# Quantal Response Statistical Equilibrium in Economic Interactions: Theory and Estimation

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## Abstract

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## 1. Introduction

The whole of the advantages and disadvantages of the different employments of labour and stock must, in the same neighbourhood, be either perfectly equal or continually tending to equality. If in the same neighbourhood, there was any employment evidently either more or less advantageous than the rest, so many people would crowd into it in the one case, and so many would desert it in the other, that its advantages would soon return to the level of other employments. This at least would be the case in a society where things were left to follow their natural course, where there was perfect liberty, and where every man was perfectly free both to choose what occupation he thought proper, and to change it as often as he thought proper. Every man’s interest would prompt him to seek the advantageous, and to shun the disadvantageous employment.(Smith, 1937, Book I, ch 10)

When the stocks of many rich merchants are turned into the same trade, their mutual competition naturally tends to lower its profit...(Smith, 1937, Book I, ch 9)

## 2. A Quantal Response Statistical Equilibrium Model Example: Adam Smith’s Theory

#### 2.1. Logit Quantal Response

#### 2.2. Impact of Actions on Outcomes

#### 2.3. Smithian Models of Competition

#### 2.4. Maximum Entropy and Statistical Equilibrium

#### 2.5. Statistical Equilibrium in Smithian Competition

#### 2.6. QRSE and Quantal Response Equilibrium

## 3. Application to Profit Rate Data

#### Model Inference

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Logit quantal response curve for $f\left[\mathrm{enter}\right|x]$ (

**a**) and $f\left[\mathrm{exit}\right|x]$ (

**b**) for various value of T and $\mu =0$.

**Figure 2.**(

**a**) Marginal frequency distributions of profit rates for U.S. publicly listed firms on a log probability scale, 1975; (

**b**) Average rate of profit 1963–2014.

**Figure 3.**Observed outcome frequency distribution $\overline{f}\left[x\right]$ and the maximum entropy predicted marginal outcome frequencies $\widehat{f}\left[x\right]$.

**Figure 4.**(

**a**) Maximum entropy conditional outcomes with observed marginal outcome frequencies $\overline{f}\left[x\right]$; (

**b**) Estimated conditional action frequencies.

**Figure 5.**Posterior distributions for T, $\mu $, $\beta $, and $\gamma $ with highest posterior estimates and $95\%$ highest posterior density (HPD) regions.

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**MDPI and ACS Style**

Scharfenaker, E.; Foley, D.K.
Quantal Response Statistical Equilibrium in Economic Interactions: Theory and Estimation. *Entropy* **2017**, *19*, 444.
https://doi.org/10.3390/e19090444

**AMA Style**

Scharfenaker E, Foley DK.
Quantal Response Statistical Equilibrium in Economic Interactions: Theory and Estimation. *Entropy*. 2017; 19(9):444.
https://doi.org/10.3390/e19090444

**Chicago/Turabian Style**

Scharfenaker, Ellis, and Duncan K. Foley.
2017. "Quantal Response Statistical Equilibrium in Economic Interactions: Theory and Estimation" *Entropy* 19, no. 9: 444.
https://doi.org/10.3390/e19090444