Family restrictions at work

Aragonés, Enriqueta Article Family restrictions at work Economies Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Aragonés, Enriqueta (2024) : Family restrictions at work, Economies, ISSN 2227-7099, MDPI, Basel, Vol. 12, Iss. 5, pp. 1-18, https://doi.org/10.3390/economies12050101 This Version is available at: https://hdl.handle.net/10419/329027 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. https://creativecommons.org/licenses/by/4.0/ Citation: Aragones, Enriqueta. 2024. Family Restrictions at Work. Economies 12: 101. https://doi.org/ 10.3390/economies12050101 Received: 23 February 2024 Revised: 22 April 2024 Accepted: 24 April 2024 Published: 26 April 2024 Copyright: © 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). economies Article Family Restrictions at Work Enriqueta Aragones 1,2 1Institut d’Anàlisi Econòmica, CSIC, Campus UAB, 08193 Bellaterra, Spain; [email protected] 2Barcelona School of Economics Ramon Trias Fargas, 25-27, 08005 Barcelona, Spain Abstract: This paper analyzes one of the causes of the current gender-unbalanced situation in the labor market: the discrimination that individuals face at work due to their commitment to unpaid care work. It aims at finding mechanisms that may induce a change from the current unbalanced situation to a world in which males and females are found in more equal shares in all professions and at all levels. I construct a formal model that includes the heterogeneity of individuals regarding their family commitments and I investigate how it affects the individual’s optimal labor market participation. The welfare of individuals with commitment to family duties is reduced for two different reasons: for not being able to participate as much in the labor market and thus receive a lower labor income and for not being able to contribute as much to their family commitments. I compare the results for the female and male sections of the society and I illustrate the observed gender gaps in terms of labor market participation, income levels, and the overall utility obtained. I find that even though the gender wage gap may be alleviated with reductions in the cost associated to unpaid care work, the gender utility gap will persist. Keywords: discrimination; labor market; unpaid care work JEL Classification: J7; J31 1. Introduction The under-representation of women with respect to men in many professions at all levels and in all professions at some levels is a fact. Given that women represent about 50 per cent of the population, it is reasonable to describe this situation as gender unbalanced. The demand for a balanced proportion of women in all professions and at all levels has been raised and its support has been increasing over time. This demand can be based on the claim that a world in which males and females are found in equal shares in all professions and at all levels would be optimal. But it also can be based on an equity claim: females and males should have the same professional opportunities. This paper analyzes one of the causes of the current gender-unbalanced situation: the discrimination that individuals face at work due to their commitment to unpaid care work. It aims at finding mechanisms that may induce a change from the current unbalanced situation to a world in which males and females are found in more equal shares in all professions and at all levels. The analysis proposed in this paper contributes to the theoretical literature on discrimination in the labor market. The existing literature is mostly devoted to explaining racial discrimination such us Peski and Szentes (2013) who analyze how racial stereotypes may determine the labor market outcomes. The theoretical models that focus on the gender gaps in the labor market assume specific strategies for the firms such as incentive contracts Albanesi and Olivetti (2009) or different types of jobs (Francois 1998;Dolado et al. 2013) and they display multiple self-fulfilling equilibria, some of which exhibit gender gaps. In addition, Cella and Manzoni (2023) analyze how discrimination in electoral contests produce fewer female candidates and fewer female elected politicians, even though in average they are more competent than their male counterparts. This paper proposes a Economies 2024,12, 101. https://doi.org/10.3390/economies12050101 https://www.mdpi.com/journal/economies Economies 2024,12, 101 2 of 18 theoretical analysis of the gender discrimination that originates in the supply side of the labor market with the aim of providing new insights that would complement the extensive empirical literature on the subject. First, we briefly state some facts about the gender gaps observed in the labor market. Across the EU, the gender employment gap (the difference between the employment rates of men and women of working age: 20–64 years) was 10.8 percentage points in 2021, meaning that the proportion of men of working age in employment exceeded that of women by 10.8 percentage points. Women tend to work less hours and they are more likely to engage in low paid and informal work and in part-time jobs. Maternity leaves have a long-run negative effect on the participation of women in the labor force (Bertrand 2020; Isen et al. 2017). The disproportionate representation of women in low paid and informal work also contributes to the observed gender earnings gap. The gender earnings gap measures the impact of the three combined factors (the average hourly earnings, the monthly average number of hours paid and the employment rate) on the average earnings of all women of working age compared with men. In 2018, the gender overall earnings gap was 36.2% in the EU. Across Member States, the gender overall earnings gap varied significantly (from 20.4% in Lithuania and Portugal to 44.2% in Austria). The current gender-unbalanced situation can be explained by causes that are related to specific gender conditions of the supply and demand in the labor market. Regarding the causes that produce a low demand for women in some professions, it is important to refer to different kinds of discrimination that originated in the decisions made by the employers that are biased in favor of men relative to women. Some kinds of discrimination are based on the fact that most employers are men and thus it is possible that the men are more likely to prefer men as employees or coworkers (taste-based discrimination). The taste-based discrimination model was first proposed by Becker (1957,1971) and Schelling (1971). Given that in the past the female labor force has been disproportionately low with respect to the male one, employers have had more experience with male employees and thus have more information about the characteristics of male labor force (statistical discrimination). The theory of statistical discrimination was pioneered by Arrow (1973), Spence (1973) and Phelps (1972). This paper focuses only on the factors that affect the female labor supply that originate in the individual’s commitment to providing unpaid care work, that is “All unpaid services provided by individuals within a household or community for the benefit of its members, including care of persons and domestic work. Common examples include cooking, cleaning, collecting water and fuel and looking after children, older persons and persons with illness or disabilities. Voluntary community work that supports personal or household care, such as community kitchens or childcare, are also forms of unpaid care work” as defined by the United Nations (UN Women 2022). The United Nations report also claims that unpaid care work is basically provided by women: “Women and girls have disproportionate responsibility for unpaid care and domestic work; globally they spend three times as much time on this work as do men and boys. Unpaid care work is one of the main barriers preventing women from moving into paid employment and better quality jobs.”. Unpaid care work is absolutely necessary for a wellbeing of the society (Folbre 2001). The care, nurture and education of children is the basis for the growth of the economy and the evolution of the society. The care of elders is becoming more important over time. Maternity is unconditionally needed to the survival of our societies. Thus, it is of great importance to study the role of unpaid care work in the economy. Unpaid care work takes up a great amount of time and its responsibility falls disproportionately on women (Samman et al. 2016). The individuals, whether male or female, that assume this responsibility will experience a restriction in the quantity and quality of their labor supply. This restriction can be thought of in terms of the amount of time that they can offer to their professional activities. The decision of how much time one should devote to unpaid care work relative to professional work is complicated for most people since both activities are considered as very relevant or even necessary (Folbre 2001). This paper aims at analyzing Economies 2024,12, 101 3 of 18 the implications of such decisions. By making the economic costs and benefits of care provision more visible, we might be able to find mechanisms that alleviate the gender gaps observed in the labor market. This paper describes a formal model of individual choice in which the choice variable is the amount of time that an individual decides to devote to the labor market. This decision affects the total amount of income that an individual may obtain and also the total amount of cost that an individual has to bear depending on her or his previous commitments to unpaid care work. I assume that different individuals have different propensities to engage in family care; thus, the society exhibits a variety of costs associated to them. However, in the real world it is evident that in general women are very much committed to such activities while men are much less committed. Thus, women are expected to be more affected by the costs originating in family commitments. The results offer a specific explanation of the lower labor market participation of women, of the salary gap that is observed in the labor market due to unpaid care work commitments and it also shows the reduction in overall welfare suffered by the female sector of the society due to both the lower income levels and the higher costs from family commitments. The results obtained also imply that the effects of unpaid care work commitments cannot be avoided. And since they are indispensable for the wellbeing of the society, the only possible way to diminish the discrimination against women and to attain a more gender-balanced labor market is to induce men to commit to take family responsibilities. The next section introduces the formal model. Section 3describes the optimal individual choices of labor participation. Section 4analyzes the effects of discrimination in the society and compares the effects of discrimination between two sections of the society: female and male. Section 5offers a discussion of the implications of the results which includes some policy implications derived from the analysis of the formal model. Finally, Section 6contains some concluding remarks and possible extensions. 2. The Model This model considers the choice of an individual about her or his participation in the labor market. Let s∈[0, 1] denote the corresponding choice variable and it is to be interpreted as follows: larger values of s denote higher levels of labor market participation, which could be thought of as time devoted to work but may include other considerations such as the quality of the time devoted to work, or the quality of the jobs that may be attained. And of course, higher values of the choice variable correspond to higher values of the individual’s labor income. Individuals are characterized by their type. The type of an individual is a measure of her or his level of commitment to activities that are related to the labor market relative to the individual’s commitment to unpaid care work. Let t∈[0, 1] denote the individual type and it is to be interpreted as the amount of time that an individual can devote to professional activities that is free of the cost derived from family commitments. This implies that lower values of t refer to individuals with strong family commitments, and higher values of t correspond to individuals with few family commitments. For an individual of type t , devoting an amount of time larger than t to the labor market implies a reduction in her or his commitment to family activities, and it represents a reduction in her or his welfare. In particular, if an individual of type t chooses to dedicate an amount of time s=t to the labor market, this individual does not suffer any additional cost. However, if an individual of type t chooses to dedicate an amount of time s>t to the labor market, this individual will suffer a cost derived from her or his family restrictions. This cost can be thought of as the amount of income that has to be devoted to pay someone else for the provision of care, or it can be thought as the loss produced due to the diminished care on other family members. It is reasonable to think that this cost is small for small deviations from the type because it produces a small distortion in the household organization. However, it is also reasonable to think that it becomes increasingly large for larger deviations from the type, because large distortions in the household organization Economies 2024,12, 101 4 of 18 may require drastic solutions. In order to formalize this argument I assume that this cost is represented by a convex function of the distance between the individual’s type and the chosen level of dedication to labor market activities. Let C(s) = γ 2(t−s)2 and γ> 0 denote the cost of labor market participation for an individual of type t. Individuals obtain their income from their participation in the labor market. We assume that this income increases with the individual’s level of participation in the labor market. The individual income is represented by W(s) = ωs where ω> 0 denotes the income per unit of time devoted to the labor market that an individual may obtain. In particular, ω also refers to the total income that the individual obtains when he or she decides on full-time participation in the labor market, that is s=1. The overall individual’s welfare is measured by a utility function that combines the individual’s income and the cost that he or she has to bear and it is represented by the following function: U(s) = W(s)−C(s) = ωs−γ 2(t−s)2 Larger values of the cost parameter relative to the income parameter imply larger reductions in welfare. Thus, γ ω can be interpreted as a measure of the level of discrimination that an individual suffers due to her or his family commitments. Since the main trade-off is determined by the relationship between the parameters ω and γ , without loss of generality I normalize the wage to be ω= 1. This implies that the interpretation of the parameter γ is the weight of the cost relative to the maximal wage and therefore we can consider γ as the discrimination index derived from unpaid care work commitments. I analyze the individual’s optimal choice of labor market participation as a function of its type and of the discrimination index γ . In particular, I am interested in the effects that γ may have on the optimal individual choice and how it affects her or his labor income and total welfare. I also analyze the effects of the discrimination index on the overall society by considering the aggregate levels of labor market participation, income and welfare that the society may obtain, and also the aggregate level of cost that the society has to bear. Finally and most importantly, I analyze these effects on two segregated sections of the society: female and male. This distinction is important according to the empirical data since it is the female section of the society that bears most of costs derived from the family commitments or more generally from the unpaid care work. Thus, I compare the effects of discrimination between the female and male sections of the society. 3. The Optimal Individual Choice In order to find the optimal level of labor market participation for an individual of type t∈[0, 1] , I solve the maximization problem of her or his utility function with respect to the choice variable s∈[0, 1]. This result is stated in the next proposition. Proposition 1. The optimal level of labor market participation of an individual of type t is s∗(t,γ)=(1 γ+t i f t ≤γ−1 γ 1i f t ≥γ−1 γ All proofs are relegated to Appendix A. Notice that the optimal level of labor market participation for all types is positive; thus, everyone chooses to devote some time to work. Individuals with larger types decide to devote more time to the labor market and only the highest types decide to fully participate in the labor market, that is s∗(t,γ)=1. Notice that γ−1 γ increases with γ . Thus, larger values of γ imply smaller proportions of individuals that choose a full labor market participation. For 0 <γ< 1, we have that all individuals decide to fully participate in the labor market and obtain the maximal wage, and thus they decide to bear all the cost that is required for it. Even if in this case all types obtain the same income, they do not obtain the same utility, because each individual has Economies 2024,12, 101 5 of 18 to bear a different cost. Thus, we also have some discrimination: lower types bear higher costs and obtain lower utility levels. However, full female participation in the labor market is not what we observe happening in the real world. Since we aim at explaining the existing gender-unbalanced features of the real labor market for most of the paper we consider that 1 <γ . In this case, we have that only some individuals, those with higher types, decide to fully participate in the labor market and obtain the maximal wage. Instead individuals with lower types opt for partial labor market participation, which is exactly what the empirical results show. Therefore, for most of the paper we assume that 1 <γ , which implies that the commitment to unpaid work is considered as very important to all individuals relative to the labor income. This assumption is relaxed towards the end of the paper, when we consider possible policies that may alleviate the existing gender gaps in the real labor market. Assumption 1. γ>1. Proposition 1 highlights the twofold relevance of the discrimination in the individual’s optimal decision about labor market participation: it determines which part of the society decides to participate full time in the labor market and it also explains the distribution of the part-time jobs derived from the individuals’ optimal choice. On the one hand, the ratio γ−1 γ characterizes all the individual types whose decision on how much to participate in the labor market is negatively affected by the cost bearing of family commitments. All types with values below γ−1 γ opt for a partial labor market participation and thus suffer from the effects of the discrimination because they are not able to obtain the full income. While all types with values above γ−1 γ opt for a full labor market participation and obtain the full income from it. Notice that larger levels of discrimination imply larger sets of types that decide on a partial labor market participation. On the other hand, the discrimination index also affects the level of participation in the labor market for those individuals that opt for partial labor market participation: their optimal choice of labor market participation decreases with γ . Thus, larger values of γ imply that most individuals decide to work less. The gains and losses produced on the individuals’ total welfare are represented by the individual’s income, cost and utility computed at the optimal level of labor market participation given by proposition 1 and they are stated in the next proposition. Proposition 2. The optimal income obtained by an individual of type t is W∗(t,γ) = (1 γ+t i f t ≤γ−1 γ 1i f t ≥γ−1 γ The optimal cost supported by an individual of type t is C∗(t,γ) = (1 2γi f t ≤γ−1 γ γ 2(1−t)2i f t ≥γ−1 γ The optimal utility obtained by an individual of type t is U∗(t,γ) = (1 2γ+t i f t ≤γ−1 γ 1−γ 2(1−t)2i f t ≥γ−1 γ Observe that the individual income and the individual utility received by all types are positive and they both increase with the type for those individuals that opt for a partial labor market participation, who in turn obtain only a fraction of the full income. Individuals with a larger type t≥γ−1 γ decide to fully participate in the labor market; they obtain Economies 2024,12, 101 6 of 18 the maximal wage ω , and their individual utility increases with their type. However, this increase becomes smaller for larger values of t. The individual cost supported by all types is also positive. The cost supported by the individuals that opt for a full labor market participation decreases with her or his type, and this reduction becomes smaller for larger types. And all those types that opt for a partial labor market participation end up supporting the exact same total amount of individual cost (see Figure 1). Figure 1: Optimal individual decisions. Figure 2: Comparing total income and total utility for female and male types. Optimal labor market participation 01 1 " 1 " − 1 " $∗% Optimal income distribution 01 1 1 " "∗# $ − 1 $ Optimal cost distribution 01 1 2# !∗" # − 1 # Optimal welfare distribution 01 1 2# 1 1 − 1 2$ $∗% # − 1 # TWF(!) ! 1 2 TUF(!) ! 1 2 1/2 3 3/8 1/8 TWM(!)TUM(!) Figure 1. Optimal individual decisions. Increases in the discrimination index γ produce decreases in the individual income, in the individual cost and in the individual utility for those types that choose a partial labor market participation. Notice that on the one hand, they decide to work less when the discrimination index increases, and that is why they obtain a lower income. However, their individual cost also increases and overall it produces a large reduction in individual utility. The individual income for the larger types, those that choose a full labor market participation, is not affected by increases in the cost parameter. However, their individual cost increases and thus their individual utility also decreases. Therefore, increases in the value of γ imply lower utility for all types. The formal description of the comparative statics is included in the proof of the proposition contained in Appendix A. 4. Discrimination in Society In this section, we present and discuss the main results. First, we focus on the aggregated economic variables for the society and then we compare these variables for the two sections of the society: female and male types. Suppose that the individual types in the society are distributed according to a uniform probability distribution function over the support [0, 1] . We compute the labor market participation of the society T∗(γ) , total income of the society TW∗(γ) , the total cost derived from labor participation TC∗(γ) and the total utility obtained TU∗(γ)as stated in the next proposition. Economies 2024,12, 101 7 of 18 Proposition 3. The labor market participation, total income, total cost and total utility for the society are: T∗(γ)=TW∗(γ)=γ2+2γ−1 2γ2 TC∗(γ)=3γ−2 6γ2 TU∗(γ)=3γ2+3γ−1 6γ2 Notice that since we have normalized the wage to be 1 we now have that the measure of the labor market participation is equal to the measure of the total income. We find that the labor market participation and total income decrease with γ because lower types choose to dedicate less time to work when its associated cost becomes more expensive. Total utility decreases with γ because an increase in the cost parameter reduces the individual utility for all types: lower types work less and higher types pay a higher cost. The total cost may increase or decrease with γ depending on the value of the discrimination. Recall that the individual cost for lower types decreases with the cost parameter while the individual cost for higher types increases with the cost parameter. Overall, we have that the total cost increases with γ if the discrimination is very low 1<γ<4 3 because in this case most individuals decide to fully participate in the labor market and thus their individual cost increases with γ . However, the total cost decreases with γ if the discrimination is more severe γ>4 3 because in this case more individuals opt for a part-time participation in the labor market and they reduce their participation when the cost parameter increases. This implies that the reduction in cost due to the reduction in part-time labor market participation compensates for the increase in cost that the larger types suffer. The reason is twofold: the cost that higher types have to pay is relatively small compared to that of lower types, and for large values of the discrimination index the proportion of types that decide to work full time is smaller. Next, suppose that the female types are represented by those t such that 0 ≤t≤1 2 and the male types are represented by those t such that 1 2≤t≤ 1. The reason is that it is a fact that women are much more committed to unpaid care work than men. I compute the labor market participation, the total income, the total cost and the total utility evaluated at the optimal individual choice for female types ( TF∗(γ) , TWF∗(γ) , TCF∗(γ) , TUF∗(γ)) and for male types ( TM∗(γ) , TWM∗(γ) , TCM∗(γ) , TUM∗(γ)) separately. I analyze how these variables are affected by changes in the discrimination index γ. Then, I compare the results obtained for each section of society in order to evaluate the extent of the effect of the gender discrimination in the labor market over participation, income, cost and utility. The next proposition shows the results obtained for the aggregated economic variables corresponding to the female types. Notice that for this analysis I have to consider two cases depending on the value of the parameter γ . For γ> 2, I have that all female types decide to work part time while for γ< 2 I have that some female types decide to work part time and some of them decide to work full time. Proposition 4. The labor participation, total income, total cost and total utility for female types are: TF∗(γ)=TWF∗(γ)=(2γ−1 2γ2i f γ≤2 4+γ 8γi f γ≥2 TCF∗(γ)=(24γ−16−γ3 48γ2i f γ≤2 1 4γi f γ≥2 Economies 2024,12, 101 8 of 18 TUF∗(γ)=(γ3+24γ−8 48γ2i f γ≤2 2+γ 8γi f γ≥2 Recall that for γ≥ 2 all female types work part time, while for γ≤ 2 some of them decide to work full time. In all cases, the total income and the total utility for female types decrease with the discrimination index, as we found in the overall society. For γ≥ 2, we have that the total cost for female types decreases with γ , because as we have seen before part-time workers decrease their labor market participation when the cost parameter increases; thus, in this case the total cost is reduced. For γ≤ 2, we have that some female types work full time. In this case, the, comparative statics for the female types resembles very much that of the society overall: the total cost for female types decreases with γ for relatively small values of the discrimination index ( γ<γ<4 3) and it increases with γ for larger values of the discrimination index. The only difference is that for values of γ such that γ<γ<4 3 the total cost of the female types increases with γ while the total cost of the society decreases with γ . The reason is that for values of γ such that γ<γ<4 3 the proportion of female types that decide to work full time relative to the female population is not large enough and thus the total cost for the female types decreases with the cost of the majority because of part-time workers that decide to work less. While for the same parameter values, the proportion of full-time workers in the society is large enough relative to the total population and thus the total cost of the society increases. Now I replicate the previous analysis for the male types. Again, for this analysis I have to consider two cases depending on the value of the parameter γ . For γ< 2, I have all male types deciding to work full time while for γ> 2 I have that some male types decide to work part time. Proposition 5. The labor participation, total income, total cost and total utility for male types are: TM∗(γ)=TWM∗(γ)=(1 2i f γ≤2 3γ2+4γ−4 8γ2i f γ≥2 TCM∗(γ)=(γ 48 i f γ≤2 3γ−4 12γ2i f γ≥2 TUM∗(γ)=(24−γ 48 i f γ≤2 9γ2+6γ−4 24γ2i f γ≥2 In this case, we also have to consider two different situations. When discrimination is low ( γ≤ 2 ) , all male types decide to work full time and the total income is not affected by changes in γ . Total cost increases with γ because all male types are working full time, and therefore their total utility decreases with γ . Of course, increases of γ beyond 2 imply that some male types decide to reduce their labor market participation. For higher values of the discrimination index ( γ> 2 ) , some male types decide to work part time and in this case we have that total income and total utility for male types decrease with γ while total cost only decreases with γ for large values of the discrimination index ( γ>8 3) . The only difference with respect to the comparative statics of the society is found for values of γ such that 4 3<γ<8 3 . For these values, the total cost of the male types increases with γ while the total cost of the society decreases with γ . The reason is that for values of γ such that 4 3<γ<8 3 the proportion of male types that decide to work full time relative to the male population is large enough and thus the total cost for the male types increases with the cost of the majority because full-time workers have to pay a larger cost. While for the same parameter values, the proportion of full-time workers in the society is not large enough relative to the total population and thus the total cost of the society decreases. Economies 2024,12, 101 15 of 18 TCF∗(γ)=R1 2 0CF∗(t;γ)=R1 2 01 2γdt =h1 2γti1 2 0=1 4γ TUF∗(γ)=1 21 γ+1 4−1 4γ=1 41 2+1 γ=2+γ 8γ Comparative statics: ∂TWF∗(γ) ∂γ =−1 2γ2<0 ∂2TWF∗(γ) ∂γ2=1 γ3>0 limγ→∞TWF∗(γ)=1 8 ∂TCF∗(γ) ∂γ =−1 4γ2<0 ∂2TCF∗(γ) ∂γ2=1 2γ3>0 limγ→∞TCF∗(γ)=0 ∂TUF∗(γ) ∂γ =−1 4γ2<0 ∂TUF∗(γ) ∂γ =1 2γ3>0 limγ→∞TUF∗(γ)=1 8. Overall, we have that: TF∗(γ)=TWF∗(γ)=(2γ−1 2γ2i f γ≤2 4+γ 8γi f γ≥2 TCF∗(γ)=(24γ−16−γ3 48γ2i f γ≤2 1 4γi f γ≥2 TUF∗(γ)=(γ3+24γ−8 48γ2i f γ≤2 2+γ 8γi f γ≥2 ∂TWF∗(γ) ∂γ <0 ∂TCF∗(γ) ∂γ <0 iff γ>γfor some 1 <γ<4 3 ∂TUF∗(γ) ∂γ <0 . Proof of Proposition 5. We consider two separate cases depending on whether γ≷2. Case 1: For γ≤2 we have γ−1 γ≤1 2and: TM∗(γ)=TWM∗(γ)=R1 1 2WM∗(γ)=R1 1 2dt =[t]1 1 2=1−1 2=1 2 TCM∗(γ)=R1 1 2CM∗(γ)=R1 1 2 γ 2(1−t)2dt =h−γ 6(1−t)3i1 1 2 =γ 48 TUM∗(γ)=1 2−γ 48 =24−γ 48 Comparative statics: ∂TWM∗(ω,γ) ∂γ =0 ∂TCM∗(ω,γ) ∂γ =1 48 >0 ∂TUM∗(ω,γ) ∂γ =−1 48 <0. Case 2: For γ≥2 we have γ−1 γ≥1 2and: TWM∗(γ)=R1 1 2WM∗(γ)=R γ−1 γ 1 21 γ+tdt +R1 γ−1 γdt = h1 γt+t2 2iγ−1 γ 1 2 +[t]1 γ−1 γ =3γ2+4γ−4 8γ2 TCM∗(γ)=R1 1 2CM∗(γ)=R γ−1 γ 1 2 1 2γdt +R1 γ−1 γ γ 2(1−t)2dt = h1 2γtiγ−1 γ 1 2 +h−γ 6(1−t)3i1 γ−1 γ =3γ−4 12γ2 TUM∗(γ)=3 8+γ−1 2γ2−3γ−4 12γ2=9γ2+6γ−4 24γ2 Economies 2024,12, 101 16 of 18 Comparative statics: ∂TWM∗(γ) ∂γ =2γ2−4γ(γ−1) 2γ2=2−γ 2γ3<0 iff γ>2 which always holds in this case. ∂2TWM∗(γ) ∂γ2=γ−3 γ4>0 iff γ>3 limγ→∞TWM∗(γ)=3 8 ∂TCM∗(γ) ∂γ =3γ−2(3γ−4) 12γ3=8−3γ 12γ3<0 iff γ>8 3 ∂2TCM∗(γ) ∂γ2=γ−4 2γ4>0 iff γ>4 limγ→∞TCM∗(γ)=0 ∂TUM∗(γ) ∂γ =4−3γ 12γ3<0 iff γ>4 3which always holds in this case. ∂2TUM∗(γ) ∂γ2=γ−2 2γ4>0 iff γ>2 limγ→∞TUM∗(γ)=3 8 Overall, we have that: TM∗(γ)=TWM∗(γ)=(1 2i f γ≤2 3γ2+4γ−4 8γ2i f γ≥2 TCM∗(γ)=(γ 48 i f γ≤2 3γ−4 12γ2i f γ≥2 TUM∗(γ)=(24−γ 48 i f γ≤2 9γ2+6γ−4 24γ2i f γ≥2 and ∂TWM∗(γ) ∂γ ≦0 if γ≷2 ∂TCM∗(γ) ∂γ <0 iff γ>8 3 ∂TUM∗(γ) ∂γ <0 for all γ. Proof of Proposition 6. We consider two separate cases depending on whether γ≷2. Case 1: For γ≤2 we have: TF∗(γ) TM∗(γ)=TWF∗(γ) TWM∗(γ)= 2γ−1 2γ2 1 2 =2γ−1 γ2∈3 4, 1<1 TCF∗(γ) TCM∗(γ)= 24γ−16−γ3 48γ2 γ 48 =24γ−16−γ3 γ3∈[3, 7] TUF∗(γ) TUM∗(γ)= 24γ−8+γ3 48γ2 24−γ 48 =24γ−8+γ3 24γ2−γ3∈h6 11 ,17 23 i and ∂TWF∗(γ) TWM∗(γ) ∂γ =2(1−γ) γ3<0; TWF∗(1) TWM∗(1)=1; TWF∗(2) TWM∗(2)=3 4 ∂TCF∗(γ) TCM∗(γ) ∂γ =48(1−γ) γ4<0; TCF∗(1) TCM∗(1)=7; TCF∗(2) TCM∗(2)=3 TUF∗(1) TUM∗(1)=17 23 ;TCF∗(2) TCM∗(2)=6 11 ∂TUF∗(γ) TUM∗(γ) ∂γ =24γγ3+2γ2−25γ+16 (24γ2−γ3)2<0 iff γ3+2γ2−25γ+16 <0 Notice that for γ=1 we have that γ3+2γ2−25γ+16 =−6<0; for γ=2 we have that γ3+2γ2−25γ+16 =−18 <0 and in addition we have that γ3+2γ2−25γ+16 decreases with γfor 1 <γ<2. Case 2: For γ≥2 we have: TWF∗(γ) TWM∗(γ)= 4+γ 8γ 3γ2+4γ−4 8γ2 =4γ+γ2 3γ2+4γ−4∈h1 3,3 4i TCF∗(γ) TCM∗(γ)= 1 4γ 3γ−4 12γ2 =3γ 3γ−4∈[1, 3] Economies 2024,12, 101 17 of 18 TUF∗(γ) TUM∗(γ)= 2+γ 8γ 9γ2+6γ−4 24γ2 =6γ+3γ2 9γ2+6γ−4∈h1 3,6 11 i and ∂TWF∗(γ) TWM∗(γ) ∂γ =−42+γ+γ2 (3γ2+4γ−4)2<0<0; TWF∗(2) TWM∗(2)=3 4; limγ→∞ TWF∗(γ) TWM∗(γ)=1 3 ∂TCF∗(γ) TCM∗(γ) ∂γ =−12 (3γ−4)2<0; TCF∗(2) TCM∗(2)=3; limγ→∞ TCF∗(γ) TCM∗(γ)=1 ∂TUF∗(γ) TUM∗(γ) ∂γ =−22+2γ+3γ2 (9γ2+6γ−4)2<0; TUF∗(2) TUM∗(2)=6 11 ; limγ→∞ TUF∗(γ) TUM∗(γ)=1 3 Overall, we have: TF∗(γ) TM∗(γ)=TWF∗(γ) TWM∗(γ)∈h1 3, 1i TCF∗(γ) TCM∗(γ)∈[1, 7] TUF∗(γ) TUM∗(γ)∈h1 3,17 23 i and ∂TWF∗(γ) TWM∗(γ) ∂γ <0; ∂TCF∗(γ) TCM∗(γ) ∂γ <0; ∂TUF∗(γ) TUM∗(γ) ∂γ <0. Proof of Proposition 7. For γ≤1 we have that s∗(t,γ)=1 and W∗(t,γ) = 1 C∗(t,γ) = γ 2(1−t)2 U∗(t,γ) = 1−γ 2(1−t)2. 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