The
exponential growth trajectory of income described by equation (4) clearly does
not present the full picture of income evolution with age. As numerous
empirical observations show (

*e.g*., Figure 1), the average income reaches its peak at some age and then starts declining. This is seen in individual income paths, for instance presented in Mincer [1974]. In our model, the effect of exponential fall is naturally achieved by setting the money earning capability*Σ*(*t*) to zero at some critical work experience,*t*=*T*._{c}
The
solution of (4) for

*t>T*then becomes:_{c}*M̃*(

_{ij}*t*) =

*M̃*(

_{ij}*T*) exp[−(1/

_{c}*Λ*)(

_{min}**γ̃**/

*L̃*)(

_{j}*t*−

*T*)] (16)

_{c}
and
by substituting (12) we can write the following decaying income trajectories
for

*t>T*:_{c}*M̃*(

_{ij}*t*) =

*Σ*{1 − exp(−(1/

_{min}Λ_{min}S̃_{i}L̃_{j}*Λ*)(

_{min}**α̃**/

*L̃*)

_{j}*T*)]exp{−(1/Λ

_{c}*) (*

_{min}**γ̃**/

*L̃*)(

_{j}*t*−

*T*)} (17)

_{c}
First
term in (17) is the level of income rate attained at

*T*. Second term expresses the observed exponential decay of the income rate for work experience above_{c}*T*. The exponent index_{c}**γ̃***represents the rate of income decay that varies in time and is different from***α̃**. It was shown in Kitov [2005a] (and also seen in Figure 1) that the exponential decay of personal income rate above*T*results in the same relative level at the same age, when normalized to the maximum income for this calendar year. This means that the decay exponent can be obtained according to the following relationship:_{c}**γ̃**

*= −lnA*

*/*(

*T*

_{A}

*–*

*T*)

_{c}*(18)*

where
A

*is the constant relative level of income rate at age**T*_{A}. Thus, when the current age reaches A, the maximum possible income rate*M̃*(for_{ij}*i*= 29 and*j*= 29) drops to A. Income rates for other values of*i*and*j*are defined by (17). For the period between 1994 and 2002, the empirical estimates of parameters in (18) are A=0*.*45 and*T*_{A}=64 years (see [Kitov, 2005a] for details).
The
critical age in (16-17) is not constant.
For example, Figure 1 demonstrates that

*T*has been increasing between 1962 and 2011. Therefore, its dependence on the driving force of income distribution - real GDP per capita - has to be one of central elements of our model since any model should match the long-term observations. To predict the increase in_{c}*T*(_{c}*τ*) we use (14): the time needed to reach some constant income level is proportional to the square root of real GDP per capita. Assuming that the peak value of the mean income is constant in relative terms, we obtain:*T*(

_{c}*τ*) =

*T*(

_{c}*τ*

_{0}) [

*Y*(

*τ*) /

*Y*(

*τ*

_{0}) ]

^{1/2}(19)

Figure
6 illustrates the growth in critical work experience,

*T*, since 1930. The curve in the left panel illustrates the time dependence and is best interpolated by a straight line with a slope of 0.28 years per year as if the real GDP per capita grows as_{c}*t*^{2}. During the last recession, the critical age dropped from 40.3 years in 2007 to 39.3 years in 2009. In the right panel, the dependence on GDP is shown for the same period.
Figure
6. Left panel: Secular increase in

*T*is driven by the growing GDP per head. Right panel: The evolution of_{c}*T*_{c}_{ }as a function of GDP growth.
Above

*T*, people can only use their earning instrument, which is growing with time, but their capability remains at zero level and income experiences an exponential decay. Formally, the size of work instrument cannot be zero since the dissipation term would be infinite. But we can easily imagine zero capability to earn money as the absence of interest to work. The model attributes positive capability to everyone in the working age population before_{c}*T*This means that each and every person in a given economy must have a nonzero income. This is not what the CPS reports – approximately 10% of the working age population reports no income from the sources included in the CPS questionnaire._{c}.
When
predicting incomes, we use the entire population. When comparing with
observations, we include the zero-income CPS population into the income bin
starting with $0 and recalculate the whole statistics like average income, the
portion of people above a given threshold, etc. According to strict guidelines
adapted in physics one should not calculate any aggregated characteristics of a
closed system using only part of it. Such estimates are always biased and
subject to fluctuations.

As
an alternative to formal introduction of zero capability, one could claim that there
exists a strong external process, which forces the exponential fall on top of
the grown related to the original capacity to earn money. This does not resolve
the problem, however, since description and explanation of these forces is needed.
In addition to the homogeneous coverage of all population these forces should include
the change in start time, i.e. should explain the growth in the age of peak
mean income. We do not know any candidate.

Initial
exponential growth and following decay, however, do not complete the model. Figure
2 shows, that our equation for income growth is not able to predict a power law
distribution. We still need to introduce
special treatment for the very top incomes that have been shown to follow the Pareto
distribution in multiple empirical studies.

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