non-cooperative versus cooperative family

J EconDOI 10.1007/s00712-014-0391-7

Non-cooperative versus cooperative family

Atsue Mizushima · Koichi Futagami

Received: 9 October 2012 / Accepted: 19 January 2014© Springer-Verlag Wien 2014

Abstract This paper focuses on strategic interaction within a family and examinesindividual decision making. We set up a two-stage game model. In the first stage ofthe game, a man and a woman who have not yet met simultaneously determine theireducation levels non-cooperatively. In the second stage, they marry and determine theirleisure time. In the second stage, we compare two decision modes, non-cooperativeand cooperative, in order to characterize the nature of cooperation within the families.In addition, we extend the basic model on the basis of a Stackelberg game. In thissetting, we consider the case in which a man acts as a leader and a woman acts as afollower. We show that the leader invests in higher education and chooses more leisuretime than the follower. This coincides with the empirical findings.

Keywords Household public goods · Cooperative game · Non-cooperative game ·Stackelberg game · Time allocation · Strategic complementarity

JEL Classification D13 · J24

1 Introduction

In traditional economic models, an individual chooses consumption and labor supplyto maximize his or her utility subject to his or her budget constraint. To reconcile

A. Mizushima (B)Faculty of Economics, Otaru University of Commerce, 3-5-21 Midori, Otaru,Hokkaido 047-8501, Japane-mail: [email protected]

K. FutagamiGraduate School of Economics, Osaka University,1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japane-mail: [email protected]

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A. Mizushima, K. Futagami

this individualistic theory of the consumer with the reality of individuals’ life in afamily setting, Samuelson (1956) and Becker (1981) created seminal models, knownas unitary models in their respective economic research on the family. They assumethat a family acts as a unit and has a single unified set of interests.1 Following theseseminal works, a considerable number of studies have been done on decision makingwithin the family. The literature on family decision making starts from the assumptionthat a family consists of two individuals who make their own decisions in the family,each according to his/her own utility. Thus, the decision mode in these models can becooperative or non-cooperative.

In non-cooperative approaches, Konrad and Lommerud (1995, 2000) and Leuthold(1968) consider the level of household public goods to be determined by the Nashequilibrium of a non-cooperative game. In the cooperative approaches, Manser andBrown (1980) and McElroy and Horney (1981) set up a model with family demandsas the solution of a cooperative bargaining game. In the cooperative bargaining game,individuals pursue their common interests given their relative bargaining positionsinside the household; these are affected by outside opportunities, the institutionalenvironment, and spouses’ relative characteristics. The collective model (Chiappori1988, 1992; Browning and Chiappori 1998) also involves a cooperative approach. Itassumes that the decisions are made such that the outcomes are Pareto efficient, andis more general than the cooperative bargaining model.2

In the analysis of family economics, many studies focus on the gains from marriagearising from the sharing of public goods. This implies the division of labor to exploitcomparative advantage. Thus, many studies have focused on substitutability in theprovision of household public goods (see, for example, Lundberg and Pollak 1996;Apps and Rees 2009), but little attention has been given to complementarity in theprovision of household public goods in the context of family economics.

In reality, many data show the synchronization of leisure activities between part-ners. For example, Sullivan (1996) uses the 1985 UK time-use survey, a diary surveywith instant enjoyment information, and finds that partners report higher levels ofsatisfaction when they synchronize their working schedules, thus maximizing thepotential time they can spend in leisure activities together. Hamermesh (1999), Hall-berg (2003), and Jenkins and Osberg (2005) find that the synchronization of leisureactivities between partners is greater than random male-female pairing would predict.This example makes it clear that married couples enjoy a higher utility from leisurespent together than leisure individually or with another person. Using the AmericanTime Use Survey, Connelly and Kimmel (2009) find that the husband’s and wife’sleisure hours appear to have a complementarity relationship.

Given these findings, in contrast to the stylized model, we set up a simple modelwith the strategic complementarities that arise in the organization of social leisure ina household production model following Becker (1965) and Corneo (2005); that is, inour model, leisure is an input in the production of the public good “time together.”

1 Such models are sometimes called common preference models or altruism models.2 See Bergstrom (1997), Lundberg and Pollak (2007), and Apps and Rees (2009) for surveys.

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In the present analysis, we employ the dynamic aspect of family decision makingdeveloped by Chiappori et al. (2009) and Konrad and Lommerud (2000) because inthe real world, a man and a woman make several decisions before getting married.

Specifically, we set up a two-stage game model in which a man and a woman simul-taneously decide their education levels in the first stage. After each agent completeshis/her education level, the man and the woman meet and become a couple. In the sec-ond stage, the husband/wife decides his/her contribution to household public goods,taking his/her education level in the first stage into account.

In our analytical model, we consider two cases for the equilibrium in the secondstage: the non-cooperative equilibrium and the cooperative one. In the non-cooperativegame, we apply the game of Konrad and Lommerud (1995, 2000) and Leuthold (1968)into our complementarity model and derive the provision of household public goodsas a solution of a Nash equilibrium. In the cooperative game, we assume that a cou-ple maximizes its joint utility function. It assumes away any influence of bargainingpower–for instance, the role of the wife’s labor force participation in bargaining. Weassume that the two spouses are equally important, and nothing is modified even ifwe take into account their relative Pareto weight. Thus, our model does not comparea general cooperative model to the non-cooperative model. The purpose in examin-ing the non-cooperative equilibrium is to characterize the cooperative equilibrium.It is important to understand the characteristics of decision modes within the familybecause of the appropriate design of the policy.

We firstly show that the leisure provision of the husband or wife depends on his orher wage rate, and it decreases with an increase in the wage rate for a given level ofthe spouse’s wage rate. This coincides with the following empirical facts. By usingthe eight waves of the European Community Household Panel-ECHP (1994–2000),García, Molina, and Navarro (2007) report that highly educated individuals have lessleisure time (or satisfaction levels) than low educated individuals. Costa (2000) alsoshows that low-wage workers reduced their market work hours relative to high-wageworkers between the 1890s and 1991.

We next show that the non-cooperative equilibrium results in overinvestment ineducation. We attempt to determine a policy to alleviate this inefficiency by applyinga labor law. Reduced working time is a challenge for our policy design, because thiswas one of the original objectives of labor law.3

The basic model also extends to a Stackelberg game. In the Stackelberg model, weshow that a leader can earn a higher wage and enjoy more leisure time. In the realworld, the census data on the gender wage gap show that men earn higher wages thanwomen in many countries. For example, in 2010, the wage level of a woman was 70.6 %of that of a man in Japan (Ministry of Health, Labour and Welfare of Japan 2011),81.2 % in the U.S. (United States Department of Labor 2011), 80.6 % in the U.K.(Office for National Statistics 2011), 81.6 % in Germany (Statistisches Bundesamt

3 According to ILO (1967), the primary technique towards achieving the reduction of working hours,mandating of limits on the hours that can be worked in each day or week, was first reflected in laws enactedin European countries in the mid-nineteenth century to reduce the working hours of children. In the mid-twentieth century, two standards were available for limiting weekly working hours, the 48-h limit of theearliest international instruments and the more recent objective of the 40-h week (ILO 2007).

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(2011)), 85.4 % in France (Ministry of the Economy, Finance, and Industry 2011),and 86.0 % in Sweden (Statistics Sweden 2011). In addition, the empirical studies ofO’Neill (2003), Weichselbaumer and Winter-Ebmer (2005), and Del Bono and Vuri(2011) identify wage gaps between couples. Further, Beblo and Robledo (2008) usethe German Socio-Economic Panel (GSOEP) and show that those who earn higherwages enjoy more leisure time than those who earn lower wages. Thus, we set thehusband as the leader of the Stackelberg model.

The remainder of this paper is organized as follows. Section 2 is divided into threesubsections: Subsect. 2.1 analyzes the second-stage outcomes, Subsect. 2.2 derivesthe education choices in the first stage, and Subsect. 2.3 examines policy. Section 3extends the basic game by using a Stackelberg game framework. Section 4 concludesthe paper.

2 The model

In this section, we develop a two-stage game that considers the strategic interactionwithin the family. For this purpose, we consider a family consisting of two agents(a husband and a wife) denoted by i = m, f . Each agent has the following payofffunction:

ui = xi + Li (l f , lm) − h(ei ), i ∈ { f, m} and i �= j. (1)

Here, Li (l f , lm) is the household production function. In this paper, we assume that itis the payoff from family leisure and that it is produced using the time of each agent:

Li (l f , lm) = li − δl2i + γ li l j , i ∈ { f, m} and i �= j, (2)

where li and l j are the leisure time of the agents, i, j = m, f .δ and γ show the marginal productivity of household production and take positive

values. Including the empirical findings of Connelly and Kimmel (2009) in our ana-lytical model, we assume that γ takes a positive value, which implies that couples’leisure time has a complementarity relationship. This corresponds to the followingsituation: traveling somewhere with her/his spouse increases an individual’s utilityabove that from traveling alone. The assumption of the household production functionimplies that a couple does household production (childcare, housework, etc.) togetherand does not allow any income sharing or transfer from one spouse to the other, noris any bargaining position variable allowed to affect the household decisions.

In addition, h(ei ) shows the effort to obtain education. It is specified as h(ei ) =βe2

i /2, where β > 0, i = f, m. For analytical simplicity, we assume that the wage isa function of education levels, ei = wi , i = f, m, to be determined in Subsect. 2.2.If an agent makes an effort to obtain an education, the education level of the agentincreases, which results in a higher wage rate.

Furthermore, xi is the individual’s consumption of private goods and is determinedas follows:

xi = wi (y − li ), i = f, m,

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where wi is the labor market wage and y is the time that can be used for work. The wageis a function of education levels, to be determined in Subsect. 2.2. Agent i allocatesher/his endowed time between working y − li and leisure li .

The model with strategic interaction within the family is formulated as a two-stagegame. We consider the situation in which a woman and a man simultaneously decidetheir education levels in the first stage. After each agent completes her/his educationlevels, the woman and the man meet and become a couple. It follows that the educationdecision is made non-cooperatively. In the second stage, each agent decides her/histime contribution to the household production function,4 taking her/his educationlevel in the first stage into account. We examine two types of game in the secondstage. One is the non-cooperative game in which each agent maximizes her/his ownutility function. The other is the cooperative game in which each agent maximizes thejoint utility function. In the cooperative game, the outcome becomes efficient. Thus,in what follows, we refer to the outcome as an efficient outcome.

2.1 Second-stage outcomes

We derive the subgame perfect Nash equilibrium by backward induction.First, let us examine the equilibrium leisure time in the non-cooperative game in the

first stage. In the game, the wife maximizes her own utility (1) given the leisure time ofher husband, and vice versa. This equilibrium must satisfy the first-order conditions:

�i (li , l j ;wi ) ≡ −wi + 1 − 2δli + γ l j = 0, i, j ∈ { f, m} and i �= j; (3)

the second-order conditions are satisfied as follows:

�i1 = −2δ < 0, i, j ∈ { f, m} and i �= j,

where �i denotes the left-hand side of (3) and �i1 ≡ ∂�i/∂li . The right-hand side of

Eq. (3) defines the reaction function of each agent as follows:

li = Ri (l j ;wi ) = 1 − wi + γ l j

2δ, i, j ∈ { f, m} and i �= j. (4)

For the interior solution, we assume that wi < 1, i = f, m. We can calculate theslope of the reaction function as follows:

∂li∂l j

= γ

2δ> 0, i, j ∈ { f, m} and i �= j.

We make the following assumption for the stability of the equilibrium. When γ > 0,the reaction curve becomes upward sloping. It shows the strategic complementaritybetween the agents; that is, when the wife increases her leisure time, the husband alsoincreases his leisure time, and vice versa.

4 In what follows, we refer to the time contribution to household production as leisure time.

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Assumption 1 γ < δ.

This assumption states that when the husband/wife has an increase of one unit ofleisure time, the increase in the leisure time of his/her spouse is smaller than one, andvice versa.

Solving (3), we have the second stage non-cooperative Nash equilibrium as follows:

l∗f = 2δ(1 − w f ) + γ (1 − wm)

(2δ)2 − γ 2 , (5)

l∗m = 2δ(1 − wm) + γ (1 − w f )

(2δ)2 − γ 2 . (6)

The second stage Nash equilibrium depends on the level of education that eachagent chooses in the first stage. Thus, we have the following proposition.

Proposition 1 When the education level of each agent increases, the equilibrium levelshifts downward.

The intuition behind this result is as follows. First, an increase in education levelraises the opportunity cost of leisure; thus, each agent decreases her/his leisure time.Second, when the strategy within the family involves strategic complementarity, adecrease in the leisure time of the wife leads to a decrease in the leisure time of thehusband, and vice versa. Therefore, when the wife decreases her leisure time becauseof an increase in her education level, the husband also decreases his leisure time, andvice versa.

Second, let us examine the equilibrium leisure time in the cooperative game. In thegame, we define social welfare as the joint utility function SW = u f +um . We assumeaway any influence of bargaining power–for instance, the role of female labor forceparticipation in bargaining. We consider the case where the two spouses are equallyimportant and nothing is modified by taking into account any relative weight in theirobjections. Because the outcome of the social welfare function is Pareto-optimal, werefer to the outcome as efficient. The efficient allocation of leisure time must satisfythe following first-order condition:

�e,i (li , l j ;wi ) ≡ −wi + 1 − 2δli + 2γ l j = 0, i, j ∈ { f, m} and i �= j; (7)

the second-order condition is satisfied as follows:

�e,i1 (li , l j ;wi ) = −2δ < 0, i, j ∈ { f, m} and i �= j,

where �e,i denotes the left-hand side of (7) and �e,i1 ≡ ∂�e,i/∂wi . From (7), we have

the following reaction function of each agent in the efficient outcome:

li = Re,i (l j ;wi ) = 1 − wi + 2γ l j

2δ, i, j ∈ { f, m} and i �= j. (8)

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Using the same procedure as in the non-cooperative game, we have the cooperativeNash equilibrium as follows:

lef (w f , wm) = 2δ(1 − w f ) + 2γ (1 − wm)

(2δ)2 − (2γ )2 , (9)

lem(w f , wm) = 2δ(1 − wm) + 2γ (1 − w f )

(2δ)2 − (2γ )2 . (10)

The equilibrium also depends on the level of education that each agent chooses inthe first stage. By comparing the equilibrium of the non-cooperative game and that ofthe efficient level, we have the following proposition.

Proposition 2 For a given level of education, in the non-cooperative game, each agentchooses less leisure time than the efficient level.

Proof

l∗i −lei = −2δ(1−wi )(3γ 2)−γ (1−w j )((2δ)2+2γ 2)

[(2δ)2−γ 2][(2δ)2−(2γ )2] < 0, i, j ∈ { f, m}, and i �= j.

In the cooperative game, the optimal level of leisure satisfies the intersection pointsof the public good of society (the sum of the spouses’ marginal rate of substitution) andmarginal cost. On the other hand, in the non-cooperative situation, each spouse onlymaximizes his/her own benefit, and does not consider the social benefit. Therefore,we have the underprovision of leisure in the non-cooperative game.

2.2 Education choice

Now, we return to the first stage. In the first stage, each agent simultaneously choosesher or his optimal education level, taking the outcome in the second stage into account.The equilibrium choice also depends on whether behaviors in the second stage aredecided in the non-cooperative way or the cooperative way. We consider these twocases.

First, we consider the education level in the non-cooperative game. Taking thesecond-stage equilibrium into account, a man and a woman simultaneously maximizethe following utility function given the education level of their expected partner.

ui = wi(y − l∗i (wi , w j )

) + l∗i (wi , w j ) − δl∗i (wi , w j )2

+ γ l∗i (wi , w j )l∗j (wi , w j ) − β

2w2

i ,

i, j ∈ { f, m}, and i �= j, (11)

where l∗f (w f , wm) and l∗m(w f , wm) are given by (5) and (6), respectively. The edu-cation choice wi , i = f, m that maximizes (11) depends on w j , j = f, m, and is

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determined by the first-order condition. Using the envelope theorem,5 we have thefirst-order condition

�i ≡ y − l∗i (w f , wm) + γ l∗i (w f , wm)∂l∗j (w f , wm)

∂wi− βwi ,

= y + −(2δ)2(2δ + γ ) + (2δ)2γw j((2δ)2 − γ 2

)2 +(

(2δ)3

[(2δ)2 − γ 2]2 − β

)wi = 0

i, j ∈ { f, m}, and i �= j (12)

and the second-order condition

�i1(w f , wm) ≡ (2δ)3

[(2δ)2 − γ 2]2 − β < 0 i, j ∈ { f, m}, and i �= j,

where �i denotes the left-hand side of (12) and �i1 ≡ ∂�i (w f , wm)/∂wi .

To guarantee the second-order condition and the stability of the equilibrium, weassume the following

Assumption 2

β > y >(2δ)2(2δ + γ )

[(2δ)2 − γ 2]2 .

Assumption 2 shows that the equilibrium exists when the marginal cost of the effortto obtain education is sufficiently high. We obtain the following reaction functiondefined by �i (w f , wm) = 0:

wi = y[(2δ)2 − γ 2]2 − (2δ)2(2δ + γ ) + (2δ)2γwi

β[(2δ)2 − γ 2]2 − (2δ)3 , i, j ∈ { f, m}, and i �= j.

From (12), we obtain the first stage Nash equilibrium in the non-cooperative game asfollows:

w∗f = w∗

j = y(2δ + γ )(2δ − γ )2 − (2δ)2

β(2δ + γ )(2δ − γ )2 − (2δ)2 . (13)

Next, let us consider the efficient education level, which is derived from the outcomein the cooperative game. Taking the second stage equilibria (9) and (10) into account,a man and a woman simultaneously maximize the following social welfare functiongiven the education level of their expected partner:

5 ∂ui∂wi

= 0 ⇔ y − l∗i + (−wi + 1 − 2δl∗i + γ l∗j )︸︷︷︸

=0,From(3)

∂l∗i∂wi

+ γ l∗i∂l∗j∂wi

− βwi = 0, i, j ∈ { f, m} and i �= j.

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SW = w f(y − le

f (w f , wm)) + le

f (w f , wm) − δlef (w f , wm)2

+ γ lef (w f , wm)le

m(w f , wm) − β

2w2

f

+wm(y − le

m(w f , wm)) + le

m(w f , wm) − δlem(w f , wm)2

+ γ lef (w f , wm)le

m(w f , wm) − β

2w2

m,

where lef (w f , wm) and le

m(w f , wm) are given by (9) and (10), respectively. Using the

envelope theorem,6 we have the first-order condition

�e,i (wi , w j ) ≡ y − lei (w f , wm) − βwi = 0,

= y − 2δ(1 − wi )+2γ (1 − w j )

(2δ)2 − (2γ )2 − βwi =0, i, j ∈ { f, m}, and i �= j

(14)

and the second-order condition

�e,i1 (wi , w j ) ≡ 2δ

(2δ)2 − (2γ )2 − β < 0,

where �e,i denotes the left-hand side of (14) and �e,i1 ≡ ∂�e,i/∂wi . Assumption

2 satisfies the second-order condition. The right hand side of Eq. (14) defines thereaction function of each agent as follows:

wi = y[(2δ)2 − (2γ )2] − (2δ + 2γ ) + 2γw j

β[(2δ)2 − (2γ )2] − 2δ, i, j ∈ { f, m}, and i �= j.

Assumption 2 also ensures the stability of the equilibrium. Solving (14), we have thefirst-stage efficient equilibrium as follows:

wef = we

m = y(2δ − 2γ ) − 1

β(2δ − 2γ ) − 1. (15)

Now, let us compare the equilibrium level of education in the non-cooperative gameand the cooperative game. This will help in determining the appropriate policy. Thefollowing proposition summarizes the comparison.

Proposition 3 In the non-cooperative game, each agent chooses a higher level ofeducation than the efficient level.

6 ∂SW∂wi

= 0 ⇔ y − lei + (−wi + 1 − 2δle

i + 2γ lej )

︸︷︷︸=0,From(7)

∂lei

∂wi− βwi + (−w j + 1 − 2δle

j + 2γ lei )

︸︷︷︸=0,From(7)

∂lej

∂wi

= 0, i, j ∈ { f, m}, and i �= j.

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Proof

w∗ − we = (β − y)γ [2δ(2δ − γ ) + γ 2][β(2δ + γ )(2δ − γ )2 − (2δ)2][β(2δ − 2γ ) − 1] > 0. (16)

The intuition of the result is summarized in a strategic effect and an income effect.First, let us consider the strategic effect. By comparing the first-order condition ofeducation in the non-cooperative game (see Eq. (12)) and that in the cooperative game

(see Eq. (14)), Eq. (12) has a strategic term, γ l∗i (w f , wm)∂l∗j (w f ,wm )

∂wi, i, j = w, f ;

on the other hand, Eq. (14) does not. The strategic term indicates that the decisionof the wife affects the decision making of the husband, and vice versa; it also has anegative effect on the marginal benefit from education. Next, let us consider an incomeeffect. Since both agents in the non-cooperative game choose lower leisure time thanin the cooperative game (see Proposition 1), the marginal income from education in thenon-cooperative game is higher than that in the cooperative game. In the equilibrium,the positive income effect dominates the negative strategic effect. Thus, in the non-cooperative game, each agent has an incentive to obtain a higher education level thanthe efficient level; that is, by playing cooperatively in a family, both spouses can enjoymore leisure time.

When the relationship between the husband and the wife is a strategic substitute,that is, γ < 0, the numerator in (16) is negative; thus, we have the opposite result.Therefore, the result crucially depends on the strategic complementarity relationshipbetween agents.

2.3 Policy

When the provision of household public goods is exhibited in the complementarityrelationship, we find that overinvestment in education occurs. In this subsection, wetry to determine the policy that alleviates this inefficiency by applying a labor law.Reduced working time is a challenge for our policy design, because this was one ofthe original objectives of labor law. We have decreasing trends in the average labortime per person in the following developed countries: Australia, Canada, France, WestGermany, Italy, Japan, the U.K., the U.S., and other OECD Countries. In particular,in Japan, government-enforced national holiday law contributes to this trend.7 Byapplying these realities to our model, we examine the effect of labor law, that is,reduced working time in equilibrium.

For this purpose, we assume that the government compulsorily sets up non-working(leisure) time. It reduces the time that can be used for work from y to y′. Noting thateach individual spends li , i = f, m time in household production and the rest of thetime, y′ − li , working, the revised individual’s budget constraint is

xi = wi (y′ − li ), i = f, m.

7 Japan has 15 national holidays a year.

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Because the household production function is a component of the time supply of eachindividual (see Eq. (2)), the policy only affects the individual’s decision making in thefirst stage. By using the same procedure as in Sect. 2.2, we obtain the Nash equilibriumin the first stage as follows:

w�f = w�

m = y′(2δ + γ )(2δ − γ )2 − (2δ)2

β(2δ + γ )(2δ − γ )2 − (2δ)2 .

The economy without the policy results in overinvestment in education (see Proposi-tion 3). Therefore, the labor law, which reduces working time, is Pareto improving.

3 Extension

In this section, we extend the basic model by employing a Stackelberg game, whichis also formulated as a two-stage game.

By doing so, we can determine the wage difference and the leisure differencebetween the spouses. We consider the situation in which a woman and a man simul-taneously decide their education level in the first stage. After each agent completesher/his education, the woman and the man meet and become a couple, and then decidetheir leisure time based on a Stackelberg game.

A wage gap may lead to different decision making within the family owing to acomparative advantage: following the census data and empirical findings,8 we assumethe husband to be a Stackelberg leader in the provision of household public goods.

It follows that the education decisions are made simultaneously in the first stage.In the second stage, each agent engages in the Stackelberg game of leisure time; thatis, the husband is the leader and the wife is the follower.

Therefore, in the second stage, the husband takes the reaction function of his wifeinto account and maximizes her utility function:

um = wm(1 − lm) + lm − δl2m + γ lm R f (lm;w f ) − β

2w2

m,

where R f (l f ;wm), which is given in (4), is the reaction function of the wife. Theequilibrium must satisfy the first-order condition

(lm;w f , wm) ≡ −wm + 1 − 2δlm + γ R f (lm;w f ) + γ lm∂ R f (lm;w f )

∂lm= 0,

= −wm + 1 − 2δlm + γ(1 − w f + γ lm

2δ

)+ γ lm

γ

2δ= 0, (17)

8 The wage level of women is lower than that of men in Japan, the U.S., the U.K., Germany, France,and Sweden. O’Neill (2003), Weichselbaumer and Winter-Ebmer (2005), and Del Bono and Vuri (2011)empirically determine the wage gap between the spouses.

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and the second-order condition is satisfied as follows:

1(lm;w f , wm) = −(2δ)2 + 2γ 2

2δ< 0,

where denotes the left-hand side of (17) and 1 ≡ ∂/∂l f . From (17), we have

lsm(w f , wm) = 2δ(1 − wm) + γ (1 − w f )

(2δ)2 − 2γ 2 . (18)

Substituting (18) into (4), we have

lsf (w f , wm) = [(2δ)2 − γ 2](1 − w f ) + 2δγ (1 − wm)

2δ[(2δ)2 − 2γ 2] . (19)

The following lemma summarizes the comparison of the equilibrium leisure timein the Stackelberg game and that in the non-cooperative game, which is derived inSubsect. 2.2.

Lemma 1 For a given level of education, each agent chooses higher leisure time inthe Stackelberg game than in the non-cooperative game.

Proof Subtracting (19) (resp. (18)) from (6) (resp. (5)), we have

l∗m − lsm = 2δ(1 − w f ) + γ (1 − wm)

[(2δ)2 − γ 2][(2δ)2 − 2γ 2]( − γ 2) < 0,

l∗f − lsf = γ (1 − wm) + 2δ(1 − w f )

2δ[(2δ)2 − γ 2][(2δ)2 − 2γ 2]( − γ 3) < 0.

�Because the marginal benefit of the leader (husband) is higher in the Stackelberg

game than in the non-cooperative game (see Eqs. (3) and (17)), the leader (husband)increases leisure time in the Stackelberg game more than in the non-cooperative game.Since the game exhibits strategic complementarity, the follower (wife) also increasesleisure time, and this leads to the results of Lemma 1.

Now, let us compare the equilibrium level of leisure in the Stackelberg game andthat in the efficient one, which is derived in Subsect. 2.2 (see Eqs. (9) and (10)).

The following proposition summarizes the characteristics of the equilibrium.

Proposition 4 For a given level of education, in the Stackelberg game, each agentchooses less leisure time than the efficient level.

Proof

lsm − le

m = −2δγ [2γ (1 − wm) + 2δ(1 − w f )][(2δ)2 − 2γ 2][(2δ)2 − (2γ )2] < 0,

lsf − le

f = γ 2[−(3(2δ)2 + (2γ )2(1 − wm)] − 2δγ (2δ)(1 − wm)][(2δ)2 − 2γ 2][(2δ)2 − (2γ )2] < 0.

�

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In the cooperative game, each individual maximizes his or her joint utility function;the marginal benefit from leisure is higher in the cooperative game. Thus, we have theresults of Proposition 4.

Now, let us return to the first stage to derive the education level. In the first stage, aman and a woman simultaneously maximize the following utility function for a giveneducation level of their partner.

ui = wi(y − ls

i (wi , w j )) + ls

i (wi , w j ) − δlsi (wi , w j )

2

+ γ lsi (wi , w j )l

sj (wi , w j ) − β

2w2

i ,

i, j ∈ { f, m} and i �= j, (20)

where lsm(w f , wm) and ls

f (w f , wm) are given by (18) and (19), respectively. Theeducation level wm that maximizes (20) depends on w f and is determined by thefirst-order condition. Using the envelope theorem,9 we have the first-order conditionof the leader (husband) as follows:

s,m(wm, w f ) ≡ y − lsm(wm, w f ) − βwm = 0,

= y − 2δ(1 − wm) + γ (1 − w f )

(2δ)2 − 2γ 2 − βwm = 0, (21)

and the sufficient condition is satisfied because of Assumption 2:

s,m1 ≡ 2δ

(2δ)2 − 2γ 2 − β < 0,

where s, f denotes the left-hand side of (21) and s, f1 ≡ ∂s, f /∂w f . The optimal

solution satisfies the following condition:

wm = y[(2δ)2 − 2γ 2] − (2δ + γ ) + γw f

β[(2δ)2 − 2γ 2] − 2δ. (22)

Similarly, the education choice of the follower w f that maximizes (20) depends onthe education level of the leader wm , and is determined by the first-order condition.Using the envelope theorem,10 we have the first-order condition of the follower (wife)as follows:

9 ∂um∂wm

= 0 ⇔ y − lsm + (−wm + 1 − 2δls

m + γ lsf + γ lm

∂lsm

∂w f)

︸︷︷︸=0,from(17)

∂lsm

∂wm− βwm = 0.

10 ∂u f∂w f

= 0 ⇔ y − lsf + (−w f + 1 − 2δls

f + γ lsm)

︸︷︷︸=0,from(3)

∂lsf

∂w f+ γ

∂lsm

∂w fls

f − βw f = 0.

123


s, f (w f , wm) = y − lsf + γ

∂lsm

∂w fls

f − βw f = 0,

= y + [(2δ)2 − γ 2](1 − w f ) + 2δγ (1 − wm)

2δ[(2δ)2 − 2γ 2](−[(2δ)2 − γ 2]

(2δ)2 − 2γ 2

)− βw f = 0,

(23)

and the sufficient condition,

s, f1 ≡ [(2δ)2 − γ 2]2

2δ[(2δ)2 − 2γ 2]2 − β < 0,

where lsf (w f , wm) and ls

m(w f , wm) are given in (18) and (19), respectively. s,m

denotes the left-hand side of (23), and s,m1 ≡ ∂s,m/∂wm . To guarantee the sufficient

condition, we assume the following:

Assumption 3

β > y >[(2δ)2 − γ 2]2 − [(2δ)2 − γ 2 + 2δγ ]

2δ[(2δ)2 − 2γ 2]2 .

Assumption 3 guarantees the sufficient condition. It also shows that the equilibriumexists when the marginal cost of the effort to obtain education is sufficiently high.

From (23), we have the following reaction function:

w f = y2δ[(2δ)2−2γ 2]2−[(2δ)2−γ 2]2−2δγ [(2δ)2−γ 2]+2δγ [(2δ)2−γ 2]wm

β2δ[(2δ)2−2γ 2]2−[(2δ)2−γ 2]2 .

(24)We have the first-stage equilibrium in the Stackelberg game; that is, the man and

the woman simultaneously choose their education levels in the first stage, and afterthey complete their education, they get married and decide their leisure time with thehusband (the wife) acting as the Stackelberg leader (follower)11 as follows:

wsf = (y2δA2 − B2)(β A − 2δ) − 2δγ B[(β − y)A + γ ]

(β2δA2 − B2)(β A − 2δ) − 2δγ 2 B, (25)

wsm = y A − (2δ + γ )

β A − 2δ+ γ

β A − 2δws

f , (26)

where A ≡ (2δ)2 − 2γ 2 and B ≡ (2δ)2 − γ 2.Now, let us examine the education level in the Stackelberg game. Since the equilib-

rium education level satisfies the first-order condition, we compare these conditions.The following proposition summarizes the results.

Proposition 5 When leisure time is decided in the Stackelberg game, both the leaderand the follower choose a higher education level than the efficient level.

11 In what follows, we refer to the game as the Stackelberg game.

123


Proof See Appendix A. �The intuition of the result is based in part on the negative strategic effect and the

positive income effect described in Subsect. 2.2.First, let us study the action of the leader. Because the leader maximizes leisure

time by taking the reaction of the follower into account, the strategic effect is absent(see Eq. (23)). In the Stackelberg game, the marginal benefit from education is higherthan that in the cooperative game; thus, the positive income effect leads to the leaders’sresults. Next, let us examine the action of the follower. In contrast with the leader, thefollower has a negative strategic effect (see Eq. (23)) in the Stackelberg game. In theequilibrium, the positive income effect dominates the negative strategic effect; thus,in the Stackelberg game, the follower has an incentive to obtain a higher educationlevel than the efficient level.

In the Stackelberg game, we can also consider the income tax policy, which wasconsidered in Subsect. 2.2. The income tax decreases the marginal benefit of educa-tion, and thus, the incentive to obtain education decreases. Thus, the policy alleviatesoverinvestment in education. Because decision making in the second stage depends onthe education level determined in the first stage, the policy also alleviates the under-provision of leisure.

Our next concern is whether the leader or the follower chooses a higher educationlevel in the Stackelberg game. The following proposition summarizes the results.

Proposition 6 In the Stackelberg game, the leader chooses a higher education levelthan the follower.

Proof See Appendix B. �The intuition of the results is summarized as follows: as can be seen in the intuition

of Proposition 5, the follower has a negative strategic effect, but the leader does not.This means that the leader can obtain a higher marginal benefit from the educationthan the follower, and this leads to Proposition 6.

By comparing the education levels in the Stackelberg game and the non-cooperativegame, we have the following corollary.

Corollary 1 In the Stackelberg game, both the leader and the follower choose a lowereducation level than in the non-cooperative game.

Proof See Appendix C.

Proposition 6 gives the result that the leader chooses a higher education level in thefirst stage. How does this result affect leisure time in the second stage? The followingproposition summarizes the results.

Proposition 7 In the Stackelberg game, the leader chooses a higher level of leisuretime than the follower.

Proof

lsm − ls

f = 1

(2δ)2 − 2γ 2

((2δ − γ )(ws

f − wsm) + γ 2

2δ(1 − ws

f )).

123


Substituting the equilibrium education levels (25) and (26) into the above equation,we have

lsm − ls

f = 1

A

[L(E K − J I ) + M F(I − E) + L

F (J G − H K − G K ) + M F

(I F − G)(K L − M F)

]

,

= (β − y)γ 2[β[(2δ)2 − 2γ 2]2 − 2δ[(2δ)2 − γ 2]](I F − G)(K L − M F)

> 0,

where E ≡ y2δA2 − B2, F ≡ β − 2δ, G ≡ 2δγ 2 B, H ≡ 2δγ B(β − y)A, I ≡β2δA2 − B2, J ≡ y A−(2δ+γ ), K ≡ β A−(2δ+γ ), L ≡ 2δ−γ , and M ≡ γ 2/2δ.Assumption 2 guarantees that the nominator is positive; thus, we have ls

m − lsf > 0.

�

Proposition 7 gives analytical support to the empirical finding of Beblo and Robledo(2008); that is, the spouse who has the higher wage enjoys more leisure time than theother spouse.

The intuition behind this result is as follows. The first-order condition of the leaderhas the positive strategic term γ l f

∂ Rm

∂l f(see Eq. (17)); however, the follower’s first-

order condition does not (see Eq. (3)). Thus, the leader has a higher marginal benefitfrom leisure than the follower, and this leads to Proposition 5.

4 Conclusion

In traditional family economics, it is postulated that gains from marriage arise fromthe sharing of public goods. This implies the division of labor to exploit a comparativeadvance, and many studies have focused on substitutability in the provision of house-hold public goods. Although a large body of empirical data shows that an increase inthe leisure time of a husband can raise the marginal benefit of the leisure time of hiswife, little attention has been paid to family economics. In this paper, we have focusedon leisure complementarity within the family and set up the model of the provision ofhousehold public goods.

The family model with a complementarity relationship was formulated as a two-stage game in which a man and a woman simultaneously complete their education,after which they meet and become a couple in the first stage. In the second stage,their time contribution to household production is decided. We derive the provision ofhousehold public goods as a solution of a Nash equilibrium in the two decision modes:cooperative and non-cooperative. The basic model also extends to a Stackelberg game,where the higher-earning spouse is a Stackelberg leader.

In examining the model, we showed that the underprovision of the household publicgood in the second stage is associated with overinvestment in education in the firststage. In the Stackelberg framework, the high wage earner (leader) chooses a highertime contribution to household public goods than the low wage earner (follower).

123


In this paper, we examined the basic model within the family. This model can beextended in several directions.12 First, the empirical test how the policy affects thehousehold leisure time can be considered. Second, we can consider how the Paretoweights are affected by the bargaining positions of the two spouses. Third, an analysisof the comparison of strategic complementarity across countries can also be consid-ered. This could help determine the social stigma and labor force participation ofmarried women. Forth, we can also consider how endogenously-formed families andfamily income pooling affect the equilibrium. In addition, we can extend the modelto allow the time allocation among labor supply, leisure, and housework. If a coupleenjoys utility from leisure together, they split the househwork and try to enjoy moreleisure time together. This also could help to study labor force participation of marriedwomen.

Acknowledgments We would like to thank Salvador Ortigueira, Kai Konrad, Giacomo Corneo, andanonymous referees for their helpful suggestions and comments on an earlier version of this paper. Thiswork was supported by Grant-in-Aid for Young Scientists (B) (23730295). All remaining errors are ourown.

Appendix A

Leader

As the equilibrium education level of the leader satisfies the first-order condition (21),we evaluate the first-order condition (21) at the efficient education level. Noting thatthe efficient education levels are symmetric between the members of the couple (see(15)) and that the equilibrium satisfies the first-order condition, the efficient educationlevel can be interpreted as

βwm = y − lem(we). (27)

Substituting (27) into s, f (w f , wm) and evaluating it at the level of w f = wm = we,we have

s, f (w f , wm)|w f =wm=we ≡ lef (w

e) − lsf (w

e) > 0 (see Proposition 3).

Evaluating this at the efficient education level, we, shows that the marginal conditionof first-order condition (21) in the Stackelberg game is positive. Thus, the educationlevel in the Stackelberg game, ws

f , is higher than that in the efficient outcome, we.

Follower

As the equilibrium wage rate of the leader in the Stackelberg game satisfies the first-order condition (23), we evaluate Eq. (23) at the efficient education level.

Using the same procedure with the leader, we have

12 We would like to thank the anonymous referees for pointing out these extensions.

123


s, f (w f , wm )|w f =wm=we

≡ lef (we) − ls

f (we) + γ lsf (we)

∂lm∂w f

∣∣∣wm=w f =we

= (1 − we)[(2γ )2((2δ)2 − γ 2) + γ (2δ)2(2δ − γ )][(2δ)2 − 2γ 2][(2δ)2 − (2γ )2]

︸︷︷︸le

f (we)−lsf (we)

− (1 − we)[(2δ)2 − γ 2 + 2δγ ]2δ[(2δ)2 − 2γ 2]

(2δγ

(2δ)2 − 2γ 2

)

︸︷︷︸γ ls

m (we)∂lm∂w f

|wm=w f =we

= (1 − we)

(2δ)2 − 2γ 22δ[(2δ)2 − 2γ 2](2δ − γ ) + 2γ [(2δ)2 + 2γ 2] + 6γ 3

[(2δ)2 − (2γ )2][(2δ)2 − 2γ 2] > 0.

This shows that the marginal condition of the first-order condition in the Stackelberggame (23) is positive by evaluating it at the efficient education level, we. Therefore,the equilibrium wage level in the Stackelberg game, ws

m , is higher than that in theefficient one, we.

Appendix B

Subtracting wsf from ws

m , we have

wsm − ws

f ≡ 1

β A − 2δ

(y A − (2δ + γ ) − (

β A − (2δ + γ ))ws

f

).

As the denominator is positive (see Assumption 2), we examine the sign of the numer-ator. Substituting (25) into the numerator, we have

−(β − y)A((β A − 2δ)

(2δA(2δ + γ ) − B2) − 2δγ 2 B + 2δγ B

(β A − (2δ + γ )

))

= −(β − y)A(β A − 2δ)[−γ 3(2δ + γ )] > 0.

Therefore, we have the result that wsm > ws

f .

Appendix C

Leader

We evaluate the first-order condition (21) in the Stackelberg game with the equilib-rium education level in the non-cooperative game. Noting that the equilibrium educa-tion levels are symmetric between the members of the couple (see (13)) and that theequilibrium satisfies the first-order condition, the equilibrium education level can beinterpreted as

βwm = y − l∗m + γ l∗m∂l∗f∂wm

. (28)

123


Substituting (28) into s,m(w f , wm) and evaluating it at the level of w f = wm =w∗, we have

s,m(w f , wm) = l∗m(w∗) − lsm(w∗) − γ l∗m(w∗)

∂l∗f∂wm

|w f =wm=w∗ ,

≡ γ 2[(2δ + γ )(1 − w∗)](2δ)2 − γ 2

−γ 2

[(2δ)2 − γ 2)][(2δ)2 − γ 2] < 0.

Evaluating this at w∗ shows that the marginal condition of first-order condition (23)in the Stackelberg game is negative. Therefore, the equilibrium wage level in theStackelberg game, ws

m is lower than that in the non-cooperative game.

Follower

The result of wsf < f ∗ is straightforward from the result of Proposition 5, ws

f < wsm

and the above result (leader): wsm < w∗.

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non-cooperative versus cooperative family

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