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Power laws, CEO compensation and inequality
Economics Letters 126 (2015) 78–80
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Power laws, CEO compensation and inequality Calvin Blackwell a,∗ , Rachel Graefe-Anderson b , Frank Hefner a , Dyanne Vaught c a
Department of Economics, College of Charleston, Charleston, SC 29407, United States
b
College of Business, University of Mary Washington, Fredricksburg, VA 22401, United States
c
Research Department, The Federal Reserve Bank of Boston, Boston, MA 02210, United States
highlights • We fit power law distributions to the distributions of overall US income and CEO compensation. • We test the hypothesis that inequality in CEO compensation mirrors inequality in overall income. • We find evidence that CEO compensation inequality is correlated with US income inequality.
article
info
Article history: Received 1 September 2014 Received in revised form 7 November 2014 Accepted 12 November 2014 Available online 24 November 2014 JEL classification: D31 J30
abstract We observe that CEO compensation and top incomes in the US have both been increasing rapidly over the last thirty years. We hypothesize that the trends in CEO compensation have been caused by the same economy-wide factors that have contributed to increases in income. We test this hypothesis by using ExecuComp and IRS tax data to estimate power law distributions and compare the behavior of these distributions over time. Using linear regression techniques, we estimate a power law distribution for CEO compensation and individual income. We find that the parameters of income distribution and the distribution of CEO compensation are correlated. © 2014 Elsevier B.V. All rights reserved.
Keywords: Inequality CEO compensation Power law distribution Pareto distribution
1. Introduction Between 1970 and 2008 pay for the average US chief executive officer (CEO) increased from $850,000 to $10.5 million (Faulkender et al., 2010). This 1135% increase in pay was matched by a relatively small 568% increase in average wages, calculated using the National Average Wage Index. Sharply increasing CEO salaries have become a pressing social concern, especially in the face of stagnating wages in a struggling economy. Many authors propose explanations for this increasing pay within the structure of the market for CEOs. Compensation structure including stock options and incentive-based pay, increased opportunities for profit skimming, changing demand for various skill sets, and growing firm sizes provide potential explanations
∗
Corresponding author. Tel.: +1 843 953 7836; fax: +1 843 953 0754. E-mail address: [email protected] (C. Blackwell).
http://dx.doi.org/10.1016/j.econlet.2014.11.012 0165-1765/© 2014 Elsevier B.V. All rights reserved.
for the increase in CEO compensation. However, we propose another explanation for the increase in CEO pay; it has not been a unique process that has increased executive compensation but the same general process by which the overall distribution of income has become more unequal. We pose the hypothesis that the distribution of CEO compensation follows trends in the distribution of income among the richest Americans. To test this hypothesis, we estimate power law distributions for CEO compensation over time. Power law distributions, also called Pareto distributions, are fat-tailed distributions commonly used to describe income and wealth. We find that changes in inequality in CEO compensation do seem to mirror changes in income. 2. Power law distributions Discovered as a representation of income distribution in nineteenth century Switzerland, the Pareto distribution has been used to describe the behavior of top incomes in many societies. A
C. Blackwell et al. / Economics Letters 126 (2015) 78–80
Pareto distribution’s density function is of the form:
f (x) =
ζ ζ xmin ζ +1
x 0,
,
x > xmin
separated into bins by income range, for each year from 1993 to 2009. (1)
1− ζ1
L(u) = 1 − (1 − u)
,
0
(2)
where u is the percentile in the income distribution. The Gini coefficient, an index of inequality, is the measure between a line of perfect equality and the observed Lorenz curve. The equation for the Gini coefficient of a power law distribution follows: G=
2ζ − 1
5. Methodology
x ≤ xmin
where greater exponent ζ values imply a distribution with a fatter tail and xmin is some lower bound for the power law distribution. Regarding the distribution of income in the US, Silva and Yakovenko (2005) show that the right tail of the distribution can be well-represented with a power law. Atkinson and Piketty (2007) estimate that for top incomes in the US, the power law exponent is approximately 1.7. The Lorenz curve is useful for understanding the amount of inequality in a given distribution. The equation for the Lorenz curve for a power law distribution is:
1
79
.
(3)
The power law exponent, ζ , can then be interpreted as a measure of inequality (Kleiber and Kotz, 2003). As can be seen from (3), the larger ζ , the smaller is G, indicating less inequality. 3. Hypothesis Four explanations of the sudden increase in pay beginning in the 1980s prevail within academic research. The theories presented offer compensation structure, opportunities for profit skimming, skill types, and firm size as possible explanations for the recent changes in CEO compensation (Gabaix and Landier, 2008; Frydman and Saks, 2010). We offer a fifth hypothesis; the trend in inequality in CEO compensation follows the trend in inequality in the distribution of income of top earners. If our hypothesis is correct, then the shape of the distribution of CEO compensation will be correlated with the shape of the distribution of income. We use the exponent of a power law distribution, ζ , (which gives information about the spread of the data) as our ‘‘shape’’ parameter. Estimating this ζ value over time for a given data set will show the trends in inequality for that data. We hypothesize that there will be a correlation between the exponent of the power law distribution of CEO compensation and the exponent for the income distribution. 4. Data To measure executive compensation, we use Standard & Poor’s ExecuComp database, which provides information on executive compensation for firms in the S&P 500, the Midcap 400, and the Smallcap 600, between 1993 and 2009. The ExecuComp data comes directly from publicly traded companies’ annual reports and includes CEOs’ salary and total compensation including restricted stock, payouts from long-term plans, benefits, and stock options valued at the grant-date using ExecuComp’s modified Black–Scholes methodology. As a measure of income, we use tax return data provided by the Internal Revenue Service’s (IRS) Tax Statistics. Using individual income tax returns, we use the size of adjusted gross income,
A standard way to look at the power law distribution is via its counter cumulative density function (P (x)) shown by Eq. (4): 1 − F (x) = P (X > x) = P (x) =
x xmin
−ζ
,
x > xmin .
(4)
The most intuitive way to recognize a power law distribution is to graph the observed data in some form of histogram. When data from a power law is graphed on log(x) and log(y) axes (where xi is the observed value and yi is the number of observations that are less than xi ), the points will form, roughly, a straight line. This visual relationship is confirmation of the mathematical fact that if we take logarithms of both sides of Eq. (4) we find: ln(P (x)) = ζ ln(xmin ) − ζ ln(x).
(5)
To make Eq. (5) an econometric model, we add a normally distributed error term, ϵ , to get Eq. (6). ln(P (x)) = β0 − ζ ln(x) + ϵ.
(6)
Eq. (6) can be estimated using linear regression, where ζ is the slope of the regression line, x is the bin value, and P (x) is the cumulative frequency. Depending on the format of the data, the rank or the cumulative frequency can be used as dependent variables. Since the IRS tax data is presented in bins with frequencies, we use the cumulative frequency approach. We use the top two bins reported by the IRS: incomes above $500,000 and incomes above $1,000,000. We do not use any smaller bins for two reasons: (1) few CEO’s are paid less that $500,000 and (2) the Pareto distribution is most appropriately used for incomes in the top 10%. (Our practice of using only the top two bins is consistent with other researchers; for example Atkinson et al. (2011) estimate ζ using data from the top 10th and 5th percentiles of the income distribution. Furthermore, Dragulescu and Yakovenko (2001) and Silva and Yakovenko (2005) suggest the Pareto distribution may be appropriate for the top 5% or less.) After obtaining an exponent value, ζ , for each year of CEO compensation and income through least-squares regression, we implement linear regression techniques to compare the trends.
ζCEO = φ0 + φ1 ζincome + ωt
(7)
where ωt is a normally distributed random error term. This approach will quantify the strength of the relationship between the distribution of CEO compensation and the distribution of income. 6. Results Using ordinary least squares regression on the log-cumulative frequency and log-bin value, we estimate equations of the form of Eq. (6) for CEO compensation and income. These regression equations yield ζ for both the distribution of income and the distribution of CEO compensation. The results of these regressions are available from the authors. We then take the estimated ζ values and estimate Eq. (7). We see the following trends in exponent values in Fig. 1. When we estimate Eq. (7) using OLS, we find that 55% of the variation in the ζ values for CEO compensation can be explained by the ζ values for income (see Table 1). A relatively low p-value suggests that we can reject the hypothesis that there is no relationship between the two series. The trends in CEO compensation and income appear to be very similar.
80
C. Blackwell et al. / Economics Letters 126 (2015) 78–80
a data set with a greater number of smaller bins would allow us to make more accurate conclusions about the distribution. A commonly accepted method of measuring inequality is by calculating the Gini coefficient. Comparing the Gini coefficient over time for each series is one alternative to comparing Pareto exponents to measure trends in inequality. This more widely used technique could provide more insight into the trends in inequality in CEO compensation and in the economy as a whole. One strength of using ζ as a measure of inequality instead of the Gini coefficient or other measures is that the power law exponent allows us to focus on the tail of the distribution. While the Gini coefficient would measure the overall inequality in the distribution of income or CEO compensation, ζ measures inequality only in the data above xmin . We are able to focus on the most relevant data by using the power law exponent. References
Fig. 1. Estimated ζCEO and ζIncome for 1993–2009. Table 1 Regression results for CEO compensation and income distribution for 1993–2009. Predictor
Coefficient
Standard error
p-value
Constant
−2.700
0.966 0.636
0.014 0.000
ζIncome 2
R
2.712 0.548
7. Discussion There is evidence to suggest that the distributions of income and the distribution of CEO compensation move together. Like all research, the methodology and data used in this paper have some limitations. For example, it is important to consider that members of the CEO sample are included in the general public; further investigation is recommended to ensure that the correlation between the trends in CEO compensation and income is not exaggerated due to this overlap. Income data was available to us only in the form of frequencies within wide income ranges. The use of linear regression poses some potential problems (see Clauset et al., 2009 for a discussion). Though our estimated ζ values are similar to those estimated in previous work (Atkinson et al., 2011),
Atkinson, A.B., Piketty, Thomas, 2007. Top Incomes over the Twentieth Century: A Contrast between European and English-Speaking Countries. Oxford University Press, Oxford, New York. Atkinson, Anthony B., Piketty, Thomas, Saez, Emmanuel, 2011. Top incomes in the long run of history. J. Econ. Literature 49 (1), 3–71. http://dx.doi.org/10.1257/jel.49.1.3. Clauset, Aaron, Shalizi, Cosma Rohilla, Newman, M.E.J., 2009. Power-law distributions in empirical data. SIAM Rev. 51 (4), 661–703. http://dx.doi.org/10.1137/070710111. Dragulescu, Adrian, Yakovenko, Victor M., 2001. Exponential and power-law probability distributions of wealth and income in the united kingdom and the united states. Physica A 299 (1-2), 213–221. http://dx.doi.org/10.1016/S03784371(01)00298-9. Faulkender, Michael, Kadyrzhanova, Dalida, Prabhala, N., Senbet, Lemma, 2010. Executive compensation: an overview of research on corporate practices and proposed reforms. J. Appl. Corp. Finance 22 (1), 107–118. http://dx.doi.org/10.1111/j.1745-6622.2010.00266.x. Frydman, Carola, Saks, Raven E., 2010. Executive compensation: a new view from a long-term perspective, 1936–2005. Rev. Financ. Stud. 23 (5), 2099–2138. http://dx.doi.org/10.1093/rfs/hhp120. Gabaix, Xavier, Landier, Augustin, 2008. Why has CEO pay increased so much? Quart. J. Econ. 123 (1), 49–100. http://dx.doi.org/10.1162/qjec.2008.123.1.49. Kleiber, Christian, Kotz, Samuel, 2003. Statistical Size Distributions in Economics and Actuarial Sciences. John Wiley & Sons, October. Silva, A. Christian, Yakovenko, Victor M., 2005. Temporal evolution of the ‘‘thermal’’ and ‘‘superthermal’’ income classes in the USA during 1983-2001. Europhys. Lett. 69 (2), 304–310. http://dx.doi.org/10.1209/epl/i2004-10330-3.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Power laws, CEO compensation and inequality Calvin Blackwell a,∗ , Rachel Graefe-Anderson b , Frank Hefner a , Dyanne Vaught c a
Department of Economics, College of Charleston, Charleston, SC 29407, United States
b
College of Business, University of Mary Washington, Fredricksburg, VA 22401, United States
c
Research Department, The Federal Reserve Bank of Boston, Boston, MA 02210, United States
highlights • We fit power law distributions to the distributions of overall US income and CEO compensation. • We test the hypothesis that inequality in CEO compensation mirrors inequality in overall income. • We find evidence that CEO compensation inequality is correlated with US income inequality.
article
info
Article history: Received 1 September 2014 Received in revised form 7 November 2014 Accepted 12 November 2014 Available online 24 November 2014 JEL classification: D31 J30
abstract We observe that CEO compensation and top incomes in the US have both been increasing rapidly over the last thirty years. We hypothesize that the trends in CEO compensation have been caused by the same economy-wide factors that have contributed to increases in income. We test this hypothesis by using ExecuComp and IRS tax data to estimate power law distributions and compare the behavior of these distributions over time. Using linear regression techniques, we estimate a power law distribution for CEO compensation and individual income. We find that the parameters of income distribution and the distribution of CEO compensation are correlated. © 2014 Elsevier B.V. All rights reserved.
Keywords: Inequality CEO compensation Power law distribution Pareto distribution
1. Introduction Between 1970 and 2008 pay for the average US chief executive officer (CEO) increased from $850,000 to $10.5 million (Faulkender et al., 2010). This 1135% increase in pay was matched by a relatively small 568% increase in average wages, calculated using the National Average Wage Index. Sharply increasing CEO salaries have become a pressing social concern, especially in the face of stagnating wages in a struggling economy. Many authors propose explanations for this increasing pay within the structure of the market for CEOs. Compensation structure including stock options and incentive-based pay, increased opportunities for profit skimming, changing demand for various skill sets, and growing firm sizes provide potential explanations
∗
Corresponding author. Tel.: +1 843 953 7836; fax: +1 843 953 0754. E-mail address: [email protected] (C. Blackwell).
http://dx.doi.org/10.1016/j.econlet.2014.11.012 0165-1765/© 2014 Elsevier B.V. All rights reserved.
for the increase in CEO compensation. However, we propose another explanation for the increase in CEO pay; it has not been a unique process that has increased executive compensation but the same general process by which the overall distribution of income has become more unequal. We pose the hypothesis that the distribution of CEO compensation follows trends in the distribution of income among the richest Americans. To test this hypothesis, we estimate power law distributions for CEO compensation over time. Power law distributions, also called Pareto distributions, are fat-tailed distributions commonly used to describe income and wealth. We find that changes in inequality in CEO compensation do seem to mirror changes in income. 2. Power law distributions Discovered as a representation of income distribution in nineteenth century Switzerland, the Pareto distribution has been used to describe the behavior of top incomes in many societies. A
C. Blackwell et al. / Economics Letters 126 (2015) 78–80
Pareto distribution’s density function is of the form:
f (x) =
ζ ζ xmin ζ +1
x 0,
,
x > xmin
separated into bins by income range, for each year from 1993 to 2009. (1)
1− ζ1
L(u) = 1 − (1 − u)
,
0
(2)
where u is the percentile in the income distribution. The Gini coefficient, an index of inequality, is the measure between a line of perfect equality and the observed Lorenz curve. The equation for the Gini coefficient of a power law distribution follows: G=
2ζ − 1
5. Methodology
x ≤ xmin
where greater exponent ζ values imply a distribution with a fatter tail and xmin is some lower bound for the power law distribution. Regarding the distribution of income in the US, Silva and Yakovenko (2005) show that the right tail of the distribution can be well-represented with a power law. Atkinson and Piketty (2007) estimate that for top incomes in the US, the power law exponent is approximately 1.7. The Lorenz curve is useful for understanding the amount of inequality in a given distribution. The equation for the Lorenz curve for a power law distribution is:
1
79
.
(3)
The power law exponent, ζ , can then be interpreted as a measure of inequality (Kleiber and Kotz, 2003). As can be seen from (3), the larger ζ , the smaller is G, indicating less inequality. 3. Hypothesis Four explanations of the sudden increase in pay beginning in the 1980s prevail within academic research. The theories presented offer compensation structure, opportunities for profit skimming, skill types, and firm size as possible explanations for the recent changes in CEO compensation (Gabaix and Landier, 2008; Frydman and Saks, 2010). We offer a fifth hypothesis; the trend in inequality in CEO compensation follows the trend in inequality in the distribution of income of top earners. If our hypothesis is correct, then the shape of the distribution of CEO compensation will be correlated with the shape of the distribution of income. We use the exponent of a power law distribution, ζ , (which gives information about the spread of the data) as our ‘‘shape’’ parameter. Estimating this ζ value over time for a given data set will show the trends in inequality for that data. We hypothesize that there will be a correlation between the exponent of the power law distribution of CEO compensation and the exponent for the income distribution. 4. Data To measure executive compensation, we use Standard & Poor’s ExecuComp database, which provides information on executive compensation for firms in the S&P 500, the Midcap 400, and the Smallcap 600, between 1993 and 2009. The ExecuComp data comes directly from publicly traded companies’ annual reports and includes CEOs’ salary and total compensation including restricted stock, payouts from long-term plans, benefits, and stock options valued at the grant-date using ExecuComp’s modified Black–Scholes methodology. As a measure of income, we use tax return data provided by the Internal Revenue Service’s (IRS) Tax Statistics. Using individual income tax returns, we use the size of adjusted gross income,
A standard way to look at the power law distribution is via its counter cumulative density function (P (x)) shown by Eq. (4): 1 − F (x) = P (X > x) = P (x) =
x xmin
−ζ
,
x > xmin .
(4)
The most intuitive way to recognize a power law distribution is to graph the observed data in some form of histogram. When data from a power law is graphed on log(x) and log(y) axes (where xi is the observed value and yi is the number of observations that are less than xi ), the points will form, roughly, a straight line. This visual relationship is confirmation of the mathematical fact that if we take logarithms of both sides of Eq. (4) we find: ln(P (x)) = ζ ln(xmin ) − ζ ln(x).
(5)
To make Eq. (5) an econometric model, we add a normally distributed error term, ϵ , to get Eq. (6). ln(P (x)) = β0 − ζ ln(x) + ϵ.
(6)
Eq. (6) can be estimated using linear regression, where ζ is the slope of the regression line, x is the bin value, and P (x) is the cumulative frequency. Depending on the format of the data, the rank or the cumulative frequency can be used as dependent variables. Since the IRS tax data is presented in bins with frequencies, we use the cumulative frequency approach. We use the top two bins reported by the IRS: incomes above $500,000 and incomes above $1,000,000. We do not use any smaller bins for two reasons: (1) few CEO’s are paid less that $500,000 and (2) the Pareto distribution is most appropriately used for incomes in the top 10%. (Our practice of using only the top two bins is consistent with other researchers; for example Atkinson et al. (2011) estimate ζ using data from the top 10th and 5th percentiles of the income distribution. Furthermore, Dragulescu and Yakovenko (2001) and Silva and Yakovenko (2005) suggest the Pareto distribution may be appropriate for the top 5% or less.) After obtaining an exponent value, ζ , for each year of CEO compensation and income through least-squares regression, we implement linear regression techniques to compare the trends.
ζCEO = φ0 + φ1 ζincome + ωt
(7)
where ωt is a normally distributed random error term. This approach will quantify the strength of the relationship between the distribution of CEO compensation and the distribution of income. 6. Results Using ordinary least squares regression on the log-cumulative frequency and log-bin value, we estimate equations of the form of Eq. (6) for CEO compensation and income. These regression equations yield ζ for both the distribution of income and the distribution of CEO compensation. The results of these regressions are available from the authors. We then take the estimated ζ values and estimate Eq. (7). We see the following trends in exponent values in Fig. 1. When we estimate Eq. (7) using OLS, we find that 55% of the variation in the ζ values for CEO compensation can be explained by the ζ values for income (see Table 1). A relatively low p-value suggests that we can reject the hypothesis that there is no relationship between the two series. The trends in CEO compensation and income appear to be very similar.
80
C. Blackwell et al. / Economics Letters 126 (2015) 78–80
a data set with a greater number of smaller bins would allow us to make more accurate conclusions about the distribution. A commonly accepted method of measuring inequality is by calculating the Gini coefficient. Comparing the Gini coefficient over time for each series is one alternative to comparing Pareto exponents to measure trends in inequality. This more widely used technique could provide more insight into the trends in inequality in CEO compensation and in the economy as a whole. One strength of using ζ as a measure of inequality instead of the Gini coefficient or other measures is that the power law exponent allows us to focus on the tail of the distribution. While the Gini coefficient would measure the overall inequality in the distribution of income or CEO compensation, ζ measures inequality only in the data above xmin . We are able to focus on the most relevant data by using the power law exponent. References
Fig. 1. Estimated ζCEO and ζIncome for 1993–2009. Table 1 Regression results for CEO compensation and income distribution for 1993–2009. Predictor
Coefficient
Standard error
p-value
Constant
−2.700
0.966 0.636
0.014 0.000
ζIncome 2
R
2.712 0.548
7. Discussion There is evidence to suggest that the distributions of income and the distribution of CEO compensation move together. Like all research, the methodology and data used in this paper have some limitations. For example, it is important to consider that members of the CEO sample are included in the general public; further investigation is recommended to ensure that the correlation between the trends in CEO compensation and income is not exaggerated due to this overlap. Income data was available to us only in the form of frequencies within wide income ranges. The use of linear regression poses some potential problems (see Clauset et al., 2009 for a discussion). Though our estimated ζ values are similar to those estimated in previous work (Atkinson et al., 2011),
Atkinson, A.B., Piketty, Thomas, 2007. Top Incomes over the Twentieth Century: A Contrast between European and English-Speaking Countries. Oxford University Press, Oxford, New York. Atkinson, Anthony B., Piketty, Thomas, Saez, Emmanuel, 2011. Top incomes in the long run of history. J. Econ. Literature 49 (1), 3–71. http://dx.doi.org/10.1257/jel.49.1.3. Clauset, Aaron, Shalizi, Cosma Rohilla, Newman, M.E.J., 2009. Power-law distributions in empirical data. SIAM Rev. 51 (4), 661–703. http://dx.doi.org/10.1137/070710111. Dragulescu, Adrian, Yakovenko, Victor M., 2001. Exponential and power-law probability distributions of wealth and income in the united kingdom and the united states. Physica A 299 (1-2), 213–221. http://dx.doi.org/10.1016/S03784371(01)00298-9. Faulkender, Michael, Kadyrzhanova, Dalida, Prabhala, N., Senbet, Lemma, 2010. Executive compensation: an overview of research on corporate practices and proposed reforms. J. Appl. Corp. Finance 22 (1), 107–118. http://dx.doi.org/10.1111/j.1745-6622.2010.00266.x. Frydman, Carola, Saks, Raven E., 2010. Executive compensation: a new view from a long-term perspective, 1936–2005. Rev. Financ. Stud. 23 (5), 2099–2138. http://dx.doi.org/10.1093/rfs/hhp120. Gabaix, Xavier, Landier, Augustin, 2008. Why has CEO pay increased so much? Quart. J. Econ. 123 (1), 49–100. http://dx.doi.org/10.1162/qjec.2008.123.1.49. Kleiber, Christian, Kotz, Samuel, 2003. Statistical Size Distributions in Economics and Actuarial Sciences. John Wiley & Sons, October. Silva, A. Christian, Yakovenko, Victor M., 2005. Temporal evolution of the ‘‘thermal’’ and ‘‘superthermal’’ income classes in the USA during 1983-2001. Europhys. Lett. 69 (2), 304–310. http://dx.doi.org/10.1209/epl/i2004-10330-3.