The Blanchard And Kahn' S Conditions In Macro-Econometric Models With Perfect Foresight

Many recent large macro-econometric models assume perfect foresight (for instance the multinational models Multimod Mark 3 and Quest 2). This choice has been made possible by the development of simulation algorithms, which are powerful and easy to use (for example the command Stacks and Lkroot in Troll and the software Dynare which works under Gauss and Matlab - see Juillard). However, the existence and the uniqueness of a solution for these models are not warranted a priori. Blanchard and Kahn established local conditions for these properties, which are easy to check, in terms of eigenvalues computed at the steady state of the model. However, these conditions can be used only on linear models, with coefficients independent of time, and with exogenous variables taking constant values after some date. Unfortunately, macro-econometric models are non-linear, their linear approximation has coefficients which change over time, in the long run many variables grow at positive and different rates, and these models may present an hysteresis. This paper explains how to overcome these difficulties, and apply the Blanchard and Kahn's conditions on this kind of models.We start by imposing a homogeneity condition on the model, which implies that it has a balanced growth path solution. Afterward, we can compute a linear approximation of the model around this path, and require that the solution of the model tends toward the balanced growth path when time increases indefinitely. We call this last condition stability in the absolute difference. However, some coefficients of the linear approximation are geometrical functions of time, and it is impossible to use the results of Blanchard and Kahn in this kind of situation (although the eigenvalues, but not the eigenvectors, of the linear approximation are independent of time).We can overcome this difficulty if we make a change of variables and put all of them on a common trend (with the same growth rate). If this trend has a zero rate, we will say that we write the model in reduced variables and that we require its stability in the relative difference. If the common growth rate is the highest balanced growth rate among all the variables, , we will say that we write the model in expanded variables, and that we require its stability in the expanded difference. In these both situations the coefficients of the linear approximation of the model are constant and we can apply Blanchard and Kahn's results. The eigenvalues of the second linear approximation are equal to times those of the first. If the Blanchard and Kahn's conditions are satisfied for the model written in reduced variables and in the expanded variable, then the model has a unique solution stable in the absolute difference.Hysteresis can be easily identified by the existence of eigenvalues equal to 1 in the model written in reduced variables. In models of endogenous growth, hysteresis is relative to the indeterminacy of the output level in the steady state. In more traditional macro-econometric models, indeterminacy concerns the price level when monetary policy follows a kind of Taylor's rule, in conformity to an old idea of Wicksell.The first section of the paper presents the results of Blanchard and Kahn, with the extension to the case of hysteresis developed by Giavazzi and Wyplosz. In the second section we present a very simple example to investigate some of the difficulties met with large econometric models, which are not taken into account by Blanchard and Kahn. The third section uses a much richer example, the endogenous growth model by Lucas, which includes all the difficulties we want to deal with in this paper. Finally, in the last section, we give a general theoretical treatment of our problem.Our results can also be applied to the study of the stability of more traditional macro-econometric models, which assume adaptive expectations, and where the current state of the economy does not depend on the future states foreseen by the model.