# 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 |

## References

- Jaynes, E.T. Where Do We Stand on Maximum Entropy? In The Maximum Entropy Formalism; Levine, R.D., Tribus, M., Eds.; MIT Press: Cambridge, MA, USA, 1978; pp. 15–118. [Google Scholar]
- Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Farjoun, E.; Machover, M. Laws of Chaos: Probabilistic Approach to Political Economy; Verso Books: New York, NY, USA, 1983. [Google Scholar]
- Foley, D.K. A Statistical Equilibrium Theory of Market. J. Econ. Theory
**1994**, 62, 321–345. [Google Scholar] [CrossRef] - Stutzer, M.J. A Bayesian Approach to Diagnostic of Asset Pricing Models. J. Econ.
**1995**, 68, 369–397. [Google Scholar] [CrossRef] - Stutzer, M.J. Simple Entropic Derivation of a Generalized Black-Scholes Option Pricing Model. Entropy
**2010**, 2, 70–77. [Google Scholar] [CrossRef] - Sims, C.A. Implications of rational inattention. J. Monetary Econ.
**2003**, 50, 665–690. [Google Scholar] [CrossRef] - Smith, E.; Foley, D.K. Classical thermodynamics and economic general equilibrium theory. J. Econ. Dyn. Control
**2008**, 32, 7–65. [Google Scholar] [CrossRef] - Toda, A.A. Existence of a statistical equilibrium for an economy with endogenous offer sets. Econ. Theory
**2010**, 45, 379–415. [Google Scholar] [CrossRef] - Toda, A.A. Bayesian general equilibrium. Econ. Theory
**2015**, 58, 375–411. [Google Scholar] [CrossRef] - Lux, T. Applications of statistical physics in finance and economics. In Handbook on Complexity Research; Rosser, J.B., Ed.; Edward Elgar: Cheltenham, UK, 2009; pp. 213–258. [Google Scholar]
- Zhou, R.; Cai, R.; Tong, G. Applications of Entropy in Finance: A Review. Entropy
**2013**, 15, 4909–4931. [Google Scholar] [CrossRef] - Rosser, J.B. Entropy and econophysics. Eur. Phys. J. Spec. Top.
**2016**, 225, 3091–3104. [Google Scholar] [CrossRef] - Yang, J. Information theoretic approaches in economics. J. Econ. Surv.
**2017**. [Google Scholar] [CrossRef] - Scharfenaker, E.; Foley, D.K. Quantal Response Statistical Equilibrium in Economic Interactions: Theory and Estimation. Entropy
**2017**, 19, 444. [Google Scholar] [CrossRef] - Michl, T.; Foley, D.K. Growth and Distribution; Harvard University Press: London, UK, 1999. [Google Scholar]
- Alfarano, S.; Milaković, M. Does Classical Competition Explain the Statistical Features of Firm Growth? Econ. Lett.
**2008**, 101, 272–274. [Google Scholar] [CrossRef] - Alfarano, S.; Milaković, M.; Irle, A.; Kauschke, J. A Statistical Equilibrium Model of Competitive Firms. J. Econ. Dyn. Control
**2012**, 36, 136–149. [Google Scholar] [CrossRef] - Jaynes, E.T. Probability Theory: The Logic of Science; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Kapur, J.N. Maximum-Entropy Models in Science and Engineering; John Wiley & Sons: Hoboken, NJ, USA, 1998. [Google Scholar]
- Von Weisäcker, C.C. A New Technical Progress Function (1962). Ger. Econ. Rev.
**2010**, 11, 248–265. [Google Scholar] [CrossRef] - Kennedy, C. Induced Bias in Innovation and the Theory of Distribution. Econ. J.
**1964**, 74, 541–547. [Google Scholar] [CrossRef] - Samuelson, P.A. A Theory of Induced Innovation along Kennedy-Weisäcker Lines. Rev. Econ. Stat.
**1965**, 47, 343–356. [Google Scholar] [CrossRef] - Drandakis, E.M.; Phelps, E.S. A Model of Induced Invention, Growth and Distribution. Econ. J.
**1966**, 76, 823–840. [Google Scholar] [CrossRef] - Foley, D. Endogenous Technical Change with Externalities in a Classical Growth model. J. Econ. Behav. Organ.
**2003**, 52, 167–189. [Google Scholar] [CrossRef] - McFadden, D.L. Quantal Choice Analaysis: A Survey. Ann. Econ. Soc. Meas.
**1976**, 5, 363–390. [Google Scholar] - McFadden, D.L. Economic choices. Am. Econ. Rev.
**1976**, 91, 351–378. [Google Scholar] [CrossRef] - Train, K. Discrete Choice Methods with Simulation; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- McKelvey, R.D.; Palfrey, T.R. Quantal Response Equilibria for Normal Form Games. Games Econ. Behav.
**1995**, 10, 6–38. [Google Scholar] [CrossRef] - Chen, H.C.; Friedman, J.W.; Thisse, J.F. Boundedly Rational Nash Equilibrium: A Probabilistic Choice Approach. Games Econ. Behav.
**1997**, 18, 32–54. [Google Scholar] [CrossRef] - McKelvey, R.D.; Palfrey, T.R. Quantal Response Equilibria for Extensive Form Games. Exp. Econ.
**1998**, 1, 9–41. [Google Scholar] [CrossRef] - Wolpert, D. Information Theory—The Bridge Connecting Bounded Rational Game Theory and Statistical Physics. In Complex Engineered Systems; Braha, D., Minai, A., Bar-Yam, Y., Eds.; Springer: Berlin, Heidelberg, 2006; pp. 262–290. [Google Scholar]
- Dos Santos, P.L. The Principle of Social Scaling. Complexity
**2017**, 2017, 8358909. [Google Scholar] [CrossRef] - Kullback, S.; Leibler, R. On information and sufficiency. Ann. Math. Stat.
**1951**, 22, 79–86. [Google Scholar] [CrossRef] - Gamerman, D.; Lopes, H.F. Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2008. [Google Scholar]
- Minh, D.D.L.; Minh, D.L. Understanding the Hastings Algorithm. Commun. Stat. Simul. Comput.
**2015**, 44, 332–349. [Google Scholar] [CrossRef] - Gelman, A.; Carlin, J.B.; Stern, H.S.; Dunson, D.B.; Vehtari, A.; Rubin, D.B. Bayesian Data Analysis, 3rd ed.; CRC Press: Boca Raton, FL, USA, 2013. [Google Scholar]
- Golan, A. Information and Entropy Econometrics—Editor’s View. J. Econ.
**2002**, 107, 1–15. [Google Scholar] [CrossRef] - Golan, A.; Judge, G.G.; Miller, D. Maximum Entropy Econometrics: Robust Estimation with Limited Data; Wiley: Hoboken, NJ, USA, 1996. [Google Scholar]
- Soofi, E.S.; Ebrahimi, N.; Habibullah, M. Information Distinguishability with Application to Analysis of Failure Data. J. Am. Stat. Assoc.
**1995**, 90, 657–668. [Google Scholar] [CrossRef] - Soofi, E.; Retzer, J. Information indices: Unification and applications. J. Econ.
**2002**, 107, 17–40. [Google Scholar] [CrossRef]

**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