# A Quantal Response Statistical Equilibrium Model of Induced Technical Change in an Interactive Factor Market: Firm-Level Evidence in the EU Economies

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Patterns of Technical Change

#### 2.1. Data

#### 2.2. Empirical Distributions of $\gamma $, $\chi $ and, $\zeta $

## 3. A Statistical Equilibrium Model of the ITC

#### 3.1. The Impact of the Cost Reduction on Adoption of a New Technology

#### 3.2. The Impact of the Adoption of a New Technology on the Rate of Cost Reduction

#### 3.3. Maximum Entropy Program of the Quantal Response ITC Model

## 4. Bayesian Estimation of the Model

#### 4.1. Model Specification

#### 4.2. Result

#### Parameter Estimation

#### Comparison of Prior and Posterior Distribution: UK 2011

#### Predicted $\tilde{p}(\zeta )$, $\tilde{p}(A|\zeta )$, and $\tilde{p}(\zeta |A)$: UK 2011

#### 4.3. Discussion of the Estimated Parameters, $\mu ,T,\beta $ and $\kappa $

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Summary Statistics of the Frequency Distributions of γ, χ, and ζ

**Table A1.**The mean of $\gamma ,\text{}\chi ,\text{}\widehat{k}$ and $\zeta $ of 7 different countries averaged from 2006–2013.

Country | UK | GER | FRA | ITA | SPA | POR | SWE |
---|---|---|---|---|---|---|---|

$\gamma $ (%) | 0.98 | 0.43 | 1.15 | −0.80 | −1.02 | 0.65 | 1.10 |

$\chi $ (%) | −6.22 | −1.13 | −1.89 | −3.38 | −6.57 | −4.43 | −2.12 |

$\widehat{k}$ (%) | 7.20 | 1.56 | 3.04 | 2.58 | 5.55 | 5.08 | 3.22 |

$\zeta $ (%) | −2.10 | −0.08 | 0.33 | −1.49 | −2.46 | −0.91 | −0.24 |

## Appendix B. Soofi ID Index

**Table A2.**The Soofi ID index of all the fitted distributions of $\tilde{p}[\zeta ]$ for all 7 countries over 8 years.

Year | UK | GER | FRA | ITA | SPA | POR | SWE |
---|---|---|---|---|---|---|---|

2006 | 0.006 | 0.069 | 0.043 | 0.006 | 0.010 | 0.010 | 0.006 |

2007 | 0.020 | 0.056 | 0.060 | 0.008 | 0.010 | 0.007 | 0.011 |

2008 | 0.061 | 0.061 | 0.029 | 0.025 | 0.010 | 0.009 | 0.015 |

2009 | 0.007 | 0.050 | 0.039 | 0.020 | 0.003 | 0.012 | 0.008 |

2010 | 0.007 | 0.027 | 0.035 | 0.008 | 0.013 | 0.018 | 0.015 |

2011 | 0.004 | 0.014 | 0.019 | 0.004 | 0.003 | 0.007 | 0.007 |

2012 | 0.008 | 0.017 | 0.018 | 0.003 | 0.002 | 0.009 | 0.004 |

2013 | 0.005 | 0.015 | 0.027 | 0.004 | 0.005 | 0.008 | 0.008 |

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**Figure 1.**The frequency distribution of the growth rate of labor and capital productivity, $\gamma ,\chi $ and the rate of cost reduction $\zeta $ of the UK from 2006–2013 with log scale on the vertical axis. The histograms are centered for each year. Both distributions exhibit a peaked tent shape. Data Source: ORIBS-AMADEUS.

**Figure 2.**The frequency distribution of the growth rate of labor and capital productivity, $\gamma $ and $\chi $, with log scale on the vertical axis. The histograms are centered for each country and each year. Both distributions exhibit a peaked tent shape.

**Figure 3.**The frequency distribution of the growth rate of cost reduction $\zeta $ with log scale on the vertical axis. The histograms are centered for each country and each year. The distribution exhibits a peaked tent shape.

**Figure 4.**A quantal response function with the behavior temperature T and $\mu =0$. The horizontal axis represents the the rate of cost reduction, $\zeta $, while the vertical axis represents the frequency of the adoption of a technology.

**Figure 5.**An innovation possibilities frontier. The bold line represents the innovation possibilities frontier (IPF), a functional relationship of the trade-off between $\chi $ and $\gamma $. The dotted line represents a tangent line of IPF at $\chi ={\chi}^{*}$. The tangent is the maximum rate of cost reduction given the unit labor and capital cost $\omega ={\omega}^{*}$ and $\pi ={\pi}^{*}$.

**Figure 6.**QRSE distributions with different parameter values of $\mu ,T,\beta $, and $\kappa $. $\mu $ determines the location of the distribution, while $\kappa $ determines its skewness. $\kappa $ close to zero represents a symmetric QRSE distribution. Predictably, T and $\beta $ determine how disperse the distribution is. The lower T and higher $\beta $ are, the more spread out the distribution becomes.

**Figure 7.**Estimated parameters $\mu ,T,\beta $, and $\kappa $ along with the 95% credible interval. The year index represents the beginning year from which the growth rate of $\zeta $ is calculated. For example, the growth rate in year 2006 is the growth of $\zeta $ between 2006–2007.

**Figure 9.**The recovered distribution of $\tilde{p}(\zeta )$, $\tilde{p}(A|\zeta )$, and $\tilde{p}(\zeta |A)$. Distributions are recovered using the mean value of estimated parameters along with 95% credible interval expressed as the scattered gray points.

**Figure 10.**The difference between the recovered $\mu $ and the observed mean of $\widehat{\zeta}$, $\mu -\widehat{\zeta}$.

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**MDPI and ACS Style**

Yang, J. A Quantal Response Statistical Equilibrium Model of Induced Technical Change in an Interactive Factor Market: Firm-Level Evidence in the EU Economies. *Entropy* **2018**, *20*, 156.
https://doi.org/10.3390/e20030156

**AMA Style**

Yang J. A Quantal Response Statistical Equilibrium Model of Induced Technical Change in an Interactive Factor Market: Firm-Level Evidence in the EU Economies. *Entropy*. 2018; 20(3):156.
https://doi.org/10.3390/e20030156

**Chicago/Turabian Style**

Yang, Jangho. 2018. "A Quantal Response Statistical Equilibrium Model of Induced Technical Change in an Interactive Factor Market: Firm-Level Evidence in the EU Economies" *Entropy* 20, no. 3: 156.
https://doi.org/10.3390/e20030156