This paper presents discussion about the importance of degree of aggregation in input-output systems for speed of convergence to the equilibrium. The basis is the hypothesis stated by Brody (1997) that the greater the dimension of flow coefficients matrix A is, the faster the economy convergences to the equilibrium because the ratio between modulus of the subdominant and dominant eigenvalue declines to zero. Since then, several papers have been published to verify mathematical aspects of the hypothesis. The development of random matrices theory provided theorems that confirm the hypothesis in the case of i.i.d. elements of flow coefficients matrix. However, in the case of deterministic input-output table, there were constructed counterexamples where the ratio between subdominant and dominant eigenvalue does not decline when the size of the matrix increases to infinity and the degree of aggregation does not influence the speed of convergence to the equilibrium.
This paper verifies how the hypothesis fits to empirical data. Analysis of different aggregation of input-output tables for European Union, Euro zone and several European countries shows that increasing the number of branches in IO table does not reduce the ratio of modulus of subdominant and dominant eigenvalues of the flow coefficients matrix A. Hence, the speed of convergence to the equilibrium does not depend on the number of sectors in IO table but rather on the size of economy and relationships between its sectors. (i.e. large economies converge faster to equilibrium than small economies).
input-output models, equlibrium, eigenvalues, Brody’s conjecture
 Białas S., Gurgul H., (1998), On a Hypothesis about the Second Eigenvalue of the Leontief Matrix, Economic Systems Research-Journal of International Input-Output Association, vol.10, no.3, 285- 289.
 Bidard C., Schatteman T., (2001), The spectrum of random matrices, Economic Systems Research- Journal of International Input-Output Association, vol. 13, 289-298.
 Brody A., (1997), The Second Eigenvalue of the Leontief Matrix, Economic Systems Research- Journal of International Input-Output Association, vol. 9, no. 3, 253-258.
 Goldberg G., Okuney P., Neumann M., Schneider H., (2000), Distribution of subdominant eigenvalues of random matrices, Methodol. Comput. Appl. Probab., 2, 137-151.
 Goldberg G., Neumann M., (2003), Distribution of Subdominant Eigenvalues of Matrices with Random Rows, SIAM Journal on Matrix Analysis and Applications, vol. 24, no. 3, 747-761.
 Gurgul H., Majdosz P., (2006), Interfund Linkages Analysis: The Case of the Polish Pension Fund Sector, Economic Systems Research-Journal of International Input-Output Association, vol. 18, no.1, 1-27.
 Leontief W. W., (1941), The Structure of American Economy 1919-1939; An Empirical Application of Equilibrium Analysis, New York.
 Marcus M., Minc H., (1964), A Survey of Matrix Theory and Matrix Inequalites, (Boston, Allyn and Bacon).
 Molnar G., Simonovits A., (1998), A Note on the Subdominant Eigenvalue of a Large Stochastic Matrix, Economic Systems Research-Journal of International Input-Output Association, vol.10, no.1, 79-82.
 Plich M., (2003), Budowa i zastosowanie wielosektorowych modeli ekonomiczno-ekologicznych, Wydawnictwo Uł (rozprawa habilitacyjna).
 Sun G.-Z., (2008), The First Two Eigenvalues of Large Random Matrices and Brody’s Hypothesis on the Stability of Large Input-Output Systems, Economic Systems Research-Journal of International Input-Output Association, vol. 20, no. 4, 429-432.
 Tomaszewicz Ł., (1994), Metody analizy input-output, Panstwowe Wydawnictwo Ekonomiczne, Warszawa 1994.