First published: december 2015
Most models used in macroeconomics, especially in the policy-making sphere, are linear DSGE – Dynamic Stochastic General Equilibrium – models. They feature rational expectations and require, for the determinacy of their equilibrium path, a set of stability conditions. In this post I would like to briefly review how one solves such models and shed light on three major questions: first what are the justifications for the stability conditions – the assumption that the macroeconomic system has to return to a steady-state ; second what are the justifications for the uniqueness of the equilibrium path ; and third what is the meaning in words of such a solution concept.
I would like to show how crazy those solution concepts – direct descendants of the rational expectations assumption – are. Just to give a little glimpse of the funniest part: in those models, the economy jumps and evolves on a unique stable path because agents who are perfectly forward looking and knows the model are able to forecast that if they sit away from the saddle path, they will make the economy explode and violate the stability conditions ; and that’s what keep the economy on track. That’s crazy!
Let’s start with a little bit of math. A DSGE can be written as a system of dynamic equations – either a system of difference equations in discrete time or a system of differential equations in continuous time. In continuous time, the model in a deterministic setup is summarized by the following matrix equation:
X’(t)=AX(t)
Where X is the vector of variables of size n, X’ is the time derivative of X and A is an n*n matrix that is independent of time. And a set of initial conditions that describes where the system starts
Y(0)=Y0
Where Y is a subvector of X: X=(Y Z)’.
Example: the neoclassical growth model
It features capital, K, and consumption, C, as variables. X=(K C)’ where K(0)=K0 is the predetermined variable of the system. Consumption which is not predetermined is called the jump variable. The linearized model is described by:
K’=aK– b C
C’=ρ C
So that it can be written as X’(t)=A X(t). The system runs from 0 to infinity.
The solution to such a system involves eigendecomposing A. The details of it can be found either here (a good treatment done by Moll) or here (by Sims if the reader looks for something more technical).
A differential equation without boundary condition admits an infinite number of solutions. Therefore, for each dynamic equation, we need a boundary condition or a stability condition. Either one needs an initial condition – this is what the predetermined variables, Y, such as capital, have – or a stability condition that imposes that the variable goes back to its steady-state. All jump variables get impose such a stability condition.
These conditions most of the time imply that the equilibrium path is determinate, i.e. unique and stable. They will force the vector of variables X to lie on the saddle path of the system, the only path that is compatible with convergence back to the steady-state. There are many other paths but all of them are divergent or explosive[1].
I would like to stress three important points. One is about the justification of the stability conditions. The second is about the justification of the uniqueness requirement. The third is about the economic intuition underlying such a solution concept.
1) The justification of the stability conditions are twofold.
The first justifications stem from the economic model itself. In many instances, stability conditions look like “natural” conditions given the specification of the model: the transversality condition for the capital stock for example prevents any upward explosion of capital and the non-negativity constraint prevents any downward explosion. However in most instances stability conditions are stronger than the requirements of the model. For example in the previous model, the transversality condition does impose that the rate of growth capital be smaller than some constant but not that it cannot explode upward. In some other instances, there are even nothing in the model that may look like a stability condition. Still we impose the stability of the solution.
The second justification is much more realistic and simple although scientifically less sound: economists like stable solutions because the actual macroeconomic system has never exploded (not yet at least). That is we restrict a priori the set of solutions not because the model we have in mind would necessarily require it but because the match with reality requires it.
2) The justification of uniqueness is a little more subtle and less robust to criticism. The main justification is purely utilitarian: if a model generates multiple equilibria, we – economists – cannot say anything sensible about the implications of a given policy in a given model. If we want to say something, we need uniqueness. Fortunately, most model spontaneously generate a unique equilibrium path.
3) The economic intuition behind the stability condition and the uniqueness of the saddle path is at the same time fascinating for its depth and worrying for the weakness of the solution’s robustness.
The reason why the economy jumps onto the saddle-path from the beginning of time until the end and cannot walk away from it is because the rational agents that populate the economy form rational expectations, so that they know the model and can perfectly forecast what is going to happen in a far remote future for any starting point of the economy. Consequently, should the economic system be away from its saddle path, the forward-looking rational agent would see that the path is divergent and would eventually violate the stability conditions, which contradicts the conditions of the model. Therefore the agent will refuse to sit on such a path. The only path on which the agent would accept to sit is one that he expects to converge to the steady-state. That’s crazy, isn’t it?
That’s rational expectations combined with stability requirement in DSGE models.
[1] Although beyond the scope of this post, there are important instances in which the stability conditions are not enough to get uniqueness of the saddle path. Such case are called “indeterminate”. We assume here that uniqueness is not an issue.