### Abstract

We solve an N ∈ N player general-sum differential game. The optimization problem considered here is based on the Uzawa-Lucas model of endogenous growth. Agents have logarithmic preferences and own two capital stocks. Since the number of players is an arbitrary fixed number N ∈ N, the model's solution is more general than the idealized concepts of the social planer's solution with one player or the competitive equilibrium with infinitely many players. We show that the symmetric Nash equilibrium is completely described by the solution to a single ordinary differential equation. The numerical results imply that the influence of the externality along the balanced growth path decreases rapidly as the number of players increases. Off the steady state, the externality is of great importance, even for a large number of players.

Original language | English |
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Pages (from-to) | 396-414 |

Number of pages | 19 |

Journal | Journal of Macroeconomics |

Volume | 30 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2008 Mar 1 |

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### Keywords

- Nash-equilibrium
- Open-loop strategies
- Ordinary differential equation
- Value function approach

### ASJC Scopus subject areas

- Economics and Econometrics

### Cite this

**The open-loop solution of the Uzawa-Lucas model of endogenous growth with N agents.** / Bethmann, Dirk.

Research output: Contribution to journal › Article

*Journal of Macroeconomics*, vol. 30, no. 1, pp. 396-414. https://doi.org/10.1016/j.jmacro.2006.09.004

}

TY - JOUR

T1 - The open-loop solution of the Uzawa-Lucas model of endogenous growth with N agents

AU - Bethmann, Dirk

PY - 2008/3/1

Y1 - 2008/3/1

N2 - We solve an N ∈ N player general-sum differential game. The optimization problem considered here is based on the Uzawa-Lucas model of endogenous growth. Agents have logarithmic preferences and own two capital stocks. Since the number of players is an arbitrary fixed number N ∈ N, the model's solution is more general than the idealized concepts of the social planer's solution with one player or the competitive equilibrium with infinitely many players. We show that the symmetric Nash equilibrium is completely described by the solution to a single ordinary differential equation. The numerical results imply that the influence of the externality along the balanced growth path decreases rapidly as the number of players increases. Off the steady state, the externality is of great importance, even for a large number of players.

AB - We solve an N ∈ N player general-sum differential game. The optimization problem considered here is based on the Uzawa-Lucas model of endogenous growth. Agents have logarithmic preferences and own two capital stocks. Since the number of players is an arbitrary fixed number N ∈ N, the model's solution is more general than the idealized concepts of the social planer's solution with one player or the competitive equilibrium with infinitely many players. We show that the symmetric Nash equilibrium is completely described by the solution to a single ordinary differential equation. The numerical results imply that the influence of the externality along the balanced growth path decreases rapidly as the number of players increases. Off the steady state, the externality is of great importance, even for a large number of players.

KW - Nash-equilibrium

KW - Open-loop strategies

KW - Ordinary differential equation

KW - Value function approach

UR - http://www.scopus.com/inward/record.url?scp=38949203874&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38949203874&partnerID=8YFLogxK

U2 - 10.1016/j.jmacro.2006.09.004

DO - 10.1016/j.jmacro.2006.09.004

M3 - Article

AN - SCOPUS:38949203874

VL - 30

SP - 396

EP - 414

JO - Journal of Macroeconomics

JF - Journal of Macroeconomics

SN - 0164-0704

IS - 1

ER -