Fertility in the absence of self-control

Mathematical Social Sciences 66 (2013) 71–86

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Mathematical Social Sciences

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Fertility in the absence of self-control

Bertrand Wigniolle ∗

Paris School of Economics, FranceUniversity of Paris 1, France

h i g h l i g h t s

• This paper studies the quantity–quality trade-off model of fertility, under hyperbolic discounting.• In a developed economy, the lack of self-control may tend to reduce fertility.• In a developing economy, it reduces investments in quality and increases fertility.• In absence of commitment, a small change of parameters may lead to a jump in fertility and education.

a r t i c l e i n f o

Article history:Received 16 June 2011Received in revised form10 August 2012Accepted 4 February 2013Available online 4 March 2013

a b s t r a c t

This paper studies the quantity–quality trade-off model of fertility, under the assumption of hyperbolicdiscounting. It shows that the lack of self-control may play a different role in a developed economy andin a developing one. In the first case, characterized by a positive investment in quality, the lack of self-control may tend to reduce fertility. In the second case, it is possible that the lack of self-control leads toboth no investment in quality and a higher fertility rate. It is also proved that if parents cannot commit ontheir investment in quality, a small change of parameters may lead to a jump in fertility and education.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

From the seminal articles of Becker and Lewis (1973) andBeckerand Tomes (1976), the benchmark theory of fertility decisionswithin the family is the quantity–quality trade-off model. Accord-ing to this model, the quality and quantity of children are bothendogenous variables. Fertility behaviors and investments in chil-dren’s human capital are consciously and jointly determined byparents. This theory explains fertility and education behaviors asan optimal choice of the household, depending on its income andon the costs of quality and quantity. One major outcome of thisliterature is to provide an explanation of education and fertilitybehaviors in developing and developed countries. In developingcountries, the cost of quantity is relatively low with respect to thefamily income, and the cost of quality relatively high. This may ex-plain a high investment in quantity and a low investment in qualityin such countries. In developed economies, the cost of quantity is

I thank two anonymous referees for helpful comments. I also thank participantsof the Public Economic Theory conference in Seoul (2008), where a preliminaryversion of this paper was presented, and particularly David de la Croix andHippolyte d’Albis. I have also benefited from helpful comments at the EQUIPPEseminar in the University of Lille 1.∗ Correspondence to: C.E.S., Maison des Sciences Economiques, 106-112,

boulevard de l’hôpital, 75647 Paris Cedex 13, France. Tel.: +33 0 1 44 07 81 98.E-mail address:[email protected].

0165-4896/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.mathsocsci.2013.02.001

relatively high with respect to the family income, and the cost ofquality relatively low. This may result in a low fertility rate and ina high investment in education.

In this paper, I argue that this theory is founded on the implicitassumption of perfect self-control of the household. Indeed, aseducation decisions are taken after the fertility decision, it is notobvious that the education decision ex-post is consistent with theeducation decision planned at the time of the fertility choice. Thisproblem of self-control exists if agents are endowed with a nonrecursive utility function, for instance if the flows of instantaneousutilities are discounted with (quasi)-hyperbolic discounting.

How can self-control problems change the trade-off betweenquantity and quality? This paper shows that the lack of self-controlmay play a different role in a developed economy and in a develop-ing one. In a developed country, the lack of self-control may tendto reduce fertility and increase education, whereas it may reduceeducation and increase fertility in a developing country. Moreover,compulsory schoolingmay play the role of a commitment technol-ogy in a developing country that allows reduction in fertility.

Recently, a growing literature has stressed the assumption of(quasi)-hyperbolic discount rates. It seems more consistent withlaboratory experiments that find a negative relationship betweendiscount rates and time delay (see e.g. Loewenstein and Thaler,1989). The consequences of quasi-hyperbolic discounting havebeen studied in various frameworks. Many articles have been con-cerned with savings behavior, mainly Harris and Laibson (2001)

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72 B. Wigniolle / Mathematical Social Sciences 66 (2013) 71–86

and Laibson (1997). Diamond and Köszegi (2003) applied hyper-bolic discounting to the early retirement pattern of workers. Barro(1999) introduced this assumption in a standard growth model.Wrede (2011) applies quasi-hyperbolic discounting to the timingand number of births, pointing out a possible postponement ofbirths.

A recent article by Salanié and Treich (2006) has made a break-through in this literature. In discrete time, quasi-hyperbolic dis-counting is introduced in the intertemporal utility function of theconsumer by adding an extra parameter β ≤ 1 that represents thebias for the present. The instantaneous flows of utility areweightedby the discount factors: 1, βδ, βδ2, βδ3, etc. The standard assump-tion of exponential consumers is obtained for β = 1. Hyperbolicconsumers have a bias for the present β < 1. In order to evaluatethe impact of self-control on behaviors, most articles have com-pared the results obtained for β < 1 with that obtained for β = 1.The point made by Salanié and Treich is that this comparison is notappropriate to isolate the effect of a lack of self-control, as β alsomodifies the preferences of the consumer. The only pertinent com-parison is between the behavior of a consumer with commitmentpower, and that of a consumer without this power.

In this paper I consider a simple model in which parents arbi-trate between the quantity and the quality of their children, as inthe benchmark models of Becker and Lewis (1973) and Becker andTomes (1976). These seminal contributions used a static microeco-nomic model of demand for quantity and quality of children. In or-der to study the problem of time consistency of the behaviors, thedynamic structure of the problemmust be introduced, here the factthat the choice of fertility is taken before the choice of education.Self-control problems may only occur in a dynamic model that in-clude (at least) three periods, with one decision in the first periodand one in the second one.

The household’s utility depends on the flows of instantaneousutility obtained during three periods. These flows are discountedwith a quasi-hyperbolic discount factor. In the first period, thehousehold chooses the quantity of children. Each Child entails acost in time for the household (mainly for the wife) and implies areduction of income. This cost comes from child rearing and theprimary education given inside the family. In the second period,the household chooses the quality level (the education level) givento each child. The education cost is proportional to the numberof children and to the level of quality. Finally, in the third periodthe flow of utility depends positively on both quantity and qualitylevels. This last assumption corresponds to the altruistic feelingof parents that value both the number and the quality of theirchildren.

The decision process within the household is based on the fol-lowing assumptions. There is no problem of coordination betweenspouses in the household, following the unitary model: at each pe-riod parentsmake a joint decision or there is a household head thatmakes all the decisions. But, as preferences are not time-consistent,there is a conflict between the objective functions of the householdin periods 1 and 2. From Strotz (1956), the usual way to solve thisproblem consists in assuming an intrapersonal game. The house-hold ismade of a sequence of selves and self i is the decision taker inperiod i. The solution resulting from the equilibrium of this game iscalled the Temporary Consistent solution (TC in abbreviated form),or the ‘‘sophisticated’’ behavior. This is the solution that is mainlystudied in the literature (see, for example, Laibson, 1997, Harrisand Laibson, 2001, or Diamond and Köszegi, 2003). Another solu-tion, called the Commitment solution (C in abbreviated form) cor-responds to the optimal choice if the first self can take all decisionsat all periods. This solution corresponds to perfect self-control. Inpractice, the household needs a ‘‘Commitment technology’’ in or-der to implement the Commitment solution, which means a wayto constrain the decision of future selves. For instance, in savings

behaviors, the access to an illiquid asset can be viewed as an im-perfect commitment technology (see Laibson, 1997).

Following Salanié and Treich, the commitment solution forfertility and education is compared to the temporary consistentsolution. Two cases are studied. In the first one, interpreted as thecase of a developed economy, the cost of quantity is assumed to berelatively high with respect to the family income, and the cost ofquality relatively low. In this case, both C and TC solutions lead to apositive investment in quality. The impact of the absence of self-control depends on the elasticity of substitution of preferences.In the case of an elasticity of substitution greater than one, theabsence of self-control implies a smaller fertility. The investmentin quality is also lower for β close to 1, but higher for a small valueof β .

In the second case, interpreted as the case of a developing econ-omy, the cost of quantity is assumed to be relatively low with re-spect to the family income, and the cost of quality relatively high.The paper focuses on the situation for which the investment inquality cancels out along the TC solution, whereas it is positive forthe C-solution. This case is thinkable for a developing economy,1 inwhich children only receive primary education. It leads to a higherfertility rate for the TC solution than for the C solution. It meansthat if the household could commit on its future education invest-ment, it would choose a lower fertility level. For instance, a policythat imposes compulsory attendance at school for children can beviewed as a commitment technology, which is expected to reducefertility.2

The influence of different parameters is considered. The timethat parents spent on their children implies a fall in family income,that can be viewed in themodel as the (opportunity) cost of fertilityw0. This opportunity costmainly corresponds to thewife’s income.Considering the TC solution and starting froma lowvalue ofw0, thefertility level is high and no education investment occurs. An in-crease of w0 reduces fertility. At some threshold value, the house-hold begins to invest in quality. At this value, fertility undergoes ajump downward and continues to decrease as w0 increases. A sec-ond parameter of interest is the cost of education τ . τ representsthe cost of formal education that corresponds to tuition fees, schoolsupplies. It can also incorporate the opportunity cost of child laborfor developing economies. Starting from a high value, the economyfeatures a high fertility levelwith no education investment. As longas no investment in quality occurs, a decrease in τ has of course noimpact on fertility. At some threshold level, the household startsto invest in quality. At this value, fertility undergoes a jump down-ward and continues to decrease as τ decreases.3

Thismodel offers twonovel featureswith respect to the existingliterature. In other words, two characteristics make it difficult toinfer directly the impact of self-control on fertility and educationfrom preceding studies of savings, retirement behaviors, etc. Thefirst characteristic is the non-linearity of the budget constraintderiving from the quantity–quality trade-off. The cost of educationis the product of quality times quantity. The second characteristiccomes from the property that no investment in quality is apossible solution. This solution represents the case of a developingeconomy, for which no investment in education is provided tochildren, except primary education.

1 A third case with no investment in quality for both solutions is not studied asit is not interesting. Indeed, for no investment in quality in both solutions, TC and Csolutions give the same value of fertility.2 It is possible to consider a more general model with more than 3 periods of

life for agents. In this case the lack of commitment implies that children stop theirstudies earlier in developing economies.3 For a high elasticity of substitution in the household preferences, it is possible

that the evolution of fertility becomes non monotonic with τ .

B. Wigniolle / Mathematical Social Sciences 66 (2013) 71–86 73

Concerning the non-linearity of the budget constraint, one con-sequence is that the lack of self-control may imply lower invest-ment in quality for β close to 1, but higher investment for a smallvalue of β . This result comes from the property that the cost ofquality depends on quantity, and quantity increases with β . Ina model with a linear budget constraint, the lack of self-controlwould have a monotonic impact.

The second novel feature comes from the case for which noinvestment in quality is reached along the TC solution. When thequality level chosen by self 2 cancels out, the optimal response forself 1 corresponds to a jump in fertility. In other words, fertilityis not continuous at the point for which quality cancels out. Thisproperty is interesting, as it means that in the neighborhood ofthis point, a small change in some parameters can lead to a bigchange in fertility. For example, a small increase in the opportunitycost of quantity can lead to a big reduction in fertility. This resultcan be explained considering the objective function of self 1, alongthe TC solution. When quality cancels out, the response functionof self 2 undergoes a discontinuity of its derivative. Whereas thisderivative is negative for a positive investment in quality (qualityis a decreasing function of quantity), the derivative is equal to zerowhen quality cancels out. As there is a discrepancy between theobjective functions of selves 1 and 2, the derivative of the self 1objective function undergoes a jumpwhen quality cancels out. Forthis reason, two levels of fertility may exist that are local maximaof the objective function of self 1. If the change of a parameter leadsto a jump from one local maximum to the other one, there is a highvariation in fertility at this point.

Few studies have been devoted to this property, that a con-tinuous change in some parameter can induce a jump of an en-dogenous variable, under quasi-hyperbolic discounting. It can betrue in all models in which the decision of a self is subject to aconstraint. Laibson (1997) was the first to point out the existenceof discontinuous optimal strategies with quasi-hyperbolic dis-counting, in a model of savings with imperfect capital markets.To avoid the difficulties related to the non-convexity of the prob-lem, he introduced a restriction on the labor income process thatruled out the possibility of corner solutions and discontinuousequilibrium strategies. Harris and Laibson (2002) have providedthe most detailed study of this question. They give an intuitionof such pathologies. They present the results of numerical sim-ulations, and conclude that such pathologies do not arise whenthe model is calibrated with empirically sensible parameter val-ues. Wigniolle (2012) remarks that the calibrations in Harris andLaibson that can eliminate the discontinuous strategies (a value ofβ close to 1, a small value for the elasticity of substitution) are alsothose that make negligible the impact of hyperbolic discounting.In other words, when hyperbolic discounting matters, it is neces-sary to deal with such pathologies. He provides a detailed study ofsuch discontinuous strategies in a simple framework that allows acomplete characterization.

These different studies point out the role of β: if β can departsignificantly from 1, the existence of discontinuous strategies mayoccur. The value of β may depend on the time horizon of decisions.If the frequency of decisions is high, a value close to 1 is expected. Ifthe interval of time between two decisions is high, a low value of βmay be relevant. For decisions concerning fertility and education,it is reasonable to assume a low frequency and a small valueof β . Therefore, it seems relevant to expect a strong impact ofquasi-hyperbolic discounting on decisions and the occurrence ofdiscontinuous strategies cannot be ignored.

This work can also be compared with the literature about col-lective choices with non-cooperative behaviors (cf. the seminalcontributions of Ulph, 1988, Woolley, 1988 and Chen and Wool-ley, 2001). This paper follows another way, considering a unitarymodel of the household: coordination problems within the house-hold do not exist in order to take decisions during a given period;

the coordination problem exists in intertemporal choices betweentwo successive selves as preferences are not time-consistent. Somesimilarities can be found between these two types of models: inboth models, non-cooperative behaviors may lead to an equilib-rium that is not Pareto-optimal, there may exist under/over provi-sion of a public good within the family. However, there exists animportant difference due to the intertemporal structure of the de-cision process: in the game between selves living at different pe-riods, the self that plays first has a strategic advantage. She/he canplay a strategy in taking into account the best response functions offuture selves. The resulting Nash equilibrium is not Pareto-optimalin general, and this inefficiency is due to time inconsistency. Theoptimal policy in this context consists in introducing a commit-ment technology that allows to implement the commitment solu-tion (C solution). The efficiency of some institutional mechanismto play the role of a commitment technology is then an impor-tant topic. In the context of the quantity–quality trade-off modelof fertility, compulsory school can be viewed as an example of acommitment technology. Compulsory school attendance impliesa minimal investment of parents in the quality of their children.Therefore, it can be viewed by the preceding self as a credible com-mitment on the future education of children, and it can lead to afertility choice that is closer to the optimal one.

Another difference between the two types of models is thatquasi-hyperbolic discounting may give birth to non-discontinuousstrategies. This property was discussed in detail above. For thenon-cooperative solution (TC), this may imply a jump of fertilitywhen the investment in education cancels out. This novel featuremay have strong economic implications. For instance, there mayexist threshold values in a policy promoting education or fertility:this type of policy may have first a smooth effect on behaviors. Atsome threshold value, a large drop in education and fertility maybe observed.

A generalization of both models could be to consider coordi-nation problems both within a given period among spouses andbetween periods among the selves of each spouse. With such as-sumptions, two sources of non-optimality in the decision processwould exist.

Section 2 presents the model. Section 3 gives the fertilitydecisions for developed and developing economies. Section 4studies how fertility and education decisions respond to changes intheir costs. Section5 concludes. A final Appendices gives theproofs.

2. The model

2.1. Basic assumptions

The household behavior derives from a utility function definedon three periods and associated with a quasi-hyperbolic discount-ing factor.4 The life-cycle of the household is represented in a verysimple way. In the first period, the household chooses the quantity(number) of children. Children entail a cost for rearing and primaryeducation in the family that is assumed to be mainly a cost in timeproportional to the number of children. In the second period, thehousehold chooses the quality (education level) of children. Qual-ity results from formal education provided by teachers and it givesrise to a financial cost for parents. Parents choose the same levelof quality for each child. In the third period, children become au-tonomous and their well-being depends on the human capital thatthey have accumulated when young.

4 An objective function defined on three periodswith one decision in the two firstperiods is the simplest framework in order to study self-control problems.


In period 1, self 1 preferences are given by the utility function:

u [c1] + βδu [c2] + βδ2mηU(q0 + q) (1)

with

u(x) =x1−

1σ

1 −1σ

, (2)

η < 1 and σ > 0. c1 and c2 are respectively the consumptionlevels of the household in periods 1 and 2.5 β and δ are two positivecoefficients not greater than 1. δ is the usual discount parameterand β is the bias for the present. m is the number (quantity) ofchildren. q0+q is the quality level of each child that is the sumof anexogenous value q0 and of the amount of education q financed byparents. U(q0 + q) is the life-cycle utility level of each child, that isassumed to depend on their human capital. This formulation relieson three assumptions that are often considered in the literature(see e.g. Becker and Barro, 1988, Becker et al., 1990 or Jones andSchoonbroodt, 2010). First, parents have altruistic feelings towardtheir children: their utility depends on the one of each child.Second, parents like having children: utility is increasing with thenumberm of children. Third, this increase is subject to diminishingreturns: an increase in m leads to an increase in the utility ofparents that is less than proportional.

As U(q0 + q) can be seen as the value function of a child, it isnatural to assume that

U(x) = λu(x)

with λ some positive constant. To keep a minimum number ofparameters of interest, it is possible to choose the units of thedifferent variables in such a way that λ = 1.6

η and σ cannot take any value to obtain a concave function thatsatisfies the three preceding properties. As it is established in Jonesand Schoonbroodt, only the following restrictions are admissible:

• either 0 < 1 − 1/σ ≤ η < 1.• or 0 > 1 − 1/σ ≥ η > −1.

To be able to deal with these two cases in the simplest frame-work, the following assumption is retained: η = 1 − 1/σ . Underthis condition, the two cases correspond to σ > 1 or 1/2 < σ < 1.The third term of the utility function can now be written

[m(q0 + q)]1−1σ

1 −1σ

.

The assumption η = 1− 1/σ is equivalent to assume that parentsvalue the total revenue of their children, which can also be viewedas a particular warm glow motivation for altruism.

The budget constraint of the family in period 1 is:

c1 = w1 + w0(1 − φm).

The family income consists of two parts: a constant part w1, anda variable part w0(1 − φm) that depends on child quantity m. φis the time cost for rearing and primary education in the family ofone child. w0 is the opportunity cost of parents to provide primaryeducation to children. It is an average of the opportunity costs of

5 In the two first periods, children consumption does not appear explicitly. Itis assumed to be part of parents’ consumption. A simple justification for thisassumption is that children consumption may be equal to a constant share ofparents’ consumption.6 Our formulation of altruism can also be viewed as a warm glow motivation of

parents to give some education to their children, following Andreoni (1989). Suchtype of formulations have been extensively used by different authors, includingGalor and Weil (2000) and De la Croix and Doepke (2003) that use the sameassumption as this paper.

parentsweighted by their relative participation to children rearing,assuming that both spouses devote time to the primary educationof children. w1 is the part of household revenues that does notdepend on the time devoted to children. As females in generaldevote more time to children, w0 can be viewed approximately asthe opportunity cost of time for the wife (wife’s income) and w1as the husband’s income. The resulting consumption level of thehousehold is w1 + w0(1 − φm).

In period 2, the family income is w2.7 The education cost ofchildren is now a financial cost, corresponding to the financing ofschool fees. τ is the unit cost for one unit of quality for one child.Therefore τmq is the cost of providing a quality q to each of the mchildren, and the resulting second period consumption level of thehousehold is

c2 = w2 − τmq.

In period 3, them children become autonomous from their par-ents. Their intertemporal utility depends on their human capitallevel (q0 + q). Parents are altruistic and value the well-being oftheir children.

Some remarks can be done on the utility function of self 1. First,consumption levels of parents do no more depend on spending fortheir children from period 3. Therefore it is not useful to includeparents consumptions in future periods as they will not play anyrole in the choice of quantity and quality of children. Second, itis assumed that the altruism term in the utility function that de-pends on quantity and quality of children appears in period 3. Itwould be possible to introduce other altruism terms in periods 1and 2, depending on fertility in period 1, and on fertility and ed-ucation in period 2. These terms would only make more compli-cated the analysis, but they would not add new features to themodel. The model is designed in order to remain tractable and ourformulation is the simplest one allowing to take into account thequality–quantity trade-off in an intertemporal setting.8 Third, it isassumed that parents choose the same level of quality for eachchild. This assumption can be proved from the concavity of the ob-jective function of the household: it is optimal to give the sameeducation level to each child.

In the literature about fertility decisions,m is usually taken as acontinuous variable, since the seminal contributions of Becker andLewis (1973) and Becker and Tomes (1976). This allows us tomakestandard optimization calculations.

In period 2, self 2 preferences are given by:

u [c2] + βδmηu(q0 + q). (3)

The discount factor between period 3 and period 2 is δ if it is com-puted by self 1, and βδ if it is computed by self 2. The parame-ter β indicates whether there is a self-control problem (β < 1)or not (β = 1).

Following Salanié and Treich (2006), the time-consistent solu-tion is compared to the commitment solution. The time-consistentsolution (TC) is the non-cooperative equilibrium obtained from thegame played by selves 1 and 2. More precisely, self 2 chooses q,mbeing given. Self 1 choosesm, taking into account the best responsefunction of self 2. The commitment solution (C) is obtained by as-suming that self 1 can choose bothm and q.

7 In period 2, education only implies a financial cost and the model is unitary.Therefore, all results will only depend on total income w2 and not on the incomeearned by each spouse.8 Unlikemost of articles that study themarginal effect of a bias for the present, the

present paper characterizes cases in which the problem of self-control may inducequalitative changes in the behavior of agents, when quality cancels out. We thenneed a formulation that allows an explicit expression for agents decisions.


2.2. Investment in quality

The best response function of self 2 for the TC solutionSelf 2 takesm as given and chooses q following his best response

function:

qTC(m) = argmax(q)

u [w2 − τmq] + βδu [m(q0 + q)]

s.t. q ≥ 0.

The solution to this program can be interior (q > 0) or not.Defining the threshold

mTC≡

(βδ/τ)σ w2

q0the best response function of self 2 is:

qTC(m) =

(βδ/τ)σ

w2m − q0

1 + (βδ)σ τ 1−σifm ≤ mTC

0 ifm ≥ mTC(4)

qTC(m) is a non-increasing function ofm.The commitment solution

Assume that self 1 can commit in period 1 on a choice of q inperiod 2. To compare this solution with the preceding one, it isuseful to split the resolution in two steps: firstly the optimal choiceof q for m given, secondly the optimal value of m, in taking intoaccount the effect of m on the optimal choice of q. For m given,defining a new threshold

mC≡

(δ/τ)σ w2

q0the optimal value of q if self 1 can commit on it in period 1 is:

qC(m) =

(δ/τ)σ

w2m − q0

1 + δσ τ 1−σifm ≤ mC

0 if m ≥ mC(5)

qC(m) is a non-increasing function of m. It is clear that, for mgiven, qC(m) ≥ qTC(m) with a strict inequality when qC(m) > 0.For a given value of fertility, self 1’s optimal investment in qualityis higher than that chosen by self 2.

Remark 1. As usual, the fertility rate is assumed to be a continuousvariable. This simplifying assumption leads to meaningless resultsfor m tending toward 0. Indeed, qC(m) and qTC(m) tend to be infi-nite when m tends toward 0, with a discontinuity in m = 0. Thus,it will be appropriate to eliminate parameter values leading to fer-tility rates close to 0.

3. Fertility decisions under quasi-hyperbolic discounting

This section studies the impact of quasi-hyperbolic discountingon fertility and education decisions. The time-consistent solutionis compared to the commitment solution. One important outputof the quantity–quality trade-off model of fertility is to explainfertility and education behaviors through standard economicvariables: the costs of quantity and quality of childrenwith respectto family income. In such a framework, the evolution of behaviorsthrough the development process can be explained by the changesin the relative costs.9 In a developing economy, the cost of quantity(φw0 in our model) is low relatively to the income of the familyas women have low opportunities on the labor market. In adeveloped economy, this relative cost is high as women havemore

9 A general presentation of this literature is given by Birdsall (1988).

opportunities to participate to the labor market. As for the cost ofeducation τ , it is high with respect to the income of the family ina developing economy, and low in a developed economy. Indeed,developed economies often feature high level of public subsidyin education. As a consequence, the quantity–quality trade-offmodel of fertility may explain why fertility is high in developingeconomies and education is low. It may also be the case thatchildren only receive primary education within the family, but noformal education in a school, that corresponds to the case q = 0 inthe model.

Following this analysis, the impact of hyperbolic discountingwill be studied in two cases. In the first case, fertility is derivedwhen it is associatedwith a positive investment in education. In thesecond case, the corner solution in which education cancels out isstudied. The first casemay occurwhen the cost of quantity (φw0) ishigh enough and the cost of education τ is low enoughwith respectto family income. The second case may occur when (φw0) is lowand τ is highwith respect to family income. The first casewill be re-ferred as the situation of a developed economy, and the second caseas the one of a developing country. A complete characterization ofthese two cases will be provided related to parameter values.

3.1. The developed economy

In this part, the case of the developed economy is studied, forwhich the cost of quantity is assumed to be relatively high withrespect to the family income, and the cost of quality relatively low.The time-consistent solution is compared with the commitmentsolution, when both are interior solutions: q > 0.The time-consistent solution

Along the time-consistent solution, self 1 choosesm, taking intoaccount the best response function of self 2 given by Eq. (4). Byassumption, m is such that qTC(m) > 0 for a developed economy.Self 1’s program is:

maxm≥0

u [w1 + w0(1 − φm)] + βδuw2 − τmqTC(m)

+ βδ2u

m(q0 + qTC(m))

.

Defining

A(β) ≡

1 + δσ βσ−1τ 1−σ

σ1 + δσ βσ τ 1−σ

σ−1 (6)

B ≡

τq0φw0

σ

(7)

the time-consistent solution is:

mTC=

(βδ)σ A(β)B (w1 + w0) − w2

τq0 + φw0 (βδ)σ A(β)B. (8)

This solution is valid only ifmTC > 0, which is satisfied if

H(β) ≡ (βδ)σ A(β)B >w2

w1 + w0. (9)

Following the preceding remark, the parameter values will berestricted in such a way that (9) will hold in what follows.The commitment solution

Along the commitment solution, self 1 chooses both m and q.This solution can be obtained using Eq. (5) with qC(m) > 0 by as-sumption. The program is:

maxm≥0

u [w1 + w0(1 − φm)] + βδuw2 − τmqC(m)

+ βδ2u

m(q0 + qC(m))

.


The commitment solution is

mC=

(βδ)σ A(1)B (w1 + w0) − w2

τq0 + φw0 (βδ)σ A(1)B. (10)

This solution is valid only ifmC > 0, which gives the condition

(βδ)σ A(1)B >w2

w1 + w0. (11)

Comparison between TC and C solutionsThe only difference between the two expressions (8) and (10)

is the term A(β) in place of A(1). Asm is increasing with respect toA,mTC < mC if and only if A(β) < A(1). It is easy to find:

d ln [A(β)]dβ

=σ(σ − 1)δσ τ 1−σ βσ−2(1 − β)

1 + δσ βσ−1τ 1−σ

1 + δσ βσ τ 1−σ .

As β < 1, A(β) < A(1) ⇐⇒ σ > 1.If σ > 1,mTC < mC: the time-consistent solution leads to a

lower fertility. As self 2 does not invest enough in education fromthe point of view of self 1, and as education choice decreases withfertility, self 1’s best response is a reduction in fertility. If self 2could commit on a higher level of quality (for instance, if he couldcommit on the behavior qC(m)), self 1 would invest more in thequantity of children.

In the opposite case σ < 1, the result is reversed. As self 2under-invests in quality, self 1 increases quantity with respect tothe commitment solution.

This result is close to the one obtained by Salanié and Treich(2006), in a model in which the decision variable of agents issavings. Applying their results to a CES utility function (2), they findthat the time-consistent solution leads to under savings iff σ > 1.

In the caseσ < 1, the lack of self-control leads to higher fertilitymTC > mC. Therefore, it also leads to a lower quality investment:as mTC > mC, qC(mC) > qC(mTC) > qTC(mTC). The absence ofcommitment implies more quantity and less quality.

In the caseσ > 1, it is not so easy to conclude onquality. Indeed,qC(m) and qTC(m) are decreasing functions, with qC(m) > qTC(m)for a given level of fertility m. But, as mTC < mC, it is not possibleyet to conclude if qC(mC) ≷ qTC(mTC). Proposition 1 proves thatparents under-invest in quality when β is close to 1, but they overinvest for a low value of β .

The different results are summarized in the following proposi-tion:

Proposition 1. Assuming an interior solution for m and q (m andq > 0),

• In the case σ < 1, the lack of self-control leads to higher fertilitymTC > mC and lower investment in education qC > qTC.

• In the case σ > 1, the lack of self-control leads to lowerinvestments in quantity mTC < mC. The investment in quality isalso lower for β close to one, but higher for a low β .

Proof. See Appendix A.

Assumption. σ > 1.

The assumption σ > 1 is retained in what follows. It corre-sponds to the case favored by Salanié and Treich (2006), in whichthe lack of self-control leads to under-savings. As a consequence ofProposition 1, forσ > 1 andβ close to 1, every commitmentmech-anism on a higher investment in quality increases fertility. For in-stance, a public policy in favor of commitment such as compulsoryschoolingwill lead to ahigher fertility level. But for a lowvalue ofβ ,there is over investment in quality. The intuition behind this resultis that, for a lowvalue ofβ , asmTC becomesweak, the cost of qualityis very low. This result is due to the non-linearity of the cost of edu-cation which depends also on quantity. This non-linearity is a par-ticular feature of the quantity–quality trade-off model of fertility.

Another consequence of the case σ > 1 is that the constraint(11) is weaker than (9). Therefore, only (9) will be retained.Existence of an interior solution

The time-consistent and commitment solutions must satisfythe following inequalities: 0 < mTC < mTC, 0 < mC < mC. AsmTC < mC, only three inequalities must be considered: 0 < mTC,mTC < mTC and mC < mC.

Condition mTC > 0 is given by (9).The inequalitymTC < mTC gives:

Z(β) <w2

w1 + w0(12)

with Z defined as:

Z(β) ≡1

φw0βσ δσ τ−σ

q0+

(φw0)σ

(τq0)σ (βδ)σ

1+βσ δσ τ1−σ

1+βσ−1δσ τ1−σ

σ .

Finally, the inequalitymC < mC gives:

G(β) <w2

w1 + w0(13)

with

G(β) ≡1

φw0δσ τ−σ

q0+

(φw0)σ

(τq0)σ (βδ)σ

. (14)

It is straightforward to see that G(β) < Z(β). Therefore thereremain two necessary conditions for the existence of an interiorsolution of the household program: (9) and (12).

3.2. The developing economy

This part focuses on the case of a developing economy, in whichthe cost of quantity is assumed to be relatively low with respectto the family income, and the cost of quality relatively high. Fromthese assumptions, the time-consistent solution may be a cornersolution with no investment in quality (qTC = 0). If the com-mitment solution is also associated with no education investment(qC = 0), it is straightforward to see that the fertility level will bethe same for the two solutions C and TC. Therefore, this case is notinteresting as the lack of self-control has no impact on decisions.

More interesting is the case in which the commitment solutionis associated with some positive education investment (qC > 0). Inthis case, the lack of commitment influences education, and thusfertility behaviors.The time-consistent solution without investment in quality

Considering the TC behavior in the corner solutionwith qTC = 0,the fertility level mTC is given by the first order condition:

−φw0 [w1 + w0(1 − φm)]−1/σ+ βδ2m−1/σ q1−1/σ

0 = 0.

The solution is denoted by mTC and is equal to:

mTC=

βδ2

σ(φw0)

−σ qσ−10 (w1 + w0)

1 +βδ2

σ(φw0)

1−σ qσ−10

. (15)

Using (4), condition mTC > mTC ensuring that qTC = 0 gives thefollowing inequality:

w2

w1 + w0< D(β) (16)

with

D(β) ≡1

φw0(βδ)σ τ−σ

q0+

(φw0)σ

(τq0)σ (δ)σ

. (17)

Comparison with the commitment solutions


By assumption, the commitment solution is associated withsome positive education investment (qC > 0). Therefore,mC is stillgiven by (10), and condition (13) must be fulfilled. The comparisonbetween mTC given by (15), andmC given by (10) gives the follow-ing result:

Proposition 2. When the lack of self-control leads to no investmentin quality for the time-consistent solution and to a positive investmentfor the commitment solution, the fertility level is higher for the firsttime-consistent solution: mTC > mC.

Proof. From (15) and (10), the inequality mTC > mC is equivalentto

G(β) <w2

w1 + w0.

This condition holds by assumption, as it corresponds to (13),which was obtained in writing the inequalitymC < mC.

This proposition shows that the lack of self-control has adifferent impact in the developing economy, as it tends to increasefertility. If self 2 could commit on some positive investment inquality, self 1would invest less in quantity. In a developed country,a policy measure that favors commitment increases fertility. In adeveloping economy, such a measure will reduce fertility.

How to understand this result? For the TC solution, while qTCremains positive, self 1 gives birth to fewer children in order toobtain more investment in quality by self 2. But, when qTC cancelsout, decreasing fertility has nomore impact on quality. The optimalresponse of self 1 is now to increase his fertility level.Conditions for a positive investment in quality

Considering the TC behavior, two solutions have been found:one interior solution associated with a positive investment inquality and one constrained solutionwith no investment in quality.The first one must satisfy the condition Z(β) <

w2w1+w0

and thesecond one w2

w1+w0< D(β). It is easy to check that Z(β) < D(β).

Therefore, three casesmay exist. If w2w1+w0

> D(β), only the interiorsolution exists. If w2

w1+w0< Z(β) only the constrained solution

exists. If Z(β) <w2

w1+w0< D(β), the problem is to choose between

the two solutions.To understand this point, it is useful to consider the first order

condition of the program of self 1. Self 1 chooses the optimalvalue of m, taking into account the best response function of self2 qTC(m):

0 = −φw0u′ [w1 + w0(1 − φm)]− τqTC(m)βδu′

w2 − τmqTC(m)

+ βδ2(q0 + qTC(m))u′

m(q0 + qTC(m))

+ βδm

dqTC(m)

dm

−τu′

w2 − τmqTC(m)

+ δu′

m(q0 + qTC(m))

.

For the commitment solution, the first order condition is thesame, except that qTC(m) is replaced by qC(m). But, for thecommitment solution, the expression −τu′

w2 − τmqC(m)

+

δu′m(q0 + qC(m))

cancels out by definition of qC(m). For the

time-consistent solution, the expression−τu′w2 − τmqTC(m)

+

δu′m(q0 + qTC(m))

is positive, as qTC(m) is implicitly defined by

−τu′w2 − τmqTC(m)

+ βδu′

m(q0 + qTC(m))

= 0. This is the

consequence of the discrepancy between the objective functions ofself 1 and self 2. For the derivative dqTC(m)/dm, there is a discon-tinuity in mTC: this derivative is negative to the left of mTC, and iszero to the right.

The consequence of this analysis is that the derivative of self1’s objective function is always continuous for the commitment

solution. But, for the time-consistent solution, the derivative ofself 1’s objective function is discontinuous at the point mTC, witha higher value to the right of mTC. It is then possible that self1 objective function admits two local maxima. The function isconcave on each interval

0, mTC

and

mTC, +∞

and continuous,

but the derivative is discontinuous in mTC.Fig. 1 presents a numerical simulationwith the following values

of parameters: σ = 2, τ = 0.5, φ = 0.17, β = 0.5, δ = 1, w2 =

2, q0 = 0.5 and w1 = 1. Different curves are obtained for differentvalues of the parameter w0.10 The objective function of self 1 withrespect to m is drawn. The value w0 = 0.951856 is such that thetwo local maxima give the same value to the utility. Forw0 smallerthan this value, the optimal behavior is to give birth to many chil-dren (mTC) and to not invest in their education. For w0 higher thanthis value, the optimal behavior is to have a small number of chil-dren (mTC) and to invest in their quality.

Considering self 1’s objective function, under the conditionZ(β) <

w2w1+w0

< D(β), this function of m has two local maxima:one associated with a positive investment in education (mTC) andone forwhich education cancels out (mTC). Therefore it is necessaryto compare the utility levels obtained for each local maximum. UTC

denotes the indirect utility level when qTC is positive and UTC theutility level when qTC is zero. The following lemma shows that theequality UTC

= UTC implicitly defines a function V (β) such thatw2

w1+w0= V (β) ⇔ UTC

= UTC. Moreover, UTC > UTC⇔

w2w1+w0

>

V (β).

Lemma 1. Assume that Z(β) <w2

w1+w0< D(β). The condition UTC

> UTC holds iff:w2

w1 + w0

φw0

τq0+ 1

1−1/σ1 +

φw0

τq0

1−σ

(βδ)σ A(β)

1/σ

>

1 +

φw0

q0

1−σ βδ2σ

1/σ

+ βδ

w2

w1 + w0

1−1/σ

. (18)

This inequality implicitly defines a function V (β) such that

UTC > UTC⇔

w2

w1 + w0> V (β), (19)

and this function satisfies:

Z(β) < V (β) < D(β).

Proof. See Appendix B.

This lemma allows characterizing the optimal solution forw2

w1+w0∈ [Z(β),D(β)]. If w2

w1+w0> V (β), the optimal TC-solution

is such that qTC > 0. If w2w1+w0

< V (β), the optimal TC-solution issuch that qTC = 0.

3.3. Existence of the different regimes

This part provides a characterization of the existence of thedifferent regimes with respect to the parameter β . It is based ona technical lemma:

Lemma 2. • H(β) is an increasing function of β , and when βgoes from 0 to 1,H(β) goes from 0 to (δτq0)σ (φw0)

−σ (1 +

δσ τ 1−σ ) > D(1). Moreover, for every β,H(β) > Z(β).

10 The same type of analysis could be carried out with respect to anotherparameter than w0 .


Fig. 1. Self 1’s objective function with respect tom.

• G, Z, V and D are such that: ∀β ∈ (0, 1),

G(β) < Z(β) < V (β) < D(β)

and

G(1) = Z(1) = V (1) = D(1) =(δτq0)σ (φw0)

−σ

1 + qσ−10 δ2σ (φw0)1−σ

.

• G increases with β and D decreases with β .

Proof. The proof results from straightforward calculations.

This lemma allows a complete characterization of the differ-ent cases. Parameters are restricted to be such that (9) holds, orw2/(w1 + w0) < H(β). In this zone, the preceding analysis hasshown that the functions G(β) and V (β) are the pertinent fron-tiers. The set of parameters can be divided into 3 sub-zones A, Band C . The following proposition gives for each zone the cor-responding expressions of fertility and education. A numericalillustration (see Fig. 2) is provided for the following values of pa-rameters: σ = 2, τ = 0.5, φ = 0.17, δ = 1, q0 = 0.5.

Proposition 3. • The plan (β, w2/(w1+w0)) can be separated intothree zones:

Zone A = (β, w2/(w1 + w0)), w2/(w1 + w0) < G(β),Zone B = (β, w2/(w1 + w0)),G(β) < w2/(w1 + w0) <V (β),Zone C = (β, w2/(w1 + w0)), V (β) < w2/(w1 + w0).

• Assuming that parameters are such that w2/(w1 + w0) < H(β),in zone A, qC = qTC = 0 and mC

= mTC,in zone B, qTC = 0, qC > 0 and mTC > mC,in zone C, qTC > 0, qC > 0 and mTC < mC.

Proof. The proof is a direct consequence of Lemmas 1 and 2.

In zone A, the optimal behavior in both cases leads to no in-vestment in quality. When investment in quality cancels out, bothsolutions are associated with the same level of fertility.

In zone B, the time consistent solution leads to a higher levelof fertility than the commitment solution, and to no investment inchildren’s quality. If self 2 could commit on a higher investment ineducation, self 1 would invest less in quantity.

Zone C corresponds to the developed economy with a positiveinvestment in quality. The temporary consistent solution leads tolower investment in quantity. If self 2 could commit on a higherinvestment in education, self 1 would invest more in quantity.


Fig. 2. Characterization of fertility and education behaviors with respect to β and w2w1 + w0 .

For a given value of w2/(w1 + w0), it is possible that all threezones A, B and C are successively reached depending on the valueof β .

Two cases may happen. In the case w2/(w1 + w0) < G(1),zone A appears for β close to 1; zone B for β such that G(β) <w2/(w1 + w0) < V (β); zone C appears only if there exist valuesof β such that V (β) < w2/(w1 + w0) < H(β). In the casew2/(w1+w0) > G(1), only zonesB andC mayexist becauseG(β) isalways smaller thanG(1) < w2/(w1+w0). These results show thatthe impact of β on the investments in quality can be ambiguous forthe TC solution. Indeed, an increase of β has a twofold effect. For agiven level of fertility, increasing β rises the investment in quality.But an increase in β also rises the investment in quantity, whichhas a negative impact on the investment in quality.

4. Impact of the costs of fertility and education

This section studies how fertility and education behaviorsrespond to changes of w0 and τ .

4.1. Effect of w0

w0 play a crucial role in education and fertility. An increase ofw0 has a twofold impact: first it increases the opportunity costof the quantity of children; second, for a given level of fertility,it increases the first period income of the family. The first effect(effect on the price) is expected to dominate the second one(effect on the revenue), as in the standard trade-offmodel betweenconsumption and leisure. Therefore, the increase of w0 is expectedto imply a fall in fertility.

In writing Eq. (8) under the form:

mTC=

(βδ)σ A(β)(τq0)σ(w1+w0)

φw0− w2(φw0)

σ−1

τq0(φw0)σ−1 + (βδ)σ A(β)(τq0)σ

it is straightforward that mTC is a decreasing function of w0 asthe numerator is decreasing and the denominator is increasing.Using the same argument,mC given by (10) is also decreasing withrespect to w0. Finally, when education cancels out, Eq. (15) can be

written again as

mTC=

βδ2

σ qσ−10

(w1+w0)φw0

(φw0)σ−1

+βδ2

σ qσ−10

which is decreasing with w0.In all cases, fertility decreaseswith respect tow0. A change ofw0

can also result in a change of regime, and a drop in fertility. Startingfrom the fertility level without education mTC, an increase of w0implies a decrease in fertility. This change of w0 may induce sucha decrease in fertility that it becomes optimal to invest in quality.At this point, there is a discontinuity in fertility that experiences afall between mTC andmTC. In the neighborhood of the frontier valueof w0, a small increase of w0 induces a great drop in fertility. Thisjump is the consequence of the discrepancy between the objectivefunctions of self 1 and self 2. Fig. 3 gives a numerical illustrationfor the following values of parameters: σ = 2, τ = 0.5, φ =

0.17, β = 0.5, δ = 1, w2 = 2, q0 = 0.5, w1 = 1. Parameters aresuch that for w0 = 0.951856 there is a discontinuity in fertility.

The frontiers between the different regimes can be character-ized with respect to w0. They cannot be deduced from Fig. 2, asthe different functions H, V and G depend on w0. The character-ization is made in the plan (w0, w1). As before, parameters areconstrained in such away thatmTC > 0, which corresponds to con-dition (9). This constraint defines in the plan (w0, w1) a zone suchthat w1 > WH(w0), with WH a function defined in Appendix C.The same method is used for condition (13): a function WG(w0)is introduced, such that the condition holds iff w1 < WG(w0). Fi-nally, the functionW V (w0) is introduced, such that condition (19)holds iff w1 < W V (w0). The three functions WH(w0),WG(w0)and W V (w0) allow us to obtain a characterization of the differentregimes in the plan (w0, w1). This characterization is equivalent tothe one given in Section 3.3 in the plan

β,

w2w1+w0

, but it shows

the role of w0 in the existence of the different regimes.The following proposition gives the complete characterizations

of the different regimes in the plan (w0, w1), using the frontiersdefined by the three functionsWH(w0),WG(w0) andW V (w0).

Proposition 4. • It is possible to define three functions WH(w0),WG(w0) and W V (w0) that are non-decreasing functions of w0,and for all w0,WG(w0) > W V (w0).


Fig. 3. Fertility mwith respect to the opportunity cost of parents w0 .

• The plan (w0, w1) can be separated in three zones:Zone A =

(w0, w1) , w1 > WG(w0)

,

Zone B =(w0, w1) ,W V (w0) < w1 < WG(w0)

,

Zone C =(w0, w1) , w1 < W V (w0)

.

• Assuming that parameters are such that w1 > WH(w0),in zone A, qC = qTC = 0 and mC

= mTC,in zone B, qTC = 0, qC > 0 and mTC > mC,in zone C, qTC > 0, qC > 0 and mTC < mC.

Proof. See Appendix C.

Zone A is obtained for a low value of w0, qC = qTC = 0 andmC

= mTC. As the opportunity cost of children is small, fertility ishigh and parents do not invest in quality.

In zone B, qTC = 0 but qC > 0 and mTC > mC. For an intermedi-ate value of w0, the TC behavior leads to no investment in quality,whereas parents invest in quality along the commitment solution.Fertility is lower for the commitment solution.

In zone C, qTC and qC are both positive andmTC < mC. For a highvalue of w0, the TC and C solutions are associated with a positiveinvestment in quality, and fertility is higher for the commitmentsolution.

A consequence of these results for the TC behavior is that fer-tility experiences a strong discontinuity for w0 =

W V

−1(w1) ≡

wl0. In the neighborhood of this value wl

0, a small increase of w0leads to a large drop in fertility.

Fig. 4 shows a numerical simulation of the different zones in theplan (w0, w1), for the same values of parameters as Fig. 3. Forw1 =

1, fertility experiences a discontinuity at the valuew0 = 0.951856.The discontinuity in the optimal strategy of self 1 is a particu-

lar feature of the model with quasi-hyperbolic discounting. In themodel with exponential discounting, a change in the value of someparameter results in a continuous effect on the choices of the agent.In themodelwith quasi-hyperbolic discounting, it is possible to ob-serve jumps that are related to the non-concavity of self one’s ob-jective. This property introduces a qualitative difference in the twomodels that may have important empirical consequences. If the

model with quasi-hyperbolic discounting is relevant, fertility be-haviors may undergo large changes for some critical values of theparameters. Thismay have consequences for the empirical analysisof fertility and for the dynamics of demographic transitions.

4.2. Effect of τ

The parameter τ is the cost of education. An increase of τchanges the optimal trade-off between quality and quantity. Thefollowing proposition summarizes the effect of τ on fertility andeducation.

Proposition 5. • mC increases when the cost of education τ in-creases, and qC decreases.

• If σ is small enough, i.e. σ ≤ 1/(1 − β),mTC increases whenthe cost of education τ increases, and qTC(mTC) decreases. If σ >1/(1−β),mTC increases with τ when τ is small (τ in a neighbor-hood of 0). But, numerical simulations show that mTC can be a nonmonotonic function of τ (see Fig. 6).

• There exists a threshold σ , with σ > 1/(1 − β), such that if σ <σ , qTC(mTC) decreases with τ .

Proof. See Appendix D.

For the commitment solution, an increase of τ reduces theinvestment in education, and increases fertility. This result isstandard in the basic model of quantity–quality trade-off. For agiven level of fertility, an increase of τ reduces the investment ineducation q. As q is lower, the cost of fertility decreases and fertilityincreases. For the TC-solution, the same effect is obtained for asmall value of σ . But if σ is very high, τ can have a non monotoniceffect.

As for w0, there exists a threshold level τ l such that educationcancels out for τ > τ l. At this point τ l, fertility undergoes a largedrop to a higher value, as education falls to zero. Figs. 5 and 6show how fertility may evolve with respect to τ . Fig. 5 uses thepreceding values for the different parameters: σ = 2, φ = 0.17,


Fig. 4. Characterization of fertility and education behaviors with respect to w0 and w1 .

Fig. 5. Fertilitymwith respect to the cost of education τ for σ = 2.

β = 0.5, δ = 1, q0 = 0.5, w0 = 1, w1 = 1, w2 = 2. The thresh-old level from which education cancels out is τ l

= 0.516066.Fig. 6 is an example of parameters leading to a non monotonic

evolution of fertility for a high value of σ . σ = 4 > 1/(1 − β) =

2, φ = 0.28, β = 0.5, δ = 1, q0 = 1.5, w0 = 1, w1 = 1, w2 = 2.The threshold level is τ l

= 0.341284. σ can be interpreted as theelasticity of substitution between consumptions. The value of σ isa controversial question in economics. A value close to one corre-sponds to the one retained in calibrated macroeconomic models.Microeconomic estimations often conclude to a smaller value. Avalue σ > 1/(1 − β) does not seem to be the most relevant as-sumption. Therefore, a nonmonotonic evolution ofmwith respectto τ is not very plausible.

An increase of τ also influences the existence of the differentregimes. This question is studied in the plan (β, w2/(w1 + w0)),

considering how the different frontiers G(β) and V (β) aremodified. For β given, it is straightforward from (14) that G(β) isan increasing function of τ . As could be expected, the region A inwhich no education occurs increases with τ . Consequently there isless space for regions B and C . The frontier between regions B andC is defined with the function V (β). Appendix E shows that V (β)increaseswith τ . A numerical experiment is provided in Fig. 7, withthe following parameters σ = 2, φ = 0.17, δ = 1, q0 = 0.5, τ =

0.5 and τ = 0.7. The case τ = 0.7 is presented with bold lines.

5. Conclusion

This paper has studied the quantity–quality fertility model un-der the assumption of quasi-hyperbolic discounting. The impactof the absence of self-control is isolated through the comparison


Fig. 6. Fertility mwith respect to the cost of education τ for σ = 4.

Fig. 7. Characterization of fertility and education behaviors for an increase of τ .

between the TC solution (sophisticated behavior) and the C solu-tion (commitment solution). The lack of self-control may have adifferent impact on fertility in a developed economy and in a de-veloping one. In a developed economy characterized by a positiveinvestment in quality, the lack of self-control tends to reduce fer-tility. In a developing economy, the lack of self-control may lead toboth no investment in quality and a higher fertility rate. It is alsoproved that if parents cannot commit on their investment in qual-ity, a small change of parametersmay lead to a jump in fertility andeducation.

This paper could be extended in different directions. First,the robustness of the results could be studied if the model wasenriched by additional assumptions: access to capital marketsfor the households, imperfect capital markets through borrowingconstraints, collective choice within the household, etc. Secondly,a technical improvement could be made by introducing more thanthree periods and more than two decisions.

Appendix A

The comparison betweenmTC andmC is made in the text. In thecase σ < 1, the comparison between qC(mC) and qTC(mTC) is sim-ple and is made in the text. It remains to compare qC(mC) and qTC

(mTC) when σ > 1.First, it appears that:

qTC(mTC) + qo =

(βδ/τ)σ

w2mTC + τq0

1 + (βδ)σ τ 1−σ

and

qC(mC) + qo =

(δ/τ)σ

w2mC + τq0

1 + (δ)σ τ 1−σ

.


From (8), we obtain:

w2

mTC+ τq0 =

(βδ)σ A(β)B [τq0 (w1 + w0) + w2φw0](βδ)σ A(β)B (w1 + w0) − w2

. (20)

From (10), we obtain:

w2

mC+ τq0 =

(βδ)σ A(1)B [τq0 (w1 + w0) + w2φw0](βδ)σ A(1)B (w1 + w0) − w2

. (21)

As A(1) = 1 + δσ τ 1−σ , it follows:

qTC(mTC) + qo < qC(mC) + qo ⇔

(β)σ

1 + (βδ)σ τ 1−σ

A(β)

(βδ)σ A(β)B (w1 + w0) − w2

<1

(βδ)σ A(1)B (w1 + w0) − w2. (22)

After rearranging and using the expression (6) of A(β), it is possibleto write this inequality:

0 < B (w1 + w0) δσ− w2f (β) (23)

with

f (β) ≡

1+δσ βσ τ1−σ

β+δσ βσ τ1−σ

σ

− 1

1 − βσ.

Firstly the inequality (23) is studied in a neighborhood ofβ = 1.In setting x = βσ , a function g is introduced such that:

g(x) ≡

1+xδσ τ1−σ

x1/σ +xδσ τ1−σ

σ

− 1

1 − x= f (β).

The limit of g when x tends toward 1 is equal to the limit of f inβ = 1. Defining a function h(x) such that:

h(x) ≡

1 + xδσ τ 1−σ

x1/σ + xδσ τ 1−σ

σ

this limit is equal to −h′(1). Taking the derivative of the logarithmof h in x = 1, we obtain:

h′(1) = h′(1)/h(1) =σδσ τ 1−σ

1 + δσ τ 1−σ−

1 + σδσ τ 1−σ

1 + δσ τ 1−σ

= −1

1 + δσ τ 1−σ.

Thus, with β = 1, (23) becomes:

0 < B (w1 + w0) δσ−

w2

1 + δσ τ 1−σ

which is satisfied as it corresponds to (11) with β = 1. It is thenproved that qTC < qC in a neighborhood of β = 1.

Secondly, the inequality (23) is studied for a low value of β .When β tends toward 0, f (β) tends to be infinite, the inequality(23) cannot be satisfied, and qTC > qC. β close to 0 is not possibleas it implies negative values for mTC and mC. The smallest possi-ble value of β corresponds to the constraint (9) ensuring mTC > 0.When β tends to this value, the left-hand side of (22) tends to beinfinite. Thus, when β is low enough, (23) cannot be satisfied, andqTC > qC.

Appendix B

In this appendix, a new notation x is introduced for the expres-sion w2

w1+w0. Assume that Z(β) < x < D(β). The equality UTC

=

UTC implicitly defines x as a function of β: f (x, β) = 0 with

f (x, β) ≡σ − 1

σ

UTC

− UTC 1

(w1 + w0)1−1/σ

=

xφw0

τq0+ 1

1−1/σ1 +

φw0

τq0

1−σ

(βδ)σ A(β)

1/σ

−

1 +

φw0

q0

1−σ βδ2σ

1/σ

− βδx1−1/σ .

First, it is proved that ∂ f /∂x > 0. The condition ∂ f /∂x > 0 isequivalent to:

φw0

τq0

xφw0

τq0+ 1

−1/σ1 +

φw0

τq0

1−σ

(βδ)σ A(β)

1/σ

> βδx−1/σ

which is equivalent to

x >1

φw0τq0

σ

(βδ)−σ+

φw0τq0

[A(β) − 1]≡ Ω(β).

From this inequality, as by assumption x > Z(β), if Z(β) > Ω(β),the property x > Ω(β) will be satisfied and ∂ f /∂x > 0.

Z(β) > Ω(β) is equivalent to:

φw0

τq0

A(β) − 1 − (βδ)σ τ 1−σ

+

φw0

τq0

σ

(βδ)−σ

1 −

1 + (βδ)σ τ 1−σ

A(β)

> 0.

From the definition of A(β), A(β) > 1 + (βδ)σ τ 1−σ⇔ 1 +

βσ−1δσ τ 1−σ > 1 + (βδ)σ τ 1−σ which is true for β < 1.Finally the property ∂ f /∂x > 0 is proved.The next step is to prove that f (Z(β), β) < 0 and f (D(β), β) >

0. These two inequalities with the property ∂ f /∂x > 0 will ensurethe existence and uniqueness of x as a function V (β) of β .

After tedious calculations, it is possible to write f (Z(β), β) < 0under the form

1 +

φw0

q0

1−σ βδ2σ

β + δσ βσ τ 1−σ

1 + δσ βσ τ 1−σ

σ−1

<

1 +

φw0

q0

1−σ βδ2σ

1/σ

×

1 +

φw0

q0

1−σ βδ2σ



σ1−1/σ

.

The following notations are introduced:

a =

φw0

q0

1−σ βδ2σ

y(β) =β + δσ βσ τ 1−σ

1 + δσ βσ τ 1−σ.

It is possible to write the preceding inequality:1 + ay(β)σ−1

σ

[1 + ay(β)σ ]σ−1 < 1 + a. (24)


The function y(β) increases from 0 toward 1 when β goes from 0to 1.

As a function of y, the expression1 + ayσ−1

σ

[1 + ayσ ]σ−1

is strictly increasing when y goes from 0 to 1, and is equal to 1 + afor y = 1.

By these two properties, it is proved that (24) is satisfied.The condition f (D(β), β) > 0 can be written after some calcu-

lations1 +

φw0

τq0

1−σ

δσ1 + δσ βσ τ 1−σ

1−1/σ

×

1 +

φw0

τq0

1−σ

δσ


σ1 + δσ βσ τ 1−σ

σ−1

1/σ

> 1 +

φw0

τq0

1−σ

δσβ + δσ βσ τ 1−σ

.

The following notations are introduced:

b =

φw0

τq0

1−σ

δσ

ξ = 1 + δσ βσ τ 1−σ

θ = β + δσ βσ τ 1−σ .

By definition, θ ≤ ξ with a strict inequality for β < 1. It is possibleto write the preceding inequality:

1 + bθσ

ξσ−1>

(1 + bθ)σ

(1 + bξ)σ−1 . (25)

Defining the function g:

g(θ) = 1 + bθσ

ξσ−1−

(1 + bθ)σ

(1 + bξ)σ−1

it is easy to check that it is strictly decreasing for θ ∈ [0, ξ ], withg(ξ) = 0. Therefore, it is proved that (25) is satisfied, and f (D(β),β) > 0.

Appendix C

Condition (13) can be written under a condition on w0 and w1.The inequality w2

w1+w0> G(β) is equivalent to:

w1 <

φδσ τ−σ w2

q0− 1

w0 +

w2(φw0)σ

(τq0)σ (βδ)σ≡ Γ (w0).

The right-hand sidemember of this inequality is a functionΓ ofw0

such that: if φδσ τ−σ w2q0

> 1, Γ is strictly increasing; if φδσ τ−σ w2q0

<

1, Γ is U-shaped, first decreasing and then increasing. As w1 can-not be negative, the negative part of Γ does not play any role. Thefunction WG is defined as

WG(w0) = max Γ (w0), 0 .

By definition, eitherWG(w0) is strictly increasing, or it is first equalto 0, and then strictly increasing.

For condition (12), the inequality w2w1+w0

> Z(β) is equivalentto:

w1 <

φβσ δσ τ−σ w2

q0− 1

w0

+w2(φw0)

σ

(τq0)σ δσ

1 + βσ δσ τ 1−σ

β + βσ δσ τ 1−σ

σ

≡ ϑ(w0).

As for the preceding example, a functionW Z is defined as

W Z (w0) = max ϑ(w0), 0 .

By definition, W Z (0) = 0, either W Z (w0) is strictly increasing, orit is first equal to 0, and then strictly increasing.

For condition (16), the inequality w2w1+w0

< D(β) is equivalentto:

w1 >

φβσ δσ τ−σ w2

q0− 1

w0 +

w2(φw0)σ

(τq0)σ δσ≡ ∆(w0).

As for the preceding examples, a functionWD is defined as

WD(w0) = max ∆(w0), 0 .

By definition, WD(0) = 0, either W Z (w0) is strictly increasing, orit is first equal to 0, and then strictly increasing.

Finally condition (9) can be written:

w1 >w2(φw0)

σ

(τq0)σ (βδ)σ


σ−11 + δσ βσ−1τ 1−σ

σ − w0 ≡ Ξ(w0).

As for the preceding examples, a functionWH is defined as

WH(w0) = max Ξ(w0), 0 .

By definition, WH(w0) is first equal to 0, and then strictlyincreasing.

Lemma 1 with Appendix B allow to define a function V (β) suchthat w2

w1+w0= V (β) ⇔ UTC

= UTC. This function V (β) dependson different parameters of the model including w0, but does notdepend on w1. Therefore, it is clear that it can be expressed underthe form:

w1 =w2

V (β)− w0.

A function W V is defined as:

W V (w0) = max

w2

V (β)− w0, 0

.

As Z(β) < V (β) < D(β), it implies that: WD(w0) < W V (w0) <W Z (w0).

To find how W V (w0) evolves with w0, it is useful to come backto the definition. When w1 = W V (w0) is positive, the function isimplicitly defined by the relation UTC

− UTC= 0. The derivative is

implicitly given by:

dW V (w0)

dw0= −

∂(UTC−UTC)

∂w0

∂(UTC−UTC)∂w1

.

If ∂(UTC−UTC)

∂w0= 0, it will prove that W V (w0) is monotonic. As

WD(w0) < W V (w0) < W Z (w0), with WD(w0) and W Z (w0) twoincreasing functions tending to +∞ when w0 → +∞, the onlypossibility will be thatW V (w0) is monotonically increasing.

It is possible to prove that ∂(UTC−UTC)

∂w0> 0. UTC is the maximum

value of self 1’s objective function when qTC(m) > 0. UTC is themaximum value of self 1’s objective function when qTC(m) = 0.The derivatives can be obtained using the envelope theorem:

∂UTC

∂w0= (1 − φmTC)

w1 + w0(1 − φmTC)

−1σ

∂UTC

∂w0= (1 − φmTC)

w1 + w0(1 − φmTC)

−1σ .

As σ > 1, the function x [w1 + w0x]−1σ is an increasing function of

x, as its logarithmic derivative isσw1 + (σ − 1)w0xx (w1 + w0x) σ

.


Consequently, the function (1 − φm) [w1 + w0(1 − φm)]−1σ is a

decreasing function ofm. As mTC > mTC, it is obtained that,

∂UTC

∂w0>

∂UTC

∂w0

which implies that the functionW V (w0) is increasing.

Appendix D

From (10),mC can be written:

mC=

x(τ ) (w1 + w0) −w2τq0

1 + φw0x(τ )

with x(τ ) ≡ (βδ)σ qσ−10 (φw0)

−σ (τ σ−1+δs). Taking the derivative

ofmC with respect to τ , it is obtained that the sign of this derivativeis the sign of the expression:x′(τ ) (w1 + w0) +

w2

τ 2q0

[1 + φw0x(τ )]

−

x(τ ) (w1 + w0) −

w2

τq0

φw0x′(τ )

= x′(τ )

(w1 + w0) + φw0

w2

τq0

+

w2

τ 2q0[1 + φw0x(τ )] > 0.

Thus,mC is an increasing function of τ .From (5), the quality level q is such that:

q0τ σ+ q(τ σ

+ δσ τ) = δσ w2

mC.

If τ increases, asmC increases, qmust decrease.From (8), the time-consistent solution can be written:

mTC=

y(τ ) (w1 + w0) −w2τq0

1 + φw0y(τ )

with y(τ ) ≡ (βδ)σ qσ−10 (φw0)

−σ τ σ−1A(β). If y′(τ ) > 0, it isknown from the preceding calculation that mTC increases with τ ,as the sign of ∂mTC/∂τ is given by

y′(τ )

(w1 + w0) + φw0

w2

τq0

+

w2

τ 2q0[1 + φw0y(τ )] > 0.

Therefore, it remains to check if τ σ−1A(β) increases with τ . Aftersome calculations, it is obtained that

∂y(τ )/∂τ

y(τ )=

d lnτ σ−1A(β)

dτ

= (σ − 1)1 − [σ(1 − β) − 1] δσ βσ−1τ 1−σ

τ1 + δσ βσ−1τ 1−σ


. (26)

If σ(1−β) < 1, y′(τ ) > 0 andmTC increases with τ . If σ(1−β) >1, it is not possible to achieve a general conclusion. For τ smallenough (τ → 0), it is possible to show from (26) that

y′(τ )

(w1 + w0) + φw0

w2

τq0

+

w2

τ 2q0[1 + φw0y(τ )] ∼

τ→0τ−2

a − bτ σ−1with a and b two positive constant parameters. Therefore, mTC

increases with τ when τ is small.Assuming that mTC increases with τ , from (4), the quality level

q is such that:

q0τ σ+ q(τ σ

+ (βδ)σ τ) = (βδ)σw2

mTC.

If τ increases, asmTC increases, qmust decrease.

Considering now qTC(mTC), Appendix A has shown that

qTC(mTC) + qo =

(βδ/τ)σ

w2mTC + τq0

1 + (βδ)σ τ 1−σ

or

qTC(mTC) + qo =(βδ)σ [τq0 (w1 + w0) + w2φw0]

τ σ + (βδ)σ τ

×(βδ)σ A(β)B

(βδ)σ A(β)B (w1 + w0) − w2.

It is easy to check that the first term(βδ)σ [τq0 (w1 + w0) + w2φw0]

τ σ + (βδ)σ τ

=

(βδ)σq0 (w1 + w0) +

w2φw0τ

τ σ−1 + (βδ)σ

is a decreasing function of τ as σ > 1.Defining z(τ ) = (βδ)σ A(β)B, the second term can be written

z(τ )

z(τ ) (w1 + w0) − w2.

This is a decreasing functionwith respect to z(τ ). Therefore, if z(τ )increases with τ , it will be obtained that qTC(mTC) is a decreasingfunction of τ .

z(τ ) can be written:

z(τ ) = (βδ)σ

q0φw0

στ σ

+ δσ βσ−1τσ

(τ σ + δσ βσ τ)σ−1 .

The sign of z ′(τ )/z(τ ) is given by the sign of:τ 2σ−1

+ τ σ δσ βσ−1 −σ 2 (1 − β) + (2 − β) σ + β

+ δ2σ β2σ−1.

A sufficient condition to have z ′(τ ) > 0 is that −σ 2 (1 − β) +

(2 − β) σ + β > 0. Considering this second degree equation in σ ,the property z ′(τ ) > 0 will hold if

σ < σ ≡2 − β +

4 − 3β2

2(1 − β).

It is easy to check that σ > 1/(1 − β) as it is equivalent to β < 1.

Appendix E

This appendix proves that V (β) increases with τ . This functionhas been defined in Appendix B as the solution x implicitly definedby the equation: f (x, β) = 0 with x =

w2w1+w0

and

f (x, β) =σ − 1

σ

UTC

− UTC 1

(w1 + w0)1−1/σ.

From this definition, it appears that

∂V (β)

∂τ= −

∂ f∂τ

∂ f∂x

.

In Appendix B, it was shown that ∂ f∂x > 0. It remains to prove that

∂ f∂τ

< 0, which is equivalent to proving that: ∂(UTC−UTC)

∂τ< 0. UTC is

the maximum value of self 1’s objective function when qTC(m) >

0. UTC is the maximum value of self 1’s objective function whenqTC(m) = 0. The derivatives can be obtained using the envelopetheorem:∂UTC

∂w0= −mTCqTCβδ

w2 − τmTCqTC

−1σ

∂UTC

∂w0= 0.

Therefore, ∂(UTC−UTC)

∂τ< 0 and V (β) is an increasing function of τ .


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