<blockquote>
<p>My OP below asks about the <em>RBC</em> model, but I am actually interested in any out-of-equilibrium macroeconomic model; CGE, DSGE or whatever the correct nomenclature is.</p>
</blockquote>
<p>The <em>Real Business Cycle</em> model is a standard equilibrium model in macro-economics. There are various <a href="http://www.sfu.ca/%7Ebkrauth/econ808/808_lec5.pdf" rel="nofollow noreferrer">versions</a>, 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.</p>
<p>The model is usually solved by specifying equilibrium conditions and using <a href="https://en.wikipedia.org/wiki/Lagrange_multiplier" rel="nofollow noreferrer">Langrange Multipliers</a> to solve for the steady-state. Sometimes, <a href="https://economics.stackexchange.com/questions/4770/simulating-real-business-cycle">technological shocks</a> are added to the model, which result in changes in the equilibrium. As discussed <a href="https://economics.stackexchange.com/questions/16434/looking-for-discussion-on-equilibrium-vs-dynamic-models-in-econometrics/">here</a>, such movements of the equilibrium are what most economists consider '<em>dynamics</em>'.</p>
<p>In this question, I am using the term '<em>dynamics</em>' differently. I would like to create a simple <em>Real Business Cycle</em> 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 <em>without</em> specifying any equilibrium conditions.</p>
<p>I realise I would need to make additional <a href="https://en.wikipedia.org/wiki/Replicator_equation" rel="nofollow noreferrer">assumptions</a> concerning the <a href="https://economics.stackexchange.com/questions/15566/can-linear-supply-demand-equilibria-be-understood-as-a-feedback-control-process">time-dynamics</a> of the model, e.g. that the rate of change of consumption is equal to the marginal utility w.r.t. consumption, <span class="math-container">$\frac{d}{dt} C = \frac{d}{dC} U$</span>.</p>
<p>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 <em>translate</em> equilibrium models to dynamic models using differential equations. Could anyone recommend an online tutorial, similar to <a href="http://faculty.econ.ucdavis.edu/faculty/kdsalyer/LECTURES/Ecn235a/Linearization/ugfinal.pdf" rel="nofollow noreferrer">this</a> but the non-equilibrium, dynamic version?</p>

CC BY-SA 4.0
2021-08-01T20:03:23.670Z
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
CC BY-SA 4.0
2021-08-02T08:21:35.623Z
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.
LBogaardt
CC BY-SA 4.0
2024-01-17T15:48:28.993Z
Basically what you described is what a phase diagram does: solving a system of differential (or difference) equations without finding the explicit solution, but just pointing out that a saddle path that leads to the steady state exists. The Ramsey model, if you just draw the phase diagram, is indeed what you are looking for.
Matteo Bulgarelli
<dynamic-optimization><real-business-cycles><difference-equations>
Out-of-Equilibrium Macroeconomic Dynamics using Differential Equations
LBogaardt
# Out-of-Equilibrium Macroeconomic Dynamics using Differential Equations

(Asked by: username_0 on 2021-07-29)

> 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](http://www.sfu.ca/%7Ebkrauth/econ808/808_lec5.pdf), 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](https://en.wikipedia.org/wiki/Lagrange_multiplier) to solve for the steady-state. Sometimes, [technological shocks](https://economics.stackexchange.com/questions/4770/simulating-real-business-cycle) are added to the model, which result in changes in the equilibrium. As discussed [here](https://economics.stackexchange.com/questions/16434/looking-for-discussion-on-equilibrium-vs-dynamic-models-in-econometrics/), 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](https://en.wikipedia.org/wiki/Replicator_equation) concerning the [time-dynamics](https://economics.stackexchange.com/questions/15566/can-linear-supply-demand-equilibria-be-understood-as-a-feedback-control-process) 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](http://faculty.econ.ucdavis.edu/faculty/kdsalyer/LECTURES/Ecn235a/Linearization/ugfinal.pdf) but the non-equilibrium, dynamic version?

**Tags:** dynamic-optimization, real-business-cycles, difference-equations

> **Comments:**
> **username_1**: 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?  
> **username_0**: 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.  
> **username_2**: Basically what you described is what a phase diagram does: solving a system of differential (or difference) equations without finding the explicit solution, but just pointing out that a saddle path that leads to the steady state exists. The Ramsey model, if you just draw the phase diagram, is indeed what you are looking for.  

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## 0 answers

