Why we insist that personal income inequality does not change

We have already reported that the personal income distribution in the USA does not change with time when normalized to the total population and total income. In other words, the relative distribution of personal income in the United States has not been changing since the start of income measurements in 1947. The accuracy of early measurements is not good enough, however, and we have to rely of the most recent results.

The US Census Bureau routinely reports income estimates obtained during the Annual Social and Economic Supplement of the Current Population Surveys. We have retrieved the population distribution over mean income in the range from $0 to $250,000 which is available only from 2000. The relevant measurements of the number of people in a given income range were carried out in $2500 bins between $0 and $100,000 and $50000 bins between $100,000 and $250,000. In order to suppress the influence of the width we have calculated the population density, i.e. the ratio of the number of people in a given bin and its width. The personal income is measured in current dollars and thus we have to reduce all incomes by the total change of the GDP deflator (one may also use CPI which gives a 20 per cent higher inflation rate) since 2000 to a given year. Figure 1 shows the result of normalization for 2000, 2005, and 2010. In relative terms, the income distribution has not been changing since 2000. At higher incomes, all three curves are practically identical. This observation is validated by the estimates of Gini ratio provided by the Census Bureau.

Figure 1. The population density function, PDF, as a function of mean income as normalized to the total personal income for a given year. At higher incomes,  the curves are practically identical.

Two more graphs on GDP in the USA

Two more graphs on real GDP in the USA. In the forth quarter of 2011, the level of real GDP was higher ($13,422 .4 billion) than that in the fourth quarter of 2007 ($13,326 billion) as Figure 1 shows. Figure 2 demonstrates that  the increasing population did not allow real GDP per capita t reach the level of 2007: $42,727 vs. $43,791. In seems to be the task for 2012 to 2014 if no recession will occur.  

Figure 1. Real GDP

Figure 2. Real GDP per capita

Real GDP and GDP deflator in 2011

Here are some quick notes on the new estimates of real GDP and GDP deflator for 2011.  Figurer 1 shows the rate of growth of real GDP, dlnGDP/dt, at annual and quarterly basis. In 2011, the rate is 0.017 1/y, i.e. 1.7% per year, despite the rate growth in the fourth quarter of 2.7% (SAAR). Previously, we predicted a small recession in 2012 and 2013. This prediction will be updated soon when the 2010 census results are published and incorporated into the so called postcensal population estimates.

Figure 2 shows the growth of total population for the purpose of per head calculations. Please notice a large step in population between 1999 and 2000 as associated with the error of closure, i.e. the difference between intercensal estimate for 2000 and the number enumerated in the 2000 census.  Figure 3 presents the rate of growth of real GDP per capita , dlnG/dt. In 2011, the rate of growth was 0.0098 1/y. This means that the total increase in population was of 0.7%.

Figure 4 shows the GDP deflator or price inflation associated with the economy as a whole. This is the most comprehensive measure of inflation as we discussed many times in this blog. For 2011, the GDP deflator is 2.1%. This is larger than we predicted but the last quarter signals about upcoming deflation, as we foresaw six years ago. Currently, the FRB also foresees very low inflation rates through 2014.

According to our concept of GDP growth,  real GDP per capita has a inertial component which is expressed in a constant annual increment, dG=const. Figure 5 updates the graph showing the evolution of real GDP per capita in the USA. One can observe a gradual return to the constant level of annual increment. This works as inertial movement in physics. However, one can expect some more years of dG less than average, dG<$490 (2011 US dollars).  It is worth noting, that there is no output gap when real GDP per cpaita is considered. The evolution of dG exactly follows it long term trend and the years after 2007 serve to return dG to the trend from its highs in the late 1990s.

We have to notice that the estimates of the nominal GDP and GDP deflator are subject to revision which may be as high as several per cent (+2.1% for 2001). However, the long term trends in all presented variables fit our concept and predictions. 

Figure 1. The growth rate of real GDP: annual and quarterly (annualized). MA(4) for the quarterly time series. For 2011, dlnGDP/dt=0.017  1/y.

Figure 2. The evolution of total resident population. Notice the jump between 1999 and 2000 – the closure error.

Figure 3. The growth rate of real GDP per capita. For 2011, the rate is (dlnG/dt=) 0.0098 1/y. 

Figure 4. Annual and quarterly (annualized) price deflator of GDP. In the last quarter of 2011 the GDP deflator dropped to 0.004 1/y. This is likely a turn to deflation.

Figure 5. The increment of real GDP per capita since 1950. As predicted, the trend returns to a zero slope. There is no output gap.

Unemployment in Spain will be increasing further

Here we revisit the rate of unemployment, ut, in Spain using its dependence on the change in labor force, lt=dLF/LFdt. There is a new estimate of 22.8% for the unemployment rate in 2011. In May 2011, we quantitatively predicted that this rate should only be growing. It may reach 29% if the link between the rate of unemployment and the rate of labor force change is correct, as has been observed since 1980.

Previously, it was found that Spain is characterized by the same relationship between unemployment and labor force as other developed countries. For Spain, we used data provided by the OECD. Figure 1 depicts unemployment and the change rate of labor force between 1960 and 2011. In line with the OECD description of the breaks in the labor force series:

Series breaks: In 2005, changes in the questionnaire and the implementation of CATI system in the field work affected the estimates. The 2005 questionnaire produced an additional increase of employment (132 000) and a decrease of unemployment (78 000). From 2001, the new unemployment definition established by the European Commission in 2000 has been introduced. From 1994, persons employed in the “Guardia Civil” are not included in the armed forces. As an indication, this category represented 59 600 people in 1994. In 1976, the lower age limit for inclusion in the Labour Force Survey was raised from 14 to 16, at the same time other modifications to the survey were introduced.

there are two spikes in the dLF/LF series near 1976 and 2001 as related to step revisions to the level. The spike around 1988 has no explanation in terms of the revisions to labor force, but is of the same amplitude. One can not exclude the opportunity that this spike is related to the processes of joining the EU in 1986.

As expected, the same functional form of dependence is valid for Spain. The estimation method is based on trial-and-error approach and seeks for the fit between annual curves. The final model is as follows

ut = -7.0lt + 0.31; t>1986

Figure 2 depicts observed and predicted curves. Before 1986, the curves diverge and a different model is likely holds. Because of high-amplitude oscillations in the original time series for the rate of labour force change, lt, we have to smooth it by MA(3). For the period after 1986, R2=0.7. Thus, the change in labor force has been driving the rate of unemployment in Spain. The negative coefficient implies that unemployment is Spain goes down when labor force starts to increase.

As has been predicted by our model, the rate of unemployment has increased in 2011. This is not the end of the sad story on unemployment in Spain. Figure 2 evidences that it will likely be growing further with the decreasing labor force.

Figure 1. Unemployment rate, u, and the rate of labor force change, l, in Spain according to the definition introduced by the OECD.

Figure 2. Prediction of inflation by labor force. Due to high variation in the estimates of labor force we have smoothed it with MA(3). For the observed and predicted curves, R2=0.7 for the period between 1986 and 2011.

Wal-Mart share in 2012

We estimated our price model for Wal-Mart Stores (NYSE: WMT) nine months ago. The model is based on the decomposition of a share price into a sum of two selected consumer price indices. This is a new model defined by the (seasonally not adjusted) index of hospital and related services (HOSP) and the price index of miscellaneous personal services (MISS), as reported by the US BLS. The former CPI component leads the share price by 10 months and the latter one evolves in sync with the price. Figure 1 depicts the overall evolution of both involved indices through December 2011. A very specific feature of both indices is their linearity over time: they are close to straight lines. 

In this post, we re-estimate the WMT share price using new data through December 2011. This allows validating the initial model and demonstrating its reliability. The previously obtained defining components are the same and provide the best fit model between June 2010 and March 2010 with only one month change in the lag for the HOST index.  All coefficients in (1) are only slightly different for the new model (see below).  The slope of the time trend is negative. The best-fit 2-C model for WMT(t) is as follows: 

WMT(t) =  0.50HOSP(t-10) + 1.42MISS(t)  - 28.39(t-1990) – 158.12 (January 2011)  (1)

WMT(t) =  0.46HOSP(t-9) + 1.49MISS(t)  - 28.03(t-1990) – 165.50 (March 2011)

WMT(t) =  0.46HOSP(t-9) + 1.30MISS(t)  - 26.06(t-1990) – 141.92 (December 2011)

where t is calendar time. The predicted curve in Figure 2 evolves in sync with the observed price. The residual error is $2.13 for the period between June 2003 and December 2011. With both indices growing along their respective trends one can expect a slight increase to the level of $60 to $65 per share in 2012Q1. Figure 3 presents the residual model error.  

Figure 1. Evolution of the price of HOSP and MISS. 

Figure 2. Observed and predicted WMT share prices.

Figure 3. Residual error of the model.

Quarterly report: Loews share price model

This is a quarterly report on the performance of our share price model for Loews Corporation (NYSE: L). The model is based on the decomposition into a weighted sum of two consumer price indices (selected from a larger number of CPIs), linear trend and constant, all coefficients and time lags to be estimated by a LSQ procedure. Here we test the previous model and make a regular update using new data. All in all, the original model is valid since October 2008 and does not show any sign of future changes. This is a reliable model valid during the past 50 months!  

A preliminary model for Loews Corp. was obtained in September 2009 and covered the period from October 2008. This old model included the index of food without beverages (FB) and the index of transportation service (TS). The most recent model also used the monthly closing prices as of April 2011 and the CPI estimates published on April 14, 2011. The defining indices were almost the same: the index of food (F) and the TS index. Figure 1 depicts the evolution of the indices which provide the best fit model, i.e. the lowermost RMS residual error, between July 2003 and December 2011.  The F index leads by 5 months and the TS index by 4 months.  When new data through December 2011 are used, the model does not show any tangible change - only coefficients have been slightly drifting:  

L(t) = -2.03F(t-5) – 2.12TS(t-4) +28.23(t-1990) + 448.98, March 2011

L(t) = -2.01F(t-5) – 2.09TS(t-4) +27.96(t-1990) +440.65, September 2011

L(t) = -2.03F(t-5) – 2.02TS(t-4) +27.65(t-1990) +431.99, December 2011      

where L(t) is the share price in US dollars, t is calendar time. The new model is depicted in Figure 2 together with high and low monthly prices as a proxy to the uncertainty bound of the share price. The predicted curve leads the observed one by 4 months. The residual error is of $2.42 for the period between July 2003 and December 2011.  In the first quarter of 2012, the model foresees essentially no change. It is worth noting that the model obtained in March 2011, accurately predicted the small fall observed in the second and third quarters of 2011.   

Figure 1. Evolution of the price indices F and TS.

Figure 2. Observed and predicted share prices.

Why the Economic Projections of Federal Reserve Board are inconsistent

The FRB members have recently projected the evolution of key macroeconomic variables including real GDP and the rate of unemployment. In our blog , we have developed a very accurate model linking the rate of unemployment in the US to the rate of real GDP (per capita) growth: (A series of posts has resulted in a working paper.) The following relationship was estimated:

du = -0.62dlnG + 1.09,  (1)

When integrated between t0 and t, equation (1) can be rewritten in the following form:

u(t) = u(t0) + bln[G/G0] +a(t-t0) + c  (2)

Without loss of generality, we assume t0=0. The intercept c≡0, as is clear for t=t0. Instead of integrating (2), we calculate cumulative sums of the annual estimates of du and lnG with appropriate initial conditions. The cumulative sum of du’s is the time series of the unemployment rate. Figure 1 depicts the measured and observed curves for the period between 1958 and 2011. The agreement is excellent and has been obtained by a formal statistical method (LSQR).

 From (1) it follows that higher rates of GDP growth decrease the rate of unemployment. The FRB has projected real GDP with the highest rates of 2.7% in 2012, 3.2% in 2013, and 4% in 2014. We reduce these rates by 1% per year to estimate the per capita rate of growth, i.e. the growth in population is 1% per year. Using (2) we calculate the rate of uneployment which will correspond to the projected real GDP.

Figure 1 also depicts these predicted rates for 2012 to 2014 by open circles. The rates of unemployment projected by the FRB are shown by red circles. There is a significant deviation between the predicted and projected rates, which likely manifests the inconsistency in the FRB member's models of unemployment.

One may check these projections in 2015. 

Figure 1. The observed and predicted rate of unemployment in the USA between 1958 and 2010.The projected rate of unemployment (middle point of the projections) is shown by red circles.