# Out-of-Equilibrium Macroeconomic Dynamics using Differential Equations

My OP below asks about the RBC model, but I am actually interested in any out-of-equilibrium macroeconomic model; CGE, DSGE or whatever the correct nomenclature is.

The Real Business Cycle model is a standard equilibrium model in macro-economics. There are various versions, but most consider a representative consumer and a representative firm. The consumer has to decide on a level of consumption, labour supply and investment so as to maximise utility. The firm maximises profits by deciding on a level of product output, labour demand and capital.

The model is usually solved by specifying equilibrium conditions and using Langrange Multipliers to solve for the steady-state. Sometimes, technological shocks are added to the model, which result in changes in the equilibrium. As discussed here, such movements of the equilibrium are what most economists consider 'dynamics'.

In this question, I am using the term 'dynamics' differently. I would like to create a simple Real Business Cycle model using differential equations. I would like to specify the behaviour of the consumer and the firm and show that the steady-state is reached without specifying any equilibrium conditions.

I realise I would need to make additional assumptions concerning the time-dynamics of the model, e.g. that the rate of change of consumption is equal to the marginal utility w.r.t. consumption, $$\frac{d}{dt} C = \frac{d}{dC} U$$.

Such simple models must surely exist. Ultimately, a dynamic model must yield the identical steady-state to the standard version. I would like to learn how to translate equilibrium models to dynamic models using differential equations. Could anyone recommend an online tutorial, similar to this but the non-equilibrium, dynamic version?

• Do you want to reach "the" steady state in the sense of an optimal state, or just some where things don't move anymore in the absence of shocks, ie. a fixed point?
– BrsG
Aug 1 at 20:03
• I guess that, under concave utility/profit curves, there is only 1 equilibrium, right? I am not necessarily looking for the socially optimal state, though I believe in simple models the socially optimal and the competitive equilibrium coincide, right? Indeed, I am looking for a fixed-point equilibrium, in the game theory sense. Aug 2 at 8:21