# An Empirical Test on Harrod’s Open Economy Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Material and Mmethods

#### 3.1. Cycles

#### 3.2. USA Recessions

#### 3.3. Groundwork on Growth and Cycle Theories

#### 3.4. The Dataset

#### 3.4.1. World GDP Data

#### 3.4.2. BEA Data

## 4. A Mathematical Specification of the Harrod’s Model

- (A)
- The desired capital is an increasing function $\mathsf{\Phi}$ of the difference between the current and the expected change of demand i.e.,$${C}_{r}=\mathsf{\Phi}\left(\frac{\dot{Y}-{\dot{Y}}_{e}}{Y}\right)=\mathsf{\Phi}\left(G-{G}_{w}\right)$$$${C}_{r}=\mathsf{\Phi}\left(G-{G}_{w}\right)=\left[{C}^{*}+\phi (G-{G}_{w})\right]$$
- (B)
- According to Alexander [40], changes in the growth rate of income depends on the difference between ex ante and ex post investments, that is$$U={I}_{j}-I={C}_{t}{\dot{Y}}_{e}-I$$$$u=U/Y={I}_{j}/Y-I/Y={C}_{t}{G}_{w}-(\mathsf{\Sigma}-x)=\mathsf{\Phi}\left(G-{G}_{w}\right){G}_{w}-\mathsf{\Sigma}+x.$$Therefore $\dot{G}$ can be expressed as a function F of u with $F\u22da0$ if $u\u22da0$ and if we assume F to be linear we obtain$$\dot{G}=F\left(u\right)=F\left(\mathsf{\Phi}\left(G-{G}_{w}\right){G}_{w}+\mathsf{\Sigma}-x\right)=\alpha \left\{\left[{C}^{*}+\phi (G-{G}_{w})\right]{G}_{w}-\mathsf{\Sigma}+x\right\},$$
- (C)
- Saving rate varies over time depending on unforeseen differences between technical progress and rate of growth and on income fluctuations:$$\dot{\mathsf{\Sigma}}=\epsilon \left({G}_{n}-{G}_{w}\right)+\delta {\dot{G}}_{w},$$
- (D)
- Changes in the ratio of the trade balance depend on ${G}_{f}$, ${G}_{n}$ and G as follows$$\frac{\dot{x}}{x}=\mathsf{\Psi}({G}_{f},{G}_{n},G)\phantom{\rule{1.em}{0ex}}\mathrm{with}\phantom{\rule{1.em}{0ex}}\frac{\partial \mathsf{\Psi}}{\partial {G}_{f}}>0,\phantom{\rule{0.166667em}{0ex}}\frac{\partial \mathsf{\Psi}}{\partial {G}_{n}}>0\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\frac{\partial \mathsf{\Psi}}{\partial G}<0.$$As usual we can assume that the mapping $\mathsf{\Psi}$ is linear and by denoting the sensitivities $\zeta ,\sigma ,\mu >0$ of the balance of trade to foreign rate of growth, technical progress and domestic growth rate, Equation (2) can be rewritten as$$\frac{\dot{x}}{x}=\mathsf{\Psi}({G}_{f},{G}_{n},G)=\left(\zeta {G}_{f}+\sigma {G}_{n}-\mu G-m\right),$$
- (E)
- The expected rate of change of aggregate demand is linear with $\gamma $ to the difference between G and ${G}_{w}$ i.e.,$${\dot{G}}_{w}=\gamma (G-{G}_{w}),$$
- (F)
- The dynamic of technological progress is described by a continuous, increasing non-linear function of share of income saved and devoted to investments$${G}_{n}={G}_{n}\left(\mathsf{\Sigma}\right)=\beta (\xi -\mathsf{\Sigma})\mathsf{\Sigma},\phantom{\rule{1.em}{0ex}}\mathrm{with}\phantom{\rule{1.em}{0ex}}\beta >1\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}0<\xi <1.$$

## 5. Results and Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Supply side policy (i.e., a policy aimed at increasing the aggregate supply through investments, deregulation, liberalization, privatization, etc.) to raise the natural growth path. When $G={G}_{w}<{G}_{n}$ there is a permanent unemployment equilibrium. Policy-makers may employ supply side policies in order to increase both: the actual growth and G and the natural growth ${G}_{n}$.

**Figure 2.**The Harrod knife-edge or unstable equilibrium. When $G={G}_{n}={G}_{w}$ there is sustainable full employment. A departure from that may lead to recession (${G}^{\prime}$) or booming periods (${G}^{\prime}$).

**Figure 3.**The business cycle can be classified into four stages: (1) expansion when economic activity grows steadily; (2) boom when the aggregate demand grows more than the aggregate output which overheats the economy; (3) recession phase when the aggregate output cool down after a peak; (4) recovery after a through. The so-called “specific cycle amplitude” corresponds to the vertical distance between the peak and the trough.

**Figure 4.**USA real gross private domestic investment ( GPDIC1 ), billions of chained 2012 dollars, seasonally adjusted annual rate. Source: FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/GPDIC1, 16 March 2019. Greyed areas correspond to periods of economic recessions (Table 1).

**Figure 5.**Time series obtained with parameters of calibration 1, that displays convergence to the long-run equilibrium. Legend: blue = rate of growth, red = expected rate of growth, yellow = share of saved income, violet: trade to income ratio. Thick line: model, normal line: data.

**Figure 6.**Time series obtained with parameters of calibration 2, that displays divergence from the long-run equilibrium. Legend: blue = rate of growth, red = expected rate of growth, yellow = share of saved income, violet: trade to income ratio. Thick line: model, normal line: data.

**Figure 7.**Time series obtained with parameters of calibration 3, that displays lightly damped oscillatory behaviour around the long-run equilibrium. Legend: blue = rate of growth, red = expected rate of growth, yellow = share of saved income, violet: trade to income ratio. Thick line: model, normal line: data.

Recessions | ||||
---|---|---|---|---|

From | To | |||

Quarter | Year | Quarter | Year | |

Q4 | 1948 | Q4 | 1949 | |

Q3 | 1953 | Q1 | 1954 | |

Q4 | 1957 | Q1 | 1958 | |

Q3 | 1960 | Q1 | 1961 | |

Q1 | 1970 | Q4 | 1970 | |

Q1 | 1974 | Q2 | 1975 | |

Q1 | 1980 | Q2 | 1980 | |

Q3 | 1981 | Q4 | 1982 | |

Q3 | 1990 | Q1 | 1991 | |

Q2 | 2001 | Q4 | 2001 | |

Q1 | 2008 | Q3 | 2009 |

# | Time Series | Data Points | Frequency | Data Range (from to) | BEA Account Code |
---|---|---|---|---|---|

1 | USA SAVE | 287 | Quarterly | 1 January 1947 to 1 July 2018 | A929RC1tnote:BEA-1 |

2 | USA GPDIC1 | 287 | Quarterly | 1 April 1947 to 1 July 2018 | A006RXtnote:BEA-2 |

3 | USA NETEXP | 287 | Quarterly | 1 January 1947 to 1 July 2018 | A019RCtnote:BEA-3 |

4 | USA GDPDEF | 287 | Quarterly | 1 April 1947 to 1 July 2018 | A191RDtnote:BEA-4 |

5 | USA GPD | 287 | Quarterly | 1 January 1947 to 1 July 2018 | A191RCtnote:BEA-5 |

Given Model | Calibration | ||||
---|---|---|---|---|---|

Cal. 1 | Cal. 2 | Cal. 3 | |||

# | Parameter | Given Value/Range | Calibrated Value | ||

1 | $\alpha $ | $0.5$ | $0.28$ | $0.29$ | $1.09$ |

2 | $\u03f5$ | $[0.2,1.31]$ | $0.13$ | $0.58$ | $0.52$ |

3 | $\sigma $ | $[2,4)$ | $1.42$ | $1.67$ | $2.45$ |

4 | ${G}_{f}$ | $0.03$ | $0.03$ | $0.00$ | $0.54$ |

5 | ${C}^{*}$ | 4 | $4.00$ | $4.00$ | $3.18$ |

6 | $\beta $ | $2.5$ | $2.50$ | $2.50$ | $2.20$ |

7 | m | $0.07$ | $0.04$ | $0.04$ | $1.23$ |

8 | $\phi $ | 15 | $15.00$ | $15.00$ | $14.89$ |

9 | $\xi $ | $0.18$ | $0.18$ | $0.18$ | $0.20$ |

10 | $\mu $ | $1.4$ | $0.78$ | $0.90$ | $2.06$ |

11 | $\gamma $ | 1 | $0.56$ | $0.57$ | $0.36$ |

12 | $\delta $ | $6.2$ | $6.20$ | $6.20$ | $5.94$ |

13 | $\zeta $ | $1.9$ | $1.06$ | $1.09$ | $2.25$ |

Value of D | 0.38 | 0.71 | 0.55 |

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**MDPI and ACS Style**

Orlando, G.; Della Rossa, F.
An Empirical Test on Harrod’s Open Economy Dynamics. *Mathematics* **2019**, *7*, 524.
https://doi.org/10.3390/math7060524

**AMA Style**

Orlando G, Della Rossa F.
An Empirical Test on Harrod’s Open Economy Dynamics. *Mathematics*. 2019; 7(6):524.
https://doi.org/10.3390/math7060524

**Chicago/Turabian Style**

Orlando, Giuseppe, and Fabio Della Rossa.
2019. "An Empirical Test on Harrod’s Open Economy Dynamics" *Mathematics* 7, no. 6: 524.
https://doi.org/10.3390/math7060524