# A generative model for age and income distribution

FOFatih Ozhamaratli  OKOleg Kitov  PBPaolo Barucca  University College London,
Gower Street, WC1E 6BT London, UK  University of Cambridge, The Old Schools,
Trinity Ln, CB2 1TN Cambridge, UK

###### Abstract

Each individual in society experiences an evolution of their income during
their lifetime. Macroscopically, this dynamics creates a statistical
relationship between age and income for each society. In this study, we
investigate income distribution and its relationship with age and identify a
stable joint distribution function for age and income within the United
Kingdom and the United States. We demonstrate a flexible calibration
methodology using panel and population surveys and capture the characteristic
differences between the UK and the US populations. The model here presented
can be utilised for forecasting income and planning pensions.

Income Dynamics,

Agent Based Model,

Pension System,

###### keywords:

Research

##  1 Introduction

A universal element of societies is the emergence of hierarchical organisation
structures within professions. People develop work experience through time and
manage to obtain jobs of increasing responsibility and increasing level of
income with time. Hence, it is a natural property of income distribution to be
correlated with work experience and age; nevertheless, most income models do
not study the relationship between income and age, and consequently between
income distribution and demographic changes. This paper introduces a model of
income, dependent on age-specific model parameters and random shocks. The
model contributes to the understanding of the relationship between age and
income and its dynamics.

Our aim is to compare the estimated parameters in the UK and the US age and
income distribution to find out similar characteristics of age and income
across states, as well as the contrasting differences. A simple age and income
model is fundamental for the development of a sustainable pension system. The
model focuses on the age and income relationship and further factors, such as
occupational levels, are not considered. The model is estimated via panel
survey data from the UK and population surveys from the USA. The data from
panel surveys track the same individuals for the duration of the survey, and
the population survey is repeated with different people each wave. The results
reflect a clear income-age relationship in the UK and US, a clear structure of
the joint distribution characterised by rapidly increasing income at younger
ages, followed by income levels stabilising near mean income but spreading
till retirement. At this point, the income decreases and concentrates around
mean retirement income. The paper demonstrates a flexible methodology to
estimate parameters from population surveys, as well as panel surveys. The
paper provides a simple generative model to evolve age-income population for
simulation and forecasting purposes, which can constitute the foundation for
future studies of financially sustainable pension systems by providing a
benchmark for capturing age and income relationship. The purpose is to have a
baseline model simple enough for isolating age and income relationship of
income dynamics. Such a model will serve for investigating the properties of a
sustainable and balanced pension system. The mean and standard deviation
statistics from the panel and population surveys on Fig. 6, Fig. 7, Fig.2 from
observed panel data and simulation results reflect a clear relationship
between age and income. More complex models, which investigate additional
factors, and profile heterogeneity of income dynamics are out of the scope of
our work.

Previous research on income have been conducted, and the research focused on
investigating and explaining wage dynamics. Champernowne explicitly introduces
a first-order Markov process to model the time-evolution of wages[1].
Following Markov process path, the validity of the first-order Markov
assumption is tested by Shorrocks[2]. Following research introduces a second-
order Markov process, yet neither of these works links individual wage
dynamics to time-evolution of the distribution of wages [3]. A different
approach focused on poverty, which deals with modelling individual data using
linear regression and transitioning to poverty (probit model)[4]. A more
comprehensive model incorporating various factors is developed to estimate
transition probability in wage quintiles conditioned on various regressors,
including education, experience and age [5], and study both intra- and inter-
group inequality. The persistence of the low pay state and on factors
affecting the low pay probability in a generalized regression model is
expressed. For modelling low income transitions the previous research use
British Panel Data for the ${}^{\prime}90$s, focus on the transition
probability and state dependence for the poverty status[6]. They define
poverty transition equation, a coarse-grained dynamics. Focus on inequality
and upward mobility between quintiles considering gender effects are
investigated [7]. The previous models in literature either incorporate
numerous external variables, distribution characteristics and functions, such
as innovation constants or limiting their scope to the investigation of
dependence on a single variable[8][9]. A more recent article by Guvenen
investigated a model for which focal variables are the human capital
consisting of education, work experience, and idiosyncratic shocks [10],
following research modelled male income for studying the impact of labour
income taxation policy on inequality [11] The referred life-cycle model’s
distribution characteristics of the pre-tax income arise from the differences
in the individual’s ability to learn new things and idiosyncratic shocks.
Previous research tried to capture the income dynamics with Markov Models,
linear autoregressive models, or by relying on econometric toolset such as
covariance matrices. We investigate a generative model with an empirical
distribution for sustainable age and income relationship in a population; we
achieve this via an income evolution model with an age-dependent parameter,
estimated from previous population and panel surveys.

In contrast to previous research, our study introduces a dynamic model that
describes the income-dependent only on age and previous income. This paper
investigates the stationary property of the income distribution dependent on
age. We provide a model in which the mean and variance of income given age are
preserved at any time point.

##  2 Methods

We introduce a simple model which focuses on age and income relationship and
differs from recent literature by not incorporating other variables such as
occupational level, level of education and skill coefficients. The model is
stationary, i.e. the mean and variance of income given age are preserved in
time. The model is utilised to represent observed panel data for gaining
empirical insights regarding age dependant, income dynamics and mobility. The
calibrated model can be utilised as a simple generative model to evolve an
age-income population for simulation purposes and it provides a theoretical
background for studies focusing on ageing and pension income of the
population. We initially assume the following model, by which $\mu(.)$ and
$\sigma(.)$ represent a function of age, income, and individual-specific
additional parameters $\theta_{i}$ or $\lambda_{i}$, for the sake of
generality. $\mu(.)$ is a function capturing mean income characteristics, and
$\sigma(.)$ captures the variational characteristics of the income. We
consider the following individual income stochastic process for an economic
agent $i$ characterised at each time step $t$ by a given age $a_{i}$ and
income $y_{i}$:

$y_{i(t+1)}=\mu(a_{i(t+1)},\,y_{it}|\mathbf{\theta}_{i})+\sigma(a_{i(t+1)},\,y_{it}|\mathbf{\lambda}_{i})\eta_{it}$
(1)

The characterising insights on Fig.2 from the panel data lead to the
assumption that the probabilistic step at time $t$ depends only on the age and
income of the preceding step.

###  2.1 Defining Income and Age Dynamics

Earnings of individual $i$ at the time step $t$ is denoted as $Y_{i,t}$ and
its logarithm is $y_{i,t}$. The parameters that describe the income process
are: age-dependent persistence parameter $q_{a}$, age-dependent mean $\mu_{a}$
and age-dependent standard deviation $\sigma_{a}$. The income shock process
consists of independent random shock $\eta_{t}^{i}$ which is normally
distributed with mean zero and variance 1, and it is applied to $\sigma_{a}$,
the model can be defined as follows:

$y^{i}_{a+1,t+1}=q_{a}y^{i}_{a,t}+\mu_{a}+\sigma_{a}\eta^{i}_{t}$ (2)

Averaging income $y$, for individuals who are $a$ years old, gives
$\bar{y}_{a}$ ,which denotes the average income for age group $a$ across all
individuals $i$ and periods $t$. Assuming that the age-dependent income
profiles are stationary, we can average incomes $y^{i}_{a,t}$ across
individuals and time to get the following equation:

$\bar{y}_{a+1}=q_{a}\bar{y}_{a}+\mu_{a}$ (3)

where $\bar{y}_{a}$ denotes the average income for age group $a$, taken across
all individuals $i$ and periods $t$. The following equation can find the
estimator for $\mu_{a}$:

$\mu_{a}=\bar{y}_{a+1}-q_{a}\bar{y}_{a}$ (4)

The income data from different waves are inflation adjusted to isolate effects
of economic growth.

###  2.2 Data

The British Household Panel Survey (BHPS)[12] from the UK and The Current
Population Survey (IPUMS CPS)[13] from the USA are used for estimating the
model parameters of the model (2) and comparing the results of simulated data
and surveys. The BHPS is a Panel Survey conducted between 1991-2008. For our
model we focus on labour income data, which captures wage, salary or self-
employment income. To investigate population characteristics, we also
incorporate other income sources and call it ”Total Income”, which
additionally captures the transfers, pensions, grants, aids, state-benefits,
dividends, capital income and rents. BHPS provides individuals specific
longitudinal weights for ensuring the representativeness of the population.
Two types of weights are provided with BHPS. The first wave is weighted for
adjusting population marginals at the households and post-stratified to the
population age by sex marginals. Consecutive waves are re-weighted to take
into account sample attrition, variables such as address change, household
region, age, sex, race, employment status, income total and composition,
educational qualifications[14]. Panel Survey is conducted via questionnaires
with tracked individuals of the initial sample. There is an extension to the
sample population in 1999.

For the USA, IPUMS CPS is used, which is annually conducted with different
samples each year. In contrast with BHPS, the Labour Income does not include
self-employed income, and the weights are cross-sectional. Income
distribution, age distribution and income-dependent age distribution from the
surveys are utilised for parameter estimation and further analysis. ${q_{a}}$,
${\mu}_{a}$, ${\sigma}_{a}$ are the key parameters estimated according to the
proposed model. Following investigation and interpretation of the estimated
parameters, these parameters are used to simulate the population’s income
transitions. The simulation is initialised using the panel data from the wave
1, and the income evolution function 2 of the model is applied transitively in
an iterative approach to the data for simulating successive waves. The
simulated data is plotted, interpreted and finally compared with the observed
data.

(a) UK Pyramid Wave 1

(b) UK Pyramid Wave 18

(c) USA Pyramid Wave 1

(d) USA Pyramid Wave 18

Figure 1: Population Pyramid for the UK & USA Income between Ages 15-100

(a) UK Labour JDF

(b) UK Total JDF

(c) USA Labour JDF

(d) USA Total JDF

Figure 2: All Waves JDF for UK and USA Income between Ages 15-100

Fig. 1 reflects the Population Pyramid in the UK and USA, and how the shape
evolved over the 18 years considered. The UK population sample from BHPS has a
relatively balanced population with a slight weight towards younger cohorts
initially in 1991, which denotes Wave 1. The UK population gradually got
older, and the population pyramid reflects mass’s shift towards older
generations, this shift happened gradually over the years. The US population
from CPS reflects a young population in Wave 1 with a notable skew towards
younger cohorts, after 17 years the US population loses this property towards
younger cohorts and gets significantly older. Both the UK and US population
get older and reflect a trend towards an ageing population, which will
significantly impact the pension system.

The shape of the population pyramid and its evolution with time from the panel
survey reflect an ageing community [15]. JDFs of Total and Labour Income in
the UK and USA are presented in Fig.2. There is a sharp increase between the
ages 15-20, which can be interpreted as the beginning of the work-life,
transitioning from part-time work to full-time work, and graduation from
higher or vocational education. The most significant difference of the UK
Total JDF for the UK Labour JDF is the tail section corresponding to the
retired population, which denotes the significant percentage of individuals
older than 55. The tail section is relatively concentrated, which can be
explained by the state pension benefit levels and mandatory social security
system. The US population reflects a surprisingly sparse older cohort for the
Total Income data, and the most significant difference to the UK is the
relatively lower income levels compared to the wage income. A higher variance
spread to a wider band, which might be caused by a non-standard retirement
system not supported by strong state pension benefit and mandatory pension
schemes during employment.

The comparison for the model simulation and observed data shows common
characteristics, as the joint distribution of age and income in logarithmic
scale is presented in Fig.2 : an initial sharp rise between the entrance to
the graph on 16 years old, the amount of 16 years old includes pocket money,
allowance and part-time or internship jobs. There is a steep increase in mean
and variance between the initial income and income at the age of 20. The
increase is sharper for the mean in comparison to the variance. The
population’s mass has similar characteristics with near 23k GBP annual income,
and for ages between 20-45. Between ages 65-75, there is a significant
decrease in income and after 75, the income converges to a certain mean. The
data and the models provide an essential tool to tackle problems related to an
ageing population and shocks introduced by technological and political
changes.

In the following sections we will focus on Labour Income and employed labour
population. Total Income covers all of the income streams including Transfer
Income such as pocket-money, Labour Income, Capital Income, and Pension
Income; these different streams might be governed by varying dynamics non-
uniform across the type of income; so we decided to focus on Labour Income,
which involves the broadest section of the population; with most significant
impact. The only other primary source of income in terms of gross value is the
Capital Income, which might be significantly affected by other factors such as
inter-generational shifts, market conditions and global financial state. In
order to focus on labour income dynamics, the other income sources are left
out of our modeling.

##  3 Data Processing and Calibration

BHPS provides a vast amount of socio-economical data for each individual and
household participating in the study. The columns of income, age data, the
individual’s statistical weight -representative of the British population and
overall survey- with the individual’s intra-wave unique identifier mostly
suffice for this paper’s scope. PID, Wxage12, wFIYR, Wxrwght fields of BHPS
are used for each wave.

The Income variable xfiyr is each individual’s annual income, including labour
income, benefits, pensions, transfer income, and investment income.
Participants were asked according to annual income in the reference year from
September in the year prior to the interview until September in which
interviewing begins [14]. The income figures are adjusted for inflation, as
part of pre-processing. During the dataset preparation, a floor wage is
determined to exclude in labour income, which denotes to excluding part-time
and short-employment income. The income data is inflation-adjusted and
transformed into log-domain.

IPUMS harmonizes the CPS and provides IPSUM CPS micro-data. The IPSUM CPS
includes a large spectrum of topics such as demographics and employment, as
well as supplemental studies such as the Annual Social and Economic Supplement
(ASEC). Each individual can be identified by ”CPSIDP”, ”INCTOT” and ”INCWAGE”
correspond to the total income and wage income, and ”ASECWT” denotes the
weights derived from ASEC Supplement. The data set is topcoded, and specific
codes are used for labelling missing and incorrect data. The ages over 80, 90
and 99 are top-coded till 2004, and after 2004, the top-coding bins are
determined as 80,85,90 and 99 by the panel data collectors[13]. Although this
dataset contains high-income individuals, there is top-coding applied, so
individuals with very high income are not included.

###  3.1 Fitting Distributions

Estimating the income evolution function parameters is the most critical part
of the research, and the decision depends on various factors such as the type
of data, bias, and assumptions. Various techniques are investigated, leading
to different results, with each having unique strengths and weaknesses.
The first method investigated is Generalized Method of Moments (GMM), which
presumes that the first three moments of the income evolution functions
provide the necessary information for approximating the underlying generative
process. The equations of the first three moments of the income evolution
equation can be solved for the parameters $q_{a}$, ${\sigma}_{a}$,
${\mu}_{a}$. Both of the BHPS from the UK and the IPUMS CPS from the USA can
be used for estimation with Generalized Method of Moments (GMM) with first
three moments.
The second method utilises the micro-data from the longitudinal surveys, which
tracks the individual for consecutive years. The parameters are approximated
to fit the income evolution function using least squares minimisation for the
individuals participating in the studies for consecutive years. The BHPS from
the UK is a suitable micro-data consisting of a panel survey, and the survey
tracks income of the same individuals over the years.

###  3.2 Estimation for Generalized Method of Moments (GMM)

The three moments of the age income evolution function are utilised to find a
polynomial; afterwards, the equation is solved for $q_{a}$, ${\sigma}_{a}$ and
${\mu}_{a}$; at this point, a observed solution for parameters is found, but
the relationship captures only the dynamics of the first three moments.
Calculations can be found in appendices. The statistical variables such as
$\bar{y}_{a}$ are found for each wave and than averaged across waves for
finding a one set of stationary variables, which can be used to estimate
$q_{a}$, ${\sigma}_{a}$, ${\mu}_{a}$. The details and derivations for the GMM
estimation technique can be found in appendix.

###  3.3 Estimation of Least Squares for Micro Data

The Least Square Method requires that an individual’s income for two
consecutive years be existent in the dataset, this restriction is fulfilled by
the BHPS, a panel survey, but the CPS IPUMS population survey does not satisfy
this condition. The income data from two consecutive years per agent is used
to estimate age-specific parameters, which characterise the income evolution
function at Eq.2. LSM tries to estimate parameters by fitting the data to the
income evolution function.

##  4 The generative model

The model can also be used for simulation and forecast, tracking income
trajectories of the individuals, providing a bench table for observing the
stylized facts and complex properties of the income dynamics. Following the
estimation of model parameters, the model is bootstrapped with data from wave
1 for initialising the simulation. Each individual from wave 1 is initialised
as an agent in our model. According to Age Income Evolution Dynamics Eq. 2 ,
the income of these agents is transitively updated at each consecutive wave
update. $\eta_{it}$ provides the random feed, which introduces variability for
the income evolution of the agents. At each wave update, a new generation of
agents consisting of 25 years old agents from the initial wave is injected.
Following each wave, distributions corresponding to the state of the simulated
population are calculated. A full calibration of the model is shown in the
Supplementary Material.

##  5 Parameters

The optimized performance of these three methods are compared and discussed in
the following sections.

No boundaries are explicitly imposed by LSM estimation.

(a) $q_{a}$ with GMM

(b) $\sigma_{a}$ with GMM

(c) $\mu_{a}$ with GMM

(d) $q_{a}$ with LSM

(e) $\sigma_{a}$ with LSM

(f) $\mu_{a}$ with LSM

Figure 3: $q_{a}$, $\sigma_{a}$ and $\mu_{a}$ Plots for UK Labour Income

The $\mu$, $\sigma$ and $q$ variables are independent of each other, but the
estimation process or data itself can introduce a slight dependence. The GMM
estimation technique results in minimal $q$ values near 0, so the estimated
parameters approximately resemble an auto-regressive model. However, despite
near 0 negligible $q$ values, the $q$ plot has a distinct shape with an
increasing trend with a small decrease between 25 and 30, has very different
characteristics depending on the estimation method. The GMM estimation method
results in minimal $q$ and the $\mu$ reflects the characteristics of
$\bar{y}$, which is in compliance with this estimation method’s nature. The
$\mu$ value increases at first and then plateaus and slightly decreases near
retirement. On the contrary LSM estimation mainly characterises the income
with an increasing $q$ parameter, so the $\mu$ parameter has limited effect
and reflects a decreasing trend. $\sigma$ values reflect a distinct trend of
initially decreasing values with a spike around the age of 34 followed by a
stable decrease and noisy plateau with a minor increase towards 55. The LSM
with bootstrap is the most accurate estimation method and reflects the
characteristics of the model clearly.

GMM
GMM estimation technique approximates the $\mu_{a}$ values to be consistently
around 10 and the $q_{a}$ values are around 0 with an initial sine-like wave
followed by a steady increase. The $\sigma_{a}$ values are around 0.8 and have
a positive trend. $q_{a}$ values display a positive trend as well. The
$\bar{y}_{a}$ and $std(y)_{a}$ plots of the simulation is similar to the
observed data, but the standard deviation plot is particularly noisy. The JDF
of the simulation on Fig. 10 is sparse, consistent; but not highly
concentrated around mean. Both of these methods depends on assumptions about
the dynamics of the income evolution function. The GMM method assumes that the
first three moments of the equation are enough for estimating the parameters
because they provide a solvable system. However individual characteristics in
an age group such as different income levels and clusters within are lost
during the moment estimation.

(a) $q_{a}$ with GMM

(b) $\sigma_{a}$ with GMM

(c) $\mu_{a}$ with GMM

Figure 4: $q_{a}$, $\sigma_{a}$ and $\mu_{a}$ Plots for USA Labour Income

LSM by individual transitions

To use LSM to approximate the parameters, one needs the individual income
transitions in consecutive years, thus identifying the same individual in
consecutive cohorts is necessary, the panel studies such as BHPS satisfy this
condition. The age-dependent income evolution function is fitted with
individual income transitions of consecutive years. The JDF of simulation has
concentrated heat regions around the mean, and the trend is decreasing unlike
the observed data. Imposing boundaries to the parameter space results in
better parameters, which results in consistent parameter plots and the
$\bar{y}_{a}$ plot of the simulation reflect similar shape with the observed
data Fig. 6 and Fig. 7. The JDF of the simulation on Fig.11 is able to reflect
the dispersion among various clusters better because unlike the other methods
heavily depending on the statistics such as mean and standard deviation of the
entire age group, the LSM utilises individual-level microdata.

The 95% confidence interval with 2000 bootstrap samples for the estimated
parameters from UK microdata by LSM can be found on Fig. 5

(a) $q_{a}$ with LSM

(b) $\sigma_{a}$ with LSM

(c) $\mu_{a}$ with LSM

Figure 5: $q_{a}$, $\sigma_{a}$ and $\mu_{a}$ Confidence Interval for UK Data
LSM Estimation

It is evident from the plots of $\bar{y}_{a}$ and $std(y_{a})$ for the
observed and simulated data that the model can capture the characteristics of
the income conditional on age distribution, and the characteristic stationary
property of this model can be observed on Fig. 6 and Fig. 7.

A close investigation of Fig. 6 and Fig. 7 on UK Labour Income Data suggests
that the GMM is most successful for reflecting the outcomes with similar mean
and standard deviation characteristics of all waves after simulation with 18
waves that were simulated with the parameters $q_{a}$, ${\sigma}_{a}$,
${\mu}_{a}$estimated by the GMM. But LSM reflects the individual trajectories,
and JDF more accurately.

(a) Observed Statistics
Figure 6: UK Labour Data Observed Statistics

(a) Simulation Statistics with GMM Estimation

(b) Simulation Statistics with LSM Estimation

Figure 7: UK Labour Data Simulation Statistics

The results showing the performance of GMM method is in Fig.8.

(a) Observed Statistics

(b) Simulation Statistics with GMM

Figure 8: USA Labour Data Observed and Simulation Statistics

A general analysis of the comparison of joint distribution of age and
inflation-adjusted income results in the following plots for weighted observed
data and simulated data in Fig.9 and Fig.10:

(a) Observed Statistics
Figure 9: UK Labour Data Observed All Waves JDF

(a) Simulation Statistics with GMM Estimation

(b) Simulation Statistics with LSM

Figure 10: UK Labour Data Simulation All Waves JDF

JDF of the simulated UK Labour data is in parallel with the expectations for
GMM Estimation method, consistent and stable, resembling a similar shape but
not concentrated for the heat regions with intense concentrations on Fig. 6
and Fig. 7.

The main differences between the observed and simulated JDFs are concentration
of the mass of the population between 23 and 50.

###  5.1 Wave-Specific analysis

(a) 1992 Observed Data

(b) 1995 Observed Data

(c) 2005 Observed Data

(d) 1992 with GMM Estimation

(e) 1995 with GMM Estimation

(f) 2005 with GMM Estimation

(g) 1992 with LSM Estimation

(h) 1995 with LSM Estimation

(i) 2005 with LSM Estimation

Figure 11: JDF Plots of Simulation for UK Labour Income

The population from wave-1 is used for bootstrapping the simulation and the
weights of the individuals are not incorporated to the simulation, because the
income evolution Func.(eq. 2) is the focus of this paper, and the main purpose
is not the perfect representativeness of the initial wave. The new agent
injection on 1999 by panel survey is reason of difference in the UK simulation
and observed JDF plots. Although the simulation’s initial state is
bootstrapped as the unweighted dataset, starting from the second wave, the JDF
of the simulated population(Fig. 11) resembles the characteristics of the JDF
from the panel survey with the weighted population, which reflects that the
model is successfully capturing income evolution dynamics.

###  5.2 A Simple Pension System

A financially sustainable pension system can be characterised by the balance
between inflow and outflow of funds. The specifics and stability of pension
system is out of the scope of this paper, and needs case specific detailed
modelling. For a general demonstration, we assume simple inflow and outflow
dynamics(Eq.5 and Eq.6), which are derived to represent statistical properties
of the savings and consumption. Figure 12 reflects the imbalance between
inflow and outflow, which results in a deficit.

Pension is assumed to be annually £16368, in light of the median net income
before housing costs for all pensioners from DWP Pensioners Income Series in
2008/2009 [16]. Constant alpha for pension saving rate is selected to be
0.0775 to 0.2, in light of OECD Pension Report statistics [17].

Outflow $O_{t}$ in a given year $t$ is characterised by constant annual
pension amount $p$, and count of people above 65 $c_{a>65}$ is assumed to be
pensioner counts.

$O_{t}=pc_{a>65}$ (5)

Inflow $I_{t}$ in a given year $t$ is characterised by constant pension
contribution rate $\alpha$ and total labour income of individuals $y_{a}$

$I_{t}=\alpha\sum_{i}(y_{a\leq 65})$ (6)

The amounts are adjusted for inflation and reflect the 2009 levels. The inflow
and outflow plots on Fig. 12 from our simplified generalisation of the pension
system reflect a deficit.

(a) Inflow & Outflow
Figure 12: UK Inflow Outflow Plot of our Simple Pension System

##  6 Discussion

The income evolution eq. 2 of the proposed model consists of the parameters
${q}_{a}$, ${\mu}_{a}$, ${\sigma}_{a}$: the persistence coefficient for the
respective age group ${q}_{a}$, determines the rate of mobility at a given
age.
Age-dependent mean income parameter ${\mu}_{a}$ expresses the expected age-
specific income evolution mean for the next income and behaves such that if
the mean parameter is high the mobility is most likely to have lower
${q}_{a}$. If the mean parameter ${\mu}_{a}$ is lower, the persistence
parameter is higher which signals a potential widening of the income gap for
the population.
${\sigma}_{a}$ captures the variability of the individuals according to
conditional distribution and incorporates randomness of the shocks.

The social safety nets, basic pension incomes and the Defined Benefit Pension
plans are financed via the working population; the ever-growing unbalance
towards ageing cohorts needs careful forecasting and planning. The demographic
shift will impact the economy’s functioning in general, introducing a heavy
burden to welfare states financing the health and pension of the retired
population, which will reflect society as taxes and benefit cuts. The best
course of action is forecasting the changes and planning in advance for the
future.

###  6.1 Interpretation

The $q_{a}$ mobility estimated by GMM reflects that UK population reflects an
initially high $q_{a}$ value in youth, followed by relative decrease, and then
a consistent increase. The $q$ values estimated by GMM fluctuate around 0 and
minimal. The income persistence variable of individuals is not captured by
GMM, which does not utilise panel survey’s tracked individual income micro-
data each year.

The LSM results in a consistently increasing $q_{a}$ value by the UK model,
with a significant jump between ages 25-30, which corresponds with a $\mu_{a}$
plot consistently decreasing with a significantly sharper decrease between
ages 25-30. $q_{a}$ and $\mu_{a}$ corresponding each other in an inverse
proportion, especially by significant changes, especially by LSM. There are
various examples of mobility that can be observed from BHPS dataset. Example
of ${q}_{a}$ mobility reflected by LSM in the UK:

  * •

One example of ${q}_{a}$ mobility is the upward mobility of age-group between
25 and 30, which is reflected by the increasing ${q}_{a}$ values and sharp
increase observed on the joint-distribution plot. This mobility can be due to
finishing higher education and internships, in addition few years of
experience, which results in a widening of income scissors. This change in
mobility is healthy for the economy and does not represent a negative effect.
One assumption should be researched further; if either this initial difference
in mobility might limit of people with lower income for upward mobility.

Example of ${\sigma}_{a}$ mobility:

  * •

An example of the ${\sigma}_{a}$ mobility is the age group of 30-35, which is
reflected by a locally sharp increase of ${\sigma}_{a}$ values. Such mobility
reflects a bidirectional movement of income for individuals, and such a
variation might arise from the short-time employment, interruption of
employment for education, temporary jobs and most importantly this mobility
might be caused by the initial differentiation according to the education of
individuals such as higher education or vocational education. This window
represents an increase in the variation of the income.

In general, the shape of the distribution can be explained in three periods;
the first period is the introduction to employment and teenagers, which
represents income from part-time and temporary jobs at the beginning and start
of full-time employment it sharply increases on Fig.2.

The age group of 25-55 denotes the main productive era of the economic life,
and the income reflects a high dispersion. All of the factors and random
shocks act together and result in dispersed but a consistent distribution.
Mobility wise this era provides opportunities for upward mobility and
possesses downward mobility risks. At the end of this period, income tends to
decrease slightly, which reflects a decrease in productivity. Another limiting
factor is the minimum wage and state benefits, which introduces a lower bound
envelope for the mass. Income sources and affecting factors of individuals in
this era vary greatly, which results in the widest dispersion in the entire
life-span. Some of the factors are education, social strata, adaptability to
innovation, total-work hours per week, experience and expertness, seniority of
the jobs and ageism. The third and final era represents the exit from the
workforce and retirement, and temporary or part-time jobs for low-income old
individuals. The income decreases gradually as the number of individuals
exiting workforce increases with time, the income stabilises, and variation
decreases significantly. Income in this era is relatively low, and the source
is usually pension benefits, state support or temporary jobs. This model’s
outcomes can be used for various purposes; the most apparent fields for
drawing consequences and planning are the works on inequality and mobility
depending on age. Characteristics of workforce entrance, work-efficiency of
individuals per age, the structure of the society, pension system, income
stability, and the taxation system are the most obvious fields.

In the paper, two main estimation techniques are investigated, and the
corresponding results from the simulated waves are presented. The first
estimation method investigated is GMM Estimation. The income regions appear
smoothed and spread. The second estimation method investigated is LSM, it
utilises the microdata and is suitable for capturing an agent’s income
evolution. The JDFs from the simulated waves have the most similar mean
characteristics to the observed data.

The LSM evidently performs better by utilizing longitudinal microdata; the GMM
estimation method can be applied to both population and panel surveys,
provides feasible distributions but with unrealistic modeling of an agent’s
individual income trajectory.

###  6.2 Conclusions

We demonstrated (1) a clear income-age relationship, which is reflected by the
data from BHPS and IPSUM CPS, as well as simulations. (2) a clear structure of
the joint age-income distribution in both the UK and USA. (3) a flexible
methodology to estimate parameters from population surveys, as well as panel
surveys. (4) a simple generative model to evolve the age-income population
with real constraints for evaluating general policy scenarios, that is
agnostic about occupation levels.

The model can be interpreted as delivering a premise that the information of
an individual’s experience and education can be encapsulated by income.
Although in early career, the income dynamics are governed by the initial
difference at the level of education and profession; the main dynamics
governing income transitioning can be reduced to the relationship between
income and age, which collectively encapsulate education and experience. These
premises can be leveraged for developing simplified models for evaluating
mobility, inequality, welfare state, and pensions.

The proposed model focuses on the evolution of age and income population and
the paper successfully demonstrates a simple model that can be calibrated for
age and income that can be used as a backbone for forecasting income and
planning pensions. Understanding the dynamics and having the ability to
forecast the age and income population is the key to the design of financially
sustainable pension systems.

There are different dimensions for the future work: one of the dimensions is
injecting random shocks to the distributions itself, which can be in the form
of new population injection or withdrawal, as well as tuning the $\eta_{it}$
with various means for simulating a global or regional shock, such as
pandemics or mass migration. Stress-testing the age and income distribution
for different labour market scenarios could lead to relevant policy
implications. The second dimension for future work is modifying the simulation
system to estimate parameters on the fly, and provide a more adaptive and
granular version of the simulation system. The third dimension for future work
is incorporating data encompassing more years and more countries and with a
higher resolution in time to investigate the role of multiple economic factors
for short, medium and long time horizons.

## References

  * [1] Champernowne, D.G.: A model of income distribution. The Economic Journal 63(250), 318–351 (1953). doi:10.2307/2227127
  * [2] Shorrocks, A.F.: Income mobility and the markov assumption. The Economic Journal 86(343), 566–578 (1976). doi:10.2307/2230800
  * [3] Shorrocks, A.: Income inequality and income mobility. Journal of Economic Theory 19(2), 376–393 (1978). doi:10.1016/0022-0531(78)90101-1
  * [4] Lillard, L.A., Willis, R.J.: Dynamic aspects of earnings mobility. Report 0898-2937, National Bureau of Economic Research (1976)
  * [5] Buchinsky, M., Hunt, J.: Wage mobility in the united states. The Review of Economics and Statistics 81(3), 351–368 (1999)
  * [6] Cappellari, L., Jenkins, S.P.: Modelling low income transitions. Journal of applied econometrics 19(5), 593–610 (2004)
  * [7] Kopczuk, W., Saez, E., Song, J.: Earnings inequality and mobility in the united states: Evidence from social security data since 1937*. The Quarterly Journal of Economics 125(1), 91–128 (2010). doi:10.1162/qjec.2010.125.1.91
  * [8] Firpo, S., Fortin, N.M., Lemieux, T.: Unconditional quantile regressions. Econometrica 77(3), 953–973 (2009)
  * [9] Firpo, S., Fortin, N.M., Lemieux, T.: Occupational tasks and changes in the wage structure. IZA Discussion Papers (2011)
  * [10] Guvenen, F.: An empirical investigation of labor income processes. Review of Economic Dynamics 12(1), 58–79 (2009). doi:10.1016/j.red.2008.06.004
  * [11] Guvenen, F., Kuruscu, B., Ozkan, S.: Taxation of human capital and wage inequality: A cross-country analysis. Review of Economic Studies 81(2), 818–850 (2014)
  * [12] University Of Essex, I.F.S., Research, E.: BHPS British Household Panel Survey: Waves 1-18, 1991-2009. UK Data Service (2018). doi:10.5255/UKDA-SN-5151-2. https://beta.ukdataservice.ac.uk/datacatalogue/doi/?id=5151#2
  * [13] Flood, S., King, M., Rodgers, R., Ruggles, S., Warren, J.R.: Integrated Public Use Microdata Series, Current Population Survey: Version 7.0. Minneapolis, MN: IPUMS (2020). doi:10.18128/D030.V7.0. https://www.ipums.org/projects/ipums-cps/d030.V7.0
  * [14] Taylor, N.B. Marcia Freed (ed). with John Brice, Prentice-Lane, E.: British Household Panel Survey User Manual Volume A: Introduction, Technical Report and Appendices. Colchester: University of Essex (2018)
  * [15] Office for National Statistics: Population Estimates for UK, England and Wales, Scotland and Northern Ireland. National Statistics (2018). https://data.gov.uk/dataset/849f9984-dfe4-46a5-8162-c5dee3f19ea4/population-estimates-for-uk-england-and-wales-scotland-and-northern-ireland
  * [16] Evans, J., Robinson, H.: The Pensioners’ Incomes Series 2008-09. Pensions, Department for Work and Pensions (2010). http://statistics.dwp.gov.uk/asd/index.php?page=pensioners_income_arc#PI_Prev
  * [17] Holzmann, R., Stiglitz, J.E.: New ideas about old age security : toward sustainable pension systems in the 21st century. World Bank (2001)

##  7 Supplementary Material

###  7.1 Model Calibration

We can define the mean and standard deviation of income at a given age $a$ as
following:

$\left(\bar{y}_{a},std(y_{a})\right)$ (7) $\langle
y_{a,t}^{i}\rangle=\bar{y}_{a}$ (8)

The standard deviation and mean has the following relation with the squared
average of incomes:

$\langle\left(y_{a,t}^{i}\right)^{2}\rangle-\left(\bar{y}_{a}\right)^{2}=\left(std(y_{a})\right)^{2}$
(9)

$\eta_{it}$ has characteristics of the standard normal distribution:

$\langle\eta_{it}\rangle=0$ (10) $\langle\eta_{it}^{2}\rangle=1$ (11)
$\langle\eta_{it}^{3}\rangle=0$ (12)

Squaring both sides of income evolution equation (2) results in following
distribution:

$\left(y_{a+1,t+1}^{i}\right)^{2}=\left(q_{a}y_{a,t}^{i}+\mu_{a}+\sigma_{a}\eta_{t}^{i}\right)^{2}$
(13)

Eq.(9) can be formalized as:

$\left(\bar{y}_{a}\right)^{2}+\left(std(y_{a})\right)^{2}=\langle\left(y_{a,t}^{i}\right)^{2}\rangle$
(14)

Placing Eq.(14) for $a+1$ and Eq.(13) results in following equation:

$\left(\bar{y}_{a+1}\right)^{2}+\left(std(y_{a+1})\right)^{2}=\langle\left(q_{a}y_{a,t}^{i}+\mu_{a}+\sigma_{a}\eta_{t}^{i}\right)^{2}\rangle$
(15)

Expanding the right side of the equation results in:

$\left(\bar{y}_{a+1}\right)^{2}+\left(std(y_{a+1})\right)^{2}=\langle\left(q_{a}y_{a,t}^{i}\right)^{2}+\left(\mu_{a}+\sigma_{a}\eta_{t}^{i}\right)^{2}+2\left(q_{a}y_{a,t}^{i}\right)\left(\mu_{a}+\sigma_{a}\eta_{t}^{i}\right)\rangle$
(16)
$\left(\bar{y}_{a+1}\right)^{2}+\left(std(y_{a+1})\right)^{2}=\langle\left(q_{a}y_{a,t}^{i}\right)^{2}+\left(\mu_{a}+\sigma_{a}\eta_{t}^{i}\right)^{2}+2\left(q_{a}y_{a,t}^{i}\right)\left(\mu_{a}+\sigma_{a}\eta_{t}^{i}\right)\rangle$
(17)
$=\langle\left(q_{a}y_{a,t}^{i}\right)^{2}+\left(\mu_{a}\right)^{2}+\left(\sigma_{a}\eta_{t}^{i}\right)^{2}+2\left(\mu_{a}\sigma_{a}\eta_{it}\right)+2\left(q_{a}y_{a,t}^{i}\mu_{a}+\left(q_{a}y_{a,t}^{i}\right)\sigma_{a}\eta_{t}^{i}\right)\rangle$
(18)

Averaging the equation by using Eq.(14), Eq.(10) and Eq.(11).

$\left(\bar{y}_{a+1}\right)^{2}+\left(std(y_{a+1})\right)^{2}=\left(q_{a}\right)^{2}\left(\left(\bar{y}_{a}\right)^{2}+\left(std(y_{a})\right)^{2}\right)+\left(\mu_{a}\right)^{2}+\left(\sigma_{a}\right)^{2}+2q_{a}\mu_{a}\bar{y}_{a}$
(19)

###  7.2 Deriving The Update Equations

For clarity $\left(\bar{y}_{a}\right)^{2}+\left(std(y_{a})\right)^{2}$ is
expressed as $(\Delta_{a})^{2}$, The number of parameters can be reduced to 2
using the third parameter of Eq.(19) by expressing $\mu_{a}$ as
$\bar{y}_{a+1}-q_{a}\bar{y}_{a}$ according to Eq.(4):

$\left(\Delta_{a+1}\right)^{2}=\left(q_{a}\right)^{2}\left(\Delta_{a}\right)^{2}+\left(\mu_{a}\right)^{2}+\left(\sigma_{a}\right)^{2}+2q_{a}\mu_{a}\bar{y}_{a}$
(20)
$\left(\Delta_{a+1}\right)^{2}=(q_{a})^{2}\left(\Delta_{a}\right)^{2}+\left(\bar{y}_{a+1}-q_{a}\bar{y}_{a}\right)^{2}+\sigma_{a}^{2}+2q_{a}\left(\bar{y}_{a+1}-q_{a}\bar{y}_{a}\right)\bar{y}_{a}$
(21)

unpacking $\Delta$:

$\left(\bar{y}_{a+1}\right)^{2}+\left(std(y_{a+1})\right)^{2}=q_{a}^{2}\left((\bar{y}_{a})^{2}+(std(y_{a}))^{2}\right)+(\bar{y}_{a+1})^{2}+\left(q_{a}\bar{y}_{a}\right)^{2}-2\left(\bar{y}_{a+1}q_{a}\bar{y}_{a}\right)+\sigma_{a}^{2}+2q_{a}\bar{y}_{a}\bar{y}_{a+1}-2(q_{a})^{2}(\bar{y}_{a})^{2}$
(22)

expressions at both sides of the equation cancel each other and simplify as
follows:

$\left(std(y_{a+1})\right)^{2}=q_{a}^{2}(std(y_{a}))^{2}+\left(\sigma_{a}\right)^{2}$
(23)

solving in quadratic equation form:

$0=q_{a}^{2}(std(y_{a}))^{2}+\left(\sigma_{a}\right)^{2}-(std(y_{a+1}))^{2}$
(24)

for
$\left(-(\sigma^{a})^{2}\left(\left(\tilde{\sigma}^{a}\right)^{2}-(\sigma^{a+1})^{2}\right)\right)>0$
and $(\sigma^{a})^{2}>0$, $\tilde{q}$ values can be solved as follows:

$\tilde{q}_{1}^{a}=\frac{\sqrt{-(\sigma^{a})^{2}\left(\left(\tilde{\sigma}^{a}\right)^{2}-(\sigma^{a+1})^{2}\right)}}{(\sigma^{a})^{2}}$
(25)
$\tilde{q}_{2}^{a}=\frac{-\sqrt{-(\sigma^{a})^{2}\left(\left(\tilde{\sigma}^{a}\right)^{2}-(\sigma^{a+1})^{2}\right)}}{(\sigma^{a})^{2}}$
(26)

Following equations are used in the method of GMM:

Using unnormalized unstandardized third moment of the Equation 2

$E\left[\left(y_{a+1}\right)^{3}\right]=E\left[\left(q_{a}y_{a}+\mu_{a}+\sigma_{a}\eta\right)^{3}\right]$
(27)

Expanding the cube equation

$\begin{array}[]{l}E\left[\left(y_{a+1}\right)^{3}\right]=\\\ \\\
E\biggl{[}\left(q_{a}y_{a}\right)^{3}+\left(\mu_{a}\right)^{3}+\left(\sigma_{a}{\eta}^{a}\right)^{3}+\left(6q_{a}y_{a}\mu_{a}\sigma_{a}\eta\right)+3\left(q_{a}y_{a}\right)^{2}\sigma_{a}\\\
+3\left(q_{a}y_{a}\right)^{2}\mu_{a}+3\left(\mu_{a}\right)^{2}q_{a}y_{a}+3\left(\mu_{a}\right)^{2}\sigma_{a}\eta+3\left(\sigma_{a}\right)^{2}\mu_{a}+3\left(\sigma_{a}\right)^{2}q_{a}y_{a}\biggr{]}\end{array}$
(28)

Using Eq.12 , $\left(\sigma_{a}{\eta}^{a}\right)^{3}$, $\eta^{3}$ equals zero

$\begin{array}[]{l}E\left[\left(y_{a+1}\right)^{3}\right]=\\\
\left(q_{a}\right)^{3}E\left[\left(y_{a}\right)^{3}\right]+\left(\mu_{a}\right)^{3}+3\left(q_{a}\right)^{2}\mu_{a}E\left[\left(y_{a}\right)^{2}\right]+3\left(\mu_{a}\right)^{2}q_{a}E\left[y_{a}\right]+\\\
3\left(\sigma_{a}\right)^{2}\mu_{a}+3\left(\sigma_{a}\right)^{2}q_{a}E\left[y_{a}\right]\end{array}$
(29) $\begin{array}[]{l}E\left[\left(y_{a+1}\right)^{3}\right]=\\\
\left(q_{a}\right)^{3}E\left[\left(y_{a}\right)^{3}\right]+\left(\mu_{a}\right)^{3}+3\mu_{a}\left(\left(q_{a}\right)^{2}E\left[\left(y^{a}\right)^{2}\right]+\left(\sigma_{a}\right)^{2}\right)+3q_{a}E\left[y_{a}\right]\left(\left(\mu_{a}\right)^{2}+\left(\sigma_{a}\right)^{2}\right)\end{array}$
(30)

Expressing $\left(\sigma_{a}\right)^{2}$ from Eq.(23) in terms of $q_{a}$

$\begin{array}[]{l}E\left[\left(y_{a+1}\right)^{3}\right]=\\\
\left(q_{a}\right)^{3}E\left[\left(y_{a}\right)^{3}\right]+\left(\mu_{a}\right)^{3}+3\mu_{a}\left(\left(q_{a}\right)^{2}E\left[\left(y_{a}\right)^{2}\right]+\left(std(y_{a+1})\right)^{2}-\left(q_{a}\right)^{2}\left(std(y_{a})\right)^{2}\right)\\\
+3q_{a}E\left[y_{a}\right]\left(\left(\mu_{a}\right)^{2}+\left(std(y_{a+1})\right)^{2}-\left(q_{a}\right)^{2}\left(std(y_{a})\right)^{2}\right)\end{array}$
(31)

Replacing $E\left[y_{a}\right]=\bar{y}_{a}$ and
$E\left[\left(y_{a}\right)^{2}\right]=\left(std(y_{a})\right)^{2}+\left(\bar{y}_{a}\right)^{2}$
from Eq.9

$\begin{array}[]{l}E\left[\left(y_{a+1}\right)^{3}\right]=\\\
\left(q_{a}\right)^{3}E\left[\left(y_{a}\right)^{3}\right]+\left(\mu_{a}\right)^{3}+3\mu_{a}\left(q_{a}\right)^{2}\left(\bar{y}_{a}\right)^{2}\\\
+3\mu_{a}\left(std(y_{a+1})\right)^{2}+3q_{a}\bar{y}_{a}\left(\mu_{a}\right)^{2}+3\mu_{a}\bar{y}_{a}\left(std(y_{a+1})\right)^{2}-3\left(q_{a}\right)^{3}\bar{y}_{a}\left(std(y_{a})\right)^{2}\end{array}$
(32)

Expressing $\mu_{a}$ from Eq.(4) in terms of $q_{a}$

$\begin{array}[]{l}E\left[\left(y_{a+1}\right)^{3}\right]=\\\
\left(q_{a}\right)^{3}E\left[\left(y_{a}\right)^{3}\right]+\left(\bar{y}_{a+1}-q_{a}\bar{y}_{a}\right)^{3}+3\left(\bar{y}_{a+1}-q_{a}\bar{y}_{a}\right)\left(std(y_{a+1})\right)^{2}\\\
+3q_{a}\bar{y}_{a}\left(\bar{y}_{a+1}-q_{a}\bar{y}_{a}\right)^{2}+3q_{a}\bar{y}_{a}\left(std(y_{a+1})\right)^{2}-3\left(q_{a}\right)^{3}\bar{y}_{a}\left(std(y_{a})\right)^{2}\end{array}$
(33)

Expressing in the form of cubic polynomial equation of $q_{a}$

$\begin{array}[]{l}0=\\\
\left(q_{a}\right)^{3}\left(E\left[\left(y_{a}\right)^{3}\right]-\left(\bar{y}_{a}\right)^{3}-3\bar{y}_{a}\left(std(y_{a})\right)^{2}\right)\\\
+\left(q_{a}\right)^{2}\left(3\bar{y}_{a+1}\left(\bar{y}_{a}\right)^{2}-6\left(\bar{y}_{a}\right)^{2}\bar{y}_{a+1}\right)\\\
+\left(q_{a}\right)\left(3\left(\mu_{a+1}\right)^{2}\bar{y}_{a}-3\bar{y}_{a}\left(std(y_{a+1})\right)^{2}+3\bar{y}_{a}\left(\bar{y}_{a+1}\right)^{2}+3\bar{y}_{a}\left(std(y_{a+1})\right)^{2}\right)\\\
+\left(\bar{y}_{a+1}\right)^{3}+3\bar{y}_{a+1}\left(std(y_{a+1})\right)^{2}-E\left[\left(y_{a+1}\right)^{3}\right]\end{array}$
(34)

This equation can be solved for $q_{a}$ corresponding each age group. Cardano
solution for cubic equations guarantees single real root to exist, the other
two complex roots that Cardano solution provides are not used. Both of the
$\sigma_{a}$ = $std(y_{a})$ and GMM estimation techniques can use the
following equations for determining the $\mu_{a}$ and $\sigma_{a}$: For
$(q_{a})_{1}$ and $(q_{a})_{2}$ according to Eq.(4):

$\mu_{a}=\bar{y}_{a+1}-q_{a}\bar{y}_{a}$ (35)

The $\sigma_{a}^{2}$ can also be expressed in terms of $q_{a}$, using Eq.(4) :

$\sigma_{a}^{2}=\left(\Delta_{a+1}\right)^{2}-q_{a}^{2}\left(\Delta_{a}\right)^{2}-\left(\bar{y}_{a+1}-q_{a}\mu^{2}\right)^{2}-2q_{a}\left(\bar{y}_{a+1}-q_{a}\bar{y}_{a}\right)\bar{y}_{a}$
(36)

###  7.3 Analysis BHPS - Joint Distribution of Age and Income for Observed and
Simulated Data

The parameters are estimated with LSM.

(a) Wave 1991 JDF of Observed Data.

(b) Wave 1991 JDF of Sim Data

(c) Wave 1992 JDF of Observed Data.

(d) Wave 1992 JDF of Sim Data

(e) Wave 1993 JDF of Observed Data.

(f) Wave 1993 JDF of Sim Data

(g) Wave 1994 JDF of Observed Data.

(h) Wave 1994 JDF of Sim Data

(i) Wave 1995 JDF of Observed Data.

(j) Wave 1995 JDF of Sim Data

Figure 13: JDF for Waves 1991-1995

(a) Wave 1996 JDF of Observed Data.

(b) Wave 1996 JDF of Sim Data

(c) Wave 1997 JDF of Observed Data.

(d) Wave 1997 JDF of Sim Data

(e) Wave 1998 JDF of Observed Data.

(f) Wave 1998 JDF of Sim Data

(g) Wave 1999 JDF of Observed Data.

(h) Wave 1999 JDF of Sim Data

(i) Wave 2000 JDF of Observed Data.

(j) Wave 2000 JDF of Sim Data

Figure 14: JDF for Waves 1995-2000

(a) Wave 2001 JDF of Observed Data.

(b) Wave 2001 JDF of Sim Data

(c) Wave 2002 JDF of Observed Data.

(d) Wave 2002 JDF of Sim Data

(e) Wave 2003 JDF of Observed Data.

(f) Wave 2003 JDF of Sim Data

(g) Wave 2004 JDF of Observed Data.

(h) Wave 2004 JDF of Sim Data

(i) Wave 2005 JDF of Observed Data.

(j) Wave 2005 JDF of Sim Data

Figure 15: JDF for Waves 2001-2005

(a) Wave 2006 JDF of Observed Data.

(b) Wave 2006 JDF of Sim Data

(c) Wave 2007 JDF of Observed Data.

(d) Wave 2007 JDF of Sim Data

(e) Wave 2008 JDF of Observed Data.

(f) Waves 2008 JDF of Sim Data

Figure 16: JDF for Waves 2005-2008