Household specialization and competition for promotion
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Bastani, Spencer; Dickmanns, Lisa; Giebe, Thomas; Gürtler, Oliver Article — Published Version Household specialization and competition for promotion Review of Economics of the Household Provided in Cooperation with: Springer Nature Suggested Citation: Bastani, Spencer; Dickmanns, Lisa; Giebe, Thomas; Gürtler, Oliver (2024) : Household specialization and competition for promotion, Review of Economics of the Household, ISSN 1573-7152, Springer US, New York, NY, Vol. 23, Iss. 1, pp. 141-163, https://doi.org/10.1007/s11150-024-09706-9 This Version is available at: https://hdl.handle.net/10419/318574 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/
Rev Econ Household (2025) 23:141–163 https://doi.org/10.1007/s11150-024-09706-9 Household specialization and competition for promotion Spencer Bastani1●Lisa Dickmanns2●Thomas Giebe3●Oliver Gürtler2 Received: 10 January 2023 / Accepted: 13 March 2024 / Published online: 30 March 2024 © The Author(s) 2024 Abstract We study how the presence of promotion competition in the labor market affects household specialization patterns. By embedding a promotion tournament model in a household setting, we show that specialization can emerge as a consequence of competitive work incentives. This specialization outcome, in which only one spouse invests heavily in his or her career, can be welfare superior to a situation in which both spouses invest equally in their careers. The reason is that household specialization reduces the intensity of competition and provides households with consumption smoothing. The specialization result is obtained in a setting where spouses are equally competitive in the labor market and there is no household production. It is also robust to several modifications of the model, such as varying the number of households, two spouses competing for promotion in the same workplace, and the inclusion of household production. Keywords Contest theory ●Gender equality ●Family ●Household ●Competition JEL classification C72 ●D13 ●J16 ●J71 ●M51 ●M52 1 Introduction A large literature in labor economics documents gender gaps in labor market outcomes, emphasizing differences in wages, working hours, employment rates, and *Thomas Giebe [email protected] 1 Institute for Evaluation of Labour Market and Education Policy (IFAU) and Department of Economics, Uppsala University; Research Institute of Industrial Economics (IFN); Uppsala Center for Fiscal Studies (UCFS); Uppsala Center for Labor Studies (UCLS); CESIfo, Uppsala, Sweden 2 Department of Economics, University of Cologne, Cologne, Germany 3 Department of Economics and Statistics, School of Business and Economics, Linnaeus University, Växjö, Sweden 1234567890();,:
occupations. Despite a strong convergence process over the last half-century, substantial gender differences in the labor market remain (Olivetti and Petrongolo, 2016). While preferences certainly might play a role in explaining this remaining gap, the literature has mostly focused on differences in the opportunities for men and women to succeed in the labor market and the desirability of policies that ‘level the playing field’. In this paper, we show how gender gaps can arise in response to labor market competitiveness even when men and women are equally competitive and have equal opportunities to succeed in the labor market. Our focus is on settings where both spouses in a household face career incentives in the sense that their work effort affects the likelihood of promotions that lead to higher pay and career advancement. Our work is motivated by the prevalence of promotion competition as an incentive system in firms and organizations (Lazear and Rosen, 1981,Waldman,2013) and the increasing number of dual career households facing such incentives (Costa and Kahn, 2000). 1 We set up a theoretical model with two identical two-earner families consisting of two identical spouses. Each spouse in the first family competes for promotion against a spouse in the second family. In this stylized but tractable model, we first show that asymmetric equilibria featuring household specialization generally can emerge. We then illustrate that the specialization equilibrium can deliver higher welfare to both households as compared to when spouses in both families adopt the same competitive effort. The intuition behind the natural emergence of household specialization is twofold. First, the asymmetric equilibrium reduces the intensity of promotion competition within each firm, implying that both households save on effort costs. Second, a situation where only one spouse exerts high effort provides smoothing of family consumption since intermediate events (where one spouse in each household gets promoted) become more likely. 2 We explore a number of extensions to highlight the robustness of the specialization result. First, we show that specialization results can be obtained even when the number of households competing at the two firms is four instead of two. Second, we show that specialization can also occur when both spouses work at the same firm. In this case, the consumption smoothing motive is replaced by a negative external effect, as one spouse’s effort reduces the other spouse’s chances of promotion, which favors specialization. Third, we show that specialization survives when households are allowed to maximize a convex combination of individual and household utility. Finally, we find that the specialization result is robust to the inclusion of a household production effort requirement. In traditional labor supply models, household specialization typically arises as long as one partner has a comparative advantage in market work, typically due to the presence of a household production sector (see Pollak 2013 for a discussion). We show that household specialization can arise due to the presence of promotion competition without modeling household production or imposing asymmetries in spouses’market skills. In particular, this implies that household specialization can arise even if all household 1 See also Green and Stokey (1983), Malcomson (1984), Baker et al. (1994a,b) Prendergast (1999), Bognanno (2001), DeVaro (2006), and DeVaro et al. (2019). 2 The consumption insurance channel has previously been highlighted in a non-tournament setting by, e.g., Blundell et al. (2018). 142 S. Bastani et al.
production is outsourced to the market (e.g., even in the presence of family-friendly policies that allow workers to combine childbearing with a career). Our paper is related to Francois (1998), who also considers ex-ante identical men and women, but focuses on explaining gender discrimination as an equilibrium outcome in a setting where men and women select into different jobs (in contrast, in our setting men and women work in identical jobs). We also add a new angle to the literature on labor market investment within families. A prominent strand of this literature discusses the “family investment hypothesis”and how credit constraints (in the case of immigrant workers) can imply labor market behavior where one “primary worker”engages in investment activities and the other partner engages in activities that finance consumption (see, e.g., Baker and Benjamin 1997 and Cobb-Clark and Crossley 2004). Section 2presents the model and derives the specialization result. Section 3 discusses the robustness of our results by exploring extensions and modifications of our baseline setting. Section 4concludes, and the appendix contains analytical results as well as numerical examples. 2 The model Following Lazear and Rosen (1981), each worker exerts effort to produce output and the worker with the highest output in the tournament is promoted and wins a prize wP, while the non-promoted worker receives wNP. The output of each worker is equal to y=e+ϵ, where eis individual effort and ϵis a random component. The ϵare assumed to be independently and identically distributed with PDF fand CDF F. The distinguishing feature of our setup is that we embed a Lazear-Rosen tournament in a household setting. More specifically, we consider two families 1 and 2, and two identical firms Aand B. In each family, one member works in firm A, while the other member works in firm B. We denote by ik the spouse in family i∈{1, 2} who works in firm k∈{A,B}, and by il the spouse in family iwho works in firm l∈{A,B}, where k≠l. Similarly, we denote by jk the spouse in family j∈{1, 2}, j≠i, who works in firm k∈{A,B}, and by jl the spouse in family jwho works in firm l∈{A,B}, k≠l. The tournament prize structure is the same in both firms. The total family income is equal to the sum of the prizes, which, given the two possible prize levels, allows four different configurations of family income given by the pairs (wP,wP), (wP,wNP), (wNP,wP), and (wNP,wNP). Adopting the unitary model of household decision making (see, e.g., Becker 1965, Boskin and Sheshinski 1983 and Kleven et al. 2009), the utility of household iis Uðbi;eik;eilÞ¼uðbiÞcðeikÞcðeilÞ;ð1Þ where bidenotes the total consumption of family i, and eik ≥0 and eil ≥0 denote the effort expended by the spouses in family i. 3 Furthermore, uis increasing and strictly concave and cis non-decreasing and strictly convex, satisfying c(0) =0 and c0ð0Þ¼0. 3 We recognize that there are other models of family decision making (see the discussion in Chiappori and Lewbel 2015), and in one of our extensions in subsection 3.3 we explore a departure from the unitary model. Household specialization and competition for promotion 143
The assumption that utility is nonlinear in consumption and depends on total household disposable income is key to our analysis as it implies a diminishing return for a family to have both spouses be highly successful in the labor market. Let Δek=eik −ejk be the effort difference in firm kand Δel=eil −ejl the effort difference in firm l. Furthermore, let ΔuP=u(2wP)−u(wP+wNP) be the consumption utility gain from going from one to two promoted family members, and ΔuNP =u(wP+wNP)−u(2wNP) be the gain from going from zero to one promoted family member. We also define Δu=ΔuP−ΔuNP =u(2wP)+u(2wNP)−2u(wP+ wNP), which is negative due to the strict concavity of u. In other words, the first promotion in the family is more valuable than the second. Family iwins the firm-ktournament against family jif eik +ϵik >ejk +ϵjk. This event can be rewritten as ϵjk <ϵik þeik ejk ð2Þ or ϵjk ϵik <e ik ejk:ð3Þ Since ϵik is i.i.d. with PDF fand CDF F, the probability of (2) can be written as ZFxþeik ejk fxðÞdx:ð4Þ Following Lazear and Rosen (1981), we define Gas the CDF of the difference ϵjk −ϵik. The probability of (3) can be stated as Ge ik ejk . Since both (2)and(3)describethe same event, we have RFxþeik ejk fx ðÞ dx ¼Ge ik ejk . In the following, we will use the latter specification because it keeps the presentation simple. 4 From the perspective of household i, there are four events. The household 1. wins both tournaments (probability G(Δek)G(Δel)) 2. wins the tournament at firm kbut not at firm l(probability G(Δek)(1 −G(Δel))) 3. wins the tournament at firm lbut not at firm k(probability (1 −G(Δek))G(Δel)) 4. wins none of the tournaments (probability (1 −G(Δek))(1 −G(Δel))) The expected utility of household ifrom both tournaments is therefore GðΔekÞGðΔelÞuð2wPÞþGðΔekÞð1GðΔelÞÞuðwPþwNPÞ þð1GðΔekÞÞGðΔelÞuðwPþwNPÞþð1GðΔekÞÞð1GðΔelÞÞuð2wNPÞcðeik ÞcðeilÞ ¼GðΔekÞGðΔelÞuð2wPÞþ GðΔekÞþGðΔelÞ2GðΔekÞGðΔelÞðÞuðwPþwNPÞ þ1GðΔekÞGðΔelÞþGðΔekÞGðΔelÞðÞuð2wNPÞcðeikÞcðeilÞ ¼GðΔekÞGðΔelÞuð2wPÞ2uðwPþwNPÞþuð2wNPÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ΔuPΔuNP 0 B B @1 C C A þGðΔekÞþGðΔelÞðÞuðwPþwNPÞuð2wNPÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ΔuNP 0 B B @1 C C Aþuð2wNPÞcðeikÞcðeilÞ ¼GðΔekÞGðΔelÞΔuPþ½GðΔekÞþGðΔelÞGðΔekÞGðΔelÞΔuNP þuð2wNPÞcðeikÞcðeilÞ: ð5Þ 4 Subsection A.5 contains the analysis for more general distribution functions. 144 S. Bastani et al.
The baseline utility from consumption is u(2wNP) and the second term above gives the increase in utility from winning one promotion, while the first term is the additional utility from a second promotion. Household ijointly chooses eik and eil in order to maximize (5). Family jfaces a problem with the same structure, where the probabilities G(Δek) and G(Δel) are replaced by G(−Δek) and G(−Δel), respectively, and the maximization is performed with respect to ejk and ejl instead. The outcome of the firm ktournament is determined by eik and ejk satisfying the first-order conditions gðΔekÞGðΔelÞΔuPþ1GðΔelÞðÞΔuNP ½¼c0ðeikÞð6Þ gðΔekÞGðΔelÞΔuPþ1GðΔelÞðÞuNP ½¼c0ðejkÞ:ð7Þ 2.1 The possibility of asymmetric equilibria The possibility of asymmetric equilibria is not immediately apparent in our fully symmetric model. However, as we will see, asymmetric household specialization equilibria can arise. We focus on such equilibria where the total effort is the same in both families. eik þeil ¼ejk þejl () eik ejk ¼ðeil ejlÞ()Δek¼Δel≠0:ð8Þ Conditional on the efforts of family j, if spouse kin household ispecializes in market work (in the sense of exerting a high effort in his/her promotion tournament), it must be the case that spouse lin household ispecializes in household work or leisure (in the sense of exerting a low effort in his/her promotion tournament). Assuming gis unimodal and symmetric around zero, using (8), we have G(Δel)=1−G(−Δel)=1−G(Δek). 5 Together with c(e)=de2with d> 0, Eqs. (6) and (7) can be rewritten as gðΔekÞ1GðΔekÞðÞΔuPþGðΔekÞΔuNP ½¼2deik ð9Þ gðΔekÞGðΔekÞΔuPþ1GðΔekÞðÞΔuNP ½¼2dejk:ð10Þ Subtracting (10) from (9) and rearranging yields: gðΔekÞGðΔekÞ1 2 ¼d ΔuΔek;ð11Þ where we recall that Δu=ΔuP−ΔuNP < 0 so the RHS is non-negative for Δek≥0. We have the following result: Proposition 1 If gis continuous on Rþand g0ðÞ>ffiffiffiffiffiffiffi d Δu q, then there exists Δe> 0 such that (11) holds. 5 A sufficient but not necessary condition for this assumption is that fis unimodal and symmetric around zero. Household specialization and competition for promotion 145
Proof. Based on (11), we define HðΔeÞ¼gΔeðÞGΔeðÞ1=2ðÞαΔewhere α¼d Δu>0. Notice that H(Δe) < 0 when Δe→∞(since G(Δe) and g(Δe) are bounded). What remains to show is that H(Δe) > 0 for some Δe> 0. We consider the point Δe=εwhere ε> 0 is small such that gðεÞ>ffiffiffi α p. We get: HðεÞ¼gεðÞR0 1 gðtÞdt þRε 0gðtÞdt 1=2 αε ¼gεðÞRε 0gðtÞdt αε ¼gðεÞgðξÞεαε >0; for some ξ∈(0, ε). The last inequality follows because gð0Þ>ffiffiffi α pand gcontinuous implies gðxÞ>ffiffiffi α pfor all x∈[0, ε]. □ 2.2 The symmetric equilibrium A symmetric equilibrium ^ esatisfies Δek=Δel=0. Insertion into either (6)or(7) yields gð0Þ1 2ΔuPþΔuNP ½¼c0ð^ eÞ:ð12Þ Since this equation has a solution, a symmetric equilibrium candidate generally exists. Note that ^ eis the level of effort chosen by both spouses in both families. 2.3 Welfare comparison between the two equilibria Whether the asymmetric or symmetric equilibrium provides higher utility to families depends on how large the expected consumption utility and total effort cost are in each equilibrium, recalling the definition of household welfare in (1). The expected utility of consumption for family iin a symmetric equilibrium is: 1 4½uð2wPÞþuð2wNPÞþ1 2uðwPþwNPÞ:ð13Þ Due to the strict concavity of u, the first promotion is more valuable than the second, i.e., we have uðwPþwNPÞuð2wNPÞ>uð2wPÞuðwPþwNPÞ>0 () 1 4½uð2wPÞþuð2wNPÞ<1 2uðwPþwNPÞ;ð14Þ which shows that the first term in (13) is smaller than the second. We will use this result in the following. In contrast, when playing the asymmetric equilibrium, rearranging (14) and again using G(Δel)=1−G(−Δel)=1−G(Δek), we get GðΔekÞGðΔelÞuð2wpÞþðGðΔekÞGðΔelÞGðΔekÞGðΔelÞþ1Þuð2wNPÞ þðGðΔekÞþGðΔelÞ2GðΔekÞGðΔelÞÞuðwPþwNPÞ ¼GðΔekÞð1GðΔekÞÞ½uð2wPÞþuð2wNPÞþ½ðGðΔekÞÞ2þð1GðΔekÞÞ2uðwPþwNPÞ: ð15Þ In this last expression, G(Δek)(1 −G(Δek)) is the probability of winning (resp. losing) both promotions, while ðGðΔekÞÞ2þð1GðΔekÞÞ2is the probability of getting exactly one promotion for the household. Since Gð0Þ¼1 2, we have GðΔekÞ>1 2, implying GðΔekÞð1GðΔekÞÞ<1 4and since probabilities add up to one, it follows 146 S. Bastani et al.
that ½ðGðΔekÞÞ2þð1GðΔekÞÞ2>1 2. It follows that, compared to (13), the first term in (15) has a smaller probability weight and the second term has a larger weight. The expression (15) will therefore be strictly larger than (13). Therefore, the asymmetric equilibrium always provides “smoothing”of total household consumption because it assigns higher probabilities to outcomes with one promoted spouse per household. Regarding total effort costs, in Appendix A.3, we show for the uniform distribution that if dis sufficiently large, and uis not too concave, then both efforts in the asymmetric equilibrium are smaller than the symmetric equilibrium effort (provided they coexist), implying that welfare is higher through both the consumption and effort channels. 6 2.4 Numerical example We now provide a numerical example to illustrate that the asymmetric equilibrium can welfare dominate the symmetric equilibrium when they both coexist (see the Appendix for details). Suppose that the noise terms are uniformly distributed on [−1/2, 1/2]. Without loss of generality, we assume that family iputs in more effort into the firm-ktournament than family j, namely, Δek>0. Table 1shows the results for c(e)=e2.Wefixu(2wNP)=2, u(wNP +wP)=4 and consider variation in u(2wP), letting it take on the three values 4.1, 4.5 and 5. Note that for u(2wP)=5 (third row), Eq. (11) has the unique solution Δek=0. In the other two examples, (11) has two solution candidates, i.e., Δek=0 and Δek> 0, although only one of the candidates is an equilibrium for u(2wP)=4.1. In the example with u(2wP)=4.5, we can compare the two equilibria as they exist simultaneously, and we see that the asymmetric equilibrium has lower total effort cost and higher expected utility from consumption, showing that the asymmetric equilibrium welfare-dominates the symmetric equilibrium. 2.5 Discussion We have derived the household specialization result in a game between households whose spouses compete for promotion, where wages (tournament prizes) are exogenously given. Thus, our analysis does not address the question of how firms’wage setting would respond to household behavior. We also do not analyze welfare consequences beyond the household, such as firm profits and social welfare. Thus, we have analyzed a ‘slice’of the labor market, focusing on household behavior as a response to given incentive systems in firms. Compared to the Lazear and Rosen (1981) tournament model, in our setup, one firm’s wage setting would affect another firm through the household’s decision making. This would lead to strategic interaction between firms at the wage-setting stage. The analysis of a larger game involving firms’wage setting is beyond the scope of our analysis. 6 In general, given asymmetric effort of household j, household ifaces asymmetric tournaments in both firms, which typically have lower effort than symmetric tournaments. On the other hand, asymmetric effort implies higher total effort cost due to convex cost functions. Household specialization and competition for promotion 147
In a situation like the second row of Table 1, both equilibria coexist for the given wages and the specialization equilibrium is strictly preferred by all households. It is not obvious whether or not it is possible and profitable for the firms to implement a symmetric equilibrium by setting wages. This is especially the case given that the firms’ wage decisions interact strategically. The symmetric equilibrium would be attractive to a firm only if the firm’sprofit net of wages in the symmetric equilibrium is greater than the profit from the specialization equilibrium. This places an upper bound on the wages that the firm would be willing to pay to induce households to abandon specialization. Wage setting as a tool to influence promotion effort would have to overcome the two benefits of specialization: consumption smoothing and effort cost saving, which is, for example, more difficult the more risk-averse households are (the more they value consumption smoothing). Note that this argument ignores household production, which may be another obstacle to moving away from specialization. 3 Extensions In this section, we consider four extensions/modifications of our model: (i) let the number of households be four instead of two, (ii) let both spouses work at the same workplace, (iii) allow households to maximize a convex combination of individual and household utility, and (iv) introduce a fixed household production effort to be performed by one of the spouses. 3.1 Four households and two firms We extend the basic setup to four households and two firms, so that the matching is similar to that in the basic model: each household competes for promotion in two different firms, each household facing competition from three other households. The tradeoffs remain similar to those in the base model: For sufficiently concave household utility (which makes the second promotion much less attractive than the first), there are again asymmetric equilibria in which households focus on winning one of the promotion contests, thereby smoothing consumption and reducing total effort costs. Analytical results and a numerical example can be found in Appendix B.1. Table 1 Numerical example symmetric equilibrium asymmetric equilibrium u(2wP)esym total cost E[u(b)] (eH,eL) total cost E[u(b)] 4.1 –– – (0.516, 0.157) 0.291 3.690 4.5 0.625 0.781 3.625 (0.595, 0.353) 0.478 3.693 5 0.750 1.125 3.750 ––– Note: Illustration of the possibility of having either only an asymmetric equilibrium (first row), only a symmetric equilibrium (third row), or both existing at the same time (second row). The total cost is equal to 2c(esym) in the case of a symmetric equilibrium, and equal to c(eH)+c(eL) in the case of an asymmetric equilibrium. The second-order conditions have been verified. 148 S. Bastani et al.
For example, assuming the Reflected Exponential distribution (A.6) with scale parameter λ=2 as well as u(2wNP)=1 and cost function c(e)=e2, a symmetric equilibrium (but no asymmetric equ.) is obtained for (u(wP+wNP), u(2wP)) =(2, 2.1) and an asymmetric ‘specialization’equilibrium (but no symmetric equ.) for (u(wP+wNP), u(2wP)) =(3, 3.1). 5.2 Extension derivations B.1 Four households and two firms Assume that four households compete for promotion in two firms, so that each household is ‘split’between two firms, i.e., the two members of each household work in different firms. This is the natural extension of the main two-by-two model, where household members also compete at different firms. Thus, there are four workers competing in each firm, and only one of them is promoted. Again, the promoted worker receives wP, while the three non-promoted workers each receive wNP. Denote the set of households by N¼f1;2;3;4gand the firms by Aand B. Household i2N wins the firm-ktournament if ioutperforms the other three households, eik þϵik >e jk þϵjk 8j2Nnfig; which we write as Δeijk +ϵik >ϵjk for all j2Nn i fg , where Δeijk ≔eik −ejk. The winning (promotion) probability for household iat firm kcan thus be stated as ZY j2Nn ifg FxþΔeijk fxðÞdx: The expected payoff for family ibecomes RQ j2Nn ifg FxþΔeijA fxðÞdx ! RQ j2Nn ifg FxþΔeijB fxðÞdx ! u2wP ðÞ þRQ j2Nn ifg FxþΔeijA fxðÞdx ! 1RQ j2Nn ifg FxþΔeijB fxðÞdx ! uw PþwNP ðÞ þ1RQ j2Nn ifg FxþΔeijA fxðÞdx ! RQ j2Nn ifg FxþΔeijB fxðÞdx ! uw PþwNP ðÞ þ1RQ j2Nn ifg FxþΔeijA fxðÞdx ! 1RQ j2Nn ifg FxþΔeijB fxðÞdx ! u2wNP ðÞ ce iA ðÞce iB ðÞ: ðB:1Þ Using ΔuPand ΔuNP (as introduced in the main text), this can be simplified to RQ j2Nn i fg FxþΔeijA fxðÞdx ! RQ j2Nn i fg FxþΔeijB fxðÞdx ! ΔuPΔuNP ðÞ þRQ j2Nn i fg FxþΔeijA fxðÞdx þRQ j2Nn i fg FxþΔeijB fxðÞdx ! ΔuNP þu2wNP ðÞce iA ðÞce iB ðÞ: ðB:2Þ Household specialization and competition for promotion 155
The first-order condition for household iregarding the contest at firm kcan be stated as ∂RQj2Nn ifgFxþΔeijk ðÞ fxðÞdx ∂eik RQ j2Nn ifg FxþΔeijl fxðÞdx ! ΔuPΔuNP ðÞ þ ∂RQj2Nn i fgFxþΔeijk ðÞ fxðÞdx ∂eik ΔuNP ¼c0eik ðÞ; ðB:3Þ which can be written as ∂RQj2Nn ifgFxþΔeijk ðÞ fxðÞdx ∂eik RQ j2Nn i fg FxþΔeijl fxðÞdx ! ΔuPΔuNP ðÞþΔuNP ! ¼c0eik ðÞ:ðB:4Þ Writing the two first-order conditions explicitly for household 1 (which chooses efforts e1Aand e1B), we have Rfðxþe1Ae2AÞFðxþe1Ae3AÞFðxþe1Ae4AÞfðxÞdx RFðxþe1Ae2AÞfðxþe1Ae3AÞFðxþe1Ae4AÞfðxÞdx RFðxþe1Ae2AÞFðxþe1Ae3AÞfðxþe1Ae4AÞfðxÞdx RFðxþe1Be2BÞFðxþe1Be3BÞFðxþe1Be4BÞfðxÞdx ΔuPΔuNP ðÞþΔuNP ¼c0e1A ðÞ ðB:5Þ and Rfðxþe1Be2BÞFðxþe1Be3BÞFðxþe1Be4BÞfðxÞdx RFðxþe1Be2BÞfðxþe1Be3BÞFðxþe1Be4BÞfðxÞdx RFðxþe1Be2BÞFðxþe1Be3BÞfðxþe1Be4BÞfðxÞdx RFðxþe1Ae2AÞFðxþe1Ae3AÞFðxþe1Ae4AÞfðxÞdx ΔuPΔuNP ðÞþΔuNP ¼c0e1B ðÞ: ðB:6Þ Again, we are interested in an asymmetric equilibrium candidate in which each household specializes, i.e., chooses effort profile (eH,eL), while in each firm there are now two high and two low efforts, competing for promotion. An arbitrary candidate effort profile of this kind is ðe1A;e1BÞ;ðe2A;e2BÞ;ðe3A;e3BÞ;ðe4A;e4BÞðÞ¼ðeH;eLÞ;ðeH;eLÞ;ðeL;eHÞ;ðeL;eHÞðÞ; ðB:7Þ i.e., each household chooses efforts eHand eL, while at each firm, we have efforts eL,eL,eH,eH. 156 S. Bastani et al.
Inserting candidate (B.7) into (B.5) and (B.6) and denoting Δ≔eH−eLwe get (where, e.g., F(x+Δ)2means F(x+Δ)⋅F(x+Δ)) RFðxþΔÞ2fðxÞ2dx þ2RFðxþΔÞfðxþΔÞFðxÞfðxÞdx RFðxΔÞ2FðxÞfðxÞdx ΔuPΔuNP ðÞþΔuNP ¼c0eH ðÞ:ðB:8Þ and RFðxΔÞ2fðxÞ2dx þ2RFðxΔÞfðxΔÞFðxÞfðxÞdx RFðxþΔÞ2FðxÞfðxÞdx ΔuPΔuNP ðÞþΔuNP ¼c0eL ðÞ:ðB:9Þ For each household, we would obtain a similar pair of first-order conditions. As c0ðeÞ¼2de, subtracting (B.9) from (B.8), we obtain 2dΔon the RHS: RFðxþΔÞ2fðxÞ2dx þ2RFðxþΔÞfðxþΔÞFðxÞfðxÞdx RFðxΔÞ2FðxÞfðxÞdx ΔuPΔuNP ðÞþΔuNP RFðxΔÞ2fðxÞ2dx þ2RFðxΔÞfðxΔÞFðxÞfðxÞdx RFðxþΔÞ2FðxÞfðxÞdx ΔuPΔuNP ðÞþΔuNP ¼2dΔ: ðB:10Þ Equation (B.10) only depends on Δ, so it allows us to use numerical methods to find candidates for asymmetric (Δ> 0) equilibria of the form described in (B.7), where eH>eL. Reinserting Δ-candidates into (B.8) and (B.9) delivers candidates for eHand eLthat can be verified numerically by replacing the, respectively, other households’efforts in (B.2)byeHand eL(according to (B.7)) and simultaneously maximizing over eiA and eiB to confirm the candidates. Asymmetric equilibrium candidate (in which all eight players choose the same effort) can be found directly from Eqs. (B.8) and (B.9) by inserting Δ=0. For example, assuming the Reflected Exponential distribution (A.6) with scale parameter λ=2 as well as u(2wNP)=1 and cost function c(e)=e2, a symmetric equilibrium (but no asymmetric equilibrium) is obtained for (u(wP+wNP), u(2wP)) =(1.2, 1.21) and an asymmetric ‘specialization’equilibrium (but no symmetric equilibrium) for (u(wP+wNP), u(2wP)) =(1.5, 1.51). B.2 Both spouses at the same workplace We now consider the case where both spouses compete for promotion in the same firm. That is, assume that households 1 and 2 work in the same firm. We denote by e11 and e12 the two efforts of the two spouses of family 1 and by e21 and e22 the corresponding efforts of family 2. We use the same indices for the random terms. In this setup, a household will have either no or one promoted spouse, associated with household consumption utilities u(2wNP) and u(wP+wNP), respectively. Two promotions per household are not possible: As in the base model and in subsection B.1, there is one promotion per firm. For each household, there are only three possible outcomes: either member 1 wins the promotion, or member 2 wins the promotion, or neither wins the promotion. Household specialization and competition for promotion 157
Household 1 wins if either member 1 or member 2 wins the tournament, that is, if either e11 þϵ11 >max e12 þϵ12;e21 þϵ21;e22 þϵ22 fg or e12 þϵ12 >max e11 þϵ11;e21 þϵ21;e22 þϵ22 fg ; which is equivalent to either ϵ12 <ϵ11 þe11 e12 AND ϵ21 <ϵ11 þe11 e21 AND ϵ22 <ϵ11 þe11 e22 ðÞ or ϵ11 <ϵ12 þe12 e11 AND ϵ21 <ϵ12 þe12 e21 AND ϵ22<ϵ12 þe12 e22 ðÞ: The winning probability of family 1 is thus given by RFxþe11 e12 ðÞFxþe11 e21 ðÞFxþe11 e22 ðÞfxðÞdx þRFxþe12 e11 ðÞFxþe12 e21 ðÞFxþe12 e22 ðÞfxðÞdx: The expected payoff for family 1 can be stated as RFxþe11 e12 ðÞFxþe11 e21 ðÞFxþe11 e22 ðÞfxðÞdxΔuNP þRFxþe12 e11 ðÞFxþe12 e21 ðÞFxþe12 e22 ðÞfxðÞdxΔuNP þu2wNP ðÞce 11 ðÞce 12 ðÞ:ðB:11Þ Likewise, the expected payoff for family 2 is RFxþe21 e22 ðÞFxþe21 e11 ðÞFxþe21 e12 ðÞfxðÞdxΔuNP þRFxþe22 e21 ðÞFxþe22 e11 ðÞFxþe22 e12 ðÞfxðÞdxΔuNP þu2wNP ðÞce 21 ðÞce 22 ðÞ: We obtain the following four first-order conditions: R∂Fxþe11e12 ðÞFxþe11e21 ðÞFxþe11e22 ðÞðÞ ∂e11 fxðÞdxΔuNP Rfxþe12 e11 ðÞFxþe12 e21 ðÞFxþe12 e22 ðÞfxðÞdxΔuNP ¼c0e11 ðÞ; Rfxþe11 e12 ðÞFxþe11 e21 ðÞFxþe11 e22 ðÞfxðÞdxΔuNP þR∂Fxþe12e11 ðÞFxþe12e21 ðÞFxþe12e22 ðÞðÞ ∂e12 fxðÞdxΔuNP ¼c0e12 ðÞ; R∂Fxþe21e22 ðÞFxþe21e11 ðÞFxþe21e12 ðÞðÞ ∂e21 fxðÞdxΔuNP Rfxþe22 e21 ðÞFxþe22 e11 ðÞFxþe22 e12 ðÞfxðÞdxΔuNP ¼c0e21 ðÞ; and Rfxþe21 e22 ðÞFxþe21 e11 ðÞFxþe21 e12 ðÞfxðÞdxΔuNP þR∂Fxþe22e21 ðÞFxþe22e11 ðÞFxþe22e12 ðÞðÞ ∂e22 fxðÞdxΔuNP ¼c0e22 ðÞ: 158 S. Bastani et al.
Notice that ∂Fxþe11e12 ðÞFxþe11e21 ðÞFxþe11e22 ðÞðÞ ∂e11 ¼fxþe11 e12 ðÞFxþe11 e21 ðÞFxþe11 e22 ðÞ þFxþe11 e12 ðÞfxþe11 e21 ðÞFxþe11 e22 ðÞ þFxþe11 e12 ðÞFxþe11 e21 ðÞfxþe11 e22 ðÞ; which means that the first of the first-order conditions becomes Rfxþe11 e12 ðÞFxþe11 e21 ðÞFxþe11 e22 ðÞfxðÞdxΔuNP þRFxþe11 e12 ðÞfxþe11 e21 ðÞFxþe11 e22 ðÞfxðÞdxΔuNP þRFxþe11 e12 ðÞFxþe11 e21 ðÞfxþe11 e22 ðÞfxðÞdxΔuNP Rfxþe12 e11 ðÞFxþe12 e21 ðÞFxþe12 e22 ðÞfxðÞdxΔuNP ¼c0e11 ðÞ: Now consider the ‘specialization’candidate e11 =e21 =eH>eL=e12 =e22, where Δ=eH−eL. The f.o.c. becomes RfxþΔðÞFxðÞFxþΔðÞfxðÞdxΔuNP þRFxþΔðÞfxðÞFxþΔðÞfxðÞdxΔuNP þRFxþΔðÞFxðÞfxþΔðÞfxðÞdxΔuNP RfxΔðÞFxΔðÞFxðÞfxðÞdxΔuNP ¼2deH: ðB:12Þ The second f.o.c. can similarly be rewritten as RfxþΔðÞFxðÞFxþΔðÞfxðÞdxΔuNP þRfxΔðÞFxΔðÞFxðÞfxðÞdxΔuNP þRFxΔðÞfxΔðÞFxðÞfxðÞdxΔuNP þRFxΔðÞFxΔðÞfxðÞfxðÞdxΔuNP ¼2deL: ðB:13Þ Subtracting (B.13) from (B.12) yields 2RfxþΔðÞFxðÞFxþΔðÞfxðÞdxΔuNP þRFxþΔðÞfxðÞFxþΔðÞfxðÞdxΔuNP þRFxþΔðÞFxðÞfxþΔðÞfxðÞdxΔuNP RFxΔðÞfxΔðÞFxðÞfxðÞdxΔuNP RFxΔðÞFxΔðÞfxðÞfxðÞdxΔuNP 2RfxΔðÞFxΔðÞFxðÞfxðÞdxΔuNP ¼2dΔ: This can be simplified to RFðxþΔÞ2FðxΔÞ2 fðxÞ2dx þ3RFðxþΔÞfðxþΔÞFðxΔÞfðxΔÞðÞFðxÞfðxÞdxΔuNP ¼2dΔ: ðB:14Þ Again, we use this equation that only depends on Δto numerically identify asymmetric equilibrium candidates with Δ> 0. Reinserting Δ-candidates into (B.12) Household specialization and competition for promotion 159
and (B.13), we get candidates for eHand eL. Replacing e21 and e22 in (B.11) with eH and eL, and simultaneously maximizing over e11 and e12, we can numerically confirm candidates. For example, assuming the Reflected Exponential distribution (A.6) with scale parameter λ=2 as well as u(2wNP)=2 and cost function c(e)=e2, a symmetric equilibrium (but no asymmetric equ.) is obtained for u(wP+wNP)=2.2 and an asymmetric ‘specialization’equilibrium (but no symmetric equ.) for u(wP+wNP)=2.8. B.3 Convex combination of joint and individual utility maximization In this extension, we assume that each spouse independently maximizes a convex combination of household utility and individual utility. Denote by αthe weight of joint utility maximization (the basic model) and by 1 −αthe weight of individual maximization. Denote a spouse’s individual payoff from being promoted by ~ uðwPÞand the individual utility from not being promoted by ~ uðwNPÞ. Note: For spouse 1A’s effort choice it does not matter whether we use αc(e1B)or c(e1B) as the other spouse’s cost term in 1A’s payoff function, as this term vanishes in the first-order condition with respect to e1A. We write down the utility of spouse 1A, i.e., the person in household 1 who chooses effort e1A: RFðxþe1Ae2AÞfðxÞdx RFðxþe1Be2BÞfðxÞdx αuð2wPÞþð1αÞ~ uðwPÞ½ þRFðxþe1Ae2AÞfðxÞdx 1RFðxþe1Be2BÞfðxÞdx αuðwPþwNPÞþð1αÞ~ uðwPÞ½ þ1RFðxþe1Ae2AÞfðxÞdx RFðxþe1Be2BÞfðxÞdx αuðwPþwNPÞþð1αÞ~ uðwNPÞ½ þ1RFðxþe1Ae2AÞfðxÞdx 1RFðxþe1Be2BÞfðxÞdx αuð2wNPÞþð1αÞ~ uðwNPÞ½cðe1AÞαcðe1BÞ: ðB:15Þ For person 1B, the utility is similar (we only switch ~ uðwPÞand ~ uðwNPÞin lines 2 and 3 below, and adjust the effort cost): RFðxþe1Ae2AÞfðxÞdx RFðxþe1Be2BÞfðxÞdx αuð2wPÞþð1αÞ~ uðwPÞ½ þRFðxþe1Ae2AÞfðxÞdx 1RFðxþe1Be2BÞfðxÞdx αuðwPþwNPÞþð1αÞ~ uðwNPÞ½ þ1RFðxþe1Ae2AÞfðxÞdx RFðxþe1Be2BÞfðxÞdx αuðwPþwNPÞþð1αÞ~ uðwPÞ½ þ1RFðxþe1Ae2AÞfðxÞdx 1RFðxþe1Be2BÞfðxÞdx αuð2wNPÞþð1αÞ~ uðwNPÞ½αcðe1AÞcðe1BÞ: ðB:16Þ The first-order condition for person 1A(resp., 1B) with respect to e1A(resp., e1B)is Rfðxþe1Ae2AÞfðxÞdx RFðxþe1Be2BÞfðxÞdx αuð2wPÞþð1αÞ~ uðwPÞ½ þRfðxþe1Ae2AÞfðxÞdx 1RFðxþe1Be2BÞfðxÞdx αuðwPþwNPÞþð1αÞ~ uðwPÞ½ Rfðxþe1Ae2AÞfðxÞdx RFðxþe1Be2BÞfðxÞdx αuðwPþwNPÞþð1αÞ~ uðwNPÞ½ Rfðxþe1Ae2AÞfðxÞdx 1RFðxþe1Be2BÞfðxÞdx αuð2wNPÞþð1αÞ~ uðwNPÞ½ ¼c0ðe1AÞðB:17Þ 160 S. Bastani et al.
and RFðxþe1Ae2AÞfðxÞdx Rfðxþe1Be2BÞfðxÞdx αuð2wPÞþð1αÞ~uðwPÞ½ RFðxþe1Ae2AÞfðxÞdx Rfðxþe1Be2BÞfðxÞdx αuðwPþwNPÞþð1αÞ~ uðwNPÞ½ þ1RFðxþe1Ae2AÞfðxÞdx Rfðxþe1Be2BÞfðxÞdx αuðwPþwNPÞþð1αÞ~ uðwPÞ½ 1RFðxþe1Ae2AÞfðxÞdx Rfðxþe1Be2BÞfðxÞdx αuð2wNPÞþð1αÞ~ uðwNPÞ½ ¼c0ðe1BÞ: ðB:18Þ Denoting Δ~ u:¼~ uðwPÞ~ uðwNPÞ, these can be simplified to Rfðxþe1Ae2AÞfðxÞdx αRFðxþe1Be2BÞfðxÞdxΔuP þ1RFðxþe1Be2BÞfðxÞdx ΔuNPþð1αÞΔ~ u¼c0ðe1AÞðB:19Þ and Rfðxþe1Be2BÞfðxÞdx αRFðxþe1Ae2AÞfðxÞdxΔuP þ1RFðxþe1Ae2AÞfðxÞdx ΔuNPþð1αÞΔ~ u¼c0ðe1BÞðB:20Þ Now consider the asymmetric ‘specialization’candidate ((e1A,e1B), (e2A,e2B)) =(( eH,eL), (eL,eH)). Denoting Δ≔eH−eL, such that e1A−e2A=Δand e1B−e2B=−Δ, and inserting into (B.19) and (B.20), the two first-order conditions can be written as RfðxþΔÞfðxÞdx αRFðxΔÞfðxÞdxΔuP þ1RFðxΔÞfðxÞdx ΔuNPþð1αÞΔ~ u¼c0ðeHÞðB:21Þ and RfðxΔÞfðxÞdx αRFðxþΔÞfðxÞdxΔuP þ1RFðxþΔÞfðxÞdx ΔuNPþð1αÞΔ~ u¼c0ðeLÞðB:22Þ Recalling c0ðeÞ¼2de, subtracting (B.22) from (B.21), we get RfðxþΔÞfðxÞdx αRFðxΔÞfðxÞdxΔuP þ1RFðxΔÞfðxÞdx ΔuNPþð1αÞΔ~ u RfðxΔÞfðxÞdx αRFðxþΔÞfðxÞdxΔuP þ1RFðxþΔÞfðxÞdx ΔuNPþð1αÞΔ~ u¼2dΔ; ðB:23Þ i.e., a single equation that depends on Δonly. This equation can be used to numerically identify asymmetric equilibrium candidates. These Δ-candidates can then be reinserted into (B.21) and (B.22) to get candidates for eHand eL. For example, assume that the weight of joint utility is 70%, i.e., α=0.7, (compared to 100% in the base model) whereas individual utility has weight 30% in each spouse’s payoff function. Now, assuming the Reflected Exponential distribution (A.6) with scale parameter λ=2 and cost function c(e)=e2, a symmetric equilibrium (but no asymmetric equ.) is obtained for (u(2wNP), u(wP+wNP), u(2wP)) =(1, 2, 2.1) and ð~ uðwNPÞ;~ uðwPÞÞ ¼ ð1;2Þ. We obtain an asymmetric ‘specialization’equilibrium (but no symmetric equ.) for (u(2wNP), u(wP+wNP), u(2wP)) =(1, 2.7, 2.71) and ð~ uðwNPÞ;~ uðwPÞÞ ¼ ð1;2:7Þ. B.4 Household production effort requirement Assuming the Reflected Exponential distribution (A.6) with scale parameter λ=2aswellasu(2wNP)=1 and cost function c(e)=e2, an asymmetric Household specialization and competition for promotion 161
‘specialization’equilibrium (but no symmetric equ.) arises for (u(wP+wNP), u(2wP)) =(3, 3.1). The equilibrium efforts in this example are (eH,eL)=(0.448735, 0.232182). We now introduce a fixed household-production effort of eHP ¼0:11 which is roughly 50% of eL, which seems to be a non-trivial amount of household production for this example. We have confirmed that the modified game has an asymmetric ‘specialization’equilibrium with promotion efforts ð~ e1A;~ e1BÞ¼ ð~ eH;~ eLÞ¼ð0:380764;0:0332649Þ¼ð ~ e2B;~ e2AÞin which spouses 1Band 2Acontribute the household production. In this example, the effort reductions relative to thebasegameareeH~ eH¼0:067971 and eL~ eL¼0:198917. In total, the reduction is larger than the amount of household production, which is intuitive given the convex effort cost functions. In percentages, eHisreducedby15% while eLis reduced by roughly 86%. References Baker, G., Gibbs, M., & Holmström, B. (1994a). The internal economics of the firm: Evidence from personnel data. Quarterly Journal of Economics,109(4), 881–919. Baker, G., Gibbs, M., & Holmström, B. (1994b). The wage policy of a firm. The Quarterly Journal of Economics,109(4), 921–955. Baker, M., & Benjamin, D. The role of the family in immigrants’labor-market activity: An evaluation of alternative explanations. American Economic Review, pages 705–727, (1997). Becker, G. S. (1965). A theory of the allocation of time. The Economic Journal,75(299), 493–517. Blundell, R., Pistaferri, L., & Saporta-Eksten, I. (2018). Children, time allocation, and consumption insurance. Journal of Political Economy,126(S1), S73–S115. Bognanno, M. L. (2001). Corporate tournaments. Journal of Labor Economics,19(2), 290–315. Boskin, M. J., & Sheshinski, E. (1983). Optimal tax treatment of the family: Married couples. Journal of Public Economics,20(3), 281–297. Chiappori, P., & Lewbel, A. (2015). Gary Becker’s a theory of the allocation of time. Economic Journal, 125(583), 410–442. Cobb-Clark, D., & Crossley, T. F. (2004). Revisiting the family investment hypothesis. Labour Economics,11(3), 373–393. Costa, D. L., & Kahn, M. E. (2000). Power couples: Changes in the locational choice of the college educated, 1940–1990. The Quarterly Journal of Economics,115(4), 1287–1315. DeVaro, J. (2006). Strategic promotion tournaments and worker performance. Strategic Management Journal,27(8), 721–740. DeVaro, J., Kauhanen, A., & Valmari, N. (2019). Internal and external hiring. ILR Review,72(4), 981–1008. Francois, P. (1998). Gender discrimination without gender difference: Theory and policy responses. Journal of Public Economics,68(1), 1–32. Goldin, C. (2014). A grand gender convergence: Its last chapter. American Economic Review,104(4), 1091–1119. Green, J. R., & Stokey, N. L. (1983). A comparison of tournaments and contracts. Journal of Political Economy,91(3), 349–364. Kleven, H. J., Kreiner, C. T., & Saez, E. (2009). The optimal income taxation of couples. Econometrica, 77(2), 537–560. Lazear, E. P., & Rosen, S. (1981). Rank-order tournaments as optimum labor contracts. Journal of Political Economy,89(5), 841–864. Malcomson, J. M. (1984). Work incentives, hierarchy, and internal labor markets. Journal of Political Economy,92(3), 486–507. Olivetti, C., & Petrongolo, B. (2016). The evolution of gender gaps in industrialized countries. Annual Review of Economics,8(1), 405–434. 162 S. Bastani et al.
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