<p>Higher order approximations such as those generated by Dynare may help a bit in terms of expanding the neighborhood in which the approximation works well, but the fundamental problem remains that the approximation is made about the steady state and deviating too far from the steady state introduces large errors.</p>

<p>Judd, Maliar and Maliar have a paper in Quantitative Economics in 2011 that details a highly nonlinear method that can be used to give pretty good policy function approximations.  This method can be finicky with large state spaces, however.  So it too suffers from a curse of dimensionality.</p>

<p>Numerically stable and accurate stochastic simulation approaches for solving dynamic economic models
KL Judd, L Maliar, S Maliar
Quantitative Economics 2 (2), 173-210</p>

CC BY-SA 3.0
2015-02-19T00:39:07.517Z
2015-02-19T00:39:07.517Z
Kerk Phillips
<p>Side note: This is one way of solving it - the alternative would be formulating a Bellman equation and iterating on that.</p>

<p>If you assume that the real economy is on or sufficiently close to the steady state, you can also infer about responses to shocks. That is, you can look at the impulse response functions to a change in whatever interests you, and see how the model economy changes. Arguing that we're close enough to that steady state will allow you to see how the economy responds to certain shocks.</p>

<p>Also, in general, you can simulate the economy with (for example TFP) shocks, and look whether the simulated economy looks similar to the real economy. Using this comparison you could judge the model.</p>

<p>This needs arguing that we are close to the steady state - or that convergence happens really fast. Generally, the growth literature around Solow has provided arguments for this.</p>

<p>But your argument is very present in most extensions of the basic RBC model: especially when it is important <em>how</em> close we are to the steady state - when models are more nonlinear. There have been many papers showing that this is the case for standard Neo Keynesian extensions of the RBC model.</p>

CC BY-SA 3.0
2015-02-14T16:26:18.363Z
Thank you! But it seems to me, in this regard, using dynamic programming to approximate a global decision rule is strictly better. After all FOC and envelope conditon for Bellman equation implies Euler equations for the original optimization problem.
Epicurus
CC BY-SA 3.0
2015-02-14T16:36:53.480Z
Big shocks can be problematic with this method even if you are **at** the steady state. It isn't really enough to assume you are close to the steady state, you also need to assume that the shocks are not so big that the non-linearity of the the problem can be ignored.
BKay
CC BY-SA 3.0
2015-02-14T17:37:34.000Z
Right, @Bkay, I think that is straightforward.
FooBar
CC BY-SA 3.0
2015-02-14T17:56:14.883Z
@Epicurus: Yes, but the more advanced models are too complex to be solved with dynamic programming methods. Be it either due to the "curse of dimensionality" or because non convexities or similar do not allow for standard dynamic programming convergence.
FooBar
CC BY-SA 3.0
2015-02-14T16:10:57.467Z
2015-02-14T17:35:47.213Z
2015-02-14T17:35:47.213Z
FooBar
<p>I'm reading <a href="https://hec.unil.ch/docs/files/80/599/plosser89.pdf">understanding real business cycle</a> by Plosser.</p>

<p>Here's my crude understanding: For a RBC model, the FOCs of lagrangian together with transversality condition usually forms a nonlinear difference equations system. The standard way of solving this system is by linearization in the neighbourhood of steadystate,  solve this problem as a linear difference equations system.</p>

<p>What insight can we get from this solution, except for gaining  some knowledge of whether the steadystate is locally stable?</p>

<macroeconomics><academic-graduate><general-equilibrium>
What is the usefulness of approximating an optimal decision rule that close enough to steadystate in RBC model?
Epicurus
# What is the usefulness of approximating an optimal decision rule that close enough to steadystate in RBC model?

(Asked by: username_0 on 2015-02-14)

I'm reading [understanding real business cycle](https://hec.unil.ch/docs/files/80/599/plosser89.pdf) by Plosser.

Here's my crude understanding: For a RBC model, the FOCs of lagrangian together with transversality condition usually forms a nonlinear difference equations system. The standard way of solving this system is by linearization in the neighbourhood of steadystate, solve this problem as a linear difference equations system.

What insight can we get from this solution, except for gaining some knowledge of whether the steadystate is locally stable?

**Tags:** macroeconomics, academic-graduate, general-equilibrium


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## 2 answers

### Answer by username_2 on 2015-02-14. Score: 5 | Accepted answer

Side note: This is one way of solving it - the alternative would be formulating a Bellman equation and iterating on that.

If you assume that the real economy is on or sufficiently close to the steady state, you can also infer about responses to shocks. That is, you can look at the impulse response functions to a change in whatever interests you, and see how the model economy changes. Arguing that we're close enough to that steady state will allow you to see how the economy responds to certain shocks.

Also, in general, you can simulate the economy with (for example TFP) shocks, and look whether the simulated economy looks similar to the real economy. Using this comparison you could judge the model.

This needs arguing that we are close to the steady state - or that convergence happens really fast. Generally, the growth literature around Solow has provided arguments for this.

But your argument is very present in most extensions of the basic RBC model: especially when it is important *how* close we are to the steady state - when models are more nonlinear. There have been many papers showing that this is the case for standard Neo Keynesian extensions of the RBC model.

> **Comments:**
> **username_0**: Thank you! But it seems to me, in this regard, using dynamic programming to approximate a global decision rule is strictly better. After all FOC and envelope conditon for Bellman equation implies Euler equations for the original optimization problem.  
> **username_3**: Big shocks can be problematic with this method even if you are **at** the steady state. It isn't really enough to assume you are close to the steady state, you also need to assume that the shocks are not so big that the non-linearity of the the problem can be ignored.  
> **username_2**: Right, @Bkay, I think that is straightforward.  
> **username_2**: @username_0: Yes, but the more advanced models are too complex to be solved with dynamic programming methods. Be it either due to the "curse of dimensionality" or because non convexities or similar do not allow for standard dynamic programming convergence.  

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### Answer by username_1 on 2015-02-19. Score: 5

Higher order approximations such as those generated by Dynare may help a bit in terms of expanding the neighborhood in which the approximation works well, but the fundamental problem remains that the approximation is made about the steady state and deviating too far from the steady state introduces large errors.

Judd, Maliar and Maliar have a paper in Quantitative Economics in 2011 that details a highly nonlinear method that can be used to give pretty good policy function approximations. This method can be finicky with large state spaces, however. So it too suffers from a curse of dimensionality.

Numerically stable and accurate stochastic simulation approaches for solving dynamic economic models
KL Judd, L Maliar, S Maliar
Quantitative Economics 2 (2), 173-210


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