Overlapping Generations Models:
Two Extensions
Jochen O. Mierau
September 4, 2007
1. Introduction . . . 5
Part I Theoretical Background 8 2. Overlapping Generations Models . . . 9
2.1 Basic ADS-OLG Model . . . 10
2.1.1 Structure and Equilibrium of the Model . . . 10
2.1.2 Brief Application of the Model . . . 18
2.2 Extensions of the ADS-OLG Model . . . 19
2.2.1 Ageing . . . 19
2.2.2 Endogenous Labour Supply . . . 22
2.2.3 Education . . . 25
2.3 Final Remarks . . . 28
3. Intra-Household Allocation . . . 29
3.1 Unitary Model . . . 31
3.2 Samuelson . . . 32
3.2.1 Functional Form Approach . . . 33
3.2.2 Society Approach . . . 35
3.3 Nash Bargaining Solution . . . 36
3.3.1 Choice Households . . . 38
3.3.2 Mandatory Households . . . 39
3.5 Final Remarks . . . 43
Part II Education, ageing and Intra-Household Allocation 44 4. Endogenous Education . . . 45
4.1 Structure and Equilibrium of the Model . . . 46
4.2 Ageing and Education in the 3 Period Model . . . 55
4.3 Final Remarks . . . 58
5. Intra-Household Allocation . . . 60
5.1 Structure and Equilibrium of the Model . . . 62
5.2 Ageing in the OLG Model with a Realistic Household Structure . . . 69
5.3 Final Remarks . . . 70
6. Conclusion . . . 72
Preface
The underlying thesis is the final product of a five year pursuit of economic theory at the
university of Groningen and the university of Uppsala. In this period I have met various people, many of which most deserve some praise and some deserve most praise for assisting
me all the way through five years of economics. Without being over-extensive I will use this part of my thesis to thank those who deserve to be thanked.
Syb van Roijen for being lenient in the grading of my university entrance exam. Gerhard Kuper, Jakob de Haan, and Richard Jong-A-Pin for showing me the first steps of doing
research, and getting it published. Ben Heijdra, my supervisor, for helping me turn a short memo devised on a quite afternoon into a full blown thesis. Jenny Ligthart, my second
supervisor, for many helpful comments on the final product.
Naturally, I also want to thank my family, friends, and girlfriend for always being
Population ageing is the key demographic and policy issue faced by countries around the world today and numerous economic models have been developed to study the consequences
of ageing. What the dynamic economic models generally fail to acknowledge is that house-holds do not consist of a single agent but, more often than not, consist of multiple agents
who differ in all kinds of aspects. The failure to introduce multi-person households into dynamic economic models means that one of the key strategic situations that agents find
themselves in, the household, has remained unnoticed. A second caveat of dynamic models dealing with ageing is that they do not take into account the initial life-cycle phase in which
agents invest in education, that is, most dynamic models merely consist of a work and re-tirement phase, hence, ignoring the interaction between education and ageing. A notable
exception are the Kotlikoff type computable general equilibrium (CGE) models (see, for instance, Fehr et al. (2004)) in which the full life-cycle of an agent is covered. However,
in the CGE models agents are inactive until they reach the working phase (i.e., there is no education) and in CGE models households typically consist of a single agent, which
is obviously not true. A further exception is Omtzigt (2006) who uses a continuous-time life-cycle model to study the interaction between longevity and investments in human
cap-ital. His model is, however, not embedded in a general equilibrium framework, so that his model does carry any implications for the macroeconomic consequences of the interaction
between ageing and education.
To tackle the problem of endogenous education we develop a three-period overlapping
generations model (OLG) in which agents live through three life-cycle periods. In the first period, agents decide how much time to spend on education given a trade-off between
how much to consume and how much to save. In the final stage of the life-cycle, agents
con-sume out of their interest income and accumulated savings. Ageing is introduced into the model by making the transition from the second to the third life-cycle stage probabilistic.
The model sheds light on the interaction between ageing and education. More specifically, the model shows that a mortality drop leads to a rise in educational investments. The key
to this effect is that ageing increases the potential time in which the benefits of education can be reaped.
The second caveat of current dynamic models is tackled by introducing a two person bargaining process in a two-period OLG model. In the bargaining process the household
members decide over the optimal allocation of consumption, leisure, and home-production. Ageing is again introduced by considering a probabilistic transition between the work and
retirement stage of the life cycle. This extension provides a genesis of household pref-erences, that is, the preferences of the household are the weighted average of the two
household members’ individual preferences, where the weights are determined by the bar-gaining power of the respective household member. The model shows that households
engage in a specialisation process that assures that the household member with the high-est earning power works more in the market and the household member with lower earning
power provides more household labour. Furthermore, the model highlights the fact that, when considering the consequences of mortality changes, it is of importance to know which
household member is ageing and how much bargaining power he/she has.
This thesis is the prelude to my PhD project “Education, Ageing, and Intra-Household
Allocation” and, as such, it is of a rather exploratory nature and does not abide to the common question-answer principle of most theses. Hence, the main focus of the thesis is
to provide two tractable models to develop intuition concerning the two above mentioned caveats in dynamic models. In a later stage we will alter and extend the models to bring
them to a more sophisticated level and to study additional questions such as the implication of differential tax treatments of household members.
brief discussion of OLG models and the relevant extensions that have been made to such
models. Furthermore, it introduces and discusses the relevant tools developed to cope with multi-person households. Part II develops, analyses and discusses the two extensions
suggested above. The final section concludes and suggests some possible extensions to the two models.
It should be mentioned at this point that the underlying thesis is constructed in a two layered fashion; a technical and a verbal layer. As the thesis is somewhat technical at
points I have tried to assure that the reader without a technical background can reap the benefits of the underlying thesis by merely considering the text and disregarding the math
Whereas most dynamic economic models rely on infinitely lived agents or dynasties of finitely lived agents, overlapping generations (OLG) models are based on the common
observation that economic agents are mortal, are confronted with different choices over the life-cycle and are not necessarily part of a tightly nit clan. This gives rise to the observation
that, at any moment in time, agents in different stages of their life-cycle coexist; generations overlap. Taking into account that agents are mortal and make different choices in the
various stages of their life-cycle adds additional dynamics to the common economic models in which agents/dynasties live forever and, regardless of the life-cycle stage, make the same
choices.
Overlapping generations models were initially devised by Samuelson (1958) and
Dia-mond (1965), and where subsequently altered and extended by various other authors. Allais (1947) also used OLG models very similar to those employed by Diamond and Samuelson,
however, as he wrote in French his version of the OLG model did not catch on and was only to be re-discovered by Malinvaud (1987). Based on the insights of Yaari (1965),
Blan-chard (1985) devised his own OLG model which differs form the Allais-Diamond-Samuelson (ADS) model in the sense that the Blanchard-Yaari (BY) model relies on continuous-time
whereas the ADS model is specified in discrete time.
Due to the different time structures of the BY and ADS OLG models, they both have
their preferred field of application. BY models are frequently used for the analysis of the impact of alternative mortality structures on the economy. Heijdra and Romp (2006),
for instance, study the effects of introducing a realistic mortality structure into the BY model. The ADS model is generally used to study the interaction between different stages
ageing, education and intra-household allocation.
2.1
Basic ADS-OLG Model
We only provide a brief overview of the basic ADS-OLG model because the model can be found in most standard texts on macroeconomics (see, for instance, Heijdra and van der
Ploeg (2002) ch. 17).
2.1.1 Structure and Equilibrium of the Model
Households
The representative household lives for two periods; youth and old age. Agents work during youth and are retired in old age. They derive utility solely from consumption, hence,
labour supply is exogenous and the household’s objective is to maximise the life-time utility function:
ΛY,it = U Cti + βU Zt+1i , (2.1)
where Ct is consumption in youth, Zt+1 is consumption in old age, β is the subjective
discount factor1, U (.) is the sub-period utility function2, ΛY is the life-time utility function
as perceived by the young, i is the individual index and t is the time index. Agents have limited resources and must set aside part of their labour income in order to consume in
old age, hence, the maximisation problem of the household is subject to the dual budget constraints:
NtiWti = Cti+ Sti, (2.2a)
Zt+1i = (1 + rt+1) Sti, (2.2b)
1β = 1
1+ρ, where ρ is the pure rate of time preference.
2The sub-utility functions are twice continuously differentiable and concave over the full domain (i.e.,
where S is savings, r is the interest rate, N is labour supply, and W is the wage rate
(see below). Because labour supply is assumed to be exogenous for now; we set N to unity. Savings take place in the form of capital accumulation because capital is the only
durable asset in the economy. Agents leave no bequests, hence, they fully consume their accumulated capital stock in old age. Therefore, the elderly are the sole owners of the
capital stock, which creates the generational linkage in the ADS-OLG; the young own labour power, the elderly own the capital stock and together they must produce output.
The two budget constraints can be consolidated to:
Wt= Cti+
Zi t+1
1 + rt+1
. (2.3)
Imposing a logarithmic utility function3, the constrained maximisation problem can be written as: max C,Z L = ln C i t+ β ln Zt+1+ λit Wt− Cti − Zt+1i 1 + rt+1 , (2.4)
where λ is the Lagrange multiplier. Assuming an interior solution, the first-order necessary conditions (FONC) are:
∂L ∂Ci t = 1 Ci t+1 − λit= 0, (2.5a) ∂L ∂Zi t+1 = β Zi t+1 − λ i t 1 + rt+1 = 0, (2.5b) and Wt = Cti + Zi t+1 1 + rt+1 . (2.5c)
The optimal levels of of savings and consumption in youth and old age, given the above optimality structure and some algebra, are:
3Notice that a logarithimic utility function adheres to all the prinicples of the above discussed utility
functions: For U (C) = ln C U0(C) = 1c; U00(C) = −c12 so that U
0(C) > 0 > U00
(C) ∀C ∈ R+. Furthermore,
Cti = Wt 1 1 + β, (2.6a) Zt+1i 1 + rt+1 = Wt β 1 + β, (2.6b) and Sti = Wt β 1 + β. (2.6c)
Even though the structure of the households is rather simplistic, one of the key paradigms of economics can be identified in the optimal consumption decisions; consumers smooth
utility over their life-cycle according to their valuation of the different life-cycle stages. This can be seen in (2.6) where the two consumption decisions are weighted according to
the subjective discount rate.
Utility smoothing reveals the fundamental building block of dynamic economic models.
If left to themselves; economic agents choose how to maximise and then optimally distribute life-time utility over their life-cycle. As a by-product of utility distribution, agents can cause
economic growth because their savings are used for productive investments. By introducing new features into the model one can analyse how the fundamental tendency of agents –
utility smoothing – is altered. The tendency of agents to maximise and distribute utility over the life-cycle is the sole engine of growth in dynamic models. All additional features
can either be catalysts or a retardant of economic growth but never an engine.
Households are not alone; they belong to a certain generation. After all, no overlapping
generations without generations in the first place. Generations are identified by their year of birth and their mass is denoted by P . Generations grow at a steady, constant,
pace p. Aggregate consumption of a certain generation can be derived by multiplying the consumption levels of representative household by the size of its generation. For instance,
aggregate consumption of generation Ptat time t is PtCti = Ct, whereas consumption of the
same generation at time t+1 is PtZt+1i = Zt+1. Naturally, generation Ptno longer consumes
Firms
Firms produce consumption goods by combining labour and capital. Firms pay wages and
interest as compensation for their use of inputs, which are provided by households. Profits of the representative firm are:
Πt= Yt− (rt+ δ) Kt− WtLt, (2.7)
where Πt are profits, Yt is output, δ is the depreciation rate, Ktis the amount of capital in
economy and Lt is aggregate labour supply. Note that the price of output (Yt) has been set
to unity for simplicity. Aggregate labour supply is determined by the aggregate number of
hours worked by the individual agents. In the current case all agents work a fixed amount of time, which is assumed to be 1, hence aggregate labour supply is simply the size of the
currently working generation:
Lt = Pt
X
i
Nti; Nti = 1 ∀ i ∈ P ⇒ Lt = Pt. (2.8)
Capital and labour are transformed into output by a production function:
Yt= F (Kt, Lt) ,
which exhibits first degree homogeneity (doubling inputs doubles output), positive but
decreasing returns to scale and essentiality of both inputs. Firms are assumed to exist forever hence we can set the number of firms to 1, which implies that output of the
representative firm equals aggregate output. Also because the derivation of the household optimality results was performed on a per capita basis it helps to redefine output in per
capita terms:
yt = F (Kt/Lt, 1) ,
which assures that interpretation of the results is not hampered by population growth. Profit maximisation requires that:
Due to the properties of the production function the optimality conditions in per capita
and aggregate terms are the same. Substituting these results into the profit function and using Euler’s theorem4 shows that in optimum pure profits are zero:
Πt ≡ Yt− (rt+ δ) Kt− WtLt
= Yt− FK(Kt/Lt, 1) Kt− FL(Kt/Lt, 1) Lt= 0.
Assuming a Cobb-Douglas production function with parameter ξ ∈ (0, 1)i.e., Yt= KtξL 1−ξ t : rt+ δ = ξkξ−1t (2.10a) Wt = (1 − ξ) ktξ. (2.10b)
From here it should be clear that firms and households interact in the labour, consump-tion goods and capital market. The interacconsump-tions in these markets are discussed in the next
section.
Equilibrium
As there are no leakages in the economy, aggregate output must equal aggregate income and this income can only be divided between consumption (Ct+ Zt) and investments:
Yt= Ct+ Zt+ It. (2.11)
Investments (It) are subdivided into replacement investments (equal to depreciation) and
new investments (∆Kt+1 = Kt+1− Kt):
It = δKt+ ∆Kt+1,
thus (2.11) can be written as:
Yt+ (1 − δ) Kt= Ct+ Zt+ Kt+1. (2.12)
4For a homogeneous function of degree λ, f (x), where x is a vector of variables it holds that: x
1fx1(x)+
Using the budget constraint in (2.2a) aggregate consumption of the young can be rewritten
as the savings residual:
Ct= PtWti− PtSti. (2.13a)
As there are no bequests, consumption of the elderly must equal their income from savings
and their total assets (remember that the elderly own the capital goods) thus:
Zt = (rt+ δ)Kt+ (1 − δ)Kt. (2.14)
Economy wide consumption is thus:
Ct+ Zt = PtWti− PtSti+ (rt+ δ)Kt+ (1 − δ)Kt
= Yt+ (1 − δ)Kt− PtSti,
where Euler’s theorem has been applied again. Substituting this into (2.12) gives:
Yt+ (1 − δ) Kt = Yt+ (1 − δ)Kt− PtSti + Kt+1
⇒ PtSti = Kt+1. (2.15)
Using the household savings function (2.6c), the wage function (2.10b) and expressing everything in per capita terms provides the difference equation that describes the full
model economy: Wt β 1 + β = Kt+1 Pt = kt+1(1 + p) ⇒ kt+1 = w (kt) = (1 − ξ) β (1 + p) (1 + β)k ξ t (2.16)
with limk→∞w0(kt) and limk→0w0(kt) = 0 therefore kt has a single unique steady state
(steady state values are denoted by a star, e.g., k∗) at:
k∗ = (1 − ξ) β (1 + p) (1 + β) 1/(1−ξ) (2.17)
and growth rate growth rates are denoted by a circumflex, e.g., ˆk:
The fundamental difference equation (2.16), together with the growth rate (2.18), and
steady state (2.17) may be considered as the trinity of dynamic economic models. Between them they explain the immediate, transitional and long-run effects of an economic shock.
For instance, if ktincreases due to some exogenous shock whilst the economy is in its steady
state then (2.17) displays that the steady state remains unaltered, (2.18) demonstrates that
the growth rate of the economy temporarily drops below unity in order to restore steady state, and the full adjustment path is traced out by (2.16).
That the model converges to a unique and stable steady state does not necessarily imply that the steady state is also optimal. Inspection of (2.17) reveals that, depending on
the combination of the model parameters, there are numerous candidates for the unique steady state. Diamond (1965, p.1129) proposes a golden-age path as the optimal stable
steady state that depends on a bi-criterion:
• all agents have the maximum possible utility; and • all agents have the same level of utility.
In the former condition, golden may be interpreted in a materialistic sense and for the
latter condition, golden may be interpreted in a reciprocal ethics sense. Diamond (1965, p. 1129) shows that the golden-age path is achieved if, in the steady state, the marginal
product of capital equals the population growth rate:
FK(K/L, 1) − δ = r = p. (2.19)
This equivalence is referred to by Samuelson (1958) as the biological optimum and the
intuition behind it is best explained in Diamond’s words:
“The shifting of one unit of consumption by an individual from his first to his second year [Period in our case, JM] is equivalent to removing one unit of
consumption from each of the living members of the younger generation and giving this total to the contemporary older generation, of whom there are n
If the biological optimum is not attained then the economy is either dynamically
inef-ficient (if r < p) due to excess-accumulation of capital or the economy has a capital stock that does not allow for full production capacity to be utilised (if r > p) due to
under-accumulation of capital. In both cases consumption is sub-par and the agents could be made better of by government intervention. In the case of dynamic inefficiency a Pareto
improvement is possible in which the government gives the elderly additional funds in the form of an unfunded social security system, see for instance Ball et al (1998) or Krueger
and Kubler (2006). In the case of under-accumulation of capital there are no free lunches and the optimal steady state cannot be reached through a Pareto policy.
As all other variables in the model depend on k∗ it is possible to derive their steady state values based on k∗. For instance, in the steady state, wages and the interest rate are:
W∗ = (1 − ξ) k∗ξ = (1 − ξ)1/(1−ξ) β (1 + p) (1 + β) ξ/(1−ξ) , (2.20) r∗ = ξk∗ξ−1− δ = ξ(1 + p) (1 + β) (1 − ξ) β − δ. (2.21)
So that for the economy to be in its golden-age path we must have that (see above):
p = FK(Kt/Lt, 1) − δ = ξ
(1 + p) (1 + β) (1 − ξ) β .
From here it is clear that there is no mechanism that drives the economy to its golden-age path and that it is only by a stroke of luck that the constellation of the model’s parameters
allow the economy to be on its golden-age path. Therefore, under- and excess-accumulation of capital are the rule rather than exception. Assuming full depreciation of capital over
one period (δ = 1) life time utility is:
Λ = ln W ∗ 1 + β + β ln β (1 + r) W∗ 1 + β = ln k∗ξ1 − ξ 1 + β + β ln k∗(2ξ−1)βξ (1 − ξ) 1 + β = (2ξβ + ξ − β) ln k + ln (1 − ξ) + β ln ξ (1 − ξ) + β ln β − (1 + β) ln (1 + β) (2.22)
by merely considering the reaction of capital. From here we can see that capital-welfare
co-movement occurs under much less strict conditions than the golden-age path.
2.1.2 Brief Application of the Model
Using the trinity of the ADS-OLG (difference equation, steady state and growth rate) we
can perform some straightforward economic analysis:
∂w (kt) ∂p , ∂k∗ ∂p , ∂ˆk ∂p < 0 (2.23a) ∂w (kt) ∂β , ∂k∗ ∂β , ∂ˆk ∂β > 0. (2.23b)
The first of these two results shows that an increase in the growth rate of the population decreases the trinity (see above) of capital per capita. Which is intuitive, as an increase
in the growth rate of the population leads to faster dilution of per capita capital. The lower result indicates that5 an increase in the valuation of utility in old age increases the
trinity. Which is intuitive because if agents cherish utility in old age more they will increase their savings in order to redistribute income from young to old age, which increases capital
in the economy. Thus, even though we started out with a rather simple version (i.e., a logarithmic utility and Cobb-Douglas production function) of the ADS-OLG we still have
an intuitively simple model in which we can perform meaningful economic analyses. Naturally, the analyses that can be performed in the realm of the ADS-OLG can be
extended in numerous directions. This is the theme of the following sections which will go through the developments of the ADS-OLG model most relevant to the current cause. For
a complete overview of all possible extensions of the ADS-OLG we refer the reader to a specialised text such as de la Croix and Michel (2002).
5Note that ∂Λ ∂β = (2ξ − 1) ln k + (2βξ + ξ − β) 1 k ∂k ∂β + ln ξ (1 − ξ) + ln β
1+β > 0; even though β surfaces
2.2
Extensions of the ADS-OLG Model
This section introduces the extensions of the ADS-OLG most relevant to our analysis. In
turn this section discusses; ageing, endogenous labour supply and education in the ADS-OLG.
2.2.1 Ageing
One of the main issues facing economies around the world is that economic constituencies
are ageing. This process is driven by two engines which both have a different impact on the economy. On the one hand, fertility is dropping and on the other hand mortality is
declining (Lee, 2003). The fertility drop can easily be translated into the language of our above developed ADS-OLG; it is a drop in the population growth parameter (i.e., a lower
p). The drop in mortality, on the other hand, is somewhat more involved. As the periods of the ADS-OLG model are fixed it is not possible to simply extend the length of the
retired phase. One way to circumvent this issue is to make the transition from the young to the old stage uncertain. This approach was introduced by Becker et al. (1990) and
further developed by Ehrlich and Lui (1991) and Blackburn and Cipriani (1998, 2002). Depending on what the interest of the underlying analysis is, the uncertain transition can
be determined endogenously or exogenously. In what follows we consider the uncertain transition between the life-stages to be exogenously given. For applications of endogenous
uncertainty, please refer to the above citations.
As agents carry capital over from the young to the old stage, the probabilistic approach
creates a leak in the economy because dead agents do not consume. Previous probabilistic transition models have modeled two-period OLG models in which agents work in both
periods and do not carry capital over from one period to the next. Thus, circumventing the leakage issue. Another way to go around the problem of leakages due to death is to
introduce a government that absorbs the funds of dying agents and spends these funds on unproductive government expenditure. As we will keep on modeling an OLG
introducing an unproductive government.
Probabilistic Ageing
In terms of the introduced ADS-OLG model the probabilistic approach alters the house-hold’s utility function by adding an additional term to the old age sub-utility function:
ΛY,it = U Cti + πβU Zt+1i ,
where π ∈ (0, 1) is the mortality rate (i.e., the inverse probability of reaching old age).
The remainder of the household maximisation problem stays the same so that the optimal consumption decisions now become:
Cti = Wt 1 1 + πβ, (2.24a) Zt+1i (1 + rt+1) = Wt πβ 1 + πβ, (2.24b) and Sti = Wt πβ 1 + πβ. (2.24c)
The probabilistic nature of the model implies that, of a generation born at time t, a
share (1 − π) Pt are still alive at time t + 1.
In order to derive the equilibrium the old age group now has to be divided into “living
old” and “dead old”. The living old consume out of their interest income and accumulated assets as before so that their aggregate consumption is:
Zt= πPtZt+1i ,
the assets of the dead old flow to the government that uses these funds for unproductive
spending:
Gt= (1 − π) PtZt+1i .
There are still no bequests so that:
The resource constraint now becomes:
Yt+ (1 − δ) Kt = Ct+ Zt+ Gt+ Kt+1, (2.25)
by substituting in economy wide consumption and investment, and using the same reason-ing as before, the trinity of the ADS-OLG becomes:
kt+1 = w (kt) = (1 − ξ) πβ (1 + p) (1 + πβ)k ξ t, (2.26a) k∗ = (1 − ξ) πβ (1 + p) (1 + πβ) 1/(1−ξ) , (2.26b) and ˆk = kt+1 kt = (1 − ξ) πβ (1 + p) (1 + πβ)k ξ−1 t . (2.26c)
From here we can see that an increase in probability of survival has the same impact as an increase in the valuation of old age utility. Because agents are more likely to make
it to retirement they will save additional funds to make old age more enjoyable. Thus, all in all, ageing should have a positive impact on the economy.
As our model economy implicitly assumes a fully funded private pension system the model does not exhibit the problems that most OECD countries have; their
pay-as-you-go pension systems cannot cope with the surge in the dependency ratio. Introducing a pay-as-you-go pension system is straightforward and the interested reader is referred to
Heijdra and van der Ploeg (2002, ch.17). The introduction of a pay-as-you-go system does not, however, dramatically change the outcomes. A drop in fertility and/or a drop in
mortality increases capital accumulation, thereby increasing output and wages, lowering the interest rate and, thus, increasing welfare. The problems of ageing faced by most OECD
countries are, thus, also more of a transitory phenomenon. This transition can, however,
take substantial amounts of (i.e., numerous decades) time and it is obvious that in order to understand the policy implications of ageing it is of importance to study the
short-and medium-term consequences of ageing. However, the focus of the underlying thesis is on the long-run consequences of ageing and, hence, we leave the analysis of short-run
It should be mentioned at this point that when studying the consequences of ageing to
optimal choices of individual agents it does not make sense to use the common interpreta-tion of infinitesimal changes to individual behaviour. After all agents live only once and in
the moment they are born they make choices so as to have an optimal live-time utility plan given their life-expectancy. Furthermore, there is no mechanism in the model that allows
agents to re-optimise their choices in case their life-expectancy turns out to be incorrect (i.e., there is no life-time uncertainty). So it is not really the agent him/herself that is
ageing but much more a group of agents. Hence, the interpretation of an infinitesimal increase in an agents mortality should not be seen from the individual’s perspective but
from the perspective of a cross-section of agents born at infinitesimal time intervals and the consequences of ageing are then the different choices that these agents of infinitesimal
age differences make.
2.2.2 Endogenous Labour Supply
In the real world agents do not only consume market goods in order to maximise their
welfare; they also consume leisure. A preference for leisure has a direct effect on the amount of labour that an agent is willing to provide because the number of hours an agent
has to spare is fixed. Thus, allowing agents to consume leisure automatically introduces endogenous labour supply into the model.
Introducing labour supply into OLG models has proved to be a substantially higher hurdle than introducing ageing. Once labour supply is endogenised there is no longer
a guaranteed unique steady state and, more often than not, the model is hampered by the existence of multiple, stable and unstable, equilibria (Gale (1973) and Grandmont
(1985)). However, Reichlin (1986) shows that endogenous cycles only arise if the interest rate elasticity of savings is less than −1/2 which seems empirically unlikely. Furthermore,
as we are using logarithmic utility functions, which assure independence between savings and the interest rate, the issue of endogenous cycles does not have an impact on our further
technical backpack, it seems that this backpack does not contain any essentialities for
survival.
Labour Supply
We introduce labour supply into the probabilistic ageing ADS-OLG model by giving agents
the option to divide their time between labour and leisure:
ΛY,it =ε ln Cti+ (1 − ε) ln1 − Nti + β ln Zt+1i , (2.27) where Nt is the part of the time endowment that is spent on working. The consolidated
budget constraint now becomes:
WtNti = C i t+ Zi t+1 1 + rt+1 or Wt= Cti+1 − N i t Wt+ Zi t+1 1 + rt+1 , (2.28)
where the second representation resembles the fact that leisure is just another consumption
good that can be bought at price W . Using the same methods as before the FONCs are:
ε Ci t − λi t= 0, (2.29a) − 1 − ε 1 − Ni t+1 + λitWt= 0, (2.29b) πβ Zi t+1 − λ i t 1 + rt+1 = 0, (2.29c) and WtNti = Cti+ Zi t+1 1 + rt+1 . (2.29d)
The equilibrium condition (2.15) remains unaltered so all that remains to be done is to
determine the savings rate of the aggregate households:
where it is obvious that individual savings remain unaltered due to the introduction of
endogenous labour supply. This arises from the fact that the household optimisation problem can be split into two stages; first, households decide how to distribute income over
the life-cycle and second, households decide how to allocate income between the options within a period. By adding an additional consumption good (i.e., leisure) the first decision
is unaltered. Thus individual savings remain the same.
Because workers do not work full time, savings per unit of labour are no longer equal
to individual savings: StA = Pt−1Sti = Pt−1Wt πβ 1 + πβ (2.31) StL = S A Lt = Pt−1Wt Pt−1ε+πβ1+πβ πβ 1 + πβ = Wt βπ ε + πβ, (2.32)
where SA is aggregate savings and SL is savings per unit of labour (remember that N = ε+πβ
1+πβ). From here we can see that strong preferences for first period consumption (i.e., high
ε) have a detrimental effect on savings, and, hence, capital accumulation in the economy.
On the other hand strong preferences for leisure have a positive effect on the economy. This is somewhat counterintuitive but it becomes logical if we consider that, even though
agents can choose between work and leisure when young, they can only consume goods when they are old. An agent with strong preferences for first period leisure must still work
in order to procure second period goods and all income earned from employment is saved,
hence boosting capital accumulation.
Besides changes in the equilibrium condition we can also analyse the impact of
endoge-nous labour supply on the demand and supply functions of the individual agents. For instance, if the mortality rate decreases (i.e., π increases) then the share of time spent
working also increases. Which is logical because the elderly only derive utility from con-sumption and, therefore, need extra savings in order to boost their concon-sumption. These
extra savings are provided by working more during youth.
However, the experience in OECD (see, for instance, Barr and Diamond
spent in the labour market. A possible explanation for this is provided by Heijdra and
Romp (2007), who point out that the pension systems of most OECD countries entail an incentive to leave the labour market between the early eligibility date and the statutory
retirement age. As these two dates have not been extended to neutralise the increase in life-expectancy, the pension systems effectively impose a 100%+ tax on those wishing to
work longer. On one hand, those who work longer forgo their pension rights (which is/used to be equal to the last earned wage), on the other hand, agents still have to pay pension
contributions and taxes. Thus, rational as they are, older agents flee from the grabbing hands of the taxman/woman even though they would want to work to accumulate
addi-tional funds. As our model does not include the perverse incentives of the pension system we only see the first-order effect of ageing, namely that in an ageing society agents will
want to work more.
2.2.3 Education
Ever since Gary Becker decided to put the whole scope of human behaviour under the
scrutiny of economic analysis, human capital has been high on the agenda of theoretical and empirical economics. Its influence has not passed the OLG literature either, and
extensions to the OLG model have been introduced in which the impact of investments in education by households can be analysed. In contrast to ageing and endogenous labour
supply, the introduction of human capital accumulation in the OLG model has led to much more discussion concerning as to who is actually investing in education (parents vs.
children) and who should pay for education (state vs private). These questions where taken up by Glomm and Ravikumar (1992).
The Glomm and Ravikumar model analyses the consequences of private versus public financing of education in an environment where parents have to provide the physical capital
for human capital investments of the young. Human capital is produced by a combination of time investments by the young, physical investments by parents and the level of human
of human capital already present in the economy. Parents invest in education of the young
for altruistic reasons and do not internalise the added benefit of educational investment to the young. Hence, if the parents’ degree of altruism is not high enough, then there may be
an under-financing of education. This is seen as the key rational for a system of mandatory education. In this respect they then also show that public financing of education leads to
faster convergence of incomes.
As the income distribution and the question of who pays for the education is not of
interest to the current analysis we only provide a simplified version of the Glomm and Ravikumar model. That is, in our model the benefits of education are solely felt by the
individuals themselves, hence, parental altruism is irrelevant. Furthermore, for simplicity we assume that educational investments only depend on the average stock of human capital
and time investments of the young.
Endogenous Education
As in the standard model agents again live for two periods, except that here the first
period is used for education and the second period is used for working and consumption. By investing in education in the first period agents can increase their human capital and,
thus, their earning power in period two. It is assumed that consumption by the young is part of the consumption expenditure of the old. The life-time utility function is given by:
ΛY,it = ln1 − Eti + β ε ln Ct+1i + (1 − ε) ln1 − Nt+1i , (2.33) where Ei is the amount of time invested in education. Human capital is accumulated
according to:
hit+1= αEtihφt, (2.34)
where hi is human capital of individual i, h is average human capital and α is the efficiency
of human capital production. It is further assumed that human capital acts as a scaling factor for the economy wide wage:
where the economy-wide wage is assumed to be given. Hence, we can model the economy
without referring to the firm as the households also do not need to save. The household budget constraint is:
hit+1Wt+1 = Ct+1i +1 − N i t+1 h i t+1Wt+1 ⇒ αEtihφtWt+1 = Ct+1i +1 − N i t+1 αE i th φ tWt+1. (2.35)
From here we can easily derive the FONCs of the household’s optimisation problem:
− 1 1 − Ei t + λitαhφtWt+1Nt+1i = 0, βε Ci t+1 − λi t = 0, −β (1 − ε) 1 − Ni t+1 + λitαEtihφtWt+1 = 0, and αEtihφtWt+1 = Ct+1i +1 − N i t+1 αE i th φ tWt+1.
By combining the FONCs we can derive the optimal investments in education by the young:
Eti = βε
1 + βε. (2.36)
The optimal investment in education shows that agents invest in education so as to balance their utility between the first and the second period. The stronger agents value utility from
consumption (the extent of βε), the more they will invest in education when young. Which again is intuitive as consumption is maximised by having a high income and the level of
this income is determined by the amount of educational investments by the young.
From (2.35) we can derive the evolution of human capital over time (as there is no
heterogeneity between agents we can drop the individual index):
ht+1 = α
βε 1 + βεh
φ t
which converges to a single unique steady state if φ ∈ (0, 1), if φ > 1 human capital will grow indefinitely and the economy will exhibit accelerating growth. From here it is
utility (the extent of βε). Thus, even though it seems that human capital is the engine of
economic growth in these models, it is really the tendency of agents to smooth utility that drives the economy. In this sense human capital accumulation is the catalyst and utility
smoothing is the engine of economic growth.
2.3
Final Remarks
This completes the discussion of the extensions to the ADS-OLG model, which are most
relevant to our analysis. This section has introduced the ADS-OLG model and described some of the extensions thereof. All along we have used a rather simple model economy,
nevertheless we where still able to derive numerous implications of human activity. The ADS-OLG literature has been extended in numerous other directions that, although
in-teresting, are not relevant at this point. The reader interested in more applications of the ADS-OLG model is referred to a specialised text such as de la Croix and Michel (2002)
and to the references given throughout the text.
An extension that we have not introduced in this section but we will use nevertheless
later on is the addition of an additional active period to the OLG. Although some authors have added additional static periods (see for instance Blackburn and Cipriani (2002)),
additional active periods are mostly neglected.
In part II, we will combine the three extensions by modeling a three-period OLG model
with education in period one, working and consumption in period two and retirement in period three. Furthermore, we will add household dynamics to a two-period OLG model by
Although used interchangeably in most economic texts (see, for instance, chapter 2 of this thesis) the concepts of households and agents are by no means equivalent. Most agents
in the real world live in multi-agent households in which agents differ in their prefer-ences and their earning power. The household members, therefore, have an incentive to
make optimal decisions in the distribution of consumption and leisure time. Furthermore, the household members have the option to explicitly spend time together, thereby
pro-ducing additional utility in the form of togetherness. Taking the household options into account, seriously undermines the rationale behind the common application of the status
quo economic model of the single decision unit household; the unitary model.
The issue of intra-household allocation was first tackled by Samuelson (1956), who
showed that for a general family of utility functions the household decisions process can be modeled as though one agent is modeled; the Gorman form. Named after Gorman
(1953), who used a similar set-up to study the properties of utility functions that allow for aggregation from the micro to the macro level. Furthermore, Samuelson uses concepts from
welfare economics to treat the household as a mini-society of two agents who act as though they are maximising a social welfare function. A natural problem with this approach is
that it provides a functional of household members’ preferences but it does not provide a theoretical basis for this function. This is where the family bargaining approach of Manser
and Brown (1980) comes in. Using the tools of game theory they derive two predictions of multi-person household behavior; the dictatorial and the Nash bargaining solution. In the
former, one of the household members dominates the other and only takes into account the other’s preferences up to the point where he/she leaves the household. In the latter,
a pooled budget. The bargaining approach has been further developed by Browning and
Chiappori (1998), who postulate a collective model of household decision making in which the household has to decide over the allocation of a collective income over private goods
and collective goods. Their model allows for additional bargaining power of household members due to contributions to the household budget and other factors that might be of
influence.
The household savings function is the key equation of dynamic economic models, hence,
we focus on the implication of intra-household allocation on the savings function of the household. In work-retirement models, savings are simply residual income after first period
consumption has been deducted, hence, we can focus on the consumption decisions in single period models to get a grasp of the implications of intra-household allocation on the savings
decisions. Furthermore, growth models generally consist of one-sector economies. Hence, we simplify the intra-household models to a single good. Thus agents can only differ in their
innate ability and their preferences toward consumption, and toward each other. Finally, in order for an intra-household allocation approach to be usable in growth models it is
of importance that consumption and leisure demand functions have interpretable closed-form solutions. Naturally, it is also possible to use intra-household approaches that do not
have a closed-form solution and then rely on the (generalised) implicit function theorem to derive meaningful implications from the models. However, if intra-household allocation
approaches do not have a closed-form solution in a simple one period static model, then the implicit solution that can be derived for a multi-period model will be of such a complex
nature that it is increasingly hard, if not impossible, to derive intuitive implications from the model. As this is a first exploration into the implications of intra-household allocation
on the macroeconomy we choose to reside with closed-form solutions so as to gain intuition from the models before, in a later stage, we use more elaborate methods that do not
have the perils of a closed-form solution. Furthermore, we focus on households with two members for simplicity (i.e., we do not consider menages a trois nor do we consider couples
with children).
the issue of multi-person households. For a complete overview of the literature on
intra-household allocation see Vermeulen (2002).
3.1
Unitary Model
To refresh the mind and set ideas we start with a brief discussion of the status quo unitary
model. This is the well known economic problem that features the introductory pages of every undergraduate economic textbooks; households seek to maximise utility but are
constrained by income. With endogenous labour supply this model can be summarised by the following optimisation problem:
max
C,LA,LBU C, 1 − L A
, 1 − LB (3.1a)
s.t WA+ WB = C + WA 1 − LA + WB 1 − LA , (3.1b) where Li i = A, B is labour supply of household member A and B, respectively, thus
(1 − Li) is leisure, Wi, and C is the consumption good. Note that we assume that house-holds pool their income. Using a logarithmic transformation of the Cobb-Douglas utility
function the optimisation problem can be solved using the standard Lagrangian method:
max C,LA,LBL = ε1ln C + ε2ln 1 − L A + (1 − ε 1 − ε2) ln 1 − LB +λWA+ WB− C − WA 1 − LA − WB 1 − LB , (3.2)
with first-order necessary conditions (FONC):
Given this optimality structure we can derive the consumption and leisure demand func-tions: C = γ1WA+ WB (3.4a) 1 − LA = γ2 WA+ WB WA (3.4b) 1 − LB = γ3 WA+ WB WB (3.4c) with γ1 ≡ (1−ε1 −ε2)ε1 (ε1+ε2)(1−ε1−ε2)+ε2ε1, γ2 ≡ ε2and γ3 ≡ (1−ε1−ε2)ε2
(1−ε1−ε2)(1−ε1)+ε2. Interpreting these results
immediately reveals that the household member that has the highest wage also gets the
least amount of leisure, which is intuitive and is often verified empirically. A fundamental problem of this approach is that the process by which the household supply and demand
functions are determined is not satisfactory; economic processes should be derived from the behaviour of individual economic agents (i.e., methodological individualism, Blaug (1992)).
3.2
Samuelson
As mentioned in the introduction, Samuelson (1956) was the first to tackle the issue of deriving household behaviour from the behaviour of the individual agents living in that
household. Samuelson used two approaches in this quest. The first showed that given certain (i.e., homothetic1) utility functions, multiple agents can be collapsed to a single
agent. The second approach, which does not rely on specific utility functions, treats the household as a mini-society and shows that what holds for social planning models at large –
income is distributed such that the marginal utilities of all agents are equal – also holds for the household. Furthermore, the society approach collapses to the functional form solution
for homothetic utility functions. However, it should be noted that Samuelson’s functional
1A function g (h (x)) is homothetic if it is strictly monotonic in h (x), and h (x) is homogeneous. The
form approach only collapses to the society approach if it is assumed that the preferences
of the household members are both equal, which is unlikely to hold for most households. Hence, after establishing that the functional form approach and the society approach not
as equivalent as is often suggested, we will focus on the society approach.
3.2.1 Functional Form Approach
The functional form approach starts from the notion that it is irrelevant (to household
demand) whether household member first aggregate their utility functions and then derive household demand, or that household members first maximise their individual utility and
then aggregate their demand. The maximisation problem of the two separate individuals is: max C,Li U C, 1 − L i (3.4d) s.t Wi = Ci+ Wi 1 − Li i = A, B, (3.4e)
using the same utility function (see, 3.2) and methods as above the consumers’ demand
functions become:
Ci = Wiεi (3.5a)
1 − Li = 1 − εi, (3.5b)
and the household demand functions become:
C = CA+ CB = εAWA+ εBWB, (3.6a)
and 1 − L = 1 − LA + 1 − LB = (1 − εA) + (1 − εB) . (3.6b)
Note that household demand for leisure is independent of prices, this is due to the fact
that leisure demand is denoted in relative terms and that – due to Cobb-Douglas utility functions in which the substitution effect and income effect of a wage increase exactly
of income. For future reference note that the ratio of consumption to leisure expenditure is: C WA(1 − LA) + WB(1 − LB) = εAWA+ εBWB (1 − εA) WA+ (1 − εB) WB (3.7)
Using the same utility functions as above we could have also first summed the utility
functions and then derive the consumption-leisure expenditure ratio:
C
WA(1 − LA) + WB(1 − LB) =
εAUA+ εBUB
(1 − εA) UA+ (1 − εB) UB
. (3.8)
Obviously, the ratios derived in the two cases are not the same, even though we are using homothetic utility functions. It should be noted that if the preferences of the two household
members are equal, then the two approaches would have been the same and the total approach would have been equivalent to the society approach (see below). This shows that
the Samuelson functional form approach only works (i.e., equality of pre- and post-utility maximisation aggregation) if household members’ preferences are assumed to be equal,
which is unlikely. To see this point more elaborately note that2 the Gorman form relies on an indirect utility function that can be written as:
vi(p,mi) = ai(p) + b(p)mi (3.9)
where i is the individual index, p is a vector of prices, m is income a (.) is an individual
component and b (.) is a component that is the same for all i. By straightforward inspection of (3.9) we can see that b (.) is the marginal propensity to consume out of income. In the
case of homothetic utility functions (i.e., the utility functions that we are using) ai(p) = 0
and the indirect utility function takes the form:
vi(p,mi) = b(p)mi.
Clearly b (.) can only be the same for all i if the preferences of each and every i are equal, which is unlikely in the case of household couples.
3.2.2 Society Approach
The second approach devised by Samuelson does not rely on certain properties of utility
functions but uses a household social welfare function to weight the utilities of the separate household members. In that sense it is much the same as a social welfare function in
which the utilities of different members of society are weighted. The natural problem of Samuelson’s second approach is to determine which weighting function is appropriate for
the specific household situation.
The simplest social welfare function would simply be the sum of the two members’
utility, this would lead to household demand functions that can be derived from (3.8). From here on the complexity of welfare functions that can be devised increases dramatically
and without further discussion we will assume that households’ use a geometric weighting function in determining the intra-household allocation of utility:
UH = f UA, UB = UAν UB1−ν,
where ν can be interpreted as A’s, static, bargaining power. Substituting in the relevant
sub-utility functions provides:
UH = CεA 1 − LA1−εA ν CεB 1 − LB1−εB 1−ν = CεAν+εB(1−ν) 1 − LA(1−εA)ν 1 − LB(1−εB)(1−ν) = Cα1 1 − LAα2 1 − LB1−α1−α2,
with α1 ≡ εAν + εB(1 − ν) and α2 ≡ (1 − εA) ν. So that Samuelson’s society approach
to intra-household allocation provides a justification for the unitary model, the demand functions derived from this model are, naturally, equal to those in (3.4), mutatis mutandis.
Although providing a justification of the usage of the unitary model Samuelson’s ap-proach still lacks two key elements of household behaviour. First, it simply assumes that
household act according to some social welfare function but it does not provide an intu-ition as to how that social welfare function is agreed upon. Second, it does not allow for
together but they do not derive utility from being together. These issues where taken up
by game theorists who stumbled upon the problems of intra-household allocation during an important stretch of the crusade of Game Theory in the 1980’s. The results of this
endeavour will pride the remainder of this section.
3.3
Nash Bargaining Solution
The Nash bargaining solution (NBS) is the one of the classical results of game theory and was initially derived by Nash (1950,1953). The NBS starts from the premise that two
agents bargain given that they have some initial endowment in utility, ¯Ui ≥ 0, i = A, B.
Through the bargaining process the agents seek to increase their utility to some level ¯
Ui ≥ 0 i = A, B. Nash proposed four axioms to which such a process must abide in order
for it to reasonably occur in practice:
1. Pareto Efficiency: Agents should be as least as well of after bargaining as they where
before; Ui ≥ ¯Ui.
2. Independence of Irrelevant Alternatives: The mutual preferences between two ele-ments should not be effected by a third element; if (X Y | X, Y ) then (X Y | X, Y, Z) ,
X, Y are some arbitrary elements and Z is irrelevant.
3. Symmetry: If the two players are exactly the same, then they should also get exactly the same pay-off from bargaining.
4. Independence of monotonic transformations: Only the curvature of the utility
func-tion matters, not its origin nor its scaling factor.
Given these four axioms the objective of the NBS is to maximise:
UA− ¯UA
UB− ¯UB , (3.10)
However, just as both household members do not necessarily have the same preferences,
household members can also have different bargaining powers. Different bargaining strengths for otherwise equal agents violate the assumption of symmetry and dropping the symmetry
assumption gives rise to the generalised NBS. Binmore and Dasgupta (1987) show that the generalised NBS may be written as:
UA− ¯UA
ν
UB− ¯UB
1−ν
. (3.11)
Simple inspection reveals that the generalised NBS is very similar to a Stone-Geary utility function in which a minimal level of consumption is necessary before it actually adds to
the general utility of agents.
As the agents are now directly coupled with each other, it is possible to introduce
a togetherness factor. This togetherness factor takes the form of home production with home labour as an input for both agents. So agents now have three alternative uses of
time, they can decide to work in the market (LM i), work at home (LHi) or enjoy private
leisure. The home labour time of the two agents is combined with each other to produce
a home good; cuddling on the couch or hard labour in the kitchen. We assume that this home production occurs according to a Cobb-Douglas production function; H = Lα
HAL 1−α HB.
The utility functions of the two individual household members then becomes:
Ui = Cεi1(1 − LHi− LM i)εi2 LαHAL 1−α HB
1−εi1−εi2
i = A, B;
where 1 − εi1− εi2 can be interpreted as the degree to which agents actually like to spend
time with each other; the higher 1 − εi1− εi2 the more the two agents like each other.
We consider two choices that can be made with respect to the agents’ initial utility, ¯U . On the one hand, we consider the situation that agents can choose to be together (i.e., a
choice household) and that their initial utility is simply the utility from being single; this can be derived by substituting (3.5) into the single person utility function (3.4d):
¯
Ui = Wiεi
εi
(1 − εi)1−εi. (3.12)
initial utility is zero. For a complete overview of the possible candidates for the initial level
of utility refer to Vermeulen (2002, p. 556).
3.3.1 Choice Households
In the case that household can choose to be together the objective function of the bargaining
problem becomes: max C,LM i,LHi i=A,B h Cε1(1 − L HA− LM A)ε2 LαHAL 1−α HB 1−ε1−ε2 − ¯UAi× h Cε1(1 − L HB− LM B)ε2 LαHAL 1−α HB 1−ε1−ε2 − ¯UBi s.t. C + WA(1 − LM A) + WB(1 − LM B) = WA+ WB
where ¯Ui is equal to (3.12), mutatis muntandis, and where we have assumed that agents
only differ with respect to their wages. The relevant FONCs are:
∂L ∂C = UA UA− ¯UA + UB UB− ¯UB ε1 C − λ = 0, ∂L ∂LHA = U A UA− ¯UA −ε2 1 − LHA− LM A + UA UA− ¯UA + UB UB− ¯UB (1 − ε1− ε2) α LHA = 0, ∂L ∂LM A = UA UA− ¯UA −ε 2 (1 − LHA− LM A) + λWA= 0, ∂L ∂LHB = UB UB− ¯UB −ε 2 1 − LHB − LM B + UA UA− ¯UA + UB UB− ¯UB (1 − ε1− ε2) (1 − α) LHB = 0, ∂L ∂LM B = U B UB− ¯UB −ε2 (1 − LHB− LM B) + λWB = 0, and WA+ WB = C + WA(1 − LM A) + WB(1 − LM B) .
Using the FONCs for market labour and consumption we can derive the marginal rate of
substitution between consumption and leisure:
ε1
ε2
= C
WA(1 − LHA− LM A) + WB(1 − LHB− LM B)
In order to derive a closed-form solution for consumption demand and labour supply, it is
necessary to derive the optimal extent of home production. This can be done by using the FONCs for home production:
LHA = (1 − LM A) UA UA− ¯UA + UB UB− ¯UB (1 − ε1− ε2) α UA UA− ¯UAε2+ UA UA− ¯UA + UB UB− ¯UB (1 − ε1− ε2) α , (3.13a) and LHB = (1 − LM B) UA UA− ¯UA + UB UB− ¯UB (1 − ε1− ε2) (1 − α) UB UB− ¯UBε2+ UA UA− ¯UA + UB UB− ¯UB (1 − ε1− ε2) (1 − α) . (3.13b)
The problem is that (3.13) has no straightforward solution for home labour. That is, home
labour is a function of all other variables in the model. Hence no closed-form consumption and labour supply functions can be derived. This implies that although the NBS with
choice households is interesting from a theoretical point of view, it is impractical when it comes to deriving a closed-form solution. Thus it is hard, if not impossible, to embed
a NBS with choice households into a growth model and still be able to derive intuitively appealing results.
3.3.2 Mandatory Households
Assuming that bargaining powers and preferences differ between agents, the objective
function of the household bargaining problem in the case of mandatory households becomes:
max C,LM i,LHi i=A,B h CεA1 (1 − L HA− LM A)ε A 2 Lα HAL 1−α HB 1−εA1−εA2iν× h CεB1 (1 − L HB− LM B) εB 2 Lα HAL 1−α HB 1−εB1−εB2i1−ν (3.14a) s.t. C + WA(1 − LM A) + WB(1 − LM B) = WA+ WB. (3.14b)
Remember that in mandatory households the agents are obliged, by some exogenous force,
to be together and that, therefore, initial utility of the household members is zero. The astute reader will have noticed that this is nothing more than the Samuelson society
in mind we can rewrite (3.14a) as: UN BS = CεA1ν+εB1(1−ν)(1 − L HA− LM A)ε A 2ν× (1 − LHB− LM B)ε B 2(1−ν) Lα HAL 1−α HB (1−εA1−εA2)ν+(1−εB1−εB2)(1−ν) (3.15a) = Cγ1(1 − L HA− LM A)γ2(1 − LHB− LM B)γ3 LαHAL 1−α HB γ4 (3.15b) with γ1 ≡ εA1ν + εB1 (1 − ν) , γ2 ≡ εA2ν, γ3 = εB2 (1 − ν) and γ4 ≡ 1 − εA1 − εA2 ν + 1 − εB 1 − εB2 (1 − ν) , naturally P 4
i=1γi = 1. γ4can be interpreted as the weighted average
of love of the household; although the household members are forced to be together they can still have preferences such that they will devote more time to being solitary than to
be confined in a union with an unwanted partner. Using the same methods as before we can derive the household demand and supply functions:
LHA = (1 − LM A) γ4α (γ2+ γ4α) , (3.16a) LHB = (1 − LM B) γ4(1 − α) (γ3+ γ4(1 − α)) , (3.16b) 1 − LM A = WA+ WB WA (γ2+ γ4α) , (3.16c) 1 − LM B = WA+ WB WB (γ3+ γ4(1 − α)) , (3.16d) and C = (WA+ WB) γ1. (3.16e)
The intuitive method of eye balling reveals that if agents start liking each other more (i.e., γ4 increases), home labour will increase and consumption will decrease. The decrease in
consumption is due to the negative pressure that love puts on market labour supply of households. Furthermore, we can see that the household member with the higher wage
spends relatively more time in the market and the agent with the lower wage spends relatively more time on home production. Naturally, this is due to home production not
3.4
Collective Model
We end this chapter with a brief discussion of the collective approach to the household.
This approach, the newest of the three here discussed approaches, has earned its merits in the empirical microeconomic literature as well as the theoretical literature. We refer the
reader to Vermeulen (2002) for a thorough, and very complete, discussion of the collective approach to household behaviour.
The collective model is due to Chiappori (1988 and 1992) who proposed that the only limitation to the household bargaining approach should be that it is Pareto efficient.
Effec-tively, the collective model is a generalisation of a household dictatorship model (Manser and Brown (1980)) in which one agent maximises his/her own utility given that the other
household member needs to attain some minimum level of utility in order to remain part of the household. The household dictatorship model boils down to a unitary model with an
additional constraint; participation of the dictated partner. The objective of the household dictator is: max C,LA,LBU C, 1 − L A s.t. U C, 1 − LB − ¯UB s.t. WA+ WB = C + WA 1 − LA + WB 1 − LA , (3.17) where we have assumed that A is the dictator and B is the dictated household member.
By varying ¯UB, the utility allowances for B, we can trace out a utility function in which all levels of utility that A can attain given B0s utility. Assuming concave utility functions (as
where λ(WA, WB) is the normalised participation constraint Lagrange multiplier (δ, see
below) of the initial household optimisation problem (i.e., λ(WA, WB) = 1/ (1 + δ)). In order to derive a closed-form solution for (3.18) it is essential to determine the functional
form of the normalised Lagrange multiplier λ(WA, WB), which can be derived from the initial optimisation problem (3.17) .
The Lagrangian of the initial optimisation problem is:
max C,Lii=A,BL = C α 1 − LA1−α +δhCα 1 − LB1−α − ¯UB i +ωWA+ WB− C − WA 1 − LA − WB 1 − LB , (3.19)
where ω is the Lagrange multiplier of the budget constraint. The FONCs of the
optimisa-tion problem are:
∂L ∂C = αC α−1 1 − LA1−α + δαCα−1 1 − LB1−α = ω, (3.20) ∂L ∂LA = (1 − α) C α 1 − LA−α = ωWA, (3.21) ∂L ∂LB = δ (1 − α) C α 1 − LB−α = ωWB, (3.22) Cα 1 − LB(1−α) ≥ U¯B; δ ≥ 0; δhCα 1 − LB1−α− ¯UBi= 0, (3.23) and WA+ WB = C + WA 1 − LA + WB 1 − LB . (3.24)
Assuming an interior solution (i.e., Cα 1 − LB1−α
= ¯UB and δ > 0), we can distill δ from
the FONCs and write it implicitly as:
WA WB (1−α)/α δ1/α = 1 + 1−αα 1 + δ WA WB 1/α δ1/a 1−α! + WA WB (1−α)/α δ1/a 1 + (1−α)δα WB WA 1/α 1 δ 1/a 1−α + δ ! +WWBA (1−α)/α 1 δ 1/a . (3.25) (3.25) clearly shows that there is no closed-form solution for δ, so that the collective model,
3.5
Final Remarks
This concludes the discussion of the most common approaches to modeling intra-household
allocation. We have seen that various models of intra-household allocation can be brought back to an adapted unitary household setting by a smart interpretation of the preference
parameters. So that the key benefit from using a realistic household structure seems to be that it provides a genesis of household preferences; household preferences are the weighted
average of the individual household members’ preferences where the weights are assigned according to individual bargaining power.
Furthermore, we have seen that the more elaborate models of household behaviour, the NBS choice model and the collective model, seem to be superior from a theoretical
point of view but are generally impossible to embed in a growth model due to the lack of a closed-form solution. This gives these models the likes of a golden hammer that cannot
strike a rusty nail.
In part II we will use the NBS mandatory household model to investigate the effects of
intra-household allocation on the macroeconomy. That is, we will embed the NBS manda-tory household model into a 2 period overlapping generations model with probabilistic
This chapter develops1, analyses and discusses the first of our two extensions of the standard overlapping generations (OLG) model introduced in chapter 2. That is, this chapter uses
the tools discussed in chapter 2 to develop an OLG with an additional period in which agents have the opportunity to invest in human capital. The extended model consists of two
types of agents that differ in their innate ability of turning educational input into human capital and live for 3 periods. Between period 2 and 3 agents face a positive probability of
death. The inter-generational link is established through the inheritance of human capital by the young from the middle aged and through the ownership of physical capital by the
old. The elderly use their physical capital in combination with the labour power of the middle aged to produce output by means of a single representative firm. The firm rewards
its inputs by distributing wages to workers and interest to capitalists. As the model unites ageing and education, we can analyse the interaction between these two intricacies of life.
As discussed in chapter 2, we are not the first to study the accumulation of human cap-ital in an OLG setting. Glomm and Ravikumar (1992) analyse the consequences of private
versus public financing of education in an environment where parents provide funding for human capital investments. Human capital is produced by combining time investments by
the young, educational investments by parents and the level of human capital in the econ-omy. Parents invest in education of the young for altruistic reasons and do not internalise
the added benefit of educational investment to the young. Thus, if the parents’ degree of altruism is not high enough, there may be under-financing of education. This is seen as
the key rational for a system of mandatory education. In this respect they also show that
1As the parts of this thesis are meant to be self-contained there is a substantial overlap, no pun intended,
