p s the unt prce and q the amount of the output produced by one. The left subfgure shows the prce, output quantty, and nput output network n year t. At the start of year t +, frms and ndvduals forecast the amount of each s prce varaton. As shown n the rght subfgure, agents adapt ther requrements, output quanttes, and producton structures to the expected prces. Eventually, the expected prces wll be realzed. Ths assumpton of self-fulflled expectaton s held n classc macroeconomcs lterature, and t wll be explaned n Sect.. Table Smple statstcal descrptons of prce and output quantty ndexes durng 998 to 007 (base year = 00) Statstcal year 998 999 000 00 00 00 00 005 006 007 Prce ndex Mean 0.968 0.9666 0.9889 0.9970.0000.057.065.00.66.000 Standard varaton 0.77 0.85 0.08 0.05 0 0.0576 0. 0.959 0.87 0.90 Quantty ndex Mean 0.97.009.07.0.0000.09.066.06.87.8 Standard varaton 0.795 0.77 0.00 0.077 0 0.055 0.05 0.90 0.96 0.56 avalable for current dollars and quantty ndexes, prces are released only n ndexes. Therefore, prces and output quanttes are analysed n the ndex form n the latter, whch can reflect the same changes from year to year relatve to the correspondng measure. As we can see from Table, the average prce of ndustres ncreases monotoncally from 998 to 007, but the average quantty of output has a downward trend n 00 and 00. Furthermore, there are more prce and output quantty fluctuatons n 006 and 007 than n other years. The nput output networks are constructed from the -by- total requrements tables, whch are one knd of Input Output matrx. These tables contan the nput estmates for each that are drectly and ndrectly requred to delver
Modellng the Evoluton of Natonal Economes Based on Input Output Networks a dollar of the output to fnal users. Smlar to prce and output quantty ndexes, there are ndustres n total. In addton, we add one addtonal column for the economy s output (goods for fnal consumpton) and one addtonal row for nput from labour nto nput output networks. Therefore, our data ncorporate the fnal consumpton expendture of each and the effects on the rest of the economy. For each year, we can construct one weghted, drected network. At frst glance, each nput output network s a nearly complete graph. On the US 00 table, for example, about 9% of lnks are nonzero. Consderng that the topologcal characterstcs are trval, we are not ready to report the statc structural propertes, such as degree dstrbuton, network densty, average connectvty, and other measurements, n ths paper. As llustrated n Fg., an nput output network descrbes the nter structure n a perod. It evolves wth tme snce the prce and output of each always change from one perod to another. Moreover, economc cycle mples a sgnfcant tendency that exhbts smultaneously n many ndustres. Therefore, uncoverng the evoluton prncple of the nput output networks mght help to understand the evolutonary dynamcs of economc evoluton. In order to dentfy the evolutonary path of the nput output networks, we consder the propensty for synchronzaton of the ndvdual ndustres prce, output quantty, and nput output nteractons. As ponted out by Chavez et al. (005), a weghtng procedure based upon the global structure of network pathways enhances synchronzaton of dentcal and non-dentcal dynamc unts. Furthermore, the synchronzablty of complex networks wll be enhanced sgnfcantly by dynamcally organzng the couplng strength and the local synchronzaton property between the nodes and ther neghbours (Zhou and Kurths 006). The evoluton of nput output networks mght follow prncples that beneft the synchronzed expansons and contractons across ndustres. Consderng that the largest egenvalue s the key quantty determnng synchronzed dynamc processes on nput output networks, we defne the dynamc mportance of nodes as the relatve change n the largest egenvalue of the nput output networks upon ther nodes removal (Restrepo et al. 006). For a gven nput output network, λ s the largest egenvalue. After the removal of node, the largest egenvalue of the network changes nto λ, and the dynamc mportance of that node s defned as I = (λ λ )/λ. Accordng to the defnton of an nput output network and the Perrot-Frobenus theorem, λ s always constant at. Therefore, we obtan I = λ. The tme-dependent property of the dynamc mportance of nodes enables us to explore the propertes more than a sngle network snapshot. Surprsngly, by applyng the above computng procedure to the ten nput output networks, we fnd that all nodes have an equally dynamc mportance. The largest egenvalue of the nput output network wll decrease to (N )/N f a node s removed, where the number of ndustres n the economy s N =. In other words, the dynamc mportance of each has the same value (I = 0.99800075 = 0.00758796998, =,...,). Furthermore, ths emprcal regularty holds for all nput output networks. Ths result seems to be unusual, but t s reasonable. Informaton about the prce and output quantty of one can quckly propagate to the whole nput output network. Industres adapt to each other so as to reach a state n whch each has the same dynamc mportance. Furthermore, we note that the second largest
Wen-Q Duan egenvalues of nput output networks are n the range of 0 5 0 8 regardless of whether one node s removed or not. Therefore, we can conjecture that the equally dynamc mportance of nodes mght reflect some knd of potental mechansm whch governs the evolutonary dynamcs of nput output networks. Economc Evoluton Model Based on Equally Dynamc Importance of Nodes Based on the prevous results, we now develop a quanttatve model to characterze the coupled dynamcs between the prce, output quantty, and nput output network. Frst, let p t and q t be the prce and output quantty, respectvely, of n year t, where =,,..., and t = 998,...,007. Consderng the mportance of choosng the rght network representaton for our problem, we use wj t, representng the nput output network n year t, whch satsfes N j= wj t qt j = q t (Butts 009). In terms of economcs, requres wj t qt outputs of j as nputs so as j to produce q t outputs. In other words, wj t / s the proporton of the output = wt j quantty of j consumed { by }. Then, let us llustrate how to obtan nput output network W t = wj t from the make and use tables provded by BEA. We can see the detaled mathematcal dervaton of the total requrements tables for nput output analyss n Horowtz and Plantng (009). Here, we quote the equaton Q t = (I D t B t ) D t e t drectly, where I s the dentty matrx. Q t ={q t } s a column vector n whch each entry shows the output quantty of each. The entres n each column of the -by-commodty matrx D t show, for a gven commodty, the proporton of the total output of that commodty produced n each. The entres n each column of the commodty-by- matrx B t show the amount of a commodty used by an per dollar of output of that. Vector e t represents the total fnal demand purchases for each commodty from the use table. The orgnal data, D t, B t, and e t, are provded by BEA, and we can obtan them drectly. Consderng that W t Q t = Q t = (I D t B t ) D t e t, we can compute W t = (I D t B t ) D t e t (Q t ). () Furthermore, our model s based on the ratonal expectaton assumpton, whch asserts that outcomes do not dffer systematcally from what people expected them to be. In other words, t s reasonable to assume agents predctons of the future prces and outputs are not systematcally wrong. Ths assumpton was often held n closed macroeconomc models n whch the solutons depend n part on the values of the expected future varables (Far 979; Muth 96). In our model, once frms and ndvduals expect that the prce p t wll change nto p t+ n the next year, they wll adapt ther nter structure wj t+ and output quantty q t+ to fulfll the expected prce. Therefore, the evolutonary dynamcs of natonal economes can be formulated as follows:
Modellng the Evoluton of Natonal Economes Based on Input Output Networks q t+ wj t+ = f (p t, q t, pt+ ) = g(wj t, pt+, q t+ p t+ = p t+ ), () where =,,...,N. There are N(N + ) varables requred to solve. In order to obtan a numercal soluton, we frst assume functon f n the form of q t+ /q t = α(t) (p t+ /p t)β(t), where α(t) and β(t) are parameters estmated from emprcal data of ndustres n year t and t +. Evdently, α(t) and β(t) can be vewed as measurements of how the prce varatons affect output varatons. Therefore, we can compute q t+ accordng to the expresson q t+ /q t = α(t) ( p t+ /p t)β(t), whch corresponds to N equatons. The functon g s determned by two condtons: () N j= wj t+ q t+ j = q t+, and () the largest egenvalue of wj t+ wll decrease to (N )/N once any node s removed, whch s equvalent to the equally dynamc mportance of all ndustres. The two condtons correspond to N and N(N ) equatons, respectvely. Now, we have obtaned N(N +) closed equatons whch help us solve p t+, q t+, and wj t+ smultaneously. Furthermore, we can obtan p t+, q t+, and wj t+ by repeatng the prevous computng procedures. Superfcally, our model looks lke fttng a posteror from emprcal data and then usng t to predct results. However, that s not true. The reason s that the estmated parameters α(t) and β(t) are ndependent of wj t and wt+ j. Consderng that we can use the computed p t+, q t+, and wj t+ to predct p t+, q t+, and wj t+ recursvely, our method has modelled the coupled dynamcs of node economc propertes and nput output network successfully. In order to mprove modellng performance, we ntroduce some degree of randomness by settng α (t) = α(t)( + ε), β (t) = β(t)( + ε), where the unformly dstrbuted random varable ε [ 0., 0.]. One run of the model conssts of = 78 closed equatons. Each computatonal experment s replcated 00 tmes, and the averaged results are reported n subsequent paragraphs. To valdate the predctablty of the proposed model, we compare the prce, output quantty, and nput output network generated by our model wth the correspondng emprcal data. Fgure shows that the smulated prce from our model well matches that of the emprcal data for the three typcal years, 999, 00, and 007. The two types of curves are very close to each other, whch suggest that our model can reproduce the evolutonary dynamcs of prce. In addton, smlar results can also be obtaned for all other years. Next, let us test whether our model can duplcate the emprcal output quantty. Smlar to the prce, we use the ndex to measure the relatve output quantty of each year, compared to the base year 00. As shown n Fg., for a gven, the output quantty generated by our model s n good agreement wth the emprcal data. These evdences show that our model can reproduce the evolutonary dynamcs of output quantty. Though we only depct the results n three typcal years, smlar qualtatve results are also observed for all other years. Consderng that t s dffcult to compare the smulated nput output network wth the emprcal one drectly, we compare ther weghts for all ndustres, whch are computed by accumulatng each row for the two networks. As we can see from
Wen-Q Duan (a).5 (b).5 (c) 999 00 007.8. smulated prce smulated prce smulated prce.6. emprcal prce.5 emprcal prce emprcal prce.....5 0.9 0.8 0.8 0.5 prce prce 0.7 prce 0 Fg. Comparson of the emprcal prces and the smulated prces of each. The emprcal prce of each n ths fgure s the nomnal prce dvded by the nomnal prce of that n year 00. The smulated prce of each s obtaned by jontly solvng the = 78 closed equatons n the above model. Correlaton coeffcents between the smulated and the real prces of 999, 00, and 007 are 0.9765, 0.8999, and 0.99, respectvely.6 (a). (b) (c) 999 00 007.5 smulated quantty.. smulated quantty smulated quantty emprcal quantty emprcal quantty emprcal quantty...5.5 0.8 0.9 quantty quantty 0.8 quantty 0.5 Fg. Comparson of the emprcal output quantty and the smulated output quantty of each. The emprcal output quantty data of each n ths fgure s the output quantty n 999, 00, and 007 dvded by the output quantty of that n 00. The smulated output quantty of each s obtaned by solvng the = 78 closed equatons n the above model smultaneously. The correlaton coeffcents between the smulated and emprcal output quanttes of 999, 00, and 007 are 0.9805, 0.905, and 0.9958, respectvely Fg., for a gven, the smulated weght s roughly dentcal to the emprcal weght. Ths agreement further confrms that our model can reproduce the evolutonary dynamcs of the nput output network. It s worth notng that the dscrepancy rate between the model-generated weghts and the emprcal ones are often much smaller than that of prce and output quantty. Ths s manly due to two reasons: () the nput output network s more stable and changes slower than the prce and the output quantty; () the accumulated effects of postve and negatve dscrepancy of ndvdual elements n the nput output network can decrease the dscrepancy rate of weghts. Combnng the results shown n Fgs.,, and, we can conclude that our model has reproduced the coupled evolutonary dynamcs of prce, output quantty, and the nput output network from the prevous year to the present year, smultaneously. As we know, the economc cycle s a common phenomenon observed n the growth of the economy, and t s always a challenge to uncover the formaton mechansms whch govern the fundamental dynamcs of economc evoluton. Therefore, we need to check whether our model can reproduce macro behavour of economc evoluton. We take 998 as the startng year, and apply the prce, output quantty, and nput output network of ths year to recursvely predct the correspondng values of year 999, 000,..., and 007. As shown by the average growth rate n Fg. 5, though
Modellng the Evoluton of Natonal Economes Based on Input Output Networks (a) (b) (c) weght of each 5 0 5 0 5 0 5 5 50 999 00 007 0 smulated weght smulated weght 0 smulated weght emprcal weght emprcal weght 5 emprcal weght 0 weght of each 0 5 0 5 0 weght of each 0 0 0 0 Fg. Comparson of the weghts of the emprcal nput output network and the smulated nput output network. For a gven, ts weght s the accumulated row elements n an nput output network. The emprcal nput output network s obtaned on the bass of Eq., drectly. The smulated nput output network s obtaned by solvng the = 78 closed equatons n the above model. The correlaton coeffcents between the smulated and real weghts of 999, 00, and 007 are 0.98, 0.98, and 0.9957, respectvely 0 8 average growth rate (%) 6 0 - smulated growth rate real growth rate - 998 000 00 00 006 008 year Fg. 5 Comparson of the growth patterns of the real economy and the model-generated economy. For each year, the growth rate s the average growth rate of ndustres. For a gven, ts growth rate s the current year s added producton value dvded by last year s producton value. Economc growth patterns observed n the real economes and the model-generated economes are smlar to each other. Therefore, ths fgure has shown that the economc evoluton model based on nput output networks can reproduce the evolutonary dynamcs of natonal economes successfully two curves do not collapse onto a sngle one, the two fluctuaton patterns of growth rate are very smlar to each other. Now, we have demonstrated that our model can reproduce the economc growth pattern up to ten years. Notably, our tests do not cover a complete economc cycle n the conventonal sense, because the U.S. went through a perod of rapd economc growth from 998 to 007, except for a slght downturn n 000. Therefore, we need more data to examne whether our economc evoluton model can be an effectve alternatve economc cycle model. Consderng the extreme complexty n modellng economc cycle, we suggest that the coupled dynamcs of prce, output quantty, and nput output network mght be a promsng mechansm to
Wen-Q Duan explan the formaton process of economc cycles, a topc deservng of much research n the future. 5 Conclusons In ths paper, we have proposed a new framework to model the evolutonary dynamcs of natonal economes, whch s based on the coupled dynamcs of prce, output quantty, and nput output network. The structure of the economy s descrbed as a weghted nput output network n whch nodes represent ndvdual ndustres and weghts represent technologcal and economc constrants between two ndustres. For a gven, we only consdered ts two essentally economc propertes, the prce and quantty of the s outputs. Based on the emprcal fndng that each node has the same mportance n determnng network dynamcs, we have shown that the coupled dynamcs of node economc propertes and nput output networks can explan the synchronzaton process n a perod of rapd economc growth. Lmted by the avalable data, our model has not been tested by data from a whole economc cycle. In the near future, we mght overcome ths data defcency problem, so that we can refne our present model to be an economc cycle model and demonstrate ts valdaton. Acknowledgements The author s very grateful to the anonymous referees for ther helpful comments and constructve suggestons. Addtonally, the author acknowledges the fnancal support from the Natonal Natural Scence Foundaton of Chna under Grant 707000. References Bernd, S. (00). Modellng the synchronzaton of sectoral nvestment cycles on the base of nformatonal externaltes. Structural Change and Economc Dynamcs, (), 5 5. Butts, C. T. (009). Revstng the foundatons of network analyss. Scence, 5(599), 6. Chavez, M., Hwang, D. U., & Amann, A., et al. (005). Synchronzaton s enhanced n weghted complex networks. Physcs Revew Letters, 9, 870. Chenkman, J. A., & Woodford, M. (99). Self-organzed crtcalty and economc fluctuatons. Amercan Economc Revew, 8(), 7. Duan, W. Q., & Sun, B. L. (009). Emprcal analyss and modellng of the global economc system. Chnese Physcs Letters, 6, 09890. Far, R. C. (979). An analyss of a macro-econometrc model wth ratonal expectatons n the bond and stock markets. Amercan Economc Revew, 69(), 59 55. Horowtz, K. J., & Plantng, M. A. (009). Concepts and methods of the nput output accounts. http:// www.bea.gov/papers/pdf/iomanual_09906.pdf. Accessed 5 Dec 009. Horvath, M. (998). Cyclcalty and sectoral lnkages: Aggregate fluctuatons from ndependent sectoral shocks. Revew of Economc Dynamcs, (), 78 808. Ior, G., De Mas, G., Precup, O., Gabb, G., & Caldarell, G. (008). A network analyss of the Italan overnght money market. J. Econ. Dyn. Control, (), 59 78. Laura, V., & Justn, W. (007). Aggregate shocks or aggregate nformaton? Costly nformaton and busness cycle co-movement. Journal of Monetary Economcs, 5(Supplement), 7 55. Muth, J. F. (96). Ratonal expectatons and the theory of prce movements. Econometrca, 9(), 5 5. Restrepo, J. G., Ott, E., & Hunt, B. R. (006). Characterzng the dynamcal mportance of network nodes and lnks. Physcs Revew Letters, 97, 090.
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