# Money and Pay-As-You-Go Pension

## Abstract

**:**

## 1. Introduction

## 2. The Model

#### 2.1. Household

#### 2.2. Firms

#### 2.3. Government

## 3. Equilibrium

## 4. Society with Fewer Children

**Proposition**

**1.**

## 5. Monetary Policy

#### 5.1. Effect of Monetary Policy

#### 5.2. Monetary Policy and Welfare

**Proposition**

**2.**

## 6. Numerical Examples

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Derivation of Signs

#### Appendix A.2. Growth Rate in Balanced Growth Path

#### Appendix A.3. Another Setting of Pension Benefit

#### Appendix A.4. Life Expectancy

#### Appendix A.5. Social Welfare Function

## References

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1. | The analyses presented in this paper assume an inelastic labor supply and that the wage tax does not distort household allocations. |

2. | Assuming ${P}_{t}$ as the price level, ${P}_{t}{s}_{t}=(1-{\tau}_{t}){P}_{t}{w}_{t}-{P}_{t}{c}_{1t}-{P}_{t}{m}_{t}$ is obtained. This real budget constraint in the young period is derived by omitting the equation by ${P}_{t}$. |

3. | The nominal budget constraint is ${P}_{t+1}{c}_{2t+1}=(1+{i}_{t+1}){P}_{t}{s}_{t}+{P}_{t}{m}_{t}+{P}_{t+1}{Z}_{t+1}$. In addition, ${i}_{t+1}$ denotes the nominal interest rate. Considering $1+{r}_{t+1}=\frac{1+{i}_{t+1}}{1+{\pi}_{t+1}}$ (Fisher equation) and omitting ${P}_{t+1}$, the budget constraint in the older period is derived. |

4. | In some papers that have described assessments of pay-as-you-go pensions, the pension benefit of a retiree depends on the wage of current workers. This setting is the same as that used for the analyses presented in this paper. However, if the pension benefit of a retiree depends on the share of wages when they are working, then (14) changes to ${Z}_{t}=\u03f5{w}_{t-1}$. Therefore, considering the government budget constraint ${N}_{t}{\tau}_{t}{w}_{t}={N}_{t-1}\u03f5{w}_{t-1}$, (15) changes to $\tau =\frac{\u03f5}{(1+n)(1+g)}$. The Appendix A presents the case of $\tau =\frac{\u03f5}{(1+n)(1+g)}$. Nevertheless, the result does not change substantially. |

5. | In Japan, the contribution rate continues increasing to $18.3\%$ by 2017. (Data: Ministry of Health, Labour and Welfare, Japan) |

6. | |

7. | See Appendix A for a detail proof. |

8. | |

9. | The Appendix A presents examination of a case of an increase in life expectancy. This paper presents derivation that no substantial difference exists between a decrease in population growth and an increase in life expectancy. |

10. | See Appendix A for a detailed derivation. |

11. | Because of ${c}_{1t}=\frac{\alpha}{1+\rho}\left(\right)open="("\; close=")">\left(\right)open="("\; close=")">1-\frac{\u03f5}{1+n}$, $\frac{d{c}_{1t}}{dg}\frac{dg}{d\mu}=0$ are derived by $\u03f5=0$. |

12. | An increase in $\u03f5$ reduces the income growth rate $1+g$. See the Appendix A for a detailed proof. |

13. | Bhattacharya et al. (2009) examines the case of the Friedman rule that the real interest rate is equal to the inverse of the inflation rate. |

14. | In the other setting, the annuity is considered. |

15. | ${\delta}^{T}$ is regarded as zero because it is extremely small. |

$\alpha $ | 0.95 |

$\rho $ | 0.3 |

$\theta $ | 0.3 |

a | 15 |

$1+n$ | 1 |

$1+\mu $ | 1.8 |

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**MDPI and ACS Style**

Yasuoka, M.
Money and Pay-As-You-Go Pension. *Economies* **2018**, *6*, 21.
https://doi.org/10.3390/economies6020021

**AMA Style**

Yasuoka M.
Money and Pay-As-You-Go Pension. *Economies*. 2018; 6(2):21.
https://doi.org/10.3390/economies6020021

**Chicago/Turabian Style**

Yasuoka, Masaya.
2018. "Money and Pay-As-You-Go Pension" *Economies* 6, no. 2: 21.
https://doi.org/10.3390/economies6020021