An attempt to understand economic inequality from a non-abstract perspective
Zhengyuan Gao
Macro-Lunch @IRES 03/21/2017
This talk is about Section 6.1, 6.2 and 7 in “Uncertainty in Economic Growth and Inequality”. Slides and (codes) available: http://rpubs.com/larcenciel/uncertainty
An Old Story
“I can calculate the motion of heavenly bodies, but not the madness of people.” — Isaac Newton's complain on South Sea Bubble
Newton's (second) law
\[ Force = mass \times acceleration \]
or say \[ y=f(x_1,x_2)=x_1 \times x_2 \]
Newton's law in social science
\[ y = f(x_1,x_2,\dots,x_N) \]
What are \( \{x_i\}_{i\leq N} \)? What is \( f(\cdot) \)? How big is \( N \)?
An Attempt to Characterize the Madness
A probabilistic counterpart of Newton's law (by focusing on uncertainty).
An economic variable \( x \): \[ \mbox{individual } i \mbox{ at time } t \mbox{ has } X_t^i. \]
The change of \( X_t^i \) follows a probabilistic law \( \mathbb{P} \): \[ X_t^i \sim \mathbb{P}(x,t). \]
The (stationary) aggregate follows the same law \( \mathbb{P} \): \[ \sum_{i=1}^N X_t^i \sim \mathbb{P}\left(\sum_{i=1}^N x^i,t\right). \]
Infinite divisible law (de Finetti, 1929)
\[ Y_t = \sum_{i=1}^N X_t^i \overset{d}{=} X_t^1 + X_t^2 + \cdots + X_t^N \] or \[ \mathbb{P}(y,t) = \mathbb{P}(x^{1},t) \otimes \mathbb{P}(x^{2},t) \otimes \cdots \otimes \mathbb{P}(x^{N},t) \]
\[ \begin{array}{l} \mathbb{P}(x^{1},t)\quad\mbox{Adam's law}\\ \mathbb{P}(x^{1},t),\mathbb{P}(x^{2},t)\quad\mbox{Adam's and Eve's laws}\\ \vdots \quad \quad \quad \vdots \quad \quad \ddots\\ \mathbb{P}(x^{1},t),\mathbb{P}(x^{2},t),\ldots,\mathbb{P}(x^{N},t)\quad\mbox{Current individuals' laws} \end{array} \]
Growth and Inequality
Growth: increases of \( x \)
Inequality: differences amongst \( \{x^i\}_{i\leq N} \)
Consider economic growth and inequality as a man-made configuration under the \( \mathbb{P} \)-law.
A growth event happens in \( \mathbb{P}(x,t) \)-law.
\( \sum_{i=1}^N X_t^i \) is distributed in \( \mathbb{P}\left(\sum^N_{i=1} x^i,t \right) \)-law.
Deeper Parameters
Parameters of individual \( \mathbb{P} \)-law may be different from those in the aggregation.
Aggregatable (Heterogenous) parameters
\( \alpha_i \): Probability (unit of time) of a significant growth event for individual \( i \)
\( c(x) \): Size of a significant growth.
Non-aggregatable (Homogenous) parameters
\( \beta \): Probability (unit of time) of a trivial growth event for everyone.
\( 1/\beta \): Size of a trivial growth.
Parameterization
\( \mathbb{P} \)-law is identifiable dynamically in terms of \( (\alpha,\beta,c(x)) \).
\[ \frac{\partial}{\partial t}\mathbb{P}(x,t)=-c(x) \frac{\partial\mathbb{P}(x,t)}{\partial x}+\alpha\left[\int\left(\mathbb{W}(x|x')\mathbb{P}(x',t)\right)dx'-\mathbb{P}(x,t)\right] \]
\( \mathbb{W}(x|x') \) contains the parameter \( \beta \).
\[ \mbox{Equilibrium solution: } \mathbb{P}(x)=\frac{\beta^{\alpha}x^{\alpha-1}e^{-\beta x}}{\Gamma(\alpha)}\sim\mbox{Gamma}(\alpha,\beta) \]
\[ \mbox{Aggregation: } X_{t}^{i} +X_{t}^{j} \sim\mbox{Gamma}(\alpha_{i}+\alpha_{j},\beta) \]
\[ \mbox{Evolution: } C\times \sum_{i=1}^{N} X_{t}^i\sim\mbox{Gamma}(\sum_{i=1}^{N}\alpha_i,\beta/C) \]
Gamma Distribution
\( \mathbb{E}[X] = \frac{\alpha}{\beta} \) and \( \mbox{Var}[X] = \frac{\alpha}{\beta^2} \)
\( \alpha = (1,2,3,4,5), \beta = (0.1,0.2,0.3,0.4,0.5),\frac{\alpha}{\beta} = 10 \)
\( \alpha= 1 \)
\( \beta = (0.5,0.4,0.3,0.2,0.1) \)
\( \frac{\alpha}{\beta} = (2,2.5,3.3,5,10) \)
\( \alpha= (1,2,3,4,5) \)
\( \beta = 0.5 \)
\( \frac{\alpha}{\beta} = (2,4,6,8,10) \)
\( \alpha= (1,1.8,1.6,3.6,1.6) \)
\( \beta = (0.1,0.09,0.08,0.09,0.04) \)
\( \frac{\alpha}{\beta} = (10,20,20,40,40) \)
Non-stationary Aggregates
For a non-linear one-to-one function \( g(\cdot) \) \[ Y_{t}=g\left( \sum_{i=1}^{N}X_{t}^{i} \right) \] is the non-stationary aggregate \( Y_t \sim \mathbb{Q}(x,t). \)
Mean field of \( Y_t \): \( m_{Y}(t)=\int y\mathbb{Q}(y,t)dy. \)
Reduce form for this aggregation
\[ \begin{cases} \frac{dm_{Y}(t)}{dt} & = \mbox{Constant}+\mbox{Coef}_{1}m_{Y}(t)+\mbox{Coef}_{2}\sigma_{Y}^{2}(t) \\ \frac{d\sigma_{Y}^{2}(t)}{dt} & =\mbox{Constant}+\mbox{Coef}_{3}m_{Y}^{2}(t)+\mbox{Coef}_{4}\sigma_{Y}^{2}(t) \end{cases} \]
U.S. GDP 1994-2015
# Mean field equaiton
GDP.ar = arima(GDP.growth, order = c(1,0,0))
sigma2 = GDP.ar$resid^2
lagGDP.growth =lag(GDP.growth,1)
lagGDP.growth[is.na(lagGDP.growth)]=0
y = cbind(GDP.growth,lagGDP.growth,sigma2)
GDP.lm = lm(y[,1]~ y[,2] + y[,3])
coef(summary(GDP.lm))
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.869442e+02 3.519560e+01 5.311578 8.939631e-07
y[, 2] 2.013500e-01 1.002543e-01 2.008392 4.785333e-02
y[, 3] -8.284303e-04 1.897569e-04 -4.365746 3.628768e-05
Call:
garch(x = GDP.lm$residuals, trace = FALSE)
Model:
GARCH(1,1)
Residuals:
Min 1Q Median 3Q Max
-3.2722 -0.5816 -0.0272 0.4895 3.7041
Coefficient(s):
Estimate Std. Error t value Pr(>|t|)
a0 4.996e+04 NA NA NA
a1 1.006e-10 NA NA NA
b1 6.293e-02 NA NA NA
Diagnostic Tests:
Jarque Bera Test
data: Residuals
X-squared = 17.798, df = 2, p-value = 0.0001365
Box-Ljung test
data: Squared.Residuals
X-squared = 0.034938, df = 1, p-value = 0.8517
# Mean field volatility
sigma.dif = diff(sigma2)
sigma.dif[is.na(sigma.dif)]=0
s = cbind(sigma.dif, lagGDP.growth^2, sigma2)
s.lm = lm(s[,1]~s[,2]+s[,3])
coef(summary(s.lm))
Estimate Std. Error t value Pr(>|t|)
(Intercept) -5.969591e+04 1.894993e+04 -3.1501911 2.277507e-03
s[, 2] 9.324156e-02 9.812031e-02 0.9502779 3.447635e-01
s[, 3] 7.303218e-01 1.141915e-01 6.3955864 9.268704e-09
Structural Aggregates
High order information can be exogenized by extracting the high order effects of \( (X_t^1,\dots,X_t^N) \).
Structural form for this aggregation
\[ \begin{cases} m_{Y}(t) & =\mathbb{E}_{t}\left[g\left(\sum X_{t}^{i}\right)\mathbb{L}(t)\right]\\ \frac{dm_{Y}(t)}{dt} & =\theta_{t}\, m_{Y}(t)+e_{t} \\ X_{t} & \sim\mathbb{P}(x,t) \end{cases} \]
where \( \mathbb{L}(t) \) is the likelihood ratio for \( Y_t \) and \( X_t \); \( \theta_{t} \) and \( e_{t} \) are time varying constants.
Income Data (Variables)
Samples: People 15 years old and over
Measurement: Total money income
Source: U.S. Census Bureau, Current Population Survey
[1] "Year Total" "Total With Income" "$1 to $2,499 or Loss"
[4] "$2,500 to $4,999" "$5,000 to $7,499" "$7,500 to $9,999"
[7] "$10,000 to $12,499" "$12,500 to $14,999" "$15,000 to $17,499"
[10] "$17,500 to $19,999" "$20,000 to $22,499" "$22,500 to $24,999"
[13] "$25,000 to $27,499" "$27,500 to $29,999" "$30,000 to $32,499"
[16] "$32,500 to $34,999" "$35,000 to $37,499" "$37,500 to $39,999"
[19] "$40,000 to $42,499" "$42,500 to $44,999" "$45,000 to $47,499"
[22] "$47,500 to $49,999" "$50,000 to $52,499" "$52,500 to $54,999"
[25] "$55,000 to $57,499" "$57,500 to $59,999" "$60,000 to $62,499"
[28] "$62,500 to $64,999" "$65,000 to $67,499" "$67,500 to $69,999"
[31] "$70,000 to $72,499" "$72,500 to $74,999" "$75,000 to $77,499"
[34] "$77,500 to $79,999" "$80,000 to $82,499" "$82,500 to $84,999"
[37] "$85,000 to $87,499" "$87,500 to $89,999" "$90,000 to $92,499"
[40] "$92,500 to $94,999" "$95,000 to $97,499" "$97,500 to $99,999"
[43] "$100,000 and Over"
Income Data (1994 - 1996)
Numbers in thousands
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
[1,] 1994 186402 18964 13353 17474 13683 14639 10750 11500 8656 9812
[2,] 1995 188073 17946 12736 16638 13553 14389 10358 11855 8691 10383
[3,] 1996 189997 16525 11984 16897 13358 14350 10551 11030 8740 10276
[,12] [,13] [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22]
[1,] 6710 8106 5081 6748 3752 5283 2991 4487 2090 2625 1687
[2,] 6864 8197 5247 7375 3976 5410 3092 4624 2264 2970 1923
[3,] 7336 8784 5339 7664 4007 5677 3320 4898 2500 3091 2015
[,23] [,24] [,25] [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33]
[1,] 2744 1336 1547 1022 1589 653 821 514 748 449 699
[2,] 2871 1350 1813 990 1709 789 1100 586 894 417 832
[3,] 3265 1485 1734 971 1956 889 1181 632 936 523 787
[,34] [,35] [,36] [,37] [,38] [,39] [,40] [,41] [,42] [,43]
[1,] 357 542 214 298 298 208 338 179 195 128
[2,] 376 601 347 333 180 356 207 212 138 72
[3,] 444 689 313 479 288 369 237 266 166 97
Income Data (2013 - 2015)
Numbers in thousands
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
[20,] 2013 222003 14466 6764 9389 10931 12784 9529 11404 7966 10987 7346
[21,] 2014 222972 14724 6613 8314 10795 13085 8898 10694 8101 11315 6894
[22,] 2015 226762 14689 6262 7657 10551 12474 8995 10672 7931 11031 6962
[,13] [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23]
[20,] 9047 5501 9462 4424 6983 4289 7824 3258 5116 3245 5939
[21,] 9215 5640 9910 4234 7785 3941 8044 3042 5348 3113 6528
[22,] 9623 5535 10399 4429 7975 3930 8091 3113 5718 3221 7130
[,24] [,25] [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34]
[20,] 2249 4018 1984 4883 1559 2535 1316 3191 1215 2353 1051
[21,] 2467 3968 1944 4806 1707 3022 1423 3227 1299 2753 1046
[22,] 2489 3834 2066 5047 1894 3289 1493 3264 1372 2922 1307
[,35] [,36] [,37] [,38] [,39] [,40] [,41] [,42] [,43]
[20,] 2213 982 1490 992 1742 725 1071 672 19108
[21,] 2571 1001 1527 791 1738 683 976 723 19063
[22,] 2725 1021 1508 856 1966 712 1090 768 20755
Income Distribution in 2015
Parameters
Parameters
Endogeneity
\[ \mathbb{E}\left[\sum_{i=1}^{N}X_{t}(\omega^{i})\right] =\frac{\alpha_{t}}{\beta_{t}}=\mbox{Cons}+\mbox{Coef}_{1}\times\ln m_{Y}(t) \]
GDP.growth.rate = diff(log(GDP.sub))
reg.pagr = lm(parm~GDP.growth.rate)
Endogeneity
\[ \hat{\varepsilon}_t = \mbox{Intercept} + \mbox{Coef} \times \frac{\alpha_t}{\beta_t} \]
reg.endo = lm(reg.pagr$resid~parm)
summary(reg.endo)
Call:
lm(formula = reg.pagr$resid ~ parm)
Residuals:
Min 1Q Median 3Q Max
-1.64393 -0.06395 0.15498 0.33365 0.63271
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -8.99869 1.04297 -8.628 3.56e-08 ***
parm 0.79041 0.09101 8.685 3.20e-08 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.5587 on 20 degrees of freedom
Multiple R-squared: 0.7904, Adjusted R-squared: 0.7799
F-statistic: 75.42 on 1 and 20 DF, p-value: 3.204e-08
Structural Estimate
\[ \ln m_{Y}(t+1)-\ln m_{Y}(t) =\theta(t)+\varepsilon_{1,t}+\frac{\varepsilon_{2,t}}{\sum_{i=1}^{N}X_{t}(\omega^{i})}. \]
# After filtering, theta is available
resid.f = GDP.growth.rate - theta
reg.res = lm(resid.f~parm)
summary(reg.res)
Call:
lm(formula = resid.f ~ parm)
Residuals:
Min 1Q Median 3Q Max
-0.044721 -0.005027 0.004239 0.008245 0.016058
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.022986 0.027492 0.836 0.413
parm -0.002483 0.002399 -1.035 0.313
Residual standard error: 0.01473 on 20 degrees of freedom
Multiple R-squared: 0.05084, Adjusted R-squared: 0.00338
F-statistic: 1.071 on 1 and 20 DF, p-value: 0.313
Structural Estimate
Filtered Growth Rate (1994 - 2015)
Some Remarks
Look for a law that some conditions, so far as being observable, can invariably be found.
Growth, inequality, and some other phenomena may be integrable.
Fundamental parameters may not be affected by superficial policies.
“Conspiracy theory” of observable “truths”.
Finally, at least I understand more about inequality.
Modeling - Mr Palomar's approach
“A model is by definition that in which nothing has to be changed, that which works perfectly; whereas reality, as we see clearly, does not work and constantly falls to pieces; so we must force it, more or less roughly, to assume the form of the model.
…
In Mr Palomar’s life there was a period when his rule was this: first, to construct in his mind a model, the most perfect, logical, geometrical model possible; second, to see if the model adapted to the practical situations observed in experience; third, to make corrections necessary for model and reality to coincide.”
Italo Calvino, 'The model of models’ in Mr Palomar (1983)