The Impact of Fiscal Policy on Intergenerational Income Distribution
Dr. Juan Carlos Medina Guirado
Lic. Ma. Aleida Ramírez Vega
Resumen
Este trabajo contribuye a la discusión actual en torno a la inequidad del ingreso entre generaciones.
Se propone una estructura teórica que establece un papel no trivial a la redistribución fiscal como
un instrumento para mitigar la brecha del ingreso intergeneracional. En particular, a través de un
modelo de generaciones traslapadas, individuos jóvenes obtienen recursos de consumo a través de
su participación en el mercado de trabajo. consumen a través del ingreso por su aportación de
fuerza laboral al mercado y los individuos viejos consumen de su tasa de retorno al capital; la
sobreinversión de este último grupo favorece la asignación de los recursos hacia el grupo más
viejo. Esto conlleva a un incremento en la inequidad del ingreso, lo que da cabida a una política
de redistribución fiscal que subsidia a individuos jóvenes a través de un impuesto al rendimiento
de capital.
Abstract
In this paper, we contribute to the debate related to income inequality between generations. In
particular, we propose a theoretical framework that clearly establishes a role for fiscal
redistribution in ameliorating this income gap. By using an overlapping generations structure in
which young individuals consume from realized labor effort and older individuals from returns to
capital; overinvestment in the latter favors resource allocation toward older individuals. As this
leads to increasing income inequality, there is a role for a redistributional fiscal policy that
subsidizes young individuals by imposing a tax on capital returns.
1
1. Introduction
In recent decades, income inequality has increased in both advanced and developing economies
(International Monetary Fund, 2014). But, when looking back, we realize that it can represent a
deeper historical condition. At least, over the past century, in The Anglo-Saxon countries, such as
the US, Argentina, China, Colombia, India, Indonesia, South Africa and, more recently, China, the
evolution (fall and rise) of income inequality -when graphically represented- forms a U-shape
(Piketty, 2013). Figure 1, shows an example of this.
Figure
1.
Income
Inequality
in
Emerging
Countries,
1910-2010.
Source:
https://philebersole.files.wordpress.com/2014/06/chart-04.jpg
This finding appears to reveal important evidence about the historically unequal concentration of
wealth that has lasted until today (Ebersole, 2014). This means that more and more income has
been accumulated in the highest top class and the remaining classes have to distribute the wealth
among themselves, which, not surprisingly, it is still lopsided.
We consider that inequality is a non-desirable condition. When present in high levels, it tends to
lower long-term growth, reduces physical and capital accumulation, and rises macroeconomic
volatility, which also impacts negatively on the economy growth (Lopez, H., & Perry, G. E., 2008).
Recent empirical work finds that high levels of inequality can be 2 | P a g e detrimental to
achieving macroeconomic stability. Others have argued that rising inequality may have been an
important contributing factor to the global financial crisis (International Monetary Fund, 2014).
2
Romer (2006) states that the highly marked gaps in the standard of living over time, across
countries, impacts population’s welfare. It is strongly associated with important differences in
nutrition, literacy, infant mortality, life expectancy and others welfare indicators. Additionally, a
high income inequality is more likely to decrease lifetime mobility dynamics (e.g. in income and
educational mobility). Populations relatively positioned in less advantageous circumstances,
remain in the same economic condition over their lives, thus the inequitable course of action is
more likely to be repeated (Lopez, H., & Perry, G. E., 2008).
Increasing inequality has been attributed to a range of factors, including the globalization and
liberalization of factor and product markets (International Monetary Fund, 2014). Others claim
that top of the income distribution comes mainly from capital income rather than unequal pay
(Pomeroy, 2014). The actual unequal ownership of assets is predicted to worsen distribution of
income away from labor and towards holders of capital (Krugman, 2014). But, moreover, much of
the difference in outcomes can be attributed directly to government.
Nonetheless, it is by public policy as well, by which we can halt and reverse the most extreme
inequal economic conditions (Krugman, 2014). One of the corrective measures that government
can implement is by establishing a redistributive policy. According to Ostry et. al (2014),
redistribution can affect economic growth in a directly way, through human capital accumulation,
political instability, etc. and indirectly through reducing net inequality. The combined direct and
indirect effects of redistribution -including the growth effects of the resulting lower inequalityare, on average, pro-growth.
Ostry et. al (2014) also state that “It would still be a mistake to focus on growth and let inequality
take care of itself, not only because inequality may be ethically undesirable but also because the
resulting growth may be low and unsustainable”. Piketty’s position declares that in democracies,
policies reflect society’s view. Therefore, the ultimate driver of inequality and policy might well
be social norms regarding fairness of the distribution of income and wealth (Pomeroy, 2014).
So, based on the foregoing, given the importance of inequality and its long-term persistence, we
can infer that inequality is composed by a generational problem. We also have the perspective that
supports the fact that government intervention is needed to rearrange population income more
effectively. Thus, through this work, we are willing to analyze inequality and income distribution
3
by means of an Overlapping Generation Model (OLG), limiting the scope of inequality to an
income problem.
Specifically, we want to know what is the mechanism through which fiscal policy (in the form of
lump-sum taxation) impacts intergenerational income distribution in an economy populated by
heterogeneous agents. In particular, in any time period, two types of agents coexist, that are born
one-period apart, making their generations overlap.
The paper is presented as follows:
In Section 2 we introduce the basic model; Section 3 gives the model specific functional forms for
preferences and production technology but no redistributional policy; Section 4 introduces income
redistribution, Section 5 concludes; Section 6 describes the derivation of results and, finally,
Section 7 presents our bibliographical references.
1. The Basic Model Setup
In this section we proceed to describe the basic setup and the environment of our modeling
framework.
As we are interested in generational inequality from a macroeconomic perspective, a natural choice
to model this phenomenon is the overlapping generations model (OLG). This type of model was
pioneered by Allais (1947), Samuelson (1958), and Diamond (1965). Because they are based on
the individual behavior, they are useful to explain issues related to welfare (Bertola et al., 2014).
OLG models contribute to the analysis of the implications of agents' behavior during their life
cycle. The model is transcendental because it gives us an insight into the need of individuals to
save during their productive years to finance consumption during retirement and the welfare of
future generations (Ostry, et al. 2014). At the same time, some authors have stated that rational
savings choices are motivated by a desire to smooth the level of consumption over time, therefore,
it should be based on households’ permanent income levels rather the current (Bertola et al., 2014).
Such an idea refers to the Permanent Income Hypothesis proposed by Friedman (1957). This also
allows us to see the interaction of individual decisions which sets the standard for incentives of the
currently generation and government policies.
4
2.1 Factor Markets
A homogenous consumption good is produced through a production technology that exhibits
constant returns to scale and uses capital 𝐾𝑡 and labor 𝐿𝑡 as inputs: 𝐹(𝐾𝑡 , 𝐿𝑡 ) and which we will
use in its labor intensive form 𝑓(𝑘𝑡 ) where 𝑘𝑡 = 𝐾𝑡 /𝐿𝑡 .The net rate of return to a holder of a unit
of capital is represented by 𝑟𝑡 and workers earn an amount 𝑤𝑡 per unit of labor effort exerted. In
this manner, factors are paid the value of their marginal contribution to the production process:
𝑟𝑡 = 𝑓′(𝑘𝑡 )
( 1)
𝑤𝑡 = 𝑓(𝑘𝑡 ) − 𝑘𝑓′(𝑘𝑡 )
( 2)
2.2 The Environment
Time is discrete and infinite indexed by = 0,1,2, … , ∞ . Individuals live for two periods of time
denoted as “young” and “old”. Because there are an infinite amount of agents born at different
time periods, at any date there are two different generations that coexist in the economy. We denote
𝑦
𝑜
a young generation born at time 𝑡 by 𝐺𝑡 , which becomes old in the following period: 𝐺𝑡+1
. We
assume that the economy starts with an initial old generation 𝐺10 as illustrated in figure 2.
Generation/Time
1
0
𝐺1𝑜
1
𝐺1
1
𝑦
2
3
…
𝑡+1
𝐺2𝑜
𝑦
𝐺2
𝐺3𝑜
⋮
⋱
𝑡−1
𝑜
𝐺𝑡−1
𝑡
𝐺𝑡−1
𝑡+1
𝑡
𝑦
𝐺𝑡𝑜
𝑦
𝐺𝑡
𝑜
𝐺𝑡+1
Figure 2. OLG Model
5
2.3 Timing of actions
We now describe the actions during the lifetime of a representative generation that starts at time 𝑡
as depicted in figure 3. First, young agents provide perfectly-inelastic labor effort for which they
earn 𝑤𝑡 per labor unit 𝑙𝑡 . After realizing their income, they consume an amount 𝑐𝑡 of a
homogeneous consumption good. Their remaining income is stored in a savings technology for
which they earn a net interest rate of 𝑟𝑡+1 per unit of the consumption good deposited. Now, during
their old stage at period 𝑡 + 1, they consume their entire savings and at the end of this period exit
the economy.
Figure 3. Sequence of actions in the economy
2.4 The consumer’s problem
The representative consumer maximizes her lifetime utility by choosing a consumption basket of
𝑦
𝑜
a homogenous when young (𝑐𝑡 ) and through savings, financing consumption when old (𝑐𝑡+1
),
𝑦
1
𝑦
𝑜
𝑜
according to preferences 𝑈(𝑐𝑡 , 𝑐𝑡+1
) = 𝑢(𝑐𝑡 ) +  𝑢( 𝑐𝑡+1
), where 𝛽 ≡ 1+𝜌 and in which 𝜌 >
𝑦
−1 is the subjective discount factor. After choosing their first period consumption 𝑐𝑡 , young
𝑦
agents’ savings are given by 𝑠𝑡 = 𝑤𝑡 𝑙𝑡 − 𝑐𝑡 so that consumption when old is dependent on the
return to savings:
𝑦
𝑜
𝑐𝑡+1
= (𝑤𝑡 𝑙𝑡 − 𝑐𝑡 ) (1 + 𝑟𝑡+1 )
(3)
6
Therefore, the consumer’s problem becomes:
𝑀𝑎𝑥
𝑀𝑎𝑥
𝑦 𝑜
𝑦
𝑜
𝑦 𝑜 𝑈(𝑐𝑡 , 𝑐𝑡+1 ) = 𝑦 𝑜 {𝑢(𝑐𝑡 ) +  𝑢( 𝑐𝑡+1 )}
𝑐𝑡 , 𝑐𝑡+1
𝑐𝑡 , 𝑐𝑡+1
(4)
And it follows that she faces a resource constraint at time 𝑡 in which the present value of
lifetime consumption expenditures equals labor income:
𝑦
𝑐𝑡
𝑜
𝑐𝑡+1
+
= 𝑤𝑡 𝑙𝑡
1 + 𝑟𝑡+1
(5)
The solution to the representative consumer’s problem implies that the optimal consumption path
𝑦∗
𝑐𝑡 𝑐𝑡𝑜∗ must follow the Euler equation:
𝑦
𝑜
𝑢′ (𝑐𝑡 ) =  (1 + 𝑟𝑡+1 ) 𝑢′ ( 𝑐𝑡+1
)
(6)
So that
𝑦
𝑢′ (𝑐𝑡 )
1 + 𝑟𝑡+1
=
.
𝑜
′
𝑢 ( 𝑐𝑡+1 )
1+𝜌
(7)
The equation above, reveals that intertemporal consumption depends on the return to savings
relative to the individuals’ cost of postponing present consumption (embodied in their impatience
parameter). For instance, if 𝑟𝑡+1 < 𝜌, the individual is relatively impatient making the ratio of
marginal utilities less than one. That is, in comparison to a situation in which the individual is
𝑦
𝑜
consuming the same amount in both periods (𝑐𝑡 = 𝑐𝑡+1
), she can increase her lifetime utility by
changing her consumption allocation: reduce consumption when young in exchange for
consumption when old. Such an exchange should continue until the ratio of marginal utilities
𝑦
equals the right hand side of equation (7). As a result, and for this particular individual, 𝑐𝑡 <
7
𝑜
𝑐𝑡+1
. By the same logic, it must be that if 𝑟𝑡+1 ≥ 𝜌, then
(1+𝑟𝑡+1 )
(1+𝜌)
𝑦
≥ 1, and thus 𝑢′ (𝑐𝑡 ) ≥
𝑦
𝑜
𝑜
𝑢′ ( 𝑐𝑡+1
), which imply 𝑐𝑡 ≤ 𝑐𝑡+1
.
2.5 Savings and income inequality
As mentioned before, young individuals must save part of their income in order to be able to
consume when old. It is this process that determines capital accumulation in the economy in such
a way that
𝑠𝑡 = 𝑘𝑡+1
(8)
that is, current savings determine the size of the capital stock in the following period, as we assume
no depreciation.
In a given time period, and as discussed before, there exists two overlapping generations: young
and old. Within this period, the consumption opportunities of young individuals are given by their
net income:
𝑦
𝑐𝑡 = 𝑤𝑡 − 𝑠𝑡
(9)
In contrast, consumption resources for old individuals in the same time period will depend on the
return to savings or equivalently, on the return to capital:
𝑐𝑡𝑜 = (1 + 𝑟𝑡 )𝑘𝑡
(10)
In this manner, we can construct a measure of consumption inequality that depends on the
individual’s savings decisions and consequently on capital accumulation. Thus, for any period 𝑡,
we will focus on the consumption resources that are available to an average old generation, relative
to those of an average young generation:
8
𝑐𝑡𝑜
𝑦
𝑐𝑡
=
(1 + 𝑟𝑡+1 )𝑘𝑡
𝑤𝑡 − 𝑠𝑡
(11)
Furthermore, as factor payments depend on the capital stock available (equation (1) and (2)), we
can rewrite the equation above such that:
𝑐𝑡𝑜
[1 + 𝑓 ′ (𝑘𝑡 )]𝑘𝑡
𝑦 =
𝑓(𝑘𝑡 ) − 𝑘𝑓 ′ (𝑘𝑡 ) − 𝑘𝑡+1
𝑐𝑡
(12)
In this way, if the above ratio increases over time, then the process of capital accumulation
increases the gap between the return to labor and that of capital accumulation, leaving
proportionally less resources to the young generations. We interpret this result as an increase in
intergenerational resource or “income” inequality.
In what follows, we will illustrate the ideas presented in this section using specific functional forms
for the individual’s utility as well as for the production technology.
2. A Baseline Model of Inequality without Income Redistribution
We start by presenting a scenario in which there is no fiscal authority that redistributes resources
among concurrent generations. Therefore, intergenerational income distribution, as defined in
I.V, in the economy will evolve solely on consumer’s lifetime savings decisions. In the next
section however, we change this by including a fiscal authority that subsidizes young agents
through resources that come from taxing capital gains of old agents. The OLG framework in
which we perform our analysis preserves the characteristics described in the previous section,
however since there is no population growth, and for simplicity, we normalize labor effort so
that 𝑙𝑡 = 1.
Production of the consumption good takes place through a (labor-intensive) Cobb-Douglas
technology:
𝑓(𝑘𝑡 ) = 𝐴𝑘𝑡 𝛼
(13)
9
Where 𝐴 is a productivity parameter an 𝛼 represents the share of capital in the production
process. Similarly, Individuals have logarithmic preferences over lifetime consumption so that
to their maximization program is represented by:
𝑀𝑎𝑥
𝑦 𝑜 ln(𝑐𝑡 ) + 𝛽 𝑙𝑛 (𝑐𝑡+1 )
𝑐𝑡 𝑐𝑡+1
(14)
𝑠. 𝑡.
𝑤𝑡 = 𝑐𝑡 +
𝑐𝑡+1
1 + 𝑟𝑡+1
(15)
The solution to the representative consumer’s problem yields the following consumption for a
young individual:
𝑤𝑡
(1 + 𝛽)
(16)
𝛽
𝑠𝑡 = 𝑤𝑡 (
)
1+𝛽
(17)
𝑦∗
𝑐𝑡 =
which implies a level of savings:
and thus consumption when old is given by:
𝑜∗
𝑐𝑡+1
=
𝑤𝑡 (1 + 𝑟𝑡+1 )[(1 + 𝛽) − 1]
1+𝛽
(18)
By equation (1) and (2), the savings equation (17) can be written as a function of the
capital stock in the economy:
10
(19)
𝑠 ∗ (𝑘) =
𝛽
[(1 − 𝛼)𝐴𝑘𝑡 𝛼 ]
1+𝛽
Where we can observe that the level of savings increases at higher levels of capital
accumulation. Now, and since 𝑠𝑡 = 𝑘𝑡+1 , then
𝑘𝑡+1 =
𝛽
[(1 − 𝛼)𝐴𝑘𝑡 𝛼 ]
1+𝛽
(20)
which shows the evolution of capital in the economy.
3.1 Inequality
Given that we now have specific functional forms, we can explicitly associate income inequality
with capital accumulation. Recalling our measure of inequality in (20), we can use equations (1),
(2) and (12) which gives:
𝑐𝑡𝑜
𝑦
𝑐𝑡
𝑘𝑡1−𝛼
𝐴 +𝛼
=
𝑠
(1 − 𝛼)(1 − 𝑤𝑡 )
(21)
𝑡
Now, we can study what happens to intergenerational inequality as capital accumulation increases.
First, we can immediately see that the numerator is increasing in 𝑘𝑡 . However, the change in the
𝑠
denominator ultimately depends on 𝑤𝑡 , the savings ratio. With the use of equations (2) and (8), we
𝑡
can see that the savings ratio is constant:
𝛽
𝑠𝑡
=
1 + 𝛽 𝑤𝑡
(22)
11
This result is due to our log-preferences specification so that savings behavior is independent of
the return to savings.
Thus, by (21) it follows that intergenerational income inequality is increasing in the capital stock
and this result drives the following proposition:
Proposition 1 (Intergenerational Inequality without Income Redistribution): In an economy
without income redistribution, income inequality increases via overinvestment in capital which
𝑐𝑜
𝜕
outweighs labor returns to younger generations in the long run: 𝜕𝑘 (𝑐𝑡𝑦 ) > 0 .
𝑡
𝑡
Which tells us that as 𝑘𝑡 increases, income inequality also increases as reflected by higher rents
from capital accumulation by the old generations.
3.2 Discussion
We have seen that in an economy without income redistribution, inequality arises as the income
gap between the old and young generations increases as the stock of capital increases. This due to
a dynamic inefficiency in which there is overinvestment in the productive asset that ultimately
makes income from capital returns superior to that from labor income earned by young individuals.
It makes sense then, to ask ourselves if by having a fiscal authority whose policy is to redistribute
income from old individuals to the young, at every period, will such action put downward pressure
on income inequality?
In order to answer this question, in the next section we develop the same model as in the present
section but we now include a fiscal authority that is able to redistribute income resources between
old and young generations that concurrently live at any time period.
12
4. The Baseline Model with Income Redistribution
In the economy, there is a government that redistributes resources forma old to young individuals.
This process implies imposing a proportional tax 𝜏 on the net returns to capital investment (capital
gains tax) to old individuals and using this tax income to subsidize young individuals via lumpsum a transfer 𝜑 every period.
Therefore, under fiscal redistribution, the consumer’s lifetime consumption when young becomes
𝑦
𝑐𝑡 = 𝑤𝑡 + 𝜑 − 𝑠𝑡
(23)
Similarly, consumption when old is:
𝑜
𝑐𝑡+1
= [1 + (1 + 𝜏)𝑟𝑡+1 ]𝑠𝑡
(24)
Consequently, the consumer’s problem is to maximize (14) subject to (23) and (24).
In this manner, the optimal consumption allocation of the consumer is now dependent on the
income redistribution parameters:
𝑦∗
𝑐𝑡 =
𝛽(𝑤𝑡 + 𝜑)
(1 + 𝛽)
𝑜∗
𝑐𝑡+1
= [1 + (1 − 𝜏)𝑟𝑡+1 ] (
(25)
𝑤𝑡 + 𝜑
)
1+𝛽
(26)
We can see that lifetime consumption is increasing in the lump-sum transfer, however, increases
in capital taxation reduce consumption only when old.
13
4.1 Inequality
Now, looking back at intergenerational inequality, equation (21) now becomes:
𝑘𝑡1−𝛼
(1
𝑐𝑡𝑜
𝐴 + − 𝜏)𝛼
=
𝑦
𝜑−𝑠
𝑐𝑡
(1 − 𝛼)(1 + 𝑤 𝑡 )
(27)
𝑡
Similarly, to the model without redistribution, income inequality ultimately depends
on the term
𝜑−𝑠𝑡
𝑤𝑡
, which can be rewritten as:
𝜑
𝜑 − 𝑠𝑡∗ 𝑤𝑡 − 𝛽
=
𝑤𝑡
1+𝛽
(28)
so that as the capital stock increases along with labor income, the denominator increases.
Therefore, income inequality is increasing in capital accumulation assuming all else constant.
However, in this case, inequality is affected by the redistribution scheme through capital taxation
on the old and subsidies to the young. By looking at (26), we can see that both taxation on capital
gains in amount𝜏, as well as the lump-sum subsidy 𝜑, work in the opposite direction of inequality.
That is, higher taxation on the old, or higher subsidies to the young are able to mitigate the
increasing resource gap between generations.
The above result leads to the following proposition:
Proposition 2 (Intergenerational Inequality and Redistribution): In an economy
in which the
government redistributes income through capital gains taxation to the old and lump-sum transfers
𝜕
𝑐𝑜
𝜕
𝑡
𝜕𝜑
to the young, 𝜕𝜏 (𝑐𝑡𝑦 ) < 0 and
𝑐𝑜
(𝑐𝑡𝑦 ) < 0.
𝑡
14
5. Conclusions and Future Work
We can conclude that, since one of the effects of highly marked gaps in income is to diminish the
standard of living among individuals, by using an overlapping generations model, we proposed a
theoretical framework that provides a structure for describing intergenerational income inequality.
Based on our model, we have that young individuals consume from realized labor effort and older
individuals from returns to capital. Thus, when sampling any period where a young and old
generation overlap, we can see how overinvestment favors resource allocation towards older
individuals, leading to increased income inequality.
In order to attend this problem, we proposed a redistributive policy that imposes a tax on capital
returns for old individuals and turns it into a subsidy for young agents. It was demonstrated that
this measure helped to smooth lifetime consumption, because higher taxation on the old, or higher
subsidies to the young are able to mitigate the increasing resource gap between generations.
This work can be further extended to test income inequality in a Steady State General Equilibrium
Analysis. With this in mind, we could discuss if there exists an optimal fiscal policy 𝜏 ∗ and 𝜑 ∗ that
maximizes intergenerational utility, which could work as guideline for a redistributive program.
6. Appendix
2.1 Factor Markets
In this section, we stated our interested in the impacts of generation inequality at individual level.
In equation 1.1 and 1.2 we describe how we obtained the factors of production per capita when all
the production function was divided by 𝐿
1
[𝑌 = 𝐴𝐾𝑡𝛼 𝐿1−𝛼
]
𝑡
𝐿 𝑡
( 3.1)
𝑓(𝑘𝑡 ) = 𝐴𝑘𝑡𝛼
(1.4)
15
In this manner, factors of production are paid the value of their marginal contribution to
the production process:
𝑟𝑡+1 = 𝑓′(𝑘𝑡 )
(1.3)
𝑤𝑡 = 𝑓(𝑘𝑡 ) − 𝑘𝑡 𝑓′(𝑘𝑡 )
(1.4)
From 1.3, we can describe our rate of returns to investments in terms of the production
function,
𝑟𝑡+1 = 𝛼𝐴𝑘𝑡𝛼−1
(1.5)
𝑤𝑡 = 𝐴𝑘𝑡𝛼 − 𝑘𝑡 (𝛼𝐴𝑘𝑡𝛼−1 )
(1.6)
𝑤𝑡 = (1 − 𝛼)𝐴𝑘𝑡𝛼
(1.6)
Similarly,
2. A Baseline Model of Inequality without Income Redistribution
Individuals have logarithmic preferences over lifetime consumption so that their maximization
program is represented by:
𝑀𝑎𝑥
𝑦
𝑜
𝑦 𝑜 ln(𝑐𝑡 ) + 𝛽 𝑙𝑛 (𝑐𝑡+1 )
𝑐𝑡 𝑐𝑡+1
(2.1)
𝑠. 𝑡.
𝑦
𝑤𝑡 = 𝑐𝑡 +
𝑜
𝑐𝑡+1
1 + 𝑟𝑡+1
(2.2)
𝑜
From 2.2 we isolate 𝑐𝑡+1
:
𝑦
𝑜
𝑐𝑡+1
= (𝑤𝑡 − 𝑐𝑡 )( 1 + 𝑟𝑡+1 )
(2.3)
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We replace 2.3 in 2.1, which gives us the objective function as follows
𝑀𝑎𝑥
𝑦
𝑦
𝑐𝑡 ln(𝑐𝑡 ) + 𝛽 𝑙𝑛 [(𝑤𝑡 𝑙𝑡 )( 1 + 𝑟𝑡+1 ) ]
(2.4)
1
1 + 𝑟𝑡+1
=0
𝑦− 𝛽
𝑦
𝑐𝑡
[(𝑤𝑡 − 𝑐𝑡 )( 1 + 𝑟𝑡+1 ) ]
(2.5)
𝑦
Then, we solve for 𝑐𝑡
𝑦
𝑦
(𝑐𝑡 )[𝛽( 1 + 𝑟𝑡+1 )] = [(𝑤𝑡 − 𝑐𝑡 )( 1 + 𝑟𝑡+1 )]
𝑦
𝑦
𝑐𝑡
(2.6)
(𝑤𝑡 − 𝑐𝑡 )( 1 + 𝑟𝑡+1 )
=
(1 + 𝛽)( 1 + 𝑟𝑡+1 )
(2.7)
1
𝑤𝑡
𝑦
𝑐𝑡 (1 + ) =
𝛽
(1 + 𝛽)
(2.8)
As a result, we obtain the optimal consumption level when young (2.9)
𝑦∗
𝑐𝑡 =
𝑤𝑡
(1 + 𝛽)
(2.9)
𝑜
We also solve for 𝑐𝑡+1
. Considering equation 2.3 we have,
𝑜
𝑐𝑡+1
= (𝑤𝑡 −
𝑤𝑡
)( 1 + 𝑟𝑡+1 )
(1 + 𝛽)
(2.10)
Thus, the optimal consumption level when old is presented below
𝑜∗
𝑐𝑡+1
=
[𝑤𝑡 (1 + 𝑟𝑡+1 )[(1 + 𝛽)
1+𝛽
(2.11)
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If we consider that savings level is denoted by 𝑠 = (𝑤𝑡 − 𝑐𝑡 ), then, by using the optimal
𝑦∗
consumption level in present time 𝑐𝑡 from 2.9 we have the optimal saving level in 2.13.
𝑠𝑡 =
𝑤𝑡 (1 + 𝛽)(1 + 𝑟𝑡+1 ) − 𝑤𝑡 ( 1 + 𝑟𝑡+1 )
(1 + 𝛽)(1 + 𝑟)
𝑠𝑡∗ = 𝑤𝑡
(2.12)
𝛽
1+𝛽
(2.13)
3.2 Inequality
After setting specific forms of preferences in production similarly to 2.1, we
develop inequality performance by solving 3.1 subject to 3.2 and 3.3.
𝑀𝑎𝑥
𝑦
𝑜
𝑦 𝑜 ln(𝑐𝑡 ) + 𝛽 𝑙𝑛 (𝑐𝑡+1 )
𝑐𝑡 𝑐𝑡+1
(3.1)
s.t.
𝑦
𝑠𝑡 = 𝑤𝑡 + 𝜑 − 𝑐𝑡
(3.2)
𝑜
𝑐𝑡+1
= [1 + (1 − 𝜏)𝑟𝑡+1 ]𝑠𝑡
(3.3)
By converting 3.2 and 3.3 into a single equation, we have the restriction function as
shown in 3.4. In 3.5 we set the latter into 3.1, denoted in 3.5.
𝑦
𝑜
𝑐𝑡+1
= [1 + (1 − 𝜏)𝑟𝑡+1 ]( 𝑤𝑡 + 𝜑 − 𝑐𝑡 )
(3.4)
𝑀𝑎𝑥
𝑦
𝑦
𝑦 𝑜 ln(𝑐𝑡 ) + 𝛽 𝑙𝑛 { [1 + (1 − 𝜏)𝑟𝑡+1 ]( 𝑤𝑡 + 𝜑 − 𝑐𝑡 )}
𝑐𝑡 𝑐𝑡+1
(3.5)
We use the first order conditions from 3.6 to 3.8 and obtain the optimal consumption
level when young in 3.9. This is a level that considers a tax 𝜏 and subsidy 𝜑.
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1
𝑦
𝑐𝑡
−𝛽
[1 + (1 − 𝜏)𝑟𝑡+1 ]
(3.6)
[1 + (1 − 𝜏)𝑟𝑡+1 ]( 𝑤𝑡 + 𝜑 −
𝑦
𝑦
𝑐𝑡 )
𝑦
𝑐𝑡 = 𝛽( 𝑤𝑡 + 𝜑 − 𝑐𝑡 )
(3.7)
𝑦
(3.8)
𝑐𝑡 (1 + 𝛽) = 𝛽(𝑤𝑡 + 𝜑)
𝑦∗
𝑐𝑡 =
𝛽(𝑤𝑡 + 𝜑)
(1 + 𝛽)
(3.9)
We proceed now to find the optimal consumption level when old in 3.11 by considering
the results in 3.9.
𝑜
𝑐𝑡+1
= [1 + (1 − 𝜏)𝑟𝑡+1 ] ( 𝑤𝑡 + 𝜑 −
𝛽(𝑤𝑡 + 𝜑)
)
(1 + 𝛽)
𝑤𝑡 + 𝜑
𝑜∗
𝑐𝑡+1
= [1 + (1 − 𝜏)𝑟𝑡+1 ] (
)
1+𝛽
(3.10)
(3.11)
The optimal savings level is determined by 𝑠 = (𝑤𝑡 − 𝑐𝑡 ), and considering 3.9, the result
is presented in 3.13
𝑠𝑡 = (𝑤𝑡 + 𝜑) −
𝛽
(𝑤𝑡 + 𝜑)
1+𝛽
1
𝑠𝑡∗ = (𝑤𝑡 + 𝜑) (
)
1+𝛽
(3.12)
(3.13)
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