# Issues on Second Order Approximation and Optimal Policies

This page describes several issues on approximation at the second order and optimal policy determination. More precisely, this page aims at:

- Providing Dynare codes that replicates Schmitt-Grohe and Uribe (2004a,2007) papers.
- Proposing illustration tools for nonlinear process (e.g. nonlinear IRF).
- Giving particular condition on consistency between welfare versus decision rules approximation order.

## Replication of Schmitt-Grohé and Uribe (2004a,2007) papers

### Schmitt-Grohé and Uribe (2004a)

#### Brief summary

The authors develop a DSGE model embedding imperfect competition and sticky prices. They compute Ramsey allocations and derive fiscal and monetary optimal decision paths. They compare simulated moments for key variables with different kinds of imperfection and different methods of simulation, namely log-quadratic and log-linear approximation for the baseline sticky-price model.

#### Outstanding points

- When running sgu_jet2004.mod without simulating the model, Dynare seems to indicate the existence of unit root in the model. This could imply that the model should be seen as non stationary. We should hence uncover this unity root in the SGU codes.
- Simulations seems to start from a given initial value for government bonds (b(-1)=.088) that could differ from the one used by Dynare.

#### Dynare code

mod file: sgu_jet2004.mod. This is a temporary file and still in progress. It should replicate results of Table 2 p. 212 in Schmitt-Groé and Uribe (2004a).

### Schmitt-Grohé and Uribe (2007)

#### Dynare code

mod file: model_sgu07_welf.mod.

## Illustration tools for nonlinear process

[To be done]

## Welfare/decision rules approximation order consistency

[To be done]

## References

Gomme, P., Klein, P. (2009) Second-order approximation of dynamic models without the use of ten- sors, *Concordia University Working Paper*, 09-004.

Lombardo, G., Sutherland, A. (2007) Computing Second-order-accurate Solutions for Rational Expectations Models Using Linear Solution Methods, *Journal of Economic Dynamics and Control*, 31(2), pp. 515–530.

Schmitt-Grohé, S., Uribe, M. (2004a) Optimal Fiscal and Monetary Policy under Sticky Prices, *Journal of Economic Theory*, 114, pp. 198-230. matlab codes

Schmitt-Grohé, S., Uribe, M. (2004b) Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function, *Journal of Economic Dynamics and Control*, 28, pp. 755-775. matlab codes

Schmitt-Grohé, S., Uribe, M. (2007) Optimal, Simple, and Implementable Monetary and Fiscal Rules, *Journal of Monetary Economics*, 54(6), pp. 1702-1725. Other version. Extended Version. matlab codes