### Abstract

In a stylised Robinson Crusoe economy, we illustrate basic dynamic programing techniques. In a first step, we define state-like and control-like variables. In a second step, we introduce the value-function-like function. While the former step reduces the number of variables that have to be considered when solving the model, the latter step reduces the dimensionality of the Bellman equation associated with the optimisation problem. The model's solution is shown to be saddle-path stable, such that the phase diagram associated with the Bellman equation has two solution branches. The simplicity of our model allows us to state both the stable and the unstable branch explicitly. We also explain the usefulness of logarithmic preferences when studying the continuous-time Hamilton-Jacobi-Bellman equation. In this case, the utility maximisation problem can be transformed into an initial value problem for an ordinary differential equation.

Original language | English |
---|---|

Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Australian Economic Papers |

Volume | 52 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 Mar 1 |

### Fingerprint

### ASJC Scopus subject areas

- Economics, Econometrics and Finance(all)

### Cite this

**Solving macroeconomic models with homogeneous technology and logarithmic preferences.** / Bethmann, Dirk.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Solving macroeconomic models with homogeneous technology and logarithmic preferences

AU - Bethmann, Dirk

PY - 2013/3/1

Y1 - 2013/3/1

N2 - In a stylised Robinson Crusoe economy, we illustrate basic dynamic programing techniques. In a first step, we define state-like and control-like variables. In a second step, we introduce the value-function-like function. While the former step reduces the number of variables that have to be considered when solving the model, the latter step reduces the dimensionality of the Bellman equation associated with the optimisation problem. The model's solution is shown to be saddle-path stable, such that the phase diagram associated with the Bellman equation has two solution branches. The simplicity of our model allows us to state both the stable and the unstable branch explicitly. We also explain the usefulness of logarithmic preferences when studying the continuous-time Hamilton-Jacobi-Bellman equation. In this case, the utility maximisation problem can be transformed into an initial value problem for an ordinary differential equation.

AB - In a stylised Robinson Crusoe economy, we illustrate basic dynamic programing techniques. In a first step, we define state-like and control-like variables. In a second step, we introduce the value-function-like function. While the former step reduces the number of variables that have to be considered when solving the model, the latter step reduces the dimensionality of the Bellman equation associated with the optimisation problem. The model's solution is shown to be saddle-path stable, such that the phase diagram associated with the Bellman equation has two solution branches. The simplicity of our model allows us to state both the stable and the unstable branch explicitly. We also explain the usefulness of logarithmic preferences when studying the continuous-time Hamilton-Jacobi-Bellman equation. In this case, the utility maximisation problem can be transformed into an initial value problem for an ordinary differential equation.

UR - http://www.scopus.com/inward/record.url?scp=84879017174&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84879017174&partnerID=8YFLogxK

U2 - 10.1111/1467-8454.12004

DO - 10.1111/1467-8454.12004

M3 - Article

AN - SCOPUS:84879017174

VL - 52

SP - 1

EP - 18

JO - Australian Economic Papers

JF - Australian Economic Papers

SN - 0004-900X

IS - 1

ER -