THE MACROECONOMIC CONSEQUENCES OF POVERTY AND
INEQUALITY
by
JEFFREY ALLEN
A DISSERTATION
Presented to the Department of Economics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2014
DISSERTATION APPROVAL PAGE
Student: Jeffrey Allen
Title: The Macroeconomic Consequences of Poverty and Inequality
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Economics
by:
Shankha Chakraborty Chair
George Evans Core Member
Bruce McGough Core Member
David Levin Institutional Representative
and
Kimberly Andrews Espy Vice President for Research & Innovation/
Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2014
ii
c© 2014 Jeffrey Allen
iii
DISSERTATION ABSTRACT
Jeffrey Allen
Doctor of Philosophy
Department of Economics
June 2014
Title: The Macroeconomic Consequences of Poverty and Inequality
This dissertation examines the macroeconomic effects of poverty and inequality.
The second chapter considers the effect of poverty and subsistence consumption
constraints on economic growth in a two-sector occupational choice model. I find
that in the presence of risk taking, subsistence consumption constraints result in a
dramatic slow down in terms of economic growth. The third chapter (joint with
Shankha Chakraborty) proposes a model in which agents face endogenous mortality
and direct preferences over inequality. I find that the greater the scale of relative
deprivation the worse the mortality outcomes are for individuals. The fourth chapter
looks at the relationship between inequality and the demand for redistribution when
individuals have social status concerns. I show that under social status concerns an
increase in consumption inequality results in higher taxation and lower growth.
This dissertation includes unpublished coauthored material.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Jeffrey Allen
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene
University of California, Riverside
DEGREES AWARDED:
Doctor of Philosophy, Economics—University of Oregon, Eugene, 2014
Bachelor of Arts Economics—University of California, Riverside, 2009
AREAS OF SPECIAL INTEREST:
Economic Growth and Development
Macroeconomics
International Economics
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow—University of Oregon, 2009-2014
v
ACKNOWLEDGEMENTS
I would like to thank Shankha Chakraborty, Bruce McGough, George Evans,
David Levin, Robert Yelle, Craig Rasmussen, the participants of the UO Macro
group, the Northwest Development Conference (2012), and the Midwest Macro
Meetings (2013) for their comments and insights. The research was supported by a
Major Research Instrumentation grant from the National Science Foundation, Office
of Cyber Infrastructure, Principal Investigator Allen Malony, “MRI-R2: Acquisition
of an Applied Computational Instrument for Scientific Synthesis (ACISS),” Grant
#: OCI-0960354.
vi
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. SUBSISTENCE CONSUMPTION, OCCUPATIONAL CHOICE, AND
THE COST OF POVERTY . . . . . . . . . . . . . . . . . . . . . . 4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Quantitative Results . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
III. INEQUALITY AS A HEALTH HAZARD . . . . . . . . . . . . . . . . 40
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
vii
Chapter Page
IV. ASPIRATIONS AND REDISTRIBUTION . . . . . . . . . . . . . . . . 72
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Computational Results . . . . . . . . . . . . . . . . . . . . . . . . 89
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
APPENDICES
A. SUBSISTENCE CONSUMPTION . . . . . . . . . . . . . . . . . . . 98
Capital Shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Robustness Check . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Social Planner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
B. INEQUALITY AS A HEALTH HAZARD . . . . . . . . . . . . . . . 107
Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . 108
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
viii
LIST OF FIGURES
Figure Page
1. The evolution of the percentage of the workforce employed in the
modern sector across time for the baseline calibration . . . . . . . . 27
2. The evolution of growth across time for the baseline calibration . . . . 29
3. The Consequences of Poverty . . . . . . . . . . . . . . . . . . . . . . . 30
4. The Gini Coefficient for income plotted across time for different levels
of subsistence consumption . . . . . . . . . . . . . . . . . . . . . . . 32
5. The Gini Coefficient for income plotted across time for different
underlying productivity distributions. . . . . . . . . . . . . . . . . . 33
6. The Effects of an Unconditional Cash Transfer . . . . . . . . . . . . . . 36
7. The Effects of a Conditional Cash Transfer . . . . . . . . . . . . . . . . 38
8. Agent i’s Decision Timeline . . . . . . . . . . . . . . . . . . . . . . . . 47
9. Labor Choice vs Aspirations Gap by Health Stock . . . . . . . . . . . . 57
10. Health Good vs Aspirations Gap by Health Stock . . . . . . . . . . . . 58
11. Percent Change in Health Stock vs Aspirations Gap by Health . . . . . 59
12. Savings vs Aspirations Gap by Health Stock . . . . . . . . . . . . . . . 59
13. Total Savings vs Aspirations Gap by Health Stock . . . . . . . . . . . . 60
14. Health Stock vs Aspirations Gap by Age . . . . . . . . . . . . . . . . . 61
15. The influence of Aspirations on Labor Supply, Health Good and the
Change in the Health Stock . . . . . . . . . . . . . . . . . . . . . . 62
16. The Evolution of Individual Asset and Health Stocks . . . . . . . . . . 63
17. Fundamental Inequality Plotted Against Income Inequality . . . . . . . 64
18. Consumption Ratios plotted against Fundamental Inequality . . . . . . 65
19. Health Stock as a Function of Income . . . . . . . . . . . . . . . . . . . 65
ix
Figure Page
20. Life Expectancy versus Inequality: Rising Income . . . . . . . . . . . . 68
21. Life Expectancy versus Inequality: Changing Medical Technology . . . 68
22. Life Expectancy Gap vs Inequality . . . . . . . . . . . . . . . . . . . . 70
23. Lifetime Utility Plotted Against the Tax Rate . . . . . . . . . . . . . . 89
24. Relationship between the level of aspirations and desired taxation for
workers and entrepreneurs . . . . . . . . . . . . . . . . . . . . . . . 90
25. Aspirations Gap vs the Desired Rate of Taxation for workers and
entrepreneurs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
26. Aspirations Gap vs the Desired Rate of Taxation for workers and the
lowest ability entrepreneur . . . . . . . . . . . . . . . . . . . . . . . 93
27. The entrepreneurial cut-off and total income differential plotted against
ability inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
28. The Growth Rate of Technology plotted against Ability Inequality for
a given tax rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
29. Aspirations Gap vs the Desired Rate of Taxation for when the median
voter is a worker and when the median voter is an entrepreneur . . 96
30. The evolution of the percentage of the workforce employed in the
modern sector across time for the baseline calibration . . . . . . . . 99
31. The evolution of the percentage of the workforce employed in the
modern sector across time for the robustness check for p . . . . . . 101
32. Policy Function: a˜it+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
33. Mean lifetime utility associated with each social planner plan . . . . . . 106
x
LIST OF TABLES
Table Page
1. Capital Shares By Income Group . . . . . . . . . . . . . . . . . . . . . 24
2. Welfare Analysis–Growth Rate Change (Baseline) . . . . . . . . . . . . 34
3. Welfare Analysis–Growth Rate Change (Cost of Poverty) . . . . . . . . 34
4. Welfare Analysis–Change in Growth Rate: Taxation . . . . . . . . . . . 39
5. Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6. Data: Life Expectancy and Inequality . . . . . . . . . . . . . . . . . . . 67
7. Model: Life Expectancy and Inequality . . . . . . . . . . . . . . . . . . 69
8. Capital Shares by Country . . . . . . . . . . . . . . . . . . . . . . . . . 98
9. Welfare Analysis–Loss in Growth Rates . . . . . . . . . . . . . . . . . . 100
10. Welfare Analysis–Loss in Growth Rates . . . . . . . . . . . . . . . . . . 101
11. Social Planner Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 105
12. Social Planner Modern Sector Allocation . . . . . . . . . . . . . . . . . 105
xi
CHAPTER I
INTRODUCTION
The topics of poverty and inequality have been of great interest to economists
for many years. This dissertation focuses on the macroeconomic effects of the
heterogenous behavior that results from poverty and inequality.
While the microeconomic consequences of poverty have been well documented in
both empirical and theoretical studies, the macroeconomic consequences, especially
on economic growth, are not as well understood. The fundamental problem is that
most growth models are highly aggregated or assume a homogenous population.
Also, these same models tend to assume away risk faced by individuals, which is of
particular concern when discussing issues faced by the poorest members of society.
Therefore, their relevance for developing countries with problems of acute poverty is
suspect at best.
In contrast to the notion of poverty, inequality is by definition a macroeconomic
outcome. The empirical and theoretical literature, typically looks at broad measures
of inequality (e.g. the Gini coefficient) to understand how unequal distributions
affect aggregate outcomes like growth, through prices and market access. These
studies neglect the impact that inequality has on individual decisions when agents
care directly about where they sit in the distribution. I endow individuals with
preferences directly over inequality and by doing so I am able to investigate how
inequality impacts macroeconomic outcomes like health, growth, and the demand
for redistribution.
1
This dissertation is organized into four chapters. The second chapter deals with
the impact poverty has on economic growth while the third and fourth investigate
how inequality influences aggregate outcomes.
The second chapter presents a two-sector occupational choice model in which the
extent and depth of poverty influence the aggregate outcome of the economy. Agents
are bound by a subsistence consumption constraint and must face a risky move to find
employment in the modern sector. The modern sector experiences learning-by-doing
productivity growth that depends on the number of workers it employs. Agents’ risk
aversion is negatively related to their distance from the subsistence constraint. While
there is no poverty trap and every one eventually ends up employed in the modern
sector, the greater the extent of poverty the longer it takes for this equilibrium to be
achieved.
The third chapter (joint work with Shankha Chakraborty) investigates a
variant of Blanchard-Yaari’s model of perpetual youth, in which agents have social
aspirations that influence their decisions and an endogenous probability of survival
that is determined by their health stock. This model is applied to the empirical
literature on health and inequality in an attempt to explain two results: relative
deprivation is detrimental to health and that the correlation between life expectancy
and inequality has been weakening over time. The aggregate results between
inequality and life expectancy show that the weakening correlation between these
two variables can be explained by increases in income.
The fourth chapter of this dissertation presents a model that examines the
relationship between inequality and the demand for redistribution. Much like the
third chapter, agents are endowed with social aspirations that are determined by
looking up the distribution of consumption. Each agent lives two periods and has a
2
single offspring. Individuals have preferences over consumption and leisure and earn
income from either running a firm as an entrepreneur or working for a wage. The
government provides services which are financed through taxation, where the tax
rate is determined by a median voter. I show that under social concerns, an increase
in consumption inequality results in a higher tax rate on entrepreneurs. This higher
tax rate results in lower aggregate growth because it reduces the returns associated
with entrepreneurship.
3
CHAPTER II
SUBSISTENCE CONSUMPTION, OCCUPATIONAL CHOICE, AND THE
COST OF POVERTY
Introduction
The consequences of poverty for an individual go far beyond having low levels
of consumption. Numerous studies have documented the adverse social, emotional,
biological, and intellectual effects of poverty. This paper focuses on one of the
behavioral consequences of poverty: increased risk aversion. Although risk is a
universal condition of humanity, its effects on an individual’s standard of living
are far from uniform. Risk and its consequences differ not only from profession to
profession, but also from country to country and between levels of income. In this
chapter I will examine how individuals’ risk preferences influence aggregate outcomes
like economic growth.
In developed countries, there are social safety nets or mechanisms that insure
individuals against bad shocks (unemployment insurance, social security, etc). These
safety nets or mechanisms are often lacking in developing countries. For instance,
lower than expected amount of rainfall can usually be dealt with in a developed
nation through the use of complex irrigation systems; in developing regions where
rainfall is the only source of water for farmers, a shortage of water can be devastating.
Negative income shocks are particularly hard on individuals who are near
subsistence. A loss of income will result in a reduction of already low levels
of consumption. This could have many consequences, ranging from a loss of
productivity due to malnutrition to outright starvation. Therefore, “gambles” taken
4
on by individuals near the subsistence level of consumption are far riskier than
for those who are not bound by this constraint. This implies that not only do
the poor face more costly risk, but also that their attitudes toward risk will most
likely differ from those in developed countries. If growth enhancing activities, like
modern production techniques, embody risk, the poor will spend less time taking
these actions.
The model presented in this chapter is a dual-sector occupational choice
framework in which the agents choose between two technologies to produce a unique
final good. The focus will be on whether the depth of poverty has any influence
over the rate of structural change out of “traditional” modes of production into
“modern” ones. It is important to note that there is no need for a relative price
in this framework because there is only a shift in the way goods are produced not
the composition of the goods that are produced. Because there is only one good
in the model the increased production from the modern sector does not result in
any relative price differences. Growth is determined endogenously by the number of
workers in the modern sector through learning-by-doing.1 This model differs from
the standard endogenous growth framework because it allows for the agent’s risk
preferences to influence the aggregate outcomes.2 The main consequence of this is
that the distribution of income plays a role in the evolution of the economy.
This dual-economy framework has a long history in the literature, originating
with Lewis’ (1954) seminal work on the transition from traditional to modern
production processes. This literature, much like this paper, contrasts a low
1This is similar to Matsuyama (1992), who depicts a two sector model with agriculture and
manufacturing where growth in the manufacturing sector is given by a learning-by-doing externality.
2While it is true that agents do not have assets to lose, the consequence of falling below the
subsistence level (death) is enough to ensure that agents do not take advantage of limited liability.
5
productivity, low growth traditional (or subsistence) sector with high productivity,
high growth modern sector. More recent examples of this include Temple (2005)
and Vollrath (2009). Both papers argue that dual-economies need to feature more
prominently in the discussion of developing countries because they can address issues
that the standard neoclassical model cannot. Also related to this paper are the
models that stem from Harris-Todaro (1970). Like this paper, Harris-Todaro look
at a binary occupational choice in terms of a migration decision in which agents
chose between rural/informal and urban/formal employment. Recently, Bryan et al
(2013) used a partial equilibrium version of the Harris-Todaro model with subsistence
requirements to examine the decision to temporarily migrate. Their experimental
evidence shows that subsistence requirements and the risks associated with migration
significantly lowered an agent’s willingness to migrate.
In the model presented below agents have preferences over their own
consumption and the size of the financial bequest given to their single offspring.
Their decisions are constrained by a subsistence consumption requirement that enters
into the utility function. Agents earn income from wages and the return from their
assets. The distribution of the work force is determined stochastically and workers
in the modern sector of production must be skilled. Therefore, in order to enter the
modern sector agents must pay a training cost, at which point they are stochastically
matched with a job. If the agent is not matched in the modern sector she returns to
production using traditional methods.3
This chapter has four main results. First, the presence of the subsistence
requirement results in a loss in the generational growth rate of between 4 and 32
3Given that this is an OLG framework, the assumption that the individual has to fall back to the
traditional sector after one failed attempt to enter the modern sector may seem extreme. However,
Banerjee (1983) provides evidence in the case of migration that an agent who takes up employment
in urban traditional sector is unlikely to move to the modern sector in the following period.
6
percentage points. This manifests in a reduction of output of 2-17% over the course
of a generation.
Secondly, in this dynamic environment, unconditional cash transfers are strictly
dominated by conditional cash transfers in terms of their impact on growth rates.
In fact, funding an unconditional transfer through taxation results in a reduction of
the generational growth rate by 2 to 9 percentage points, while funding the same
transfer through aid dollars results in an increase of between 0.9 and 1.8 percentage
points. In contrast, conditional cash transfers have a positive impact on generational
growth rates of between 2 and 6 percentage points.
Third, if the modern sector is operating, it will eventually absorb all
employment. However, the speed of convergence to this equilibrium depends
negatively on the distance between the agents’ consumption and the subsistence
level. Moving to the modern sector is a risky endeavor, therefore the poorer agents
are, the less likely they are to undertake it.
Note that the implication of this result is that there is no poverty trap. This
finding is in contrast to a large literature that shows poverty traps can arise in
the presence of incomplete borrowing markets, warm-glow bequest motives, and
indivisible investments.4 My model avoids a poverty trap by making the indivisible
investment affordable to all agents and allowing for technological change that
influences wages in both sectors. This technological progress allows for poor agents
to increase their incomes enough so that they find it optimal to pay the fixed cost.
Finally, an increase in the depth of poverty (poverty gap) measured by the
average consumption gap increases convergence time to full employment in the
4For examples see: Galor and Zeira (1993), Ghatak and Jiang (2002), and Mookherjee and Ray
(2003).
7
modern sector. When there are fewer “rich” individuals, there will be fewer people
working in the modern sector, which slows technological advancement.
Overall, the results of the model confirm this paper’s premise: if the acquisition
of technology (or any growth enhancing activity) is risky, poorer agents will undertake
it at a lower rate. Or in a more aggregate sense, the extent and depth of poverty is
negatively related to growth outcomes.
This paper is related to four literatures: subsistence consumption, risk-taking,
barriers to technological adoption and structural change. Quite a bit of the literature
dealing with subsistence consumption deals with what Schultz (1953) referred to
as the “food problem.” Two papers here are particularly relevant. Donovan
(2012) uses subsistence consumption and exogenous idiosyncratic productivity shocks
to examine cross-country agricultural productivities. The presence of subsistence
consumption results in decreasing relative risk aversion which causes agents to use
fewer intermediate inputs in their production process. His model accounts for two-
thirds of the difference in intermediate income shares and the presence of risk
increases per capita income differences between the richest and poorest countries
by almost eighty percent. In contrast to Donovan, this paper looks at growth rates,
and not only allows agents to choose their risk exposure, but to avoid it completely
if they so desire.
The second paper in the subsistence consumption literature relevant here is
Chatterjee and Ravikumar (1999). In their paper the authors evaluate the effect
that subsistence consumption has on economic growth and the evolution of the
wealth distribution. They show that when a subsistence constraint is included
into a CRRA utility function, the inter-temporal elasticity of substitution is no
longer constant. Instead, as consumption increases, the inter-temporal elasticity
8
of substitution increases. The results of their paper regarding economic growth
line up well with those presented later in this chapter. First, in both models the
presence of the subsistence consumption constraint does not influence the steady-
state growth rate, but it does cause the economy to converge asymptotically rather
than immediately. The second similarity is that larger subsistence constraints result
in longer transition paths to the steady-state. Differently from Chatterjee and
Ravikumar, this chapter includes risk and shows how it can prolong this asymptotic
convergence.
Moving onto the literature on risk aversion, the paper that most closely
resembles this one is Sadler (2000) who finds that poverty traps are eliminated in the
presence of a risk taking technology. Sadler assumes that there are two production
technologies: traditional and modern. The modern production technology requires
a large entry cost that is greater than the individual resources of an agent in the
traditional sector. Therefore, Sadler introduces an actuarially fair lottery which
allows a few agents to win enough so that they can pay the entry cost. Sadler goes
on to show that as long as there is an infinitesimally small probability of entering the
modern sector, agents will choose to engage in this lottery. My paper builds upon
Sadler’s model by adding a subsistence consumption constraint and making the cost
of entry to the modern sector affordable to all agents.
Several examples in the development economics literature show that the poor
engage in costly activities in order to avoid risk. Banerjee and Duflo (2009) and
Binswanger and Rosenzweig (1993) both show that the poor engage in activities
that limit their exposure to risk at the cost of lowering their incomes.5 On the macro
5There are several experimental papers that show the importance of social networks in the
process of technology adoption. These papers find that agents can minimize their exposure to risk
by learning from the actions of others. For examples see Bandiera and Rasul (2006), Conley and
Udry (2010), and Karlan et al (2013).
9
side, Acemoglu and Zilibotti (1997) consider how an inability to completely diversify
risk affects economic growth. They use indivisible projects that keep agents from
diversifying to show that market-incompleteness can hinder capital accumulation and
growth.
Several barriers to adoption have been identified in the literature including:
education (Nelson and Phelps, 1966, and Caselli, 1999), political resistance (Parente
and Prescott, 1994), and inappropriate technologies (Atkinson and Stiglitz, 1969
and Acemoglu and Zilibotti, 2001). This research adds to the literature by including
behavioral factors stemming from risk aversion close to subsistence.
The last related literature is that of structural change. Chanda and Dalgaard
(2008) show that the relative efficiency between two sectors is determined by
constraints on the distribution of resources and the relative level of technology. Both
are features of the model in this chapter. This chapter is also related to those papers
who consider the transition from the Malthusian growth regime to a modern one.6
This literature, as well as this chapter, considers the transition from traditional
production, low productivity methods to modern production.
This chapter proceeds as follows: section 2 presents the production/occupation
side of the economy. Section 3 discusses the agent’s preferences, while section 4
presents some analytical results. Computational results are found in section 5 and
section 6 concludes.
6For examples of this literature see Lagerof (2003) and Galor and Weil (2000).
10
Production
Production Functions
The economy produces a unique final good using two different production
processes. To fix ideas, these processes will be referred to as traditional and modern.
The two sectors differ in three distinct ways: labor (endowed with idiosyncratic
productivities), technology, and capital intensity. The traditional sector only requires
unskilled workers to produce, while the modern sector uses skilled workers who
have paid a training cost to enter the production process. Because the two sectors
differ in their skill intensity, they also differ in their labor augmenting productivity.
Specifically, they differ in the growth of their productivity. In the traditional sector
it is assumed that all productivity increases have been realized and the technology
level is fixed throughout time. However, in the modern sector, the skilled workers are
assumed to be able to improve their productivity through learning-by-doing. Finally,
the traditional sector is assumed to be less capital intensive as the modern sector.
Explicitly, if θ is the capital share in the traditional sector and α is the capital share
in the modern sector under competitive markets, I assume that α ≥ θ. Equations
(2.1) and (2.2) give the production functions.
Y Tt = Ω(K
T
t )
θ(BΦTt )
1−θ (2.1)
Y Mt = Ω(K
M
t )
α(AtΦ
M
t )
1−α (2.2)
where the superscripts M and T denote the modern and traditional sectors and
Φjt for j ∈ {T,M} denotes the aggregate stock of human capital.
7 For simplicity
7The parameter Ω is used to calibrate the model to fit the data.
11
in what follows I will define: ΦTt = φ¯
T
t Lt and Φ
M
t = φ¯
M
t Ht where φ¯
j
t , j ∈ {T,M},
denotes the average productivity and Lt and Ht denote the stock of workers in the
traditional and modern sector, respectively. Since the focus is on the behavior of
individuals who live in developing countries, it is quite possible that the traditional
sector requires very little capital. Note that as θ → 0, the traditional production
function collapses to BLt. Aggregate production is given by the sum of outputs:
Yt = Y
M
t + Y
T
t
Labor Markets
Assume that each individual has a single offspring, which implies a fixed
population N . The population can be decomposed into skilled labor (Ht) and
unskilled labor (Lt).
N = Lt +Ht (2.3)
In order to simplify computation, I will define λt as the proportion of workers who are
employed in the modern sector. Solving for λt will implicitly provide the allocation
of labor between the two sectors.
λt =
Ht
N
, 0 ≤ λt ≤ 1
Risk is introduced through the labor market: the labor markets for modern
and traditional sectors differ in their ability to efficiently match workers to jobs.
It is assumed that the labor market for the traditional sector is well developed,
which means the matching technology in this sector is efficient and any individual
seeking a job will find one. In contrast, the labor market in the modern sector
12
is underdeveloped: not all workers seeking a job find one. Agents who are seeking
employment in the modern sector must pay a training cost because the modern sector
only employs skilled labor. Once the training cost is paid, agents are stochastically
matched in the modern sector. If an agent fails to find a match to a production
unit or firm in the modern sector, she returns to the traditional sector. As more
individuals enter the modern sector, the labor market becomes more developed and
the probability of a successful match will increase. This is given by:
p(λt−1) = max{p, λ
ξ
t−1}, p > 0 (2.4)
This formulation implies that even if there is no one working in the modern sector at
t − 1 there is still some chance that agents will successfully match. The parameter
ξ > 0 determines the influence that congestion in the labor market and network
effects have on the probability of successfully matching in the modern sector, and its
value depends upon on beliefs as to when congestion is the biggest problem. If ξ < 1
so that the matching technology is concave, the congestion effect will dominate. On
the other hand if ξ > 1 so that the matching technology is convex, positive network
externalities are important. In the simulations that follow ξ will be chosen so that
the matching technology is convex. Given the context of a developing country, it is
likely that early on (when λ is small) the modern sector will be underdeveloped and
therefore it will be much more difficult to find employment in, as opposed to later
in the economy’s history when the modern sector is running smoothly. It should be
noted that the choice of ξ does not have qualitative effects on the results presented
later.
13
Wages, Interest Rates, and Arbitrage
Since there are two sectors of production in this model, it implies that there are
two different wages and two different rates of return on capital. As usual, both labor
and capital earn their marginal products, given by:
wMt = (1− α)Ω(K
M
t )
αA1−αt (φ¯
M
t Ht)
−α (2.5)
wTt = (1− θ)Ω(K
T
t )
θB1−θ(φ¯Tt Lt)
−θ (2.6)
RMt = αΩ(K
M
t )
α−1(Atφ¯
M
t Ht)
1−α (2.7)
RTt = θΩ(K
T
t )
θ−1(Bφ¯Tt Lt)
1−θ (2.8)
Arbitrage in investment in the two sectors pins down a unique rate of return as long
as both production sectors are active (if one sector is inactive the rate of return is
given by either equation (2.7) or (2.8)). Since capital is fully mobile, RMt = R
T
t .
Setting equation (2.7) equal to (2.8), I can solve for the ratio of the capital allocated
to the two sectors.
(KTt )
1−θ
(KMt )1−α
=
θ
α
B1−θ
A1−αt
(φ¯Tt )
1−θ
(φ¯Mt )1−α
(1− λt)1−θ
λ1−αt
Nα−θ (2.9)
Defining, Kt = KMt +K
T
t and using (2.9), I can write the amount of capital allocated
to each sector as:
(Kt −KMt )
1−θ
(KMt )1−α
=
θ
α
B1−θ
A1−αt
(φ¯Tt )
1−θ
(φ¯Mt )1−α
(1− λt)1−θ
λ1−αt
Nα−θ (2.10)
14
which implicitly solves for KTt (a closed-form solution does not exist for α 6= θ).
It is clear from equation (2.10) that increases to At and λt result in an increased
allocation of capital to the modern sector.
Technology
As mentioned above the two production sectors differ in their labor augmenting
technology. The traditional sector has a constant technology of B, while the modern
sector experiences technological growth and the time t stock is given by At. The
learning-by-doing externality is proportional to the percentage of the labor force
employed in the modern sector and the average productivity of those workers.
Technological growth is given by:
At+1 − At
At
= g(λt, φ¯Mt ), g(λ, 0) = 0, g(0, φ¯
M
t ) = 0 g1(·, ·) > 0, g2(·, ·) > 0 (2.11)
In the simulations that are presented in section 5, I assume the following functional
form for (2.11):
g(λ, φ¯M) = ηλω(φ¯M)1−ω
where η pins down the long-run growth of the economy when λ = 1 and 0 < ω ≤ 1
determines the relative importance of the average skill of the workforce in the modern
sector.
Households
This economy is populated by a large number of one-period households that
transfer financial bequests and occupational skills to their single off-spring. While
the transfer of occupational skills in the traditional sector is not controversial because
15
the labor market is perfectly efficient, the ability of parents in the modern sector to
do so requires explanation since their labor market is not perfect. The assumption
of transferability is made for two reasons. First, skilled parents tend to raise skilled
children (meaning that their training cost is lower if not zero) and second because
the parent works in the modern sector they have better connections and are able to
secure jobs for their children.8
Preferences
Altrusitic households have preferences over own consumption and the size of the
financial bequest passed on to the next generation. Preferences are given by:
Uit ≡ (1− γ)u(cit − c¯) + γv(ait+1) (2.12)
where c¯ > 0 is the subsistence consumption constraint, cit is stochastic consumption,
and ait+1 is agent i’s bequest level. The subsistence consumption constraint, c¯, can
be thought of as the expenditures on food, clothing, and housing that are essential
for survival. Both u(·) and v(·) are assumed to take the CRRA functional form:
u(cit − c¯) =



(cit−c¯)1−σ
1−σ if cit ≥ c¯
−∞ otherwise
v(ait+1) =
a1−σit+1
1− σ
(2.13)
Before moving onto the household’s optimization problem, I briefly note how
the subsistence consumption constraint influences risk aversion and choices. Typical
8Banerjee (1983) shows that a good deal of employment in the urban modern sector comes about
because of interpersonal connections.
16
measures of risk aversion include relative (rR) and absolute (rA). When using the
standard CRRA preference structure, these measures are rR = σ and rA = σ/c.
However, once a subsistence constraint is included in the preference structure, these
measures of risk aversion change to:
rA(c) =
σ
c− c¯
, rR(c) =
σc
c− c¯
The main difference between these measures of risk aversion and the standard ones
is that rR(c) depends on the level of consumption. In fact, relative risk aversion
approaches infinity as consumption goes to the subsistence level. This implies that
as agents approach destitution, they will be more conservative with the risks that
they take. Decreasing relative risk aversion is consistent with the evidence presented
by Ogaki and Zhang (2001).
Budget Constraint
Agents earn income from two sources, wages (wjt ) j ∈ {T,M} and the return
on their parents’ financial bequest (Rtait). This income is allocated towards
consumption, financial bequests and the training cost. Agents can costlessly remain
in the sector that employed their parents. This can be thought of as children
inheriting some sort of sector-specific human capital, or as a social network effect.
For instance, the children of agents who are already employed in the modern sector
may already know the employers, and therefore, do not have to pay a cost to signal
their desire to enter. This is similar to Song et al’s (2011) assumption that children
inherit the entrepreneurial skills of their parents. This assumption has no meaningful
effect on both the analytic and computational results presented in this paper. As
17
noted above, moving to the modern sector from the traditional sector requires a
training cost that I will denote, x¯ (going the opposite direction is costless). This
implies that the agent has effectively two choices to make: switching to the other
sector and the size of bequest left to her offspring. Therefore, the budget constraint
can be written as:
cit = φiw
j
t +Rtait − hitx¯− ait+1 (2.14)
where
hit =



1 if the agent attempted a move to the modern sector
0 otherwise
(2.15)
and φi is agent i’s idiosyncratic productivity.
Finally note that an agent born at time t first decides whether or not to attempt
to a switch into the modern sector, then realizes the outcome of her job search, after
which she sets her level of bequest and consumes the remainder of her income.
Optimization
Using the functional form of (2.13) and the budget constraint given in (2.14),
I get the following optimization problem for agent i with labor productivity φi and
initial assets ait is:
max
hit,ait+1
Et(1− γ)
(φiw
j
t +Rtait − hitx¯− ait+1 − c¯)
1−σ
1− σ
+ γ
a1−σit+1
1− σ
(2.16)
18
First consider the bequest decision. There are four different types of households.
Defining:
Γ ≡



(
γ
1−γ
) 1
σ
1 +
(
γ
1−γ
) 1
σ



the first order conditions for ait+1 are:
– aMit+1 = Γ(φiw
M
t +Rtait− c¯) for those households already in the modern sector.
– aMXit+1 = Γ(φiw
M
t +Rtait−x¯− c¯) for those households who are new to the modern
sector.
– aTXit+1 = Γ(φiw
T
t +Rtait− x¯− c¯) for those households who failed to move to the
modern sector.
– aTit+1 = Γ(φiw
T
t + Rtait − c¯) for those households who did not attempt a move
to the modern sector
Moving onto the occupation decision. Note that for an agent to be willing to
switch to the modern sector, φiwMt − x¯ > φiw
T
t , therefore any agent whose parent
is in the modern sector would have lower income if she moved to the traditional
sector. This means that it is never optimal for an agent to switch from the modern
sector to the traditional. Therefore, the only choice that needs to be examined is the
move from the traditional to the modern sector. Since occupational choice is not a
continuous variable, I will solve a linear programing problem to compare the expected
utility from changing occupations, US, to the expected utility from remaining in the
same occupation, UR. Substituting in the appropriate first order conditions for ait+1,
the expected lifetime utility can be written as:
19
EtU
S
it =
(
(1− γ)u
(
(1− Γ)(φiw
M
t +Rtait − x¯− c¯)
)
+γv
(
Γ(φiw
M
t +Rtait − x¯− c¯)
))
p(λt−1)
+
(
(1− γ)u
(
(1− Γ)(φiw
T
t +Rtait − x¯− c¯)
)
+γv
(
Γ(φiw
T
t +Rtait − x¯− c¯)
))
(1− p(λt−1))
(2.17)
and
URit = (1− γ)u
(
(1− Γ)(φiw
T
t +Rtait − c¯)
)
+ γv
(
Γ(φiw
T
t +Rtait − c¯)
)
(2.18)
Agents will attempt to move from the traditional to the modern sector if USit > U
R
it .
Analytical Results
Since a closed-form solution for the optimal allocation of capital does not exist
(except for the special case α = θ), I will rely on computational methods. However,
something can be said about the evolution of capital and the output growth rate
without special assumptions. The bequests are invested and become the capital
used in period t: Kt =
∑N
i=1 ait, which can be split up into financial bequests given
by parents in the modern sector and those given by parents in the traditional sector.
After making this separation, it is possible to average the bequests across individuals
in each sector. Therefore the aggregate capital stock can be written as:
Kt = Ht−1a
M
t + Lt−1a
T
t
20
where ajt = Γ(φ¯
j
tw
j
t−1 + Rt−1a
j
t−1 − c¯) for j = T,M.
9 Because production is given by
a constant returns to scale function, Kt can be written as:10
Kt = Γ(Yt−1 − c¯N) = ΓN(yt−1 − c¯) (2.19)
where y denotes aggregate output per capita. Using equation (2.19) the growth of
the aggregate capital stock is given by:
Kt+1
Kt
=
Yt − c¯N
Yt−1 − c¯N
=
yt − c¯
yt−1 − c¯
(2.20)
Equation (2.20) implies that the growth of the capital stock is determined by the
distance between per capita income and the subsistence level.
Now I consider two potential steady-states: λ∗ = 0 and λ∗ = 1. Starting with
λ∗ = 0, aggregate production can be written as:
Yt = (Kt)
θ(Bφ¯N)1−θ
This implies that the growth factor of output can be given by:
Yt+1
Yt
=
(
Kt+1
Kt
)θ
9This formulation requires that no one is attempting a switch. This assumption is inconsequential
because the following discussion will only consider steady-states.
10This can be seen by recognizing that Ht−1aMt−1+Lt−1a
T
t−1 = Kt−1 = K
M
t−1+K
T
t−1. Distributing
the capital accordingly and noting that Rt−1 = RMt−1 = R
T
t−1, we can write the capital equation as:
Kt = Γ
[
Ht−1φ
M
t−1w
M
t−1 +R
M
t−1K
M
t−1 + Lt−1φ
T
t−1w
T
t−1 +R
T
t−1K
T
t−1 − c¯N
]
= Γ[YMt−1 + Y
T
t−1 − c¯N ] = Γ[Yt−1 − c¯N ]
21
Substituting in (2.20) into the above equation yields:
Yt+1
Yt
=
(
Yt − c¯N
Yt−1 − c¯N
)θ
Defining χt+1 =
Yt+1
Yt
. Then, the above equation can be written as:
χt+1 =
(
χt − c¯NYt−1
1− c¯NYt−1
)θ
It is clear that χt+1 = χt = 1 solves the above equation. In order for this to be a
steady-state in λ the additional constraint that no one desires to switch production
technologies at time t must also be imposed. Under this constraint, no one attempts
a switch in t and because the growth rate of output is zero (hence the wage and rate
of return are constant as well), no one will attempt a switch in t+ 1.
For the other steady-state where λ∗ = 1, aggregate production is given by:
Yt = (Kt)
α(Atφ¯N)
1−α
Using the steps outlined above, I can write down the following dynamic system in χ:
χt+1 =
(
χt − c¯NYt−1
1− c¯NYt−1
)α
(1 + ηφ¯1−ω)1−α
where ηφ¯1−ω is the growth rate of technology. In a balanced growth path as Yt−1 →
∞, the dynamic system collapses to:
χt+1 = χ
α
t (1 + ηφ¯
1−ω)1−α
22
This system is solved by the constant value: χ = (1 + η)φ¯1−ω. In order to achieve
this steady-state level of growth, output needs to be increasing over time, therefore,
outside of the limiting case, χt must satisfy:
χt >
1− c¯NYt−1
(1 + ηφ¯1−ω)
1−α
α
+
c¯N
Yt − 1
where the RHS is less than 1.11 This condition will most likely hold, because if an
economy with zero technological growth can achieve constant output, it is reasonable
to think that an economy with constant technological growth will be able to achieve
it as well. It is trivial to show that λ∗ = 1 is a steady-state because, by construction,
once a household is in the modern sector, they will not switch back.
Quantitative Results
Computational methods are necessary to understand the dynamics of the model.
This section presents some numerical results that center on four questions:
– What role does subsistence consumption play in the movement of agents from
the traditional sector to the modern one?
– What role does the distribution of income play in convergence to full
employment in the modern sector?
– What are the welfare consequences of the subsistence constraint?
– Which policies bring about the fastest rate of convergence to full employment
in the modern sector?
11This condition is found by setting the dynamic system for output growth equal to 1.
23
Parameters
There are seventeen parameters that need to be calibrated. The first set of
parameters are the capital shares associated with production in the traditional and
modern sectors. For these I rely on parameter estimates from three sources: Caselli
and Feyrer (2007), Gollin (2002), and Guerriero (2012).12 Starting with the Caselli
and Feyrer paper, the authors make the argument that the capital share that is
normally reported in the literature does not accurately represent the capital share
used in theoretical models. Their concern is that the measure of capital used in the
theoretical literature is mainly of reproducible capital, while the empirical estimates
include both reproducible and non-reproducible capital.13 Therefore the authors
adjust a country’s overall capital to include only reproducible capital and report
these estimates for 53 countries. Using their method, I adjust the estimates found
in Gollin, Guerriero, and the OECD data website so that they only reflect the share
of income associated with reproducible capital. The complete data set is available
in the appendix. Table 1 shows the averages grouped by the World Bank income
classifications.
Caselli & Feyrer Gollin OECD Guerriero
Low Income 0.029 (1) —– —– —–
Low-Middle Income 0.152 (10) 0.094 (4) —– 0.124 (8)
High-Middle Income 0.193 (15) 0.183 (4) 0.256 (5) 0.176 (25)
High Income Non-OECD 0.219 (4) 0.3 (1) 0.246 (1) 0.172 (6)
OECD 0.196 (23) 0.201 (14) 0.235 (24) 0.175 (21)
The number of countries in each group is in parentheses
TABLE 1. Capital Shares By Income Group
12There is a fourth source from the OECD data website. The data available from this source
does not include developing countries, therefore it is only used for a robustness check.
13This would conceivably result in large estimated capital shares for developing countries whose
primary means of income revolve around natural resources.
24
Given the set-up of the model, the capital shares reported in table 1 are clearly
an amalgamation of the shares in the traditional and modern sectors and do not
directly correspond to either θ or α. The model does imply that poorer countries
have lower capital intensities, therefore using the capital shares on the lower-end of
the income distribution for the traditional sector and the OECD estimates for the
modern sector should provide a good proxy for reality.
Since only one estimate is available for a county in the lowest income bracket, I
set the share of capital in the traditional sector to estimates for low-middle income
countries. Likewise the capital share for the modern sector is given by the OECD
estimate. Because three of the studies provide capital shares for both the low-
middle income countries and OECD countries each pair will be used in the baseline
simulations.14
Several parameters do not have empirical counter-parts. The population, N , is
a scaling parameter and set at 10,000 in order to avoid any small sample problems. p
is set at 0.2 which allows for a rather large success probability for a nascent modern
sector while the matching technology is assumed to be convex with ξ = 2.15 The
preference share γ is set so that the bequests are large enough to ensure cit > c¯
and the training cost is constant across simulations and is set so that even the
poorest agent can afford to pay it and still satisfy her subsistence constraint (the
parameterization implies a value of 28 rupees). The labor augmenting technology
in the traditional sector is normalized to 1 and the initial technology in the modern
sector is set at 1.2. As for the growth rate of technology, η is set so that the long-run
14The simulations are also run using a fixed capital share for the modern sector (the estimate
from the OECD dataset), the results do not differ qualitatively from those presented in the paper.
15Appendix B provides a robustness check for ξ and p.
25
generation growth rate annualizes to 2% and ω is set at .75.16 Finally, Chatterjee
and Ravikumar (1999) use estimates from Ogaki and Atkeson (1997) and Rosenzweig
and Wolpin (1993) to show that subsistence consumption makes up between 58% and
80% of total consumption.17 Therefore the subsistence constraint is set using these
estimates and the model is calibrated (Ω is set) to match the consumption data
presented in Townsend (1994). In the results presented in this paper, I assume σ = 1
which is close to the value used in Chatterjee and Ravikumar.18 Finally, the length
of a generation is assumed to be 35 years.
The last step is to set the distribution for the idiosyncratic productivities.
Because I must ensure that all agents can satisfy their subsistence constraint, a
Pareto distribution is used. The minimum productivity is set at 2.65 with the spread
parameter set at 7.5. These parameters are chosen because increasing the spread any
larger causes the poor agents’ incomes to fall below the line in which they are able
to afford both their subsistence constraint and the training cost.
Baseline Results
The simulation results presented in this section are calculated using four different
values for c¯. Two of these values come from the empirical literature and correspond
to 58% and 80% of consumption. The third value corresponds to 69% of consumption
and is chosen to to determine whether the results are linear in the constraint or if
they are more extreme for larger values of c¯. Finally, the value c¯ = 0 is used to
16The results are robust to a wide variety of choices for these parameters.
17It should be noted that Ogaki and Atkeson (1997) state that the consumption data used to
estimate the subsistence parameter do not include housing and transportation.
18The qualitative results are robust to different choices for σ.
26
contrast the effect of subsistence consumption. Figure 1 presents the results for λ in
these baseline simulations.
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(a) Caselli & Feyrer
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(b) Gollin
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(c) Guerriero
FIGURE 1. The evolution of the percentage of the workforce employed in the modern
sector across time for the baseline calibration. Each panel represents different capital
share estimates for the modern and traditional sectors.
c¯ = 0: Solid/Black, c¯ = 58%: Dashed/Red, c¯ = 69%: Dashed-Dot/Green, c¯ = 80%:
Dotted/Blue
Regardless of the values chosen for the capital shares, the closer agents are
to their subsistence constraint, the longer it takes for the economy to reach full
employment in the modern sector. The time to convergence is increasing with the
level of subsistence because those agents whose constraint represents 80% of their
total consumption will be significantly more risk adverse than the agents whose
constraint is only 58%, which results in less risk taking on average. Before looking at
the impact that the subsistence requirement has on growth, I want to further discuss
the effect that the capital intensities have on convergence to full employment in the
modern sector.19 The pictures in figures 1a and 1c are quite similar in terms of the
path to convergence, and looking back at table 1 this should not be surprising because
the estimates found in Caselli/Feyrer and Guerriero are quite similar. Figure 1b,
however exhibits a much more direct path to convergence than the other simulations.
The only difference between the three simulations is that ratio of capital shares
19Note that given the parameter choices and functional forms described above, the economy
always converges to a unique stationary equilibrium where only the modern sector is active.
27
is significantly higher with the Gollin estimates. This affects the transition path
because the ratio of capital shares influences the capital allocated to each sector.
A larger capital share ratio not only results in more capital being allocated to the
modern sector (resulting in higher wages), it also mitigates the effect that capital
leaving the traditional sector has on traditional wages. In other words, the capital
shares under the Gollin estimates result in higher wages in both the modern and
traditional sectors, thus making a switch optimal sooner.
Figure 1 also shows the implications of incomplete markets: that agents carefully
diversify their idiosyncratic risk. In a two-sector model without a subsistence
constraint and risk, the model would predict a path similar to the black-solid line
in the graphs shown in figure 1. This would lead the researcher to conclude that
modernization is both imminent and immediate. In contrast, this model implies that
there is a prolonged build up towards full fledged industrialization (full employment
in the modern sector).
The evolution of λ is not the only variable of interest. Figure 2 shows the growth
rates of output for the different estimations of the capital shares and subsistence
requirements.20 There are a couple of noteworthy results. First, during the transition
to full employment in the modern sector, output growth overshoots the long-run
growth rate. This happens because soon after a complete switch the economy not
only gets a boost from the increased technology associated with skilled workers, but
also the added increase from the subset of workers who are employed in the modern
methods in the current period, but used traditional techniques in the previous period.
This is an empirically appealing result of the model, as countries that are rapidly
20The initial decline in output growth is a result of the economy adjusting from the initial
condition.
28
developing often have growth rates above what is considered sustainable in the long-
run. A prime example of such a pattern may be China.
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
Generations
Out
put
Gro
wth
(a) Caselli & Feyrer
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
Generations
Out
put
Gro
wth
(b) Gollin
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
Generations
Out
put
Gro
wth
(c) Guerriero
FIGURE 2. The evolution of output growth across time for the baseline calibration.
Each panel represents different capital share estimates for the modern and traditional
sectors.
c¯ = 0: Solid/Black, c¯ = 58%: Dashed/Red, c¯ = 69%: Dashed-Dot/Green, c¯ = 80%:
Dotted/Blue
While these spikes in output growth are an interesting feature of the model, just
about any two-sector framework will produce a growth rate that is higher than the
long-run during the transition phase. What makes this model different is the delay
in this spike. Looking at figure 2, it is clear that the simulation that does not impose
a subsistence constraint immediately produces this spike in output growth, while the
peak is significantly delayed under the constraint.
Consequences of Poverty
At this point it is natural to ask: what happens when the extent of poverty
increases? In other words, how do the results from the previous section change when
the standard deviation of the idiosyncratic productivities decreases? Answering this
question will allow me to determine what happens when two economies do not differ
in their α, θ, or c¯ but in their wealth (income) distributions.
29
In order to determine the consequences of having a larger proportion of the
population in poverty, the productivity will be drawn from a Pareto distribution that
maintains the same mean as the baseline simulation, but whose standard deviations
represent 85, 70, and 56 percent of the baseline simulation’s standard deviation. This
will push a higher percentage of the population toward the subsistence constraint.
Figure 3 plots the evolutions of λ and the growth rate of output for the Guerriero
parameter estimates and c¯ = 80%.21
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(a) λ
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
Generations
Ou
tpu
tG
row
th
(b) Output Growth
FIGURE 3. The Consequences of Poverty. Each line represents a different spread
of the underlying idiosyncratic productivity distribution. The reported standard
deviations are given as a percentage of the baseline.
Std: 1–Solid/Black, Std: 85%–Dashed/Red, Std: 70%–Dashed-Dot/Green, Std:
56%–Dotted/Blue
Ultimately, the results presented in figure 3 should not be that surprising. It
is clear that the result of having a higher density of poverty is a lower proportion
of individuals in the modern sector, which drives the low growth outcomes. This is
exactly what one would expect from this model. The greater the population density
around the poor income level, the fewer the number of agents who are willing to take
on the risk of moving to the modern sector. As suggested by the previous results,
21These parameters were chosen because they clearly illustrated the effect poverty has on the
transition path. The results do not differ qualitatively with other parameterizations.
30
this will result in a prolonged time to convergence to full employment in the modern
sector. The results so far imply that the chief determinants for the length of the
transition period are: proximity to the subsistence constraint, the capital shares in
both sectors, and the distribution of income.
Inequality
Consider next the implications for inequality. First, in figure 4, I present the
results by subsistence level with each line representing a different concentration of
poverty. Figure 4 clearly depicts a Kuznets curve with inequality increasing during
the transition period only to return to a lower level once everyone is employed in
the modern sector. The more important part of these graphs is that inequality
is uniformly lower for economies with less heterogeneity. In other words, those
economies who have a smaller dispersion of individual productivities, also experience
a smaller increase in inequality during the transition from traditional to modern
production methods.
Figure 5 looks at the results in a slightly different manner. Rather than looking
across subsistence levels, figure 5 looks at the Gini coefficient across concentrations
of poverty with each of the lines representing a different subsistence constraint.
There is a distinct pattern in figure 5 which shows that inequality is lower for
those economies that start further away from their subsistence constraint regardless
of the concentration of poverty. This is because economies with lower subsistence
requirements do not have as many agents “stuck” with traditional methods because
their risk profiles do not allow for an attempted switch. 22
22The results presented in figures 4 and 5 line up well with those found in Atolia et al (2012).
31
0 5 10 150.0
0.1
0.2
0.3
0.4
0.5
Generations
Gin
i
(a) c¯ = 0
0 5 10 150.0
0.1
0.2
0.3
0.4
0.5
Generations
Gin
i
(b) c¯ = 58%
0 5 10 150.0
0.1
0.2
0.3
0.4
0.5
Generations
Gin
i
(c) c¯ = 69%
0 5 10 150.0
0.1
0.2
0.3
0.4
0.5
Generations
Gin
i
(d) c¯ = 80%
FIGURE 4. The Gini Coefficient for income plotted across time for different levels
of subsistence consumption. Each line represents a different spread of the underlying
idiosyncratic productivity distribution. The reported standard deviations are given
as a percentage of the baseline.
Std: 1–Solid/Black, Std: 85%–Dashed/Red, Std: 70%–Dashed-Dot/Green, Std:
56%–Dotted/Blue
Overall the inequality results help explain how countries can experience rapid
economic growth with only a slight increase in the level of inequality. It could be that
these countries that are an apparent affront to the Kuznets curve, do not experience
the rise in the Gini coefficient because they are either (a) relatively homogenous, (b)
further from their subsistence constraints, or (c) some combination of (a) and (b).
Welfare
To quantify the welfare implications of subsistence consumption requirements,
I will use the metric of loss in average generational growth rates. The case with
32
0 5 10 150.0
0.1
0.2
0.3
0.4
0.5
Generations
Gin
i
(a) Std : 100%
0 5 10 150.0
0.1
0.2
0.3
0.4
0.5
Generations
Gin
i
(b) Std : 85%
0 5 10 150.0
0.1
0.2
0.3
0.4
0.5
Generations
Gin
i
(c) Std : 70%
0 5 10 150.0
0.1
0.2
0.3
0.4
0.5
Generations
Gin
i
(d) Std : 56%
FIGURE 5. The Gini Coefficient for income plotted across time for different
underlying productivity distributions (where the standard deviations are reported
as a percentage of the baseline). Each line represents a different subsistence
consumption level.
c¯ = 0: Solid/Black, c¯ = 58%: Dashed/Red, c¯ = 69%: Dashed-Dot/Green, c¯ = 80%:
Dotted/Blue
c¯ = 0 is considered the benchmark and table 2 reports the difference between the
benchmark and each of the subsistence consumption levels.23
The loss in generational growth ranges from 4 to 32 percentage points. This
implies that over the course a generation, the existence of a subsistence constraint
results in total output being 2-17% lower than the benchmark case. It should be
noted that the Caselli/Feyrer and Guerriero estimates for the capital shares are
fairly close and looking only at those numbers implies a reduction of total output of
between 5-17% over the course of a generation.
23The elements in table 2 are calculated using: gc¯=j − gc¯=0 for j ∈ {58%, 69%, 80%}.
33
α & θ Values c¯ = 0 c¯ = 58% c¯ = 69% c¯ = 80%
Caselli & Feyrer 0 -0.1186 -0.1872 -0.3059
Gollin 0 -0.0362 -0.0796 -0.1999
Guerriero 0 -0.0979 -0.1636 -0.3144
TABLE 2. Welfare Analysis–Growth Rate Change (Baseline): This table gives the
generational change in the growth rate in comparison to the baseline with c¯ = 0
using the three sets of estimates for the capital shares and the baseline calibration
parameters.
The above discussion of growth rate loss can be applied to the situation in which
the income distribution varies across subsistence levels. Table 3 reports the growth
loss associated with an increase in poverty (the parameter set is the same as in section
2.5).
Standard
Deviation c¯ = 0% c¯ = 58% c¯ = 69% c¯ = 80%
100% 0 0 0 0
85% 0 0 -0.0128 -0.0127
70% 0 -0.0095 -0.0261 -0.0473
56% 0 -0.0115 -0.0406 -0.0885
TABLE 3. Welfare Analysis–Growth Rate Change (Cost of Poverty): This table gives
the generational change in the growth rate in comparison to the baseline with c¯ = 0
using the three sets of estimates for the capital shares and the baseline calibration
parameters.
It shows that the growth rate loss ranges from negligible to 9 percentage points.
This implies a reduction of total output of between 0-5.6% over the course of a
generation. These results are quite intuitive as the consequences of poverty increase
as subsistence consumption becomes a larger share of total consumption. Consider
the benchmark (c¯ = 0) case, one would expect the consequences of poverty to be zero
when there is no notion of poverty in the model. While the reduction of growth rates
for either the benchmark case or the simulations with different income distributions
34
are not exceptionally large, they are economically significant when one considers the
the dismal growth rates in many LDCs.
Policy
Given the welfare loss associated with delayed structural change, I conclude by
discussing the possible policy implications. I show computationally in the appendix
that the social planner would prefer to allocate all workers to the modern sector in the
first period. It is only the agent’s proximity to her subsistence constraint that limits
her movement. Therefore if policy can be designed to provide more income to those
agents in the traditional sector, it may speed up convergence to full employment in
the modern sector.
Before discussing tangible policies, I will frame this problem in terms of an
optimal cash transfer. There are two different approaches that can be taken in
terms of financing the cash transfer: aid and taxation. The goal of this exercise
is to judge the relative effectiveness of conditional (CCT) and unconditional cash
transfers (UCT). Starting with the UCT, under this policy every agent will receive
the same amount through the transfer and in the case of taxation every agent will
be taxed at the same rate. Therefore under taxation, every agent will receive:
zt = τ(φ¯
M
t w
M
t λt + φ¯
T
t w
T
t (1− λt))
where τ is the tax rate. If the unconditional cash transfer is financed by aid the
agents will receive: ζ × x¯, where 0 ≤ ζ ≤ 1 and x¯ is the cost of switching.
Figure 6 presents the results of the simulations using an unconditional cash
transfer. The results for this simulation are quite interesting. First, if the UCT
35
is financed through taxation, this policy actually slows down convergence to full
employment in the modern sector. The reason for this result is two fold: first,
taxation lowers the return on switching to the modern sector by depressing the take
home income. Secondly, the UCT raises the benefit from not attempting a switch
because the agent receives the transfer regardless of her occupation decision. When
the UCT is financed by aid, the results change so that the policy results in a (slightly)
positive impact on both convergence to full employment in the modern sector and
economic growth.
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(a) λ: Taxation
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Generations
Ou
tpu
tG
row
th
(b) Output Growth: Taxation
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(c) λ: Aid
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Generations
Ou
tpu
tG
row
th
(d) Output Growth: Aid
FIGURE 6. The Effects of an Unconditional Cash Transfer: This figure depicts the
time paths of λ and output growth under an unconditional cash transfer financed by
both taxation and aid.
Taxation–solid/black: τ = 0%, red/dotted: τ = 5%, blue/dashed: τ = 7.5%,
green/dashed-dot: τ = 10%.
Aid–solid/black: τ = 0%, red/dotted: τ = 50%, blue/dashed: τ = 75%,
green/dashed-dot: τ = 100%.
36
Moving away from the UCT, the conditional cash transfer will only be
distributed to those agents who successfully find employment in the modern sector.24
The advantage of designing the transfer in this way is that occupational location is
largely observable and by conditioning on a successful employment I remove the
potential for moral hazard problems. Under taxation, this policy is financed by
imposing a proportional tax on the wages of those individuals who are already
operating in the modern sector. Explicitly this is given by:
zt = τw
M
t φ¯
M
t−1
(
λt−1
λt − λt−1
)
where the variables are as defined above. The CCT financed by aid is distributed in
the same manner as the UCT.
Figure 7 presents the results of the CCT financed by both aid and taxation.
The results for CCT are less surprising than the UCT, under both taxation and aid,
convergence to the modern sector is faster than under the baseline.
The welfare effects of both types of cash transfers can be quantified using the
same metric as above. Table 4 shows the change in the growth rates in relation to
the baseline results. Clearly the CCT dominates the UCT regardless of the type of
financing used. Comparing the two sets of results, the CCT results in growth rates
that are at least 2.5 percentage points higher than the UCT.
This discussion on cash transfers shows results that are in-line with Mookherjee
and Ray (2008) who compare unconditional cash transfers to conditional cash
transfers. They argue that conditional cash transfers not only raise per capita output
24A CCT policy that does not depend on successfully finding employment in the modern sector
was also analyzed (not reported). Such a policy results in a larger improvement in growth rates,
but does open up the issue of moral hazard.
37
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(a) λ: Taxation
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Generations
Ou
tpu
tG
row
th
(b) Output Growth: Taxation
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(c) λ: Aid
2 4 6 8 10 120.0
0.2
0.4
0.6
0.8
1.0
Generations
Ou
tpu
tG
row
th
(d) Output Growth: Aid
FIGURE 7. The Effects of a Conditional Cash Transfer: This figure depicts the time
paths of λ and output growth under an unconditional cash transfer financed by both
taxation and aid.
Taxation–solid/black: τ = 0%, red/dotted: τ = 5%, blue/dashed: τ = 7.5%,
green/dashed-dot: τ = 10%.
Aid–solid/black: τ = 0%, red/dotted: τ = 50%, blue/dashed: τ = 75%,
green/dashed-dot: τ = 100%.
but also have a positive welfare effect. In the unconditional setting, agents get stuck
on “welfare” and do not have the incentive to invest in productive activities, the
conditional transfer, however, counteracts this underinvestment by providing the
right incentives. My paper highlights the need to consider poverty in a dynamic
sense, showing that when individual decisions affect aggregate growth outcomes,
unconditional transfers result in worse growth outcomes. This is particularly relevant
given the on going discussion in the field as to the efficiency of unconditional cash
transfers.
38
Taxation Baseline τ = 5% τ = 10% τ = 15%
UCT 0 -0.0248 -0.0493 -0.0921
CCT 0 0.0226 0.0347 0.0426
Aid Baseline ζ = 50% ζ = 75% ζ = 100%
UCT 0 0.0091 0.0166 0.0176
CCT 0 0.0346 0.0570 0.0664
TABLE 4. Welfare Analysis–Change in Growth Rate: Taxation
Conclusion
This chapter has contributed to the literature on dynamic poverty in two ways.
First, it has shown that agents behave differently when they are near their subsistence
constraint, especially in terms of risk taking. When risky activities are the same ones
that are necessary for economic growth, poverty results in a substantial loss in output
over the course of a generation.
The second contribution deals with how we interact with the poor in an attempt
to lift them out of poverty. Conventional wisdom would suggest that unconditional
cash transfers would be enough to move the poor far enough away from their
subsistence level so that they would be willing to take on risk. However, this paper
has shown that only under a system of conditional transfers does handing out cash
to those in poverty actually result in the desired effects.
There are plenty of avenues of research for this topic. The main ones include:
endogenizing the probability of a successful match through the entry and exit of
firms. This would allow for a more in-depth study of the development of the modern
sector. Another interesting possibility is to look at the role of education in greater
detail and allowing for the training cost to be more involved than a simple cash
payment.
39
CHAPTER III
INEQUALITY AS A HEALTH HAZARD
This chapter is based on joint work with Shankha Chakraborty. In its current
form, I am responsible for the formalization and execution of the model, including
both the analytical and computational results. Dr. Chakraborty is responsible for
the idea that drives the paper as well as editorial and thematic guidance.
Introduction
Our perception of and tolerance for income inequality are shaped by how it
affects our choices, behavior and welfare. While the economics literature in this area
is deep, it has generally shied away from the notion that individuals care directly
about inequality. They may do so because of social aspirations that are profoundly
affected by how they fare relative to others. A body of research that does include such
positional concerns is the literature on status seeking (sometimes called aspirations),
though often, researchers assume identical decision-makers or behavior contingent
on a common level of aspirations, typically the economy-wide average wealth or
consumption.
This chapter starts from the premise that socially aware individuals care directly
about inequality but departs from the literature in assuming social aspirations differ
across individuals. More concretely, we assume that individuals have upward-looking
aspirations: they pursue the living standards of those who are economically better
off than them and this is as true of the rich as of the poor. We build on the
Blanchard-Yaari model of “perpetual youth” (Blanchard, 1985; Yaari, 1965) and
Grossman’s (1972) work on health production. Individuals earn wage income from
40
labor and annuitized returns on investment and they differ in their intrinsic labor
productivity. They also differ in their social aspirations which is based on the
reference group consisting of all individuals with consumption levels higher than
theirs. An individual’s utility depends negatively on how far below this reference
group his current consumption falls.
We use this framework to study the main thesis of Wilkinson’s (2002) book The
Unequal Society, that inequality has a first-order effect on personal and population
health since it promotes unhealthy behavior. To do so we assume that the production
of health capital requires time investment in the form of leisure and consumption of
health goods. Hence the decision to supply labor not only influences an individual’s
wage earnings and utility from leisure but also the evolution of his health stock. This
health stock in turn determines the probability of surviving onto the next period.
We establish two main results. First, an individual’s health declines as the
measure of his relative deprivation (aspirations gap) increases. The further below
his aspirations level an individual is, the more effort he exerts on the labor market
to increase his relative income and consumption. The increased labor activity
directly translates into lower health investment from less leisure – more generally
the compounding effects of added stress, longer work-hours and an unhealthy
lifestyle. This effect is only partially attenuated by higher consumption of the health
good. Hence a higher relative deprivation results in a lower life expectancy for the
individual. Secondly we show that despite this relationship at the individual level, in
the aggregate, the effect of economic inequality on life expectancy is ambiguous. The
somewhat weak negative correlation between the two weakens still when income goes
up. An increase in income weakens the relationship because the biological constraints
41
on life expectancy ensure that a one unit increase in income does not translate into
a one unit increase in survival probability.
In a series of work, the British epidemiologist Richard G. Wilkinson (Wilkinson
1992, 2002, Wilkinson and Pikett, 2009) argues that income inequality is itself a
health hazard. Wilkinson documents this health-inequality connection by relying on
evidence on mortality and income inequality in the OECD countries. For his sample,
he finds a distinct negative relationship between inequality and life expectancy at
the aggregate level. Subsequent studies have cast doubt on the robustness of this
relationship.1
The disaggregated evidence is, however, clearer and robust. The Whitehall
studies on British civil servants have found, for example, a strong inverse correlation
between position in the administrative hierarchy and mortality rates. “Men in the
lowest grade had a death rate three times higher than that of men in the highest
grade” which was related to “higher risk of heart disease . . . chronic lung diseases,
gastrointestinal disease, depression, suicide, sickness absence from work, back pain
and self reported health” (Wilkinson and Pikett, 2009). The direct effect of income
on health choices seems to explain only a third of such higher mortality risk, the
residual presumably explained by the direct effect inequality has on an individual’s
health (Smith et al., 1990).
Using panel data on reported health and inequality within the United States,
Deaton (2001) shows that, when controlling for ethnic make-up, between-state
inequality has no effect on observed health outcomes but within-state inequality does.
Using a measure of relative deprivation similar in spirit to this paper’s aspirations
gap, Deaton shows that an increase in relative deprivation results in worse reported
1See Deaton (2003) for an overview of the literature and Judge (1995) for one of the earliest
critiques.
42
health. Eibner and Evans (2005) confirm Deaton’s result for a larger range of health
outcomes including mortality. They find that relative deprivation has a particularly
large impact on deaths linked to smoking and coronary heart diseases, both of which
are tied to behaviors brought on by stress and excessive work. The theoretical results
that we establish in this paper affirm the generality of Deaton’s (2001) and Eibner
and Evans (2005)’s empirical findings. They indicate quantitatively how significant
the aspirations gap can be: in our model, relative deprivation can account for a gap
in conditional life expectancy of at least ten years. At the same time our work
shows that what is true at the individual level is not necessarily matched by a
similar aggregate picture as the lack of a firm correlation between inequality and
life expectancy in industrialized countries indicates.
Our work is also related to the sizable literature on status-seeking and
aspirations. While much of that work is not directly related to ours, a few are. We
utilize Abel (1990) and Gali’s (1994) specification of aspirations in the form of relative
consumption levels, though we include income and aspirations heterogeneity. Among
more recent works on social aspirations, two are closely connected to our paper.
Genicot and Ray (2010) discuss how various forms of aspirations – common (as used
in the status-seeking literature), stratified, upward-looking and local aspirations –
are shaped by different moments of the income distribution. They embed the first
two types of aspirations into a simple two-period growth model to illustrate the
possibility of income polarization. Bogliacino and Ortoleva (2011) also use common
aspirations in a two-period model to show the existence of multiple equilibria,
including polarization. Both these papers use a logistic function to formalize the
effect of aspirations failure. We rely, instead, on a concave specification and assume
aspirations are defined with respect to consumption levels which are likely more
43
easily observed than income levels in the two aforementioned papers. There is no
possibility of polarization in our model.
The paper proceeds as follows. Section 2 presents the dynamic model. The
individual’s decision problem is formalized and partially analyzed in Section 3.
Computational work in Section 4 studies the implication of aspirations and inequality
for individual and aggregate health. We conclude in Section 5.
Model
A discrete time infinitely-lived economy is populated by individuals who
potentially live forever. Individuals are born with a labor productivity draw θ, initial
assets a0 and health capital H0. Time is indexed by t = 0, 1, . . .∞.
Health Production
Much like the Grossman (1972) model, agents accumulate a stock of health
by making purposeful investments. Unlike the Grossman model they do not face a
deterministic length of life that is dictated by a minimum health stock. Rather, the
model incorporates the perpetual youth framework from Yaari (1965) and Blanchard
(1985) in assuming that agents face a positive probability of death each period. The
Grossman and Blanchard-Yaari frameworks are combined by allowing the agent’s
health capital Ht at the beginning of time t to positively affect his probability of
surviving in that period.
Health capital depreciates at the rate δ ∈ (0, 1). For individual i the stock
of health at time t + 1 depends on the stock of undepreciated capital and health
investment at time t. In other words, health evolves much like physical capital in
44
the standard neoclassical model:
Hit+1 = (1− δ)Hit + Iit. (3.1)
Health investment at time t depends upon the amount of leisure, 1 − lit, lit being
agent i’s labor supply out of a unit time endowment, and consumption of a market-
provided health good, qit, such as visits to the doctor, drugs, vitamins, etc.. The
relative price of this good is taken to be one and we do not explicitly model its
production.2 Given these two inputs, the amount of health invested is given by:
Iit = I(lit, qit),
an increasing and concave function of leisure and the health good, that is, I1(·, ·) < 0,
I11(·, ·) < 0, I2(·, ·) > 0, and I22(·, ·) < 0. Using (1) the health stock in period t + 1
can be expressed in terms of initial health and past investments:
Hit+1 = (1− δ)
t+1Hi0 +
t∑
s=0
(1− δ)sI(lit−s, qit−s). (3.2)
The next step is to relate this health stock to agent’s i decision problem.
Individual i’s survival probability, φit, depends upon agent i’s stock of health in
time t through an increasing concave function
φit = φ(Hit), (3.3)
2This is easily done by assuming q is produced solely from labor with labor productivity of χ
normalized to unity.
45
where φ′(·) > 0, φ′′(·) ≤ 0, φ(0) = 0 and limH→∞ φ(H) = 1. To ensure that the agent
is alive in the initial period we also assume that φi0 = 1. Given this formulation,
φit+1 is the probability of being alive in t+ 1 conditional on being alive in period t.
Finally note that the cumulative probability of being alive until period t is
Φit =
t∏
n=0
φin. (3.4)
Health capital has no other effect on i’s decision problem except through the survival
rate. In particular, it does not directly affect his labor productivity.
Budget Constraint
Agent i’s labor productivity θi is time invariant, drawn at the beginning of his
life from the distribution Γ(θ) with finite support. We assume that the wage rate per
efficiency unit of labor w is constant and exogenous. The return on investment R˜it,
on the other hand, is endogenous and individual-specific. Since individuals die over
time, we need to ensure their assets are accounted for, and following Yaari (1965)
we assume a perfect annuities market in the form of an insurance company. Under a
perfectly competitive insurance market, the zero profit condition implies equilibrium
annuitized return on investment is R˜it = R/φit, R being the constant market return
on investment. This brings us to agent i’s period t budget constraint:
cit + qit + ait+1 = wθilit + R˜tait, (3.5)
where a denotes assets and c the consumption good. Figure 8 shows the ordering of
events for each period.
46
Begin Period t
Choose lit , qit , ait+1
Realize Hit+1
Begin Period t+1
Reach Period t+1
wê Prob fHHit+1 L
FIGURE 8. Agent i’s Decision Timeline
Preferences
These health and mortality behavior are embedded in an economy where
individuals are “socially minded”. Specifically we assume preferences are defined
over both consumption and leisure. Where we depart from the standard neoclassical
paradigm is in the formulation of utility from consumption which depends on an
individual’s relative position in the consumption distribution. While it may seem
reasonable to have agents care about inequality measures like the Gini coefficient
or the Kuznet’s ratio, these capture the whole distribution when it is not clear
that people care about those who are worse off then themselves in the same way
they care about those who are doing better than themselves. The former is usually
labeled “pride” in the literature, the latter variously as “envy”, “status seeking” and
“upward-looking aspirations” (see Hopkins 2008 for an excellent discussion of these
alternatives). The macro literature in this area commonly assumes mean dependence,
that is, individuals care about their status relative to the economy-wide average
consumption, income or wealth.
We assume upward-looking aspirations in that agents care only about how worse-
off they are relative to those who are better off than themselves. More precisely agents
form their aspirations by taking the average of the consumption of every agent who
consumes at least as much as they do. This ensures that the highest-consumption
agent remains an aspirant, using his own consumption level to form that aspiration.
47
Hence, individual i’s aspirations level is set according to
C¯it =
∑N
j=1 1(cjt ≥ cit)cjt
∑N
j=1 1(cjt ≥ cit)
(3.6)
where 1(cjt ≥ cit) is an indicator function that takes on the value 1 if true and 0
otherwise.
The exact formulation of how this influences the individual’s overall utility
is borrowed from the status-seeking and Keeping-up-with-the-Joneses (henceforth
KUWJ) literature, particularly Gali (1994):
uit ≡ U(cit, C¯it, lit) =
c1−σit
1− σ
C¯ψσit + γ
(1− lit)1−σ
1− σ
(3.7)
where σ > 0 and 0 < ψ < 1. This specification is also motivated by Alpizar et
al.’s (2005) survey-experimental evidence that relative consumption of non-positional
goods matter as much as positional goods; we do not distinguish between the two
types of consumption. The parameter restrictions are such that for a given level of
consumption, cit an increase in the aspirations will result in strictly lower utility.
A final point about the period utility function. Note that when σ > 1,
U(cit, C¯it, lit) < 0. To ensure that utility from being alive always exceeds that from
death, we normalize utility from dying to U such that
U < inf
{
U(cit, C¯it, lit)
}∞,N
t=0,i=1
48
which means a complete specification of the period utility function is given by
U(cit, C¯it, lit) =



c1−σit
1−σ C¯
ψσ
it + γ
(1−lit)1−σ
1−σ if agent i is alive
U if agent i is dead
Decision Problem
With the economic environment described above, individual i faces the decision
problem of maximizing his expected lifetime utility
E0
∞∑
t=0
βt
[
Φit
{
c1−σit
1− σ
C¯ψσit + γ
(1− lit)1−σ
1− σ
}
+ (1− Φit)U
]
, (3.8)
β ∈ (0, 1) being the subjective discount rate, subject to the health transition equation
(3.1), budget constraint (3.5) and initial conditions (ai0, Hi0).
We reformulate this decision problem as a dynamic programming problem.
First, we assume that the individual takes into account how its health choices affect
the annuity return R˜ that it receives. The rationale for this is that people often base
their insurance decisions on actuarial tables. This assumption has the advantage of
reducing the state space since the annuity return does not have to be considered part
of it, cutting down on computation time. In addition, results are very similar for
price taking behavior. Thus individual i faces four state variables (θi, ait, Hit, C¯it) and
three controls (ait+1, lit, qit) and his optimization decision is specified by the Bellman
equation
V (θit, ait, Hit, C¯it) = max
lit,ait+1,qit+1
u
(
cit, C¯it, lit
)
+
{
βφ(Hit+1)V (θit+1, ait+1, Hit+1, C¯it+1)
+(1− φ(Hit+1))U}
(3.9)
49
subject to
cit = wtθitlit +
Rt
φ(Hit)
ait − qit − ait+1.
Optimization
The optimal choices of ait+1, lit, and qit can be solved for by maximizing (3.9)
subject to (3.5), and (3.1). Starting with ait+1, taking the derivative and using the
envelope condition yields the Euler equation:
cit+1
cit
= (βRt)
1
σ
(
C¯it+1
C¯it
)ψ
. (3.10)
Since prices are exogenous in this environment, to ensure a stable invariant
distribution we impose the restriction that R = 1/β under which the Euler equation
simplifies to:
cit+1
cit
=
(
C¯it+1
C¯it
)ψ
. (3.11)
Equation (3.11) will provide useful information regarding the distribution of
consumption. Before looking at the distribution in general, we will make the following
assumption (A1):
A1: When agents die they are replaced by identical agents.
This assumption removes aggregate uncertainty for all agents. Aggregate
uncertainty is problematic because a single agent’s decision will be conditioned on
her aspirations which are determined by the distribution of consumption. Therefore,
because this model has agents exiting through mortality, it is extremely difficult
to pin down steady-state dynamics. Assumption A1 solves this problem because it
constrains the consumption of the new agents to be the same as the recently deceased
and because the new agents’ decisions are governed by the same Euler equation the
50
aggregate system will evolve as if there is no mortality. Using A1 it is possible to
derive Lemma 1.
Lemma 1
Suppose that A1 holds. Order agents in terms of increasing levels of
consumption such that for individuals i− 1 and i, ci−1t < cit. If for i > j
cit+1
cit
= 1 then
cjt+1
cjt
= 1.
For proof, see the appendix. Lemma 1 says that if all agents with consumption
greater than agent j are enjoying their stationary consumption level, then so will
agent j. This is a rather intuitive result. Because all agents i > j are experiencing
a constant consumption level, it implies that their upward-looking aspirations are
constant as well. This in turn means agent j’s aspiration is unchanging, causing her
to settle into her relative position in the consumption distribution. It follows then
that
Proposition 2
If A1 holds, the distribution of consumption is stable.
The proof is simply an application of Lemma 1 after showing that the agent with
the highest level of consumption necessarily has zero consumption growth. However,
proposition 2 can be established analytically only under A1, so the question becomes:
how restrictive is this assumption?
Suppose we relax assumption A1 so that entering agents can have any
consumption level (specifically they are allowed to draw their own productivity but
51
start with ai0 = 0 and Hi0 = 35) in the distribution. How does this influence
the distribution of consumption, or more importantly, how does this change the
aspirations of the surviving agents? The answer is not much when the model is
calibrated to empirically plausible mortality rates and a large population. Consider
the gross mortality rate in 2010 for the United States: 0.8%3. For a large N ≥ 500
this will clearly have very little effect on the aspirations of agents because the role of
new agents will be quite small. Therefore the following relationship is approximately
true:
C¯it ≈ C¯it+1 (3.12)
and will prove useful when we solve the model computationally. Specifically, the
recursive form that (3.12) provides will allow for the use of aspirations as a state
variable.
Next turn to optimal choices of the other two control variables. Equations (3.13)
and (3.14) show the first order conditions for lit and qit respectively.
wθic
−σ
it C¯
ψσ
it − γ(1− lit)
−σ + β
∂Hit+1
∂lit
[φ′(Hit+1)[V (θit+1, ait+1, Hit+1, C¯it+1)− U ]
+ φ(Hit+1)V2(θit+1, ait+1, Hit+1, C¯it+1)] ≤ 0
(3.13)
−c−σit C¯
ψσ
it + β
∂Hit+1
∂qit
[φ′(Hit+1)[V (θit+1, ait+1, Hit+1, C¯it+1)− U ]
+φ(Hit+1)V2(θit+1, ait+1, Hit+1, C¯it+1)] ≤ 0
(3.14)
3Source: The Centers for Disease Control–http://www.cdc.gov/nchs/fastats/deaths.htm
52
Define
Ωit+1 ≡ φ
′(Hit+1)[V (θit+1, ait+1, Hit+1, C¯it+1)−U ]+φ(Hit+1)V2(θit+1, ait+1, Hit+1, C¯it+1),
the common term on the left hand side of equations (3.13) and (3.14). Using this,
from (3.14) it follows that
Ωit+1 =
c−σit C¯
ψσ
it
β [∂Hit+1/∂qit]
. (3.15)
Substituting (3.15) into (3.13) yields:
(
C¯ψit
cit
)σ (
wθi +
∂Hit+1/∂lit
∂Hit+1/∂qit
)
= γ(1− lit)
−σ. (3.16)
To make further progress, we assume a functional form for investment in health
capital. We assume that leisure time in health production and consumption of health
goods are complementary inputs according to a Cobb-Douglas technology:
I(lit, qit) = Q(1− lit)
αqρit,
where Q > 0 is the productivity of health investment and 0 < α + ρ ≤ 1. Equation
(3.16) can now be written as:
(
C¯ψit
cit
)σ
(1− lit)
σ−1
(
wθi(1− lit)−
α
ρ
qit
)
= γ. (3.17)
53
Since
(
C¯ψit/cit
)σ
is necessarily positive, in order for this equality to hold it must be
that
wθi(1− lit) >
αqit
ρ
.
Our goal is to understand how aspirations failure, more precisely relative deprivation,
affects an individual’s mortality. So consider the following comparative statics
exercise. Suppose (3.17) holds with equality and agent i experiences an increase
in her aspirations level, that is, C¯it. Since there is an increase in the first term on the
left hand side of (3.17) and the right hand side is constant, this means individual i
will have to adjust his labor supply or consumption of the health good. In the case
of the labor supply, (3.17) shows that an increase in labor supply (holding all else
constant) will unambiguously push the equation toward equality.
Consumption of the health good, however, has an ambiguous effect on (3.17).
An increase in qit will cause the second term on the left hand side to decline, but it will
also cause consumption to fall leading to a further increase in the aspirations gap. For
little more insight, consider the special case of γ = 0 for which wθi(1− lit) = αqit/ρ.
In this case, labor supply and the consumption of the health good have a negative
relationship suggesting that the increase in the labor supply in the previous example
(with γ > 0) would lead to a decline in qit. The negative relationship between
lit and qit would necessarily be true in the general case if the increase in labor
supply due to the increase in the aspirations gap caused the following to be true:
wθi(1 − lit) < αqit/ρ, which would necessitate a drop in the consumption of the
health good in order for (3.17) to hold with equality. From this reasoning it is clear
that an increase in aspirations level will raise labor supply and, quite likely, lower
consumption of the health good.
54
Simulations
Differently from the Ramsey model with KUWJ preferences, the entire
consumption and wealth distributions, not just their means, matter for households’
choices in this economy. Hence we rely on numerical methods to identify the
individual and aggregate consequences of upward-looking aspirations. We make the
parametric assumption that
φ(Hit) =
(
1−
1
Hit
)τ
whose curvature is determined by τ ∈ (0, 1).
In order to solve this problem, we will use dynamic programming. The agent
enters period t with an idiosyncratic labor productivity, financial assets, health
capital, and an aspirations level. These variables constitute the state-vector for
any individual and for ease, the Bellman system is rewritten as
V (θ, a,H, C¯) = max
l,a′,q
{
u+ β[φ(H ′)V (θ′, a′, H ′, C¯ ′) + (1− φ(H ′))U ]
}
c+a′ + q = θwl + R˜a
H ′ = (1− δ)H +Q(1− l)αqρ
C¯ ′ = C¯
θ′ = θ
R˜′ =
R
φ(H ′)
(3.18)
Baseline Results
Table 5 presents the parameters used in these simulations. The length of a
period is chosen to be a year, so the discount rate is set at 0.96 similar to the
55
business cycles literature. The shares of leisure and the health good in the health
production process are set at 0.85 and 0.15, respectively.
Parameter Value Description Source
α 0.85 Leisure Parameter in Health
Accumulation Equation
Match Health Investment-GDP
ratio. He, Huang, Hung (2014)
β 0.96 Discount Rate
σ 2 Elasticity of Substitution Carroll, Overland, Weil (1997)
Q 0.0602 Health Investment Parameter Match Health Investment-GDP
ratio. He, Huang, Hung (2014)
τ 0.1 Shape Parameter for Probability
of Survival
Match Health Investment-GDP
ratio. He, Huang, Hung (2014)
w 20 Wages
Hi0 35 Initial Stock of Health Capital
ρ 0.15 Health good Parameter in Health
Accumulation Equation
Match Health Investment-GDP
ratio. He, Huang, Hung (2014)
γ 0.5 Weight of Leisure Match Labor Supply. He,
Huang, Hung (2014)
ψ 0.5 Strength of Reference Level of
Consumption
Free
δ 0.03 Depreciation of Health Capital He, Huang, Hung (2014)
N 200 Size of the Population Scale
R 1β Rate of Return on Savings
TABLE 5. Parameter Values
For the aggregate simulations we will need to draw from different idiosyncratic
productivities. The state space for θ is discretized and agents are endowed with
productivities from the set Θ = {1, 20 : 0.01}. The weights are truncated Pareto
where the probability associated with observations greater than 20 is redistributed
over each point on the range 1 to 20 using the geometric formula G(Θ). The mean
and the functional form for G(·) were chosen to maximize inequality.
Before a detailed presentation of the computational results it will be helpful to
keep in mind the main result from the analytical section that agents with larger
aspirations gaps C¯it/cit will unambiguously supply more labor and most likely
consume less of the health good.
56
The existence of four state variables makes it difficult to present the policy rule
for all possible realizations.4 Because our primary goal is to understand the effect of
inequality, all of the choice variables are plotted against the individual’s aspirations
gap. Each choice variable will be presented in three graphs that correspond to
health stocks of 5, 10, and 15 and, unless otherwise noted, the individual’s savings
level will be zero. This choice does not have qualitative effects on the results shown
in this section, but it does allow for the clearest picture of how choices differ across
productivity levels.
Starting with the choice variables that influence the health stock, we see in figure
9 that, as indicated by our analytical results, the labor supply of an individual is
increasing in her aspirations gap. This confirms the basic intuition of the model: as
the gap between actual and desired consumption increases, agents will supply more
labor. The main issue with this outcome is that, ceteris paribus, an increase in the
labor supply results in lower health. However, before we can say anything definitive
about the evolution of the health stock we need to examine how consumption of the
health good varies. These are shown in figure 10.
1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
Lab
orS
upp
ly
(a) Health Stock: 5
1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
Lab
orS
upp
ly
(b) Health Stock: 10
1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
Lab
orS
upp
ly
(c) Health Stock: 15
FIGURE 9. Labor Choice vs Aspirations Gap by Health Stock
Solid: θ = 1, Dashed: θ = 3, Dotted: θ = 12
4The policy rules are plotted by calculating the aspirations gap, labor supply, health good, and
savings for a given exogenous level of aspired consumption, health stock, and savings.
57
Much like figure 9, figure 10 confirms the basic premise of the model. Those
agents with the largest aspiration gaps forgo consumption of the health good in
order to make up the difference. It is clear from these figures that an increase in the
aspirations gap results in fewer inputs into the health production function, resulting
in a higher probability of mortality. While the existence of the gradient between the
health inputs and the aspirations gap is independent of the agent’s health stock, the
level of investment is not. Both figures 9 and 10 show that as the agent’s health stock
deteriorates, she invests more in health production at the expense of consumption.
1 2 3 4 5 60
5
10
15
20
25
Aspirations Gap
Hea
lthG
ood
(a) Health Stock: 5
1 2 3 4 5 60
5
10
15
20
25
Aspirations Gap
Hea
lthG
ood
(b) Health Stock: 10
1 2 3 4 5 60
5
10
15
20
Aspirations Gap
Hea
lthG
ood
(c) Health Stock: 15
FIGURE 10. Health Good vs Aspirations Gap by Health Stock
Solid: θ = 1, Dashed: θ = 3, Dotted: θ = 12
Figure 11 shows the relationship between the change in the health stock and the
aspirations gap. As the results seen in figure 9 and 10 indicated, the health stock
is declining in the aspirations gap. Figure 11 also confirms the level effect shown in
figures 9 and 10: agents invest more in their health as their health stock falls.
The agent’s savings decision looks very similar to the that of the health good.
Figure 12 shows that as the aspirations gap increases, the agent chooses to hold less
in savings. Beyond that, the savings decision exhibits two interesting patterns. As
the health stock declines, first, the gradient between the aspirations gap and the level
of savings gets flatter and second, the absolute amount of savings increases. This
seems a little counter intuitive, because a declining health stock would seemingly
58
1 2 3 4 5 6
- 0.025
- 0.020
- 0.015
- 0.010
- 0.005
0.000
Aspirations Gap
%Ch
ang
ein
H it
(a) Health Stock: 5
1 2 3 4 5 6
- 0.028
- 0.026
- 0.024
- 0.022
- 0.020
Aspirations Gap
%Ch
ang
ein
H it
(b) Health Stock: 10
1 2 3 4 5 6
- 0.029
- 0.028
- 0.027
- 0.026
- 0.025
- 0.024
- 0.023
Aspirations Gap
%Ch
ang
ein
H it
(c) Health Stock: 15
FIGURE 11. Percent Change in Health Stock vs Aspirations Gap by Health Stock
Solid: θ = 1, Dashed: θ = 3, Dotted: θ = 12
cause agents to substitute away from savings towards the health good and leisure.
However, in this case savings acts as a way to increase health in the future. By
transferring wealth from today to tomorrow, agents can not only increase the amount
of the health good purchased but also their consumption. In this way, savings allows
an agent who is below her aspirations today to move closer in the future.
1 2 3 4 5 60
2
4
6
8
10
Aspirations Gap
Sav
ings
(a) Health Stock: 5
1 2 3 4 5 60
2
4
6
8
10
Aspirations Gap
Sav
ings
(b) Health Stock: 10
1 2 3 4 5 60
2
4
6
8
10
Aspirations Gap
Sav
ings
(c) Health Stock: 15
FIGURE 12. Savings vs Aspirations Gap by Health Stock
Solid: θ = 1, Dashed: θ = 3, Dotted: θ = 12
In this model, savings goes beyond the accumulation of financial assets, agents
can also save through investment in health. Therefore figure 13 plots the relationship
between total savings and the aspirations gap. Total savings is monetized by adding
59
the time value of leisure w(1−li) to the consumption of the health good and financial
assets.5
1 2 3 4 5 60
50
100
150
200
Aspirations Gap
Tota
lSa
ving
s
(a) Health Stock: 5
1 2 3 4 5 60
50
100
150
200
Aspirations Gap
Tota
lSa
ving
s
(b) Health Stock: 10
1 2 3 4 5 60
50
100
150
200
Aspirations Gap
Tota
lSa
ving
s
(c) Health Stock: 15
FIGURE 13. Total Savings vs Aspirations Gap by Health Stock
Solid: θ = 1, Dashed: θ = 3, Dotted: θ = 12
Figure 13 clearly shows a negative relationship between total savings and the
aspirations gap. Again this follows the logic that those with greater aspirations will
focus more on closing the gap and less on future consumption. The other striking
result from this figure is that a large proportion of savings is in the form of health.
Apparently, the return agents get on financial investment is not nearly as high as
the return on health which prompts them to invest more in health than conventional
savings instruments.
Next consider the evolution of an agent’s health stock. In the preceding
paragraphs it was made clear those agents that suffered from the greatest amounts of
relative depravation, invested the smallest amount in health. If agents are relatively
deprived for long periods of time, one would expect this lack of investment to manifest
in lower health stocks and shorter life expectancy. Figure 14 plots the health stock
against the aspirations gap for an aggregate simulation.
5This definition of total savings is not unlike Becker et. al ’s (2005) approach of valuing longevity
gains into their definition of a “full income”.
60
1.0 1.5 2.0 2.520.0
20.2
20.4
20.6
20.8
21.0
Aspirations Gap
He
alt
h
Sto
ck
(a) Age 20
1.0 1.5 2.0 2.5 3.0
11.6
11.8
12.0
12.2
12.4
12.6
12.8
13.0
Aspirations Gap
He
alt
h
Sto
ck
(b) Age 40
1.0 1.5 2.0 2.5 3.0 3.5 4.07.0
7.5
8.0
8.5
9.0
Aspirations Gap
He
alt
h
Sto
ck
(c) Age 60
FIGURE 14. Health Stock vs Aspirations Gap by Age. Lighter dots indicate
simulations with lower inequality.
The relationship depicted in figure 14 is easy to understand and, importantly,
confirms one of the empirical results reported in Deaton (2001). Clearly mortality
depends upon relative deprivation through upward-looking aspirations. What is
interesting about this result is that the gap between the “healthiest” and least healthy
individual increases with age (that is, it is higher in older cohorts), which means
that being exposed to inequality over long periods has a compounding effect on an
individual’s health. A back-of-the-envelope calculation for conditional life expectancy
suggests that the gap increases from 8 years at age 20 to 13 years at age 60 for the
parameter values reported in table 5.
What does upward-looking aspirations add to the model? Figure 15 presents
results of labor supply, consumption of the health good and the change in the health
stock plotted against the aspirations gap for the baseline model and a version of the
model without aspirations (equivalent to setting ψ = 0).
The inclusion of aspirations has two main effects. First, labor supply and
consumption of the health good strongly respond to the consumption gap when
individuals care about their relative position. Secondly, aspirations seriously impacts
the accumulation of health capital. Figure 15 also informs us about the effect
of aspirations on individual welfare. It can result in a significant loss of health,
61
1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
Lab
orS
upp
ly
(a) Labor Supply
1 2 3 4 5 60
5
10
15
Aspirations Gap
Hea
lthG
ood
(b) Health Good
1 2 3 4 5 6
- 0.04
- 0.03
- 0.02
- 0.01
0.00
Aspirations Gap
%Ch
ang
ein
H it
(c) Percent Change in Health
Stock
FIGURE 15. The influence of Aspirations on Labor Supply, Health Good and the
Change in the Health Stock.
Solid: Baseline, Dashed: No Aspirations
significantly increase labor supply that directly reduces utility through reduced
leisure and, of course, directly impact welfare through the aspirations gap.
Aggregate Effects
Now that we have characterized the individual’s problem, we can look at the
aggregate impact of aspirations on mortality. Given that we are looking at aggregate
simulations in this section we need to deal with the replacement of dead agents. We
assume that all incoming agents draw their own productivity level and start with
initial conditions: ai0 = 0 and Hi0 = 35. The zero assets assumption is in keeping
with the perfect annuities market, where the assets of dead agents are the property
of the risk neutral firm. In addition to the replacement of dead agents it is important
to address the issue of convergence. We check the issue of convergence by looking
at the time paths of average consumption, labor, and the Gini coefficient. Typically
simulations imply that these three variables reach stability after 100 periods. In what
follows each of the simulations were run for 125 periods with a “burn-in” period of
125 (these observations were dropped from the sample). In this section we will
presenting several aggregate results that depict the relationship between inequality
62
and life expectancy, but first we would like to show the time paths of the health
stock and asset holdings of the typical agent. Figure 16 shows these results.
0 20 40 60 80
0
5
10
15
20
25
30
35
Age
He
alth
Sto
ck
(a) Health Stock
0 20 40 60 80
0
2
4
6
8
10
Age
Sav
ing
s
(b) Savings
FIGURE 16. The Evolution of Individual Asset and Health Stocks
As can be seen in figure 16 the agent’s stock of health is monotonically decreasing
over time, while the asset stock monotonically increases until settling around the
maximum asset holdings. Not surprisingly, figure 16 implies that as agents get older
they are more likely to die in any given period.
The first step in looking at how aspirations effect mortality is to understand
how aspirations influence the inequality outcomes of the model. In other words, how
does the consumption and income inequality that result from the model relate to the
fundamental inequality (the inequality present in ability)? Milton Friedman posited
in Capitalism and Freedom (1962) that inequality is motivating for individuals as it
compels them to strive for something better, thus attenuating the effects of inequality.
In order to test this prediction, we remove health from the model and look at the
relationship between consumption/income inequality for the model with aspirations
against the model without aspirations. The results are in figure 17.
As seen in figure 17, social aspirations do not result in lower amounts of
inequality, which is contradictory to Friedman’s conjecture. The next question is:
63
ææ ææææ æææ
ààààààà
ààà
ììììòò òòò òòòòôôôô
ôôôôççççççáááá
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.1
0.2
0.3
0.4
0.5
Income Inequality HΨ= 0L
Inc
om
eI
neq
ual
ity
HΨ>
0L
(a) Income Inequality
ææ ææææ
ææ
àààààà
à
ìììììòòòòò òòòòòôôôô ôôôôôçççççççááááá
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.1
0.2
0.3
0.4
0.5
Consumption Inequality HΨ= 0L
Co
nsu
mp
tio
nI
neq
ual
ity
HΨ>
0L
(b) Consumption Inequality
FIGURE 17. Fundamental Inequality Plotted Against Income Inequality.
Solid: 45◦ Line, Markers: Simulations
why do social aspirations cause greater amounts of inequality? The reason could
be that social aspirations do not have a uniform impact on the distribution. In
particular, if aspirations cause higher productivity individuals to work relatively
harder than poorer individuals, it would be possible to see a spreading of the
consumption/income distribution, resulting in greater inequality. Figure 18 looks
at this possibility by plotting two ratios: median consumption/consumption by the
bottom 10% and consumption by the top 10%/median consumption.
Figure 18 shows that aspirations do have a differential effect on the distribution.
It is clear that the increase in the consumption ratio due to social aspirations is
greater for the top decile. The other result that can be ascertained from figure 18 is
that consumption is higher under aspirations, but it is important to note that this
increase in consumption come at the cost of lower health.
The next step in fully understanding the aggregate cost of social aspirations
is to look at how health production depends on income (sum of wage and rental
income). Because agents live for multiple periods, the relationship between income
and health production is not a static object that can be plotted. However, we can
look at the relationship between income and the health stock at various ages. In
64
*******
+
++++++
0.0 0.1 0.2 0.3 0.4 0.51.0
1.2
1.4
1.6
1.8
2.0
Fundamental Inequality
Co
nsu
mp
tio
nR
atio
HMe
dia
nB
ott
on
10%L
(a) Median/Bottom 10%
*******
+
+
+++++
0.0 0.1 0.2 0.3 0.4 0.51
2
3
4
5
6
Fundamental Inequality
Co
nsu
mp
tio
nR
atio
HTop
10%
Me
dia
nL
(b) Top 10%/Median
FIGURE 18. Consumption Ratios plotted against Fundamental Inequality.
This figure shows the consumption ratio of the median and the bottom decile and
the top decile and median plotted against inequality for both the model without
aspirations and the one with.
Blue/Plusses: Aspirations
Black/Stars: No Aspirations
figure 19, we fit a nonlinear model to the data produced by four different social
aspirations preferences (ψ ∈ {0, 0.25, 0.5, 0.75}).
20 40 60 80 100 120 140 16020.0
20.5
21.0
21.5
22.0
22.5
Income
Hea
lthS
tock
(a) Age: 20
20 40 60 80 100 120 140 16015.0
15.5
16.0
16.5
17.0
17.5
18.0
Income
Hea
lthS
tock
(b) Age: 30
20 40 60 80 100 120 140 160
11.5
12.0
12.5
13.0
13.5
14.0
14.5
Income
Hea
lthS
tock
(c) Age: 40
FIGURE 19. Health Stock as a Function of Income.
Solid/Black: ψ = 0, Red/Dotted: ψ = 0.4,Dashed/Black: ψ = 0.5, Purple/Dotted:
ψ = 0.6
In figure 19, regardless of the magnitude of the social concerns, aspirations
significantly lowers health production. The effect is clearly non-linear, as social
aspirations increases (going from ψ values of 0 to 0.4 versus 0.50 to 0.6) the impact
on health production worsens. One interesting aspect of this figure is that social
aspirations results in a greater loss of health production for richer individuals than
65
poorer ones. This result is most likely driven by the fact that richer individuals have
more to give up in terms of health expenditures. No matter how poor an agent is,
her health can only deteriorate by the depreciation rate every period. Therefore even
if aspirational concerns are so high that no one invests in health production there
is still a floor on what health can be for any age. Intuitively, richer individuals are
further away from this floor, which implies that an increase in aspirational concerns
will cause them to move more than their poorer counterparts.
Now that the inequality effects and the health costs of social status have been
characterized, it is important to examine the empirical results found in the literature
regarding inequality and health. As noted in the introduction, Wilkinson (1992)
originally found a significantly negative relationship between inequality and life
expectancy while Deaton (2001) found no such significant relationship.6 In what
follows we attempt to identify one potential reason why the empirical evidence is
mixed. First, we report a few empirical correlations between inequality and life
expectancy in table 6.7
The most significant result presented in table 6 is that the correlation weakens
over time.8 It is not obvious why this would be so. We explore two possibilities:
increase in income and change in the health production function. Looking again at
table 6, the split sample shows that after 2000 income growth had a significant effect
on life expectancy, so it could be that increases income are causing the weakening
6Subramanian and Kawachi (2004) survey the empirical literature on income inequality and
health and find mixed evidence.
7The data set was constructed using Gini data from the OECD, CIA World Fact Book, Deininger
and Squire Dataset. The life expectancy and income data are from the OECD website. The data
covers the period 1974-2010.
8This result is robust to splitting the sample at 1985, 1990,1995, 2000, and 2005.
66
Full Sample Before 2000 After 2000
Gini −9.386∗∗ −13.167∗∗∗ −8.831∗∗
(-2.486) (-2.853) (-1.993)
Full Sample Before 2000 After 2000
Gini −9.234∗∗ −13.791∗∗∗ −7.302∗
(-2.477) (-3.0179) (-1.735)
GDP Growth −0.167 0.055 −0.391∗∗∗
(-1.637) (0.425) (-3.555)
Full Sample Before 2000 After 2000
Gini −7.370∗∗ −12.928∗∗∗ −8.393∗∗
(-2.058) (-2.862) (-8.393)
Mean GDP Growth −0.308∗ 0.135 −0.662∗∗∗
(-1.932) (0.573) (-4.012)
t-stat in Parentheses. Significance: ***: 1%, **:5%, *:10%
TABLE 6. Data: Life Expectancy and Inequality
relationship. Figure 20 plots the relationship between life expectancy and inequality
with wages of 15, 20, and 25.9
Figure 20 shows that increasing incomes decreases the gradient between life
expectancy and inequality. This result is not immediately obvious, however one
would expect that an increase in income would result in increased health expenditure
(in absolute terms). Due to biological constraints, life expectancy can only increase so
much as a result of an increase in income, the reason for this stems from the concavity
of the survival function. Therefore the impact of a uniform increase income will be
lower for those economies with already high life expectancy/low inequality than those
economies with low life expectancy/high inequality. The empirical estimates of the
simulations in table 7 show the magnitude of this effect.
9In comparison to the baseline, the wage of 25 represents an 8-10% increase in GDP, while the
wage of 15 represents a 1-3% reduction.
67
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0.25 0.30 0.35 0.40 0.45 0.50
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FIGURE 20. Life Expectancy versus Inequality: Rising Income.
Solid/Black/Circles: w = 20, Dashed/Red/Squares: w = 15,
Dotted/Blue/Diamonds: w = 25
Table 7 confirms what was shown in figure 20, an increase in income results
in smaller gradient between life expectancy and inequality. The second option for
the weakening correlations was a change in the health production function. For
instance, improvements in medical science could result in greater health production
for a given set of inputs. Figure 21 depicts the relationship between inequality and life
expectancy for the baseline parameters (Q = 0.08) and the comparison parameters
(Q = 0.12).
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Gini HIncomeL
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FIGURE 21. Life Expectancy versus Inequality: Changing Medical Technology.
Solid/Black/Circles: Baseline, Dashed/Red/Squares: α = 0.7, ρ = 0.3
68
Regression 1 Regression 2
Inequality Gini Gini GDP Per Capita
w = 15 Income −18.72∗∗∗ −22.439∗∗∗ −0.303
(-5.214) (-5.073) (-1.4278)
Consumption −13.579∗∗∗ −15.867∗∗∗ −0.264
(-5.182) (-4.987) (-1.262)
w = 20 Income −12.831∗∗∗ −12.534∗∗∗ 0.151
(-3.558) (-3.448) (0.739)
Consumption −9.380∗∗∗ −9.151∗∗∗ 0.166
(-3.234) (-3.136) (0.807)
w = 25 Income −9.113∗∗∗ −9.75678∗∗∗ −0.212∗∗
(-3.568) (-3.872) (-2.377)
Consumption −7.146∗∗∗ −7.973∗∗∗ −0.227∗∗
(-3.318) (-3.740) (-2.517)
t-stat in Parentheses. Significance: ***: 1%, **:5%, *:10%
TABLE 7. Model: Life Expectancy and Inequality
Figure 21 shows that improvements in the medical technology result in greater
life expectancy, but do not explain why the correlation between life expectancy
and inequality is weakening over time. This result is rather intuitive. First,
improvements in health care would undoubtably result in better life expectancy.
However, this improvement in medical science does not change the marginal cost of
leisure (increased distance from aspired consumption), therefore we would expect to
see same gradient between life expectancy and inequality.
One final aspect of the model to look at is the role of the survival function’s
curvature. In order to examine the impact of changing the curvature, we will look at
health inequality in terms of the life expectancy gap. Specifically, we will look at the
difference in life expectancy for the top and bottom deciles of the distribution. Figure
22 plots the life expectancy plotted against inequality for τ = 0.1 and τ = 0.25.
Figure 22 shows that an increase in the curvature of the survival function
(decrease in τ) results in a steeper gradient between the life expectancy gap and
69
**
*
*
*
**
++
+
++++
0.25 0.30 0.35 0.40
0
2
4
6
8
Inequality HConsL
L
if
e
E
xp
ec
ta
nc
y
G
ap
FIGURE 22. Life Expectancy Gap vs. Inequality.
Black-Solid Line/Black Stars: Baseline, Blue-Dashed Line/Blue Plusses: τ = 0.25
inequality. Intuitively this due to the fact that for lower values of τ there is a steeper
drop-off in survival probability for those individuals with low health stocks. Given
that figure 19 showed low incomes were also associated with low health stocks, an
increase in the curvature of the survival function will lower life expectancy of those
in the bottom decile faster than those individuals in the top decile.
Conclusion
This chapter has used a model of upward-looking aspirations and endogenous
health to study the effect of inequality when social aspirations matter. The model
shows that relative deprivation within a reference group is an important determinant
for mortality outcomes. In addition, we showed that social aspirations act as
a motivating force which causes income and consumption inequality to be lower
than fundamental inequality. However, this motivation comes at a cost as social
70
aspirations drastically shift down the health production function, resulting in worse
health outcomes. Finally, we provided an explanation for why the correlation between
inequality and life expectancy has been declining over time. Clearly, increases in
income result in better health outcomes, however these improvements in health are
not equally shared across levels of inequality. Due to biological constraints those
individuals who live in low inequality societies with already high life expectancy
do not receive the same improvements in health that their counterparts in high
inequality/low life expectancy economies do.
A helpful extension to this work would be to explore the role of government.
Redistributive taxation may be able to improve health outcome by making
individuals feel relatively less deprived. The provision of public health may also
explain the weakening correlation between inequality and life expectancy because
health decisions would then be partially outside of the control of individuals, less
responsive to an individual’s inequality aversion.
71
CHAPTER IV
ASPIRATIONS AND REDISTRIBUTION
Introduction
The notion that individuals have social status concerns or care about their
relative standing has long been around in economics. Both Veblen (1898) and
Duesenberry (1949) stressed the importance of status concerns in examining the
individual’s decision problem. Recently, experimental evidence has shown that most
people do in fact care about relative income.1 2
Hopkins (2008) identifies three reasons why people could care about their relative
position within society. First, there could be rivalry: when others do better an
individual may worry that their success will elevate them to a position of power.
Secondly, it may reveal information—for example, witnessing others have success
may indicate that a change in an individual’s behavior is needed. The third reason
has to do with perception—the only way to know if an individual is doing well is to
compare herself to the success of others.
In this chapter, I take it as given that people exhibit these behavioral patterns
and assume people are status conscious. Status seeking is then used to address the
relationship between inequality and growth. The premise is that when individuals
feel deprived relative to the rest of the society, they may undertake actions that
are not necessarily in the best interest of the economy as a whole. In the case of
this chapter, these actions take the form of demanding higher redistribution, which
reduces entrepreneurship and innovation.
1See Johansson-Stenman et al. (2002), Solnick and Hemenway (1998), and Alpizar et al (2005).
2For an overview of the literature regarding social status see Weiss and Fershtman (1998).
72
Socially oriented preferences are formalized through the use of individual-specific
consumption benchmarks, referred henceforth as aspirations. Each agent forms
aspirations by taking the average of consumption by those agents above them in
the distribution. Therefore agents are upward-looking and, in terms of the Hopkins
(2008) classifications, motivated by feelings of rivalry.3
Agents inhabit a two-period OLG model and have preferences over two types of
goods, referred to broadly as consumption and leisure. Consumption is subject to
social status concerns while leisure is not.4 I assume that enjoyment of leisure time
requires requires certain marketed goods. Individuals are endowed with market time
and leisure time, which are both supplied inelastically to their respective activities.
Agents become workers or entrepreneurs based upon their endowed entrepreneurial
ability. Workers earn a wage determined in the market for labor, while entrepreneurs
produce a market good and earn profits net of taxation. The tax rate is determined
by a median voter and the proceeds are used to finance government services, which
act as a redistributive force that mitigates the negative effects of social concerns.
This can be thought of as the government providing services that could otherwise
not be afforded by an agent, making them feel relatively less deprived.
I present three main results. First, an economy with upward-looking aspirations
desires higher redistribution than one without. The intuition behind this result lies
in the relationship between government services and aspirations. Upward-looking
aspirations make the median agent feel relatively deprived, therefore in order to
mitigate these feelings of relative deprivation the median agent sets a higher tax rate
and hence higher government services.
3For a discussion of aspiration formation see Genicot and Ray (2010).
4See Hirsh (1976) and Frank (1985a, 1985b, 1999) for a discussion of the varying positionality
of goods.
73
Second, the tax rate is increasing in consumption inequality. The intuition here
again relies on the counteracting effects of government services on aspirations. An
increase in consumption inequality results in greater relative deprivation, thus the
median agent demands more of the government service.
Finally, the effect of an increase in ability (fundamental) inequality on the
equilibrium tax rate depends upon the occupation of the median agent. If the median
agent is a worker, an increase in ability inequality results in higher taxes, for reasons
similar to the first two results. However, if the median agent is an entrepreneur, an
increase in ability inequality results in a lower equilibrium tax rate, as the erosive
power of taxation is strong for an entrepreneur. This suggests the effect of inequality
on growth is not unambiguously negative.
This chapter is related to the vast literature on social status. Most relevant
are papers which deal with growth, taxation, and inequality. Two papers on the
linkage between inequality and growth have been influential. Persson and Tabellini
(1994) and Alesina and Rodrik (1994) both present theoretical and empirical evidence
depicting the negative relationship between inequality and growth. Their mechanism
by which this relationship arises is redistributive taxation, which lowers capital
accumulation and thus economic growth.
Within the social status literature the topics of growth and optimal taxation
have garnered significant attention. Theoretical support has been mixed regarding
the impact that social status concerns have on growth. Several papers show that
growth is higher under social status concerns, for example Corneo and Jeanne,
henceforth CJ, (1997, 1999a, 2001) use asset holdings to determine social status. The
measure of social status is slightly different in each of these paper, one uses asset held
relative to the average (1997), another uses total assets held (1999a), and the last
74
paper uses position within the wealth distribution (2001). In each of these papers the
authors use Arrow-Romer production externalities and showed that growth is higher
under social status. Peng (2008) builds a model similar to CJ (2001) but instead of
relative position, the author uses relative deprivation and confirms the result found
in CJ (2001). The interesting thing about these results is that just because social
status results in higher growth, the same is not necessarily true for increases in the
level of inequality. Corneo and Jeanne (2001) and Peng (2008) show that when
individuals care about relative position within the asset distribution, an increase in
inequality lowers the marginal benefit of increasing their position, thus they invest
less in productive activities. Fershtman et al. (1996) show that if social status is
attached to growth enhancing occupations, low-ability/high-wealth individuals will
acquire enough schooling to obtain these jobs. This in turn lowers the average ability
in these occupations thus lowering the growth rate.
The social status literature as it relates to taxation has largely focused on
optimal tax policies as set by a welfare optimizing government. Aronsson and
Johansson-Stenman (2008) construct a model where consumption is positional and
leisure is not. They show that taxation can result in better outcomes as it corrects for
overspending on the positional good. Ireland (2001) uses non-linear taxation on labor
in order to show that positive impact that taxation has on the economy. Much like
Aronsson and Johnasson-Stenman the reason for the corrective impact of taxation
is that it lowers the incentive to consume positional goods that have no beneficial
purpose. Other examples of taxation and social status that follow the same pattern
include Dodds (2012), Ljungqvist and Uhlig (2000), and Aronsson and Johansson-
Stenman (2010). One paper that studies social status concerns in a median voter
model is Alesina and Angeletos (2005), where multiple equilibria can arise based
75
upon a society’s view of fairness and how much of individual income is attributed to
luck rather than effort and ability.
My contribution to this literature is two-fold. First, I show that, in contrast to
the literature on social status and growth, social concerns (regardless of inequality) do
not necessarily result in better growth outcomes. The difference between this paper
and prevailing literature is that in my paper the agents do not have social concerns
over those goods which are necessarily growth enhancing. In the papers discussed
above, agents were given an incentive to hold more capital through social status
concerns, which given the Arrow-Romer externalities in the production function
results mechanically in a higher growth rate. In contrast, social concerns in this
paper are over consumption and the steps taken to satisfy these concerns actually
lower the growth rate. It should be noted however, that while social status concerns
result in higher taxation and lower growth, they do not necessarily result in lower
welfare. I show that the tax rate has a Laffer curve effect on welfare.
The second contribution of this paper is theoretical. It uses a dynamic median
voter economy compared to static models with an optimizing government commonly
used in the literature. Moreover, rather than peg aspirations to a common economy-
wide average, I use individual specific benchmarks.
This paper proceeds as follows. Section 2 discusses the model set-up, section 3
presents some analytical results, section 4 discusses the computational results, and
section 5 concludes.
76
Model
Household’s Problem
The economy is populated with a continuum of two-period OLG households
which consist of a parent and a single offspring on the unit measure. Each household
is endowed with an entrepreneurial ability drawn from an invariant distribution G(a),
which is bounded below by amin > 0. A subset of households use their entrepreneurial
ability to start a monopolistically competitive firm and earn profits. The rest find
employment in the firms of the entrepreneurial agents. A household’s income is
allocated toward consumption of the goods produced by the firms and production of
leisure. In this model leisure enjoyment requires the purchase of non-market goods.
The reason for this distinction between what the literature refers to as positional
(consumption) and non-positional (leisure) goods is that agents need face a trade-off
in order to respond to changes in the consumption distribution, otherwise the entirety
of income would be allocated to consumption. Lifetime utility is given generally by
equation (4.1).
Uit = u(cit, C¯it, gt) + v(Xit) (4.1)
where c is a consumption of the composite good, C¯ is the agent’s aspired
consumption, g government spending, and X is the amount of leisure consumed.
Aspirations are upward-looking and are formed using the average level of
consumption of the composite good by those agents who consume more than
individual i. The partials for the above preferences are given by:
∂U
∂c
> 0,
∂2U
∂c2
< 0,
∂U
∂g
> 0,
∂U
∂X
> 0,
∂2U
∂X2
< 0,
∂2U
∂c∂C¯
> 0,
∂2U
∂c∂g
< 0
77
The first five partials follow the standard assumptions within the literature.
It should be noted that the partial ∂U∂C¯ is left unrestricted because it is not clear
what impact an increase in aspired consumption would have on overall utility. An
argument can be made that an increase in aspired consumption makes an individual
better off because their prospects are better, while on the other hand an increase in
aspired consumption may make an individual worse off because she is farther away
from her goals.
Before looking at the budget constraint, I want to draw attention to how the
marginal utility of consumption depends on the level of aspirations and government
services. First, the cross-partial with respect to aspirations is positive: an increase
in aspirations makes consumption more valuable, thus inducing the household to
consume more. Second, the cross-partial with respect to government services is
negative. The effect of this is to attenuate the impact that aspirations has on the
level of consumption. In other words, when an increase in aspirations pushes the
agent further below her desired consumption, an increase in government spending
allows her to feel less relatively deprived.
Households are endowed with a unit of work time which they supply inelastically
and a unit of leisure time. If the agent is an entrepreneur, she produces a unique
final good in a monopolistically competitive market. Entrepreneurs must also pay
taxes on their profits which are used to provide the government service. Agents also
have a unit leisure time whose enjoyment requires the market inputs through the
production function:
H(x) = x.
78
The budget constraint is given by equation (4.2).
∫
j∈Υt
pjtc
j
itdj + p
x
t xit = yit
yit =



wt if i’s a worker
(1− τt)piit if i’s an entrepreneur
(4.2)
where at time t, Υt is the set of goods available for purchase, c
j
t is the consumption
variety j, pjt is the price of consumption variety j, wt is the wage paid to workers, xt
is the amount of leisure inputs purchased, px is the price of the leisure input, pit is
profits, and τt is the economy-wide tax rate.
The composite consumption good is given by
ct =
(∫
j∈Υt
(cjt)
−1
 dj
) 
−1
(4.3)
where  ∈ [1,∞) is the elasticity of substitution between varieties. Individual i’s
aspired consumption level is:
C¯t =
∫∞
ct
xdFt(x)
1− Ft(ct)
. (4.4)
Ft(·) is the time t distribution of consumption.
It is important to note that this aspirations level is defined as the average
of all consumption greater than or equal to their own. This implies that the
level of aspirations for the agent with the highest of consumption will be her own
consumption.
79
From the utility maximization problem, the demand for variety j by individual
i is
cjt =
(
pt
pjt
)
c
where the price index is defined by
pt =
(∫
j∈Υt
(
pjt
)1−
dj
) 1
1−
.
I normalize the budget constraint by the price level, which yields:
cit + Ptxit = yit (4.5)
where Pt =
pxt
pt
.
Entrepreneur’s Problem
As stated above, those agents who become entrepreneurs use their ability
endowment to produce a unique variety of consumption good in a monopolistically
competitive environment. Using the individual’s demand for good j from above, I
can write down the total demand faced by a firm as a function of their price and
total consumption. This expression is given by equation (4.6).
Cjt =
(
1
pjt
)(∫ 1
0
citdi
)
≡
(
pt
pj
)
Ct (4.6)
where C is aggregate demand. I assume that this is a closed economy so that total
production equals total demand.
Y jt = C
j
t (4.7)
80
Production technology is AK and it depends on the entrepreneurial ability of
the individual (a), the aggregate level of technology (A), and the amount of labor
employed (l). The expression for the production function is given by equation (4.8).
Y jt = Ata
j
t l
j
t (4.8)
Given the expression for total demand and the production function, I can solve
for the optimal price for each good j. The price is given by equation (4.9).
pjt =

− 1
(
wt
ajtAt
)
(4.9)
which is the standard result that price is simply a constant mark-up over marginal
cost.
Using the expression for the optimal price, I can write down the expressions
for profits (equation 4.10) and labor demand (equation 4.11) as a function total
consumption, aggregate productivity, entrepreneurial ability, and wages.
pijt = Ct
(
ajtAt
wt
)−1(
− 1

)( 1
− 1
)
(4.10)
ljt = Ct(a
j
tAt)
−1
(
− 1
wt
)
(4.11)
The next step in solving this model is to examine the occupational decision.
Note that ∂pijt/∂a
j
t > 0. Hence, I anticipate that there is a threshold ability level a
such that agents with a ≥ a become entrepreneurs and the rest enter the workforce.
81
In order to solve for a, I need to determine the value of a for which the after
tax profits from being an entrepreneur are equal to the wage received from working.
Explicitly this is given as:
(1− τt)pi(at) = wt
Solving the above expression gives:
at ≡
(wt
)
1
−1
At(− 1)C
1
−1
t (1− τt)
1
−1
(4.12)
It is assumed that the aggregate technology evolves according to a learning-
by-doing externality. This externality is determined by the share of the population
that are entrepreneurs. That means I can write down the evolution of the aggregate
technology stock as a function of the entrepreneurial cut-off value (given in equation
4.13).
At+1 = (1 + ν(1−G(at)))At (4.13)
Equation (4.13) shows the implicit dependency of aggregate technology growth
on taxation. This relationship poses a problem for growth outcomes if the tax rate
is positively related to the cut-off. In this case, an increase in the rate of taxation
actually has a negative impact on aggregate output.
When taking the derivative of a with respect to τ it is important to recognize the
implicit relationship between wages and taxes. If a change in taxation does influence
the cut-off value, this will effect wages because they are determined endogenously
through labor demand which is set by a. The derivative given by equation (4.14)
takes this implicit relationship into account.
82
∂at
∂τt
=
C
1
1−
t (1− τt)
− −1 (w(τt))
1
−1 (w(τt) + (1− τt)w′(τt))
At(− 1)2w(τt)
(4.14)
The sign of this derivative is not immediately apparent, it depends on:
(1− τt)
w′(τt)
w(τt)
T −1 (4.15)
To make further progress, I adopt specific functional forms.
Analytical Results
The analytical results will require placing structure on the agent’s utility and
the distribution from which entrepreneurial ability is drawn. Equation (4.16) gives
the assumed functional form.
Uit =
c1−σit
1− σ
C¯ψσit g
γσ
t + η
σ x
1−σ
it
1− σ
(4.16)
In order to satisfy the derivative given above the following parameter restrictions are
imposed:
0 < ψ < 1, γ < 0, σ > 1.
For convenience, I set the distribution of entrepreneurial ability to be Pareto
with location parameter amin = 1 and the shape parameter equal to α. I make the
additional assumption that  < 1 + α in order to ensure that the demand for labor
is positive.
83
Production
On the production side of the economy I am interested in obtaining closed form
solutions for two variables: wages and the entrepreneurial cut-off ability.
The first step in determining the equilibrium wage is to write down the
expression for aggregate labor supply and labor demand. Starting with aggregate
labor supply, note that because there is a continuum of agents I can write the
integral over the distribution of productivities rather than agents. The expression
for aggregate labor supply is given by equation (4.17).
LSt ≡
∫ at
0
1dG(x) = G(at) = 1− (− 1)
α
α
1−Aαt (1− τt)
α
−1C
α
−1
t w
α
1−
t (4.17)
from this alone, the effect of the tax rate on labor supply is unclear because wages
also depend on τ . However, equation (4.17) shows that an increase in wages, holding
everything else constant, does result in an increase in labor supply. In the same
manner, I can write down the aggregate labor demand by integrating over the firm
level demand. The equation for aggregate labor demand is given by (4.18).
LDt ≡
∫ ∞
at
Ct
(
− 1
wt
)
A−1t x
−1dG(x) =
α(− 1)α+1
α
1−Aαt (1− τt)
α−+1
−1 C
α
−1
t w
α
1−
t
α− + 1
(4.18)
Much like labor supply, I cannot determine the impact that an increase in the
tax rate has on the labor demand, but it is clear that an increase in wages results in
a lower demand for labor. Equations (4.17) and (4.18) can be combined in order to
determine the equilibrium wage rate (given by equation 4.19).
84
wt =
(− 1)
−1
 A
−1

t C
1

t
(
α(−τt)+(1−τt)(1−)
α−+1
) −1
α
(1− τt)
α−+1
α

(4.19)
Taking the derivative of the wage rate with respect to the tax rate yields:
∂wt
∂τt
= −
(− 1)
−1
 A
−1

t C
1

t (− τt)(1− τt)
− (α+1)(−1)α
(
α−+1
α−+1 − τt
) −1
α −1
2
< 0, (4.20)
which is unambiguously negative. In order to understand why this is the case let’s
look at the expression for the ability cut-off for entrepreneurs. When equation (4.19)
is substituted in equation (4.12) simplifies to:
at =
(
−τt(α− + 1) + (α− 1)+ 1
(1− τt)(α− + 1)
) 1
α
(4.21)
whose derivative with respect to the tax rate is given by:
∂at
∂τt
= −
(− 1)
(
1− α(−1)(τt−1)(α−+1)
) 1
α
(τt − 1)(−(α + 1)τt + (α + τt − 1) + 1)
.
This is positive if:
1 > −
α(− 1)
(1 + α− )(1− τt)
.
This is unambiguously true based on the assumption that 1+α > . This means that
an increase in the tax rate results in fewer entrepreneurs, which explains why the
wage rate falls. As the tax rate increases, the entrepreneurial cut-off value increases
which decreases labor demand while simultaneously increasing labor supply, thus
decreasing the wage rate.
85
The next step is to determine if the median agent is a worker or an entrepreneur.
This is straightforward to determine, because the median agent will be endowed with
the median ability draw from the assumed Pareto distribution, which is given by:
aM = 2
1
α
Therefore if a > (<)aM then the median agent is a worker (entrepreneur) and
the tax rate will be determined accordingly. Under what conditions is the median
agent a worker?
(
−τt(α− + 1) + (α− 1)+ 1
(1− τt)(α− + 1)
) 1
α
> 2
1
α
Simplifying yields:
1 + α− 
1 + α− 
> 2− τt
Note that the right hand side of the above expression is at its maximum when τt = 0.
Hence, sufficient conditions (by setting τ = 0) for the median agent to be a worker
are:
α > 1−−2 if  > 2
α < 1−−2 if  < 2
.
Given the conditions above, if  > 2 then the median agent has to be a worker
because α > 1 (under the conditions for a Pareto Distribution).
If either of the conditions are violated, then the median agent is a worker only
if the tax rate is sufficiently high. The condition on the tax rate is given by:
τt > 2−
1 + α− 
1 + α− 
86
Household’s Problem
The economy-wide tax rate is set through a median voter, therefore I need
to determine the desired tax rate for each individual. It is important to note that
because all the workers earn the same wage they will desire the same tax rate, hence if
the median voter is a worker the economy-wide tax rate will be given by an arbitrary
worker’s first order condition. In contrast, if the median agent is an entrepreneur,
the economy-wide tax rate will be given by the agent with ability aM .
Each household has effectively two choice variables: enjoyment of leisure and
their desired tax rate. While the choice of leisure is straightforward, the agent’s
desired tax rate is not. In choosing the desired tax rate, every agent acts as if she
was the median voter, thereby internalizing the effect of her desired tax rate on
aggregate variables like wages and government spending. The first order condition
for leisure is given by equation (4.22).
xit =
η
C¯ψitP
1
σ
t g
γ
t + ηPt
yit (4.22)
Given the parameter assumptions at the start of this section, it is clear that an
increase in the level of aspirations results in a decline in the consumption of leisure.
This makes sense because the increase in aspirations results in an increase in the
marginal value of the composite good. In contrast, an increase in the provision
of government services increases the consumption of leisure because g lowers the
marginal value of consumption.
The desired tax rate on the other hand implicitly sows the first order condition
given by equation (4.23).
87
γσ
(
P
1
σ
t C¯
ψ
itg(τt)
γ
P
1
σ
t C¯it
ψg(τt)γ + Pη
)
g′(τt)
g(τt)
= (σ − 1)
y′(τt)
y(τt)
(4.23)
where g(·) is the level of government services and y(·) is the household income. The
government is assumed to run a balanced budget:
g(τ) = τ
∫ ∞
a
pi(x)dG(x),
which implies that g(0) = 0 and g(1) = 0, the latter a direct result of the expression
for a. Also, because τ and pi(x) are necessarily non-negative: g(x) > 0 for 0 < x < 1.
This means that the derivative of g′(τ) is ambiguous over the range of possible tax
rates.
Because the relationship between taxation and government services is non-
monotonic, it stands to reason that the relationship between the tax rate and life
utility is non-monotonic as well. In figure 23 I look at the relationship between social
welfare (as measured by the Benthamite social welfare function, that is, a weighted
sum of the individual lifetime utilities) and the tax rate.
Figure 23 shows that despite the fact an increase in the tax rate lowers the
growth rate of the economy it is not necessarily welfare reducing. The Laffer curve
presented in figure 23 presents an interesting relationship between the tax rate and
total social welfare. For a large range of tax rates, the marginal impact of increasing
taxation on welfare is quite small. This suggests that within this range of tax rates the
decline in wages from increased taxes is almost completely balanced by the increase
in government services. At the extremes of the possible tax rates, the Laffer curve
shows that the marginal impact of changing taxation is quite high.
88
ΤT
ot
al
S
oc
ia
l
W
el
fa
re
FIGURE 23. Lifetime Utility Plotted Against the Tax Rate
To make further progress I use computational methods to understand the
relationship between desired taxation and aspirations.
Computational Results
The first step is to look at the relationship between the level of aspirations
and desired taxation. In this section, I will not be looking at dynamics but rather
the static relationship between inequality and taxation. Figure (24) shows this
relationship for both workers and entrepreneurs.5
Two things are to be noted in this figure. First an increase in the level
of aspirations results in a greater demand for taxation by both workers and
5In the figures presented in this section, the following parameters are used: technology and
aggregate demand are scaling parameters and are set at 1 and 100, respectively. The preferences
parameters are set: σ = 2, η = 4, γ = −0.5, and ψ = 0.5. The distribution parameters are amin = 1
and α = 3.25. The elasticity of substitution is set at 1.75 or 2.1 depending on the figure. Finally,
the price of the leisure good is set at 1.5. The results presented in this section are robust to a wide
range of parameter values.
89
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Aspirations
D
es
ire
d
Ta
xa
tio
n
FIGURE 24. Relationship between the level of aspirations and desired taxation for
workers and entrepreneurs.
Workers: Black/Solid, Entrepreneurs: Red/Dashed
entrepreneurs. This make sense, as agents would desire to reduce the impact of
higher aspirations by demanding more government services. The parameter choices
of γ and ψ determine the level and shape of the curves in figure (24). As γ → 0, the
level of desired taxation converges to zero because the effect of government services
on utility diminishes. Likewise, as ψ → 0, the demand of redistribution flattens out
with respect to the level of aspirations because they play less of role in determining
the agent’s overall level of utility.
Figure (24) does not reveal how inequality influences the agent’s choice. One
could have an extremely rich but relatively equal economy with a high level of
aspirations. Therefore, I need a notion of an aspirations gap, the ratio of an
individual’s aspired consumption to their actual consumption. This measure captures
how relatively deprived agents are in comparison to their reference group.
90
The first order condition for leisure implies that this aspirations gap is:
C¯it
cit
=
C¯1−ψit
(
P
1
σ
t C¯
ψ
itg
γ
t + Ptη
)
P
1
σ
t g
γ
t yit
This expression is used in figure (25) to plot the relationship between the
aspirations gap and the desired rate of taxation for four possible situations: Median
Agent (MA) is a worker and  < 2, MA: Worker and  > 2, MA: Entrepreneur and
 < 2 and MA: Entrepreneur and  > 2.
2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
De
sire
dT
axa
tio
n
(a) Median Agent: Worker ( < 2)
2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
De
sire
dT
axa
tio
n
(b) Median Agent: Worker ( > 2)
2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
De
sire
dT
axa
tio
n
(c) Median Agent: Entrepreneur ( < 2)
2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
De
sire
dT
axa
tio
n
(d) Median Agent: Entrepreneur ( > 2)
FIGURE 25. Aspirations Gap vs the Desired Rate of Taxation for workers and
entrepreneurs. If the median agent is a worker, the ability of the shown entrepreneur
is a. If the median agent is an entrepreneur, her ability is aM .
Black/Solid: Worker, Red/Dashed: Entrepreneur
91
The first thing to note is that figure (25d) is blank because, as discussed above,
the median agent will never be an entrepreneur if  > 2 so the results pertaining to
this case are irrelevant. Figures (25a) and (25b) show the desired tax rates for the
workers and the entrepreneur with ability a. In both of these figures, the economy-
wide tax rate will be given by the solid-black line. In figure (25c) the median tax rate
will be given by the entrepreneur with ability a = 2
1
α (depicted by the red/dashed-
line). There are a couple of things to note from this figure. First, regardless of
the type of worker the desired tax rate is increasing in the agent’s aspirations gap.
This implies that an increase in consumption inequality results in an increase in the
demand for redistribution. This is means an increase in taxation will shrink the pool
of entrepreneurs, resulting in lower growth of output per capita.
It is clear that the elasticity of substitution between varieties of consumption
goods has a rather meaningful impact on the desired tax rate. As can be seen
in figures (25a) and (25b) as  increases the gap between entrepreneurs and workers
increases and the tax rate as determined by the median voter increases. This happens
because an increase in the elasticity of substitution makes various goods closer
substitutes, which erodes profits and increases the entrepreneurial cut-off ability,
thus lowering wages. Workers optimally deal with the loss of wages by substituting
towards government services, while entrepreneurs attempt to stave off further losses
by demanding lower taxation.
The next question that needs to be addressed is the relationship between
inequality and the desired tax rate. Figure (25) shows that for an invariant
distribution of entrepreneurial ability an increase in consumption inequality will
increase the desired tax rate. To see this consider the situation where the median
agent is a worker. Because the workforce is homogenous and an agent’s aspirations
92
are inclusive of her own consumption, her aspirations will simply be average
consumption, cˆ. Therefore, her aspirations gap is simply cˆ/cM . As this measure
increases, so does inequality, which implies an increase in consumption inequality
results in greater taxation. However, it does not explain what happens when the
inequality in the underlying distribution of ability increases. Figure (26) plots the
relationship between the aspirations gap and the desired tax rate for three different
distributions of ability (the parameter α is varied).
5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
De
sire
dT
axa
tio
n
(a) MA: Worker/ DT:Worker
5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
De
sire
dT
axa
tio
n
(b) MA: Worker/ Entrepreneur
5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
De
sire
dT
axa
tio
n
(c) MA: Entrepreneur/ DT:Worker
5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
De
sire
dT
axa
tio
n
(d) MA: Entrepreneur/ DT:Entrepreneur
FIGURE 26. Aspirations Gap vs the Desired Rate of Taxation for workers and
the lowest ability entrepreneur. The median voter is a worker and there are three
different underlying distributions of ability.
MA: Worker—Black/Solid: Gini=0.5, Red/Dashed: Gini=0.35, Blue/Dotted:
Gini=0.2
MA: Entrepreneur—Black/Solid: Gini=0.2, Red/Dashed: Gini=0.1, Blue/Dotted: Gini=0.05
Figure (26) presents an interesting set of results. The effect that increases
in inequality have on the desired tax rate depends upon which type of agent is
93
being examined. If the agent under examination is a worker, an increase in ability
inequality results in a greater demand for redistribution. However, if the agent is
an entrepreneur the increase in inequality results in a lower demand for taxation.
To gain further understanding of this behavior, I want to examine two variables:
(1) the entrepreneurial cut-off level and (2) the difference in income accruing to
entrepreneurs and workers.
0.0 0.2 0.4 0.6 0.8 1.00
1
2
3
4
5
Gini
a
(a) Entrepreneurial Cut-Off Value
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Gini
Dif
fer
enc
eIn
Pro
fits
(b) Profits vs. Wages
FIGURE 27. The entrepreneurial cut-off and total income differential plotted against
ability inequality.
a: Black/Solid, aM : Red/Dashed
The first thing to note is that in both the case of the cut-off value and the
difference in accrued income are both increasing in the level of inequality. Figure
(27a) implies that both the profits of the marginal entrepreneur and wages are
increasing inequality—in order for the cut-off value to increase the costs faced by
the entrepreneur (wages) must increase as well. The results presented in figure (26)
make more sense when interpreted in the light of figure (27b). As inequality increases,
workers face a larger gap between their income and the income of those above them
in the distribution, thus in order attenuate this impact they demand greater amounts
of redistribution. Entrepreneurs, on the other hand, face a greater loss of profits to
taxation as inequality increases thus they demand a lower amount of redistribution.
94
Therefore the impact that increases in inequality on the rate of taxation depends
upon the occupational location of the median voter. That being said, figure (27a)
shows that, except in the case of extremely low levels of inequality, the marginal
entrepreneur is above the median agent in the distribution, therefore it is likely that
the tax rate will be determined by a worker.
The next thing to look at in regards to ability inequality is how it affects the
growth rate. Figure (28) plots the growth rate of technology against ability inequality
for a given tax rate (ν = 1).
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Gini
G
ro
w
th
Ra
te
of
Te
ch
no
lo
gy
FIGURE 28. The Growth Rate of Technology plotted against Ability Inequality for
a given tax rate.
Not surprisingly the increase in ability inequality results in lower growth. The
reason for this is that an increase in ability inequality raises the entrepreneurial
cut-off ability, which limits the amount of learning-by-doing within an economy.
Finally, I want to look at the impact that aspirations has on the economy-wide
tax rate, or in other words, how do the outcomes this model differ from a model with
no aspirational considerations. Figure (29) plots the economy-wide tax rate for both
the aspirations and no aspirations case.
95
2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
De
sire
dT
axa
tio
n
(a) Median Agent: Worker
2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
Aspirations Gap
De
sire
dT
axa
tio
n
(b) Median Agent: Entrepreneur
FIGURE 29. Aspirations Gap vs the Desired Rate of Taxation for when the median
voter is a worker and when the median voter is an entrepreneur.
Aspirations: Black/Solid, No Aspirations: Red/Dashed
Figure (29), shows two results. First, the presence of aspirations causes a
gradient between the economy-wide tax rate and where the median agent sits in
the consumption distribution. This is rather intuitive, in the case where the agent
does not have preferences over where she sits in the distribution, her desired tax rate
does not depend upon her location. The second result is that aspirations result in
uniformly higher rates of taxation. This is of concern if entrepreneurship is directly
related to taxation and growth. In this case, societies with strong aspirational
concerns would experience lower growth rates due to their higher rates of taxation,
which is in direct contrast to past studies of social status and growth. The reason for
this contrast is that in past studies, social status either directly or indirectly induces
agents to do things that are growth enhancing. While in this paper agents seek out
actions that are growth reducing in an effort to satisfy their relative concerns.
Conclusion
In this chapter, I present a dynamic model of social status and growth. I show
that, in contrast to the existing literature, social status results in an unambiguously
96
lower growth rate. Second, an increase in consumption inequality results in greater
redistribution and lower economic growth. And finally, an increase in ability
inequality has an ambiguous impact on redistribution and economic growth.
Future work on this chapter will include five avenues. First, a better motivation
for the trade-off between positional and non-positional goods is necessary. Second, I
will simulate the model in order to examine the evolution of inequality over time and
the relationship between fundamental inequality and consumption inequality. Third,
I will study the empirical implications regarding inequality and taxation. Fourth, it
would be helpful to look at the possibility that entrepreneurs may use their wealth to
capture de jure political power and influence tax policy. Finally, it will be interesting
to look at the situation where entrepreneurship depends on wealth, for example due
to credit constraints, besides ability.
A further extension of the work presented here would be to apply it to a
developing country in order to understand how social concerns and redistributive
taxation effect the decision to move from the informal sector to the formal sector.
In this context agents would have to choose between producing/working in the low
productivity informal sector with no taxation or the higher productive formal sector
with taxation.
97
APPENDIX A
SUBSISTENCE CONSUMPTION
Capital Shares
Caselli Guerriero Caselli Guerriero
Country & Feyrer Gollin OECD LS6 Country & Feyrer Gollin OECD LS6
Algeria 0.125 0.122 Japan 0.256 0.246 0.298 0.168
Argentina 0.267 Jordan 0.251
Australia 0.182 0.189 0.232 0.131 Latvia 0.300 0.247 0.164
Austria 0.220 0.227 0.169 Malaysia 0.162 0.238
Barbados 0.142 Mauritius 0.329 0.254 0.359
Belgium 0.200 0.197 0.239 0.223 Mexico 0.251 0.318 0.240
Bolivia 0.081 0.092 0.062 Morocco 0.231 0.137
Botswana 0.327 0.303 Namibia 0.130
Brazil 0.109 Netherlands 0.240 0.233 0.230 0.218
Bulgaria 0.223 0.235 New Zealand 0.121 0.190 0.118
Burundi 0.029 Nicaragua 0.080
Canada 0.157 0.189 Norway 0.216 0.197 0.240 0.177
Chile 0.163 0.135 Panama 0.149 0.160
Colombia 0.120 0.131 Paraguay 0.187 0.077
Congo 0.173 Peru 0.216 0.167
Costa Rica 0.108 0.120 Philippines 0.209 0.173 0.199
Cote d’Ivoire 0.062 0.061 Portugal 0.202 0.182 0.237 0.079
Denmark 0.204 0.218 Korea 0.265 0.230 0.170 0.174
Dominican Rep. 0.176 Moldova 0.120
Ecuador 0.079 0.062 Romania 0.192 0.163
Egypt 0.101 0.264 Russia 0.153
El Salvador 0.277 Singapore 0.379 0.307
Estonia 0.257 0.249 0.199 South Africa 0.209 0.214
Fiji 0.095 Spain 0.240 0.254 0.211
Finland 0.197 0.181 0.243 0.190 Sri Lanka 0.136 0.056
France 0.189 0.206 0.233 0.196 Sweden 0.163 0.160 0.239 0.170
Gabon 0.093 Switzerland 0.183 0.264 0.099
Germany 0.235 0.231 0.197 Thailand 0.096
Greece 0.146 0.240 0.237 Trin. and Tob. 0.080 0.087
Hungary 0.139 0.230 0.208 Tunisia 0.188 0.238
India 0.052 Turkey 0.318 0.292
Iran 0.083 UK 0.178 0.156 0.220
Ireland 0.178 0.270 0.191 United States 0.177 0.175 0.231
Israel 0.222 0.260 0.170 Uruguay 0.182 0.178
Italy 0.215 0.209 0.239 0.222 Venezuela 0.126 0.100
Jamaica 0.256 0.278 0.077 Zambia 0.063
TABLE 8. Capital Shares by Country
Robustness Check
As discussed above, the form of the matching technology depends upon the
nature of the externalities associated with the labor market. In this appendix I will
show that the results above hold regardless of the value chosen for ξ. The matching
technology I will consider can be described generally in three ways: convex, linear,
and concave. I will check the impact that the shape has on the baseline results
using different values for ξ (which are taken from {0.5, 1.0, 1.5, 2.0}, where 2.0 is
the baseline number). Figure 30 shows the evolution of λ for each of the matching
technologies.1
1The capital shares used in figure 30 are from Guerriero.
98
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(a) c¯ = 0
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(b) c¯ = 58%
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(c) c¯ = 69%
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(d) c¯ = 80%
FIGURE 30. The evolution of the percentage of the workforce employed in the
modern sector across time for the baseline calibration. Each panel represents different
subsistence estimate.
ξ = 2.0: Solid/Black, ξ = 1.5: Dashed/Red, ξ = 1.0: Dotted/Blue, ξ = 0.5: Dashed-
Dot/Green
In the figure above it is clear that regardless of the level of subsistence,
the concave matching technology results in the fastest rate of convergence to full
employment in the modern sector. This result is to be expected, given that the
concave technology results in higher probabilities of successfully matching into the
modern sector. That being said, even under the concave matching technology it can
take up to six generations of relative stagnation before there is a rapid transition out
of the traditional sector. The last thing to consider is the welfare effect of subsistence
under each of these technologies. Table 9 presents the loss in generational growth
rates.
Despite the fact that concavity causes a faster transition to full employment
in the modern sector, Table 9 shows that under eight of the nine parameter
combinations ξ = 0.5 results in the greatest reduction in growth rates when compared
to the zero subsistence baseline. The reason for this is that the head start the
economy is given under c¯ = 0 is amplified with a concave technology. In other
words, because λξ > p happens fastest under c¯ = 0, this parameterization results in
relatively faster convergence when compared to other subsistence levels.
The final robustness check is to examine the impact that p has on the results
above. Using the baseline calibration (with ξ = 2.0), figure 31 shows the evolution
of λ for five different values of p.
Figure 31 shows that as p increases, the rate of convergence to full employment
in the modern sector increases as well. This is driven by the increased probability
of successfully matching when the modern sector is underdeveloped. Even though
the speed of convergence increases, the basic pattern remains: the existence of
subsistence constraints results in slower growth. Table 10 shows the loss in
generational growth rates across the different values of p.
As is expected the growth rate effect is muted for larger p. But subsistence still
has a non-zero effect on the generational growth rate. While the reduction of output
growth by 0.0126 may not seem like a large number, it manifests itself as a reduction
99
Caselli
& Feyrer Gollin
c¯ = 58% c¯ = 69% c¯ = 80% c¯ = 58% c¯ = 69% c¯ = 80%
ξ = 0.5 0.1395 0.2299 0.3729 0.0805 0.1190 0.2182
ξ = 1.0 0.1343 0.2050 0.3160 0.0506 0.0942 0.2244
ξ = 1.5 0.1275 0.1911 0.3024 0.0430 0.0918 0.2124
ξ = 2.0 0.1149 0.1840 0.2883 0.0360 0.0746 0.2041
Guerriero
c¯ = 58% c¯ = 69% c¯ = 80%
ξ = 0.5 0.1531 0.2038 0.3567
ξ = 1.0 0.1208 0.1948 0.3319
ξ = 1.5 0.1054 0.1818 0.3271
ξ = 2.0 0.1025 0.1669 0.3121
TABLE 9. Welfare Analysis–Loss in Growth Rates: This table gives the annualized
growth rate loss (positive number denotes a reduction in the growth rate) in
comparison to the baseline with c¯ = 0 using the three sets of estimates for the
capital shares. This table provides a robustness check for the matching technology,
checking different values for ξ.
in output of 4% over the course of the transition path to full employment in the
modern sector.
Social Planner
In order to solve the social planner’s problem I will need to move away from
idiosyncratic productivities. The reason for this is that the presence of heterogeneity
makes it impossible to determine the transition equation for the aggregate capital
stock, therefore homogeneity is needed to solve the dynamic programming problem.2
In this set-up, the social planner’s goal is to maximize the total utility of all agents
over an infinite horizon by choosing the agent’s bequests and λt ∀t, taking into
account the evolution of technology. Explicitly, the social planner solves:
∞∑
t=0
N∑
i=1
βt[(1− γ) log(cit − c¯) + γ log(ait+1)]
s.t. cit + ait+1 + hitx¯ = φ¯w
j
t +Rtait
At+1 = (1 + ηλt)At
(A.1)
2It is assumed that every individual is endowed with the mean of the distribution from the
baseline simulations.
100
0 10 20 30 40 50 600.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(a) p = .1
0 2 4 6 8 10 12 140.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(b) p = .2
0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(c) p = .3
0 2 4 6 80.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(d) p = .4
0 1 2 3 4 5 6 70.0
0.2
0.4
0.6
0.8
1.0
Generations
Λ
(e) p = .5
FIGURE 31. The evolution of the percentage of the workforce employed in the
modern sector across time for the robustness check for p. Each panel represents
different p.
c¯ = 0%: Solid/Black, c¯ = 58%: Dashed/Red, c¯ = 69%: Dotted/Blue, c¯ = 80%:
Dashed-Dot/Green
c¯ = 58% c¯ = 69% c¯ = 80%
p = .1 0.4780 0.5808 0.7293
p = .2 0.1018 0.1639 0.3012
p = .3 0.0556 0.0637 0.1417
p = .4 0.0543 0.0765 0.1077
p = .5 0.0126 0.0215 0.0671
TABLE 10. Welfare Analysis–Loss in Growth Rates: This table gives the annualized
growth rate loss (positive number denotes a reduction in the growth rate) in
comparison to the baseline with c¯ = 0 using the three sets of estimates for the
capital shares. This table provides a robustness check for the matching technology,
checking different values for p.
where β is the social discount rate. Unfortunately, solving for an analytical solution
to this problem is not possible because there does not exist a closed form solution
for the allocation of capital. However, it is possible to determine the solution to the
social planner’s problem using computational methods. The computational methods
described in this section will require a bit of finesse. I will use dynamic programming
to solve this problem, though doing so is not completely straightforward. The reason
for this is that the discrete choice between the two sectors does not lend itself to
a recursive formulation. That being said, the decentralized solution shows that all
agents will eventually switch to the modern sector, and it stands to reason that this
will be the goal of the social planner as well. I can rely on the fact that there are
no idiosyncratic differences between agents, which implies that in the long-run all
agents will bequeath the same amount to their offspring. This is the starting point
for the social planner solution. Using the associative property, the objective function
101
can be written as:
max
{ait,λt}∞t=0
N∑
i=1
∞∑
t=0
βt[(1− γ) log(cit − c¯) + γ log(ait+1)]
Consider period τ , for which all t > τ the following is true:
λt = 1, At+1 = (1 + ηφ¯
1−ω)At, ait = at
The objective function can be written as:
max
{ait,λt}τt=0
N∑
i=1
τ∑
t=0
βt[(1− γ) log(cit − c¯) + γ log(ait+1)]
+
N∑
i=1
max
{at}∞t=τ+1
∞∑
t=τ+1
βt[(1− γ) log(cit − c¯) + γ log(at+1)]
Let
V (at) = max
at+1
{(1− γ) log(cit − c¯) + γ log(at+1) + βV (at+1)}
Using the V (·) function above, I can write the objective function as:
max
{ait,λt}τt=0
N∑
i=1
[
τ∑
t=0
βt[(1− γ) log(cit − c¯) + γ log(ait+1)]
]
+ βτ+1V (at) (A.2)
The V (·) is in the form necessary for dynamic programming and satisfies the
necessary recursive properties, while the first term of (A.2) is a straightforward finite
horizon problem that can be solved using the standard numerical techniques. There
is one problem that needs to be dealt with prior to solving for the optimal solution:
at is not stationary because At is growing over time. To solve this problem it will
be necessary to normalize the variables in the model. The obvious choice for doing
so is to divide by At. This is straightforward for t > τ . Starting with the budget
constraint we get:
cit
At
+
at+1
At
=
φ¯wMt
At
+Rt
at
At
Letting tilde denote normalized variables, I can the write the above expression as:
c˜it +
At+1
At
a˜t+1 = φ¯w˜
M
t +Rta˜t
102
Because I am considering period t > τ , I can substitute 1 + ηφ¯1−ω in for the growth
of technology, which yields the following budget constraint:
c˜it + (1 + ηφ¯
1−ω)a˜t+1 = φ¯w˜
M
t +Rta˜t (A.3)
where
w˜Mt =
wMt
At
=
(1− α)Kαt A
1−α
t (φ¯N)
−α
At
= (1− α)kαt (φ¯N)
−α
Rt = αk
α−1
t (φ¯N)
1−α
(A.4)
kt =
Kt
At
=
N∑
i=1
a˜it = Na˜t
Notice that substituting the expression for normalized capital into the expression for
wages and the rate of return, yields:
w˜Mt = (1− α)(Na˜t)
α(φ¯N)−α = (1− α)φ¯−αa˜αt
Rt = α(Na˜t)
α−1(φ¯N)1−α = αφ¯1−αa˜α−1t
Substituting in these expressions for income into the budget constraint and
simplifying yields:
c˜it + (1 + ηφ¯
1−ω)a˜t+1 = φ¯
1−αa˜αt (A.5)
The next step is to normalize the utility function. This can be done by adding the
following term: log(At)− log(At) + log(At+1)− log(At+1). Simplifying the objective
function yields:
max
at
N∑
i=1
∞∑
t=0
βt[(1− γ) log(c˜it − cˆt) + γ log(a˜t+1)] +Qt (A.6)
where Qt is a time determined constant that does not depend on the individual’s
choice. The variable cˆt is normalized subsistence consumption and it follows the
recursion:
cˆt =
cˆt−1
1 + ηφ¯1−ω
, cˆ0 =
c¯
A0
Using (A.5) and (A.6) it is fairly straightforward to set-up the dynamic programming
problem with two states (a˜t, cˆt) and one control (a˜t+1). Figure 32 shows the policy
rule for cˆt = 0. The next step is to solve the finite horizon problem in which the
terminal value of the utility function is given by V (aτ+1). In order to solve this
problem, I need to write down the associated budget constraint. Taking the original
constraint given in (A.1) and normalized by At yields (using the same notation as
103
0 50 100 150 200
0
50
100
150
200
at
a t
+1
FIGURE 32. Policy Function: a˜it+1
before):
c˜it + (1 + ηλ
ω
t φ¯
1−ω)a˜t+1 + hitxˆt = φ¯w˜
j
t +Rta˜t (A.7)
where xˆt is the normalized cost of switching and the growth of technology includes
λ because it is no longer 1 for all t. Normalized wages and rates of return are given
by:
w˜Mt =
wMt
At
=
(1− α)(KMt )
αA1−αt (λtφ¯N)
−α
At
= (1− α)
(
kMt
)α
(λtφ¯N)
−α
w˜Tt =
wTt
At
=
(1− θ)(KTt )
θB1−θ((1− λt)φ¯N)−θ
At
= (1− α)
(
kTt
)α
Bˆ1−θt ((1− λt)φ¯N)
−θ
RMt = α
(
KMt
)α−1
A1−αt (λtφ¯N)
1−α = α
(
kMt
)α−1
(λtφ¯N)
1−α
RTt = θ
(
KTt
)θ−1
B1−θ((1− λt)φ¯N)
1−θ
(
At
At
)1−θ
= α
(
kTt
)θ−1
Bˆ1−θt ((1− λt)φ¯N)
1−α
where kjt for j ∈ {M,T} and Bˆt are the normalized values of capital in each sector
and traditional technology, respectively. Arbitrage implies that the rates of return
must be equal in both sectors. This allows me to solve for the capital allocated to
each sector for a given λ and Bˆ.
kTt
kMt
=
θBˆ1−θt ((1− λt)φ¯N)
1−θ
α(λtφ¯N)1−α
(A.8)
The next step is the objective function, which is quite similar to (A.2) but instead
of a single a the social planner chooses a bequest for each agent.
N∑
i=1
τ∑
t=0
βt[(1− γ) log(c˜it − cˆt) + γ log(a˜it+1)] + β
τ+1V (a˜iτ+1) (A.9)
104
The final step in this solution technique is to establish a rule by which the social
planner allocates individuals to the modern sector. It makes sense that the social
planner would not want to remove an agent from the modern sector once that
individual takes up employment because doing so would result in an unambiguous
loss of utility. Therefore, it is reasonable to assume that the social planner would
follow some sort of additive rule in which the stock of agents in the modern sector
is augmented by χN each period (where 0 ≤ χ ≤ 1). I can write the transition
equation for λ as:
λt = min(λt−1 + χ, 1) (A.10)
Now I can solve the social planner’s problem by maximizing (A.9) subject to (A.7),
(A.8), and (A.10). The parameter values for these simulations are given in table 1.
Parameter Value Parameter Value
β .96 γ 1/4
c¯ 211 α .175
θ .124 N 100
B 1 η 1φ¯1−ω
χ ∈ {.01, .025, .05, .1}
TABLE 11. Social Planner Parameters
The social planner’s problem is solved using the four values of χ listed in
the table above. It is important to note that traditional wages have a negative
dependence on the modern technology stock. This is important because as the length
of time needed to converge to full employment in the modern sector increases, the
traditional wages move quickly toward zero. Practically, this poses a problem for
values of χ < .05 because the traditional wage gets so small that the agent’s income
drops below the machine precision effectively making it zero. Therefore the starting
points for the simulations must be tailored such that the issues with machine precision
do not come into play. The simulations for the social planner’s problem are run at
.01 increments for the initial value of λ from table ?? (for instance if λ0 ∈ {.5, 1}
there would be a simulation for λ0 = .5, λ0 = .51, . . . , λ0 = 1).
Step Size Initial λ
χ = .01 λ0 ∈ {.8, 1}
χ = .025 λ0 ∈ {.525, 1}
χ = .05 λ0 ∈ {0, 1}
χ = .1 λ0 ∈ {0, 1}
TABLE 12. Social Planner Modern Sector Allocation
Figure 33 shows the simulation results for the social planner’s problem.
105
0.0 0.2 0.4 0.6 0.8 1.0
110
115
120
125
130
135
Initial Λ
Me
an
Ut
ilit
y
FIGURE 33. Mean lifetime utility associated with each social planner plan. Each
point on the x-axis represents a different λ0. Solid: χ = .1, Dashed: χ = .05, Dotted:
χ = .025, Dashed-Dot: χ = .01
Though I am unable to run the simulations for every initial value of λ the pattern
is quite clear: mean utility is decreasing in the length of time it takes to reach full
employment in the modern sector. The result that mean utility is maximized for
λ0 = 1 should not be surprising. A large λ results in high technology growth,
which drives unnormalized wage growth in the modern sector causing utilities to
rise. Therefore, a social planner would want to move everyone in the modern sector
to take advantage of this wage growth.
106
APPENDIX B
INEQUALITY AS A HEALTH HAZARD
Proof of Lemma 1
Proof. Choose agent m s.t. ∀i > j:
cit+1
cit
= 1
From the FOC and the definition of the aspirations term I know that:
cjt+1
cjt
=


1
I−J
(
cjt+1 +
∑N
h=j+1 cht+1
)
1
I−J
(
cjt +
∑N
h=j+1 cht
)


ψ
=
(
cjt+1 +
∑N
h=j+1 cht+1
cjt +
∑N
h=j+1 cht
)ψ
Define: gjt+1 =
cjt+1
cjt
Dividing the above first order condition by 1cjt yields:
gjt+1 =


gjt+1 +
∑N
h=j+1 cht+1
cjt
1 +
∑N
h=j+1 cht
cjt


ψ
Let Ajt =
∑N
h=j+1 cht
cjt
. Given the assumption that cit+1 = cit, it is clear that: Ajt+1 =
Ajt. Making the appropriate substitutions yields:
gjt+1 =
(
gjt+1 + Ajt
1 + Ajt
)ψ
(L.1)
Clearly gjt+1 = 1 solves this equation. Note that:
If gjt+1 > 1→ gjt+1 <
(
gjt+1 + Ajt
1 + Ajt
)ψ
If gjt+1 < 1→ gjt+1 >
(
gjt+1 + Ajt
1 + Ajt
)ψ
Therefore the solution gjt+1 = 1 is also unique.
107
Proof of Proposition 2
Proof. The final step is to show that for I, consumption is constant. Taking the
Euler equation for N and applying the definition of the aspirations term, yields:
cIt+1
cIt
=
(
cIt+1
cIt
)ψ
The only solution to this equation is cIt+1cIt = 1. The result is achieved by applying
Lemma X.1.
108
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