Analyzing the effects of minimum wages: a microeconomic approach

Thielen, Clemens; Weinschenk, Philipp Article — Published Version Analyzing the effects of minimum wages: a microeconomic approach Economic Theory Provided in Cooperation with: Springer Nature Suggested Citation: Thielen, Clemens; Weinschenk, Philipp (2024) : Analyzing the effects of minimum wages: a microeconomic approach, Economic Theory, ISSN 1432-0479, Springer, Berlin, Heidelberg, Vol. 79, Iss. 3, pp. 945-991, https://doi.org/10.1007/s00199-024-01607-3 This Version is available at: https://hdl.handle.net/10419/323260 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. 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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/ Economic Theory (2025) 79:945–991 https://doi.org/10.1007/s00199-024-01607-3 RESEARCH ARTICLE Analyzing the effects of minimum wages: a microeconomic approach Clemens Thielen1·Philipp Weinschenk2 Received: 18 October 2022 / Accepted: 9 September 2024 / Published online: 22 November 2024 © The Author(s) 2024 Abstract We use a microeconomic approach to analyze the effects of minimum wages. Agents are allowed to have different productivities at different principals as well as different costs of working. We obtain several new and interesting effects. Minimum wages could influence the generated surplus when leaving employment unaffected, and destroy jobs that generate relatively high levels of surplus when affecting employment. Furthermore, minimum wages could harm agents even if these stay employed, while principals could benefit from them. We provide a complete characterization of the effects and show that these hold independently of the specific bargaining procedure and information structure. Keywords Minimum wages ·Principal-agent model ·Costs of working ·Welfare effects JEL Classification C78 ·D21 ·J31 ·J38 ·K31 1 Introduction The insight that workers’ occupational decisions depend not only on monetary compensation, but also on nonmonetary job characteristics, is widely accepted.1Employers might take these nonmonetary characteristics into consideration when making wage 1This insight—already conceived by Smith (1937)—is, for example, the key assumption in the literature on compensating wage differentials and has recently gained much attention, cf. Kaplan and Schulhofer-Wohl (2018). BPhilipp Weinschenk [email protected] Clemens Thielen [email protected] 1Campus Straubing for Biotechnology and Sustainability, Technical University of Munich, Am Essigberg 3, 94315 Straubing, Germany 2Department of Business and Economics, University of Kaiserslautern-Landau, Gottlieb-Daimler-Str., 67663 Kaiserslautern, Germany 123 946 C. Thielen, P. Weinschenk offers.2Still, these aspects have not been fully considered when analyzing minimum wages, which is one of the most controversial, yet important topics in economics.3We contribute to closing this gap by considering a simple microeconomic model where an agent (interpreted as a worker) can not only have different productivities at different principals (potential employers), but also different costs of working to which principals can adjust their wage offers. Agents are thus allowed to have preferences regarding their employment, and these preferences can be taken into account by principals when offering wages. This is important since job opportunities usually differ in their nonmonetary characteristics, e.g., how satisfying, demanding, engaging, hazardous, and flexible they are, and what commuting costs they cause.4 Our model replicates the standard results we know from the existing literature in case each agent faces the same costs at all principals: Minimum wages redistribute income from principals to agents, such that principals suffer, while agents who stay employed benefit, and the efficiency is only affected if employment is decreased. Accordingly, the effects of minimum wages are a simple trade-off between efficiency (which is only affected if employment is lowered) and redistribution (directed from employers to workers). This is illustrated in the following example.5Consider a situation with two principals and one agent. Let the agent Ahave productivity θ1=11 when working for principal P1, and productivity θ2=9 when working for principal P2. The agent’s costs care 5 at both principals. Principals can make offers to the agent, who can P1 P2 A θ1=11 c=5 θ2=9 c=5 2Despite the fact that truck drivers earn well above the national average salary, the U.S. truck industry has problems finding drivers (CNN 2015;CNBC2018), which indicates that relatively low-skilled workers also care for nonmonetary job characteristics. As a reaction, Walmart has increased truck driver salaries to $87,500 a year on average (CNBC 2019). 3According to the (International Labour Organization (ILO) 2017), 90% of ILO member states have minimum wages. In the U.S., for example, there is a controversial debate not only about the minimum wage at federal level, but also about the minimum wage in states and cities (New York Times 2021a,b,c). There is also a discussion in the UK about the potential rise of the minimum wage and living wage (Financial Times 2022). 4Bonhomme and Jolivet (2009) empirically show that the type of work, the working conditions, the working times, the job security, and the distance to work are indeed important job characteristics for workers. Card et al. (2018) show that allowing workers to have idiosyncratic tastes for different workplaces is helpful to understand many well-documented empirical regularities on labor markets. 5In this example as well as in the main part of the paper, we examine the single-agent case. As we show in Appendix C, our results carry over to the multi-agent case with an arbitrary number of agents if there are no (binding) capacity constraints for the principals, and the observed effects of minimum wages still arise when capacity constraints are considered. 123 Analyzing the effects of minimum wages 947 then accept one of the offers or rejects all offers. For simplicity of exposition, we do not model the agent as a strategic player, but assume the agent to accept the utilitymaximizing offer.6In the Nash equilibrium without a minimum wage, the agent works for P1for a wage of w=9, which results in utility w−c=4 for the agent and a surplus of θ1−c=6. Every binding minimum wage wthat leaves the agent employed, i.e., w∈(9,11], increases the agent’s utility (to w−5) while leaving the surplus unaffected. Higher minimum wages w>11 cause the agent to become unemployed and lower the surplus. The assumption that each agent faces the same costs at all principals—which might belong to different economic sectors or regions—is strong and generically violated.7 We, therefore, allow agents to have different costs and obtain new results that, in particular, invalidate the simple trade-off perspective on minimum wages. The main results are the following. First, a minimum wage can adversely affect the generated surplus even if the employment level is unchanged. This effect arises in our model since a minimum wage could destroy job opportunities with relatively high levels of surplus while simultaneously maintaining opportunities with relatively low levels of surplus. These welfare losses may be hard to detect for outside observers (e.g., politicians or econometricians), since employment levels stay constant and wages increase. We can thus speak of “hidden costs” of minimum wages. Second, an agent can also suffer from a minimum wage when remaining employed. There are two different reasons for this effect to emerge: (i) the minimum wage could force an agent to relocate to another principal, which we show to decrease his8utility generically, or (ii) the minimum wage could allow an agent to stay at the same principal, but cause a reduction of the wage payment. This shows that minimum wages—which are usually intended to help agents—can actually harm agents also when they stay employed. Some of the basic results are illustrated in the following example (see figure on the next page), which is identical to the previous example except that the agent’s costs are different (e.g., due to different commuting costs or working conditions). Without a minimum wage, the agent works for P2for a wage of 7,9which results in the utility w−c2=4 and the surplus θ2−c2=6. Any minimum wage w∈(9,11)causes the agent to stay employed (by relocating to P1), but lowers the agent’s utility to w−7 and the surplus to θ1−c1=4. Interestingly, not only agent’s utility lowers, but also aggregate profits. Relaxing the seemingly innocent assumption that each agent has identical costs at all principals thus drastically changes the effects of minimum wages. The results are robust and hold in a variety of different settings; see Sections 5and 6and the appendices for numerous extensions and robustness checks. Our analysis provides further robust effects. 6In case of indifference, we let the agent behave according to natural tie-breaking rules that, as we show in Appendix A.1, exactly model the agent’s behavior in any subgame-perfect Nash equilibrium. 7Sorkin (2016,2018), for instance, documents that nonpay characteristics differ substantially between economic sectors. 8We follow the standard convention and talk about female principals and male agents. 9P1will offer a wage of 11 so that P2has to offer at least 7 to attract the agent. 123 948 C. Thielen, P. Weinschenk P1 P2 A θ1=11 c1=7 θ2=9 c2=3 First, wages may overshoot. That is, equilibrium wages may increase strictly above the imposed minimum wage. Second, principals may benefit from minimum wages. This effect arises when the minimum wage (i) forces an agent to relocate to another principal or (ii) shrinks the set of competing principals if no relocation occurs. Case (i) arises in the example above, where P1benefits from any minimum wage w∈(9,11). Case (ii) arises in the example if the agent’s cost at P1is changed to c1=4. Here, the agent works for P1for a wage of 10 without a minimum wage, but any minimum wage w∈(9,10)decreases the wage paid by P1to wsince P2will no longer compete. Third, when causing unemployment, minimum wages may destroy jobs that generate relatively high levels of surplus. Accordingly, the jobs that are lost due to a minimum wage are not necessarily the ones that generate only marginal levels of surplus, and the effect of minimum wages on efficiency is not only of second order. Fourth, minimum wages can cause an increase in equilibrium productivities by relocating agents from low to high productivity jobs. The productivity gains are, however, generically overcompensated by higher costs. Thus, the productivity gains caused by minimum wages are generically accompanied by efficiency losses. We start our analysis with the basic bargaining procedure where the principals make offers. This simple procedure is standard in the principal-agent literature but quite specific. Therefore, we also consider alternative bargaining procedures as well as stable outcomes, which abstract from how parties bargain and what they know. We show that the set of stable outcomes is given by the convex combinations of the outcome obtained when principals make offers and the outcome when agents make offers. Interestingly, except for the boundary case corresponding to the situation where agents make offers, the effects of a minimum wage on stable outcomes are qualitatively exactly the same as in the case where the principals make offers.10 Therefore—and this is important—all effects of minimum wages we identify when principals make offers are the consequence of stability and not the consequence of the specific bargaining procedure or information structure. 1.1 Related literature The empirical literature on minimum wages is extensive; see (Neumark and Wascher 2008; Manning 2021) for overviews. Early studies summarized by Brown et al. (1982) 10 For the boundary case where the stable outcome coincides with the case where agents make offers, the same effects arise as in case where the principals make offers, except that agents can never benefit from a minimum wage. 123 Analyzing the effects of minimum wages 949 mainly found negative employment effects of minimum wages. This view was challenged by the famous study of Card and Krueger (1994), who find no indication for a reduction of employment. The debate on the employment effect of minimum wages is ongoing and there is still no consensus (Neumark 2019; Neumark et al. 2014b,a; Allegretto et al. 2011,2017; Dube et al. 2010; Clemens and Wither 2019; Fang and Lin 2015; Addison et al. 2013; Liu et al. 2016; Thompson 2009; Muravyev and Oshchepkov 2016; Meer and West 2016; Wolfson and Belman 2019; Kreiner et al. 2020; Caliendo et al. 2018; Cengiz et al. 2019; Harasztosi and Lindner 2019; Aaronson et al. 2018). In the conclusion, we identify several factors that influence the employment effects of minimum wages, which might help us to understand the large variety of the empirical findings. There is also a rich theoretical literature on minimum wages; see Neumark and Wascher (2008); Flinn (2010) for overviews. The traditional view is based on the supply-demand model, also referred to as the neoclassical competitive model (Mankiw 2017; Card and Krueger 1995). In its basic form, a unique equilibrium wage obtained at the intersection of the labor supply and demand curves is paid to all workers. A minimum wage is only effective if it is above this equilibrium wage. Such an effective minimum wage lowers employment (due to a decreased labor demand), causes unemployment (since labor demand falls short of labor supply), and lowers the generated surplus (by causing a deadweight loss). However, only the jobs that generate the lowest levels of surplus are destroyed. Since only marginal workers lose there jobs, a minimum wage has only a second-order effect on efficiency (Lee and Saez 2012, page 739). Overall, a minimum wage benefits workers who stay employed, but harms employers and workers who become unemployed. Another class of models takes into account that employers could have market power. As explained by Robinson (1933) and Stigler (1946), a monopsonist optimally chooses an employment level below the competitive equilibrium in order to reduce its total wage payment. A minimum wage removes the monopsonist’s incentive to keep employment artificially low and, thus, increases the wage (to the minimum wage) and the employment. This is beneficial for the workers and the generated surplus (since the deadweight loss is reduced) but harmful for the employer. In the related setting of monopsonistic competition studied by Bhaskar and To (1999), a minimum wage also increases firmlevel employment, but may at the same time cause an exit of firms, thus leading to ambiguous effects on the aggregate employment level. A third class of models is based on search theory. In these models, the wage offer distribution emerges as the equilibrium of a noncooperative wage search and wage posting game between workers and employers, and a minimum wage changes the game. See Flinn (2010) for a comprehensive overview. In a seminal paper, Burdett and Mortensen (1998) show that a minimum wage increases employment and shifts the equilibrium wage offer distribution to the right. Hence, the common result of the existing literature, according to which employed agents benefit from a binding minimum wage, also holds here. This type of model is typically referred to as “dynamic monopsony” since search-related frictions induce monopsony-like behavior (see Neumark and Wascher (2008, Chapter 3.2.2)). Heterogeneous preferences over job characteristics are considered by Bhaskar et al. (2002); Bhaskar and To (1999,2003) to analyze minimum wages. These authors 123 950 C. Thielen, P. Weinschenk assume that firms are unable to make individual offers to workers that depend on the workers’ specific preferences. Moreover, their work focuses more on the aggregate firm and industry employment effects of minimum wages, while our paper distinctly emphasizes the potential heterogeneity of wage, employment, and welfare outcomes. In our model, depending on the specific scenario, a (higher) minimum wage may lead to agents continuing to work for the same principals and accruing a higher or lower utility, relocating to different principals and experiencing a lower utility, or ending up out of work entirely—and these effects might be heterogeneous among agents. Thus, minimum wages can have a rich set of effects in our model, and we provide a complete characterization of them. 2 Model description We now introduce our model for the case of a single agent, which is extended to multiple agents in Sect. 6. Consider an agent (interpreted as a worker) who can work for one of nprincipals (interpreted as potential employers). If the agent works for principal i∈N={1,...,n}, his utility is ui=wi−ci, where wiis the wage paid by principal iand ciis the agent’s cost when working for i.11 The cost depend on the job characteristics at principal i, e.g., commuting costs, type of work, working conditions, working times, and job security. If the agent does not work for any principal, we say the agent is unemployed and his utility is u0=0.12 The cost ciis thus the agent’s opportunity cost, i.e., the payment for which he is indifferent between working for principal iand not working. We analyze both the case of unrestricted wages and the case of restricted wages, where a minimum wage wrequires that wi≥w. We underline all variables in case of restricted wages. The profit of principal iif the agent works for her is πi=θi−wi, where θiis the agent’s productivity13 at principal i, while the profit is normalized to zero if the agent does not work for her. We consider only principals for which the 11 We can also allow for non-linear utility functions u(wi,ci)=h(wi)−ci(or some monotone transformation of it). This is equivalent to the case ui=wi−ciwhen transforming the productivities via h. Interestingly, if the utility function is concave in wi, some of the possible negative effects of a minimum wage on the agent’s utility and on the surplus might become even stronger. 12 The case where the agent has a nonzero reservation utility is equivalent to the case with zero reservation utility and all costs increased by the initial reservation utility. 13 The productivity θican be interpreted as principal i’s gross profit if the agent works for her. In case productivity is stochastic, θiis interpreted as expected gross profit. Similarly, if the agent’s cost is stochastic, cican be interpreted as expected cost. 123 Analyzing the effects of minimum wages 951 productivity exceeds the cost (i.e., θi>cifor all i)14 and let θmax :=maxiθidenote the maximum productivity. The surplus generated when the agent works for principal iis si=πi+ui=θi−ci. If the agent does not work for any principal, the surplus is s0=0. We call the principal the agent works for the winning principal and let i∗denote her index. If the agent does not work for any principal, we set i∗=0. The remaining principals N\{i∗}are referred to as the losing principals. The maximum surplus is smax :=maxisiand Nmax :={i∈N:si=smax}is the set of principals where this surplus can be generated. In case of a minimum wage, we denote the principals who can afford the minimum wage without making a loss by N:={i∈N:θi≥w}. The maximum surplus among these principals is smax :=max{0,si:i∈N}. The set of principals who can afford the minimum wage and yield this surplus is Nmax :={i∈N:si=smax}. Recognize that different combinations of productivity and cost may not only stem from principals who are possibly located in different geographic regions or industrial sectors, but also from the possibility of choosing working hours, investments in working conditions, or efforts; see Appendix D.2. In all these scenarios, a higher productivity is naturally associated with a higher cost, but not necessarily with a higher surplus. 3 Basic bargaining procedure We start by considering the basic bargaining procedure where the principals make takeit-or-leave-it offers. This procedure is predominant in the agency literature (cf. Laffont and Martimort (2002)) and allows for a simple and intuitive characterization. As we show later, the effects of minimum wages we obtain by using this procedure are robust. The procedure is as follows: First, each principal ioffers a wage wi∈Rto the agent or makes no offer. In case a minimum wage wis imposed, the offered wages must satisfy wi≥w. After receiving all offers, the agent either accepts one of the offers or rejects all offers, and the payoffs are realized. We assume that the principals know the parameters of the model, which enables each principal to determine her best response to the other principals’ offers. This assumption is not needed when interpreting the equilibrium of the basic bargaining procedure as the result of ascending offers made by principals (cf. Appendix A.5). It is not needed either when examining stable outcomes (cf. Appendix B.2), where one abstracts from how the parties bargain and what they know. For simplicity of exposition, we do not model the agent as a strategic player that acts after the principals have made their offers. Instead, rationality of the agent is modeled by assuming that he always accepts an offer that maximizes his utility or no offer if all offers provide negative utility. In case of indifference, we let the agent 14 Principals ifor which θi≤ciare redundant since employing the agent can never yield a positive profit/utility for one of the two parties without yielding a negative utility/profit for the other party. 123 952 C. Thielen, P. Weinschenk behave according to the following tie-breaking rules that, as we show in Appendix A.1, exactly model the agent’s strategic behavior in any subgame-perfect Nash equilibrium of the extensive form game. First, in case of indifference, the agent prefers to work, i.e., accept an offer instead of no offer. Second, if several offers maximize the agent’s utility, he chooses one that maximizes the surplus among these offers. Third, if several offers maximize the agent’s utility and the surplus, the agent chooses an offer according to an arbitrary deterministic tie-breaking rule. We concentrate on Nash equilibria in undominated strategies.15 For some results, we require the tie-breaking rule used in case that several offers simultaneously maximize the agent’s utility and the surplus to satisfy the property of independence of irrelevant alternatives: Definition 3.1 A tie-breaking rule satisfies independence of irrelevant alternatives if the following holds for any two subsets N ⊆N⊆N: If the tie-breaking rule selects principal jamong the principals in N, it also selects jamong the principals in N whenever j∈N. We next characterize the pure-strategy Nash equilibria both for unrestricted and restricted wages and then analyze the effects of minimum wages. 3.1 Unrestricted wages The following proposition establishes the existence of a Nash equilibrium. In the specified equilibrium, the principals where the maximum surplus can be generated offer wages equal to the second-highest surplus plus the agent’s cost,16 while the remaining principals offer wages equal to the productivity. Proposition 3.1 For unrestricted wages, the strategy profile wi=max{0,θj−cj:j∈N\{i}} + cifor i ∈Nmax θifor i /∈Nmax constitutes a Nash equilibrium yielding the maximum surplus smax. Proof Observe that, for all i∈Nmax and all i/∈Nmax, the offers in Proposition 3.1 yield ui=wi−ci=max{0,θj−cj:j∈N\{i}} ≥ θi−ci=wi−ci=ui. Hence, ui≥0i∈Nmax and the agent cannot do better than accepting an offer from a principal i∗∈Nmax such that the maximum surplus smax is generated. It remains to show that no principal can improve by changing her offer. Each principal i/∈Nmax—i.e., each principal where the agent cannot generate the maximum 15 Weakly dominated strategies are discussed in Appendix A.2. 16 In case of n≥2 principals, an alternative interpretation is that each principal iwhere the maximum surplus can be generated offers a wage equal to the wage offered by a best competitor j(i.e., a principal with the second-highest surplus) plus the cost differential ci−cj. 123 Analyzing the effects of minimum wages 959 Fig. 1 Effects on the paid wage Theorem 4.3 Suppose there is a minimum wage w≤θmax. (I) If i∗=i∗and (1) w>w i∗, the agent’s utility increases, i.e., ui∗>ui∗. (2) w=wi∗, the agent’s utility remains unchanged, i.e., ui∗=ui∗. (3) w<w i∗, the agent’s utility decreases weakly, i.e., ui∗≤ui∗. (II) If i∗= i∗, the agent’s utility decreases weakly, i.e., ui∗≤ui∗. When the tie-breaking rule satisfies independence of irrelevant alternatives, Case (I) applies if and only if w≤θi∗, and Case (II) applies otherwise. Proof (I) (1) Follows since w>w i∗and wi∗≥wsuch that wi∗>w i∗as well as i∗=i∗. (2) If w=wi∗, then w−ci∗=wi∗−ci∗. Since i∗=i∗and, by Proposition 3.2, wi∗−ci∗=max{0,θj−cj:j∈N\{i∗}}, this yields w−ci∗=max{0,θj−cj:j∈N\{i∗}}. Because the maximum is no smaller than max{0,θj−cj:j∈N\{i∗}}, maxw−ci∗,max{0,θj−cj:j∈N\{i∗}}=max{0,θj−cj:j∈N\{i∗}}. By Propositions 3.2 and 3.4, this is equivalent to ui∗=ui∗. (3) If w<w i∗, the same argumentation as in the proof of (2) shows that w−ci∗<max{0,θj−cj:j∈N\{i∗}}. Since max{0,θj−cj:j∈N\{i∗}} ≤ max{0,θj−cj:j∈N\{i∗}}, Propositions 3.2 and 3.4 imply that ui∗≤ui∗. 123 960 C. Thielen, P. Weinschenk (II) Proposition 3.2 and i∗= i∗yield ui∗=max{0,θj−cj:j∈N\{i∗}} ≥ θi∗−ci∗≥wi∗−ci∗=ui∗, where the second inequality follows since the paid wage wi∗cannot exceed θi∗. The claim that, when the tie-breaking rule satisfies independence of irrelevant alternatives, Case (I) applies if and only if w≤θi∗, while Case (II) applies otherwise, follows directly from the definition of independence of irrelevant alternatives and the fact that i∗∈Nmax if and only if w≤θi∗. Theorem 4.3 shows that a minimum wage increases the agent’s utility when the minimum wage exceeds the initial wage and there is no relocation. Otherwise, the agent’s utility decreases at least weakly. This also implies that the agent can never benefit from wage overshooting, i.e., the wage increase can never compensate the agent’s cost increase. We show next that a minimum wage could cause a strict decrease of the agent’s utility. Corollary 4.4 Suppose there is a minimum wage w≤θmax. (I) If i∗=i∗, the agent’s utility decreases (i.e., ui∗<ui∗) if and only if w<w i∗and (EBC)holds. (II) If i∗= i∗, the agent’s utility decreases (i.e., ui∗<ui∗) if and only if |Nmax|=1or i∗/∈argmax{θj−cj:j∈N\{i∗}}.(UD) Proof (I) Since i∗=i∗, the agent’s cost stay constant such that the agent’s utility decreases if and only if the paid wage decreases. By Corollary 4.3, this happens if and only if w<w i∗and max{0,θj−cj:j∈N\{i∗}} + ci∗<max{0,θj−cj: j∈N\{i∗}} + ci∗, where the latter condition reduces to (EBC) since i∗=i∗. (II) A decrease in the agent’s utility occurs if and only if at least one of the two inequalities in Case (II) of the proof of Theorem 4.3 is strict, which happens exactly if |Nmax|=1ori∗/∈argmax{θj−cj:j∈N\{i∗}}. A minimum wage can thus reduce the agent’s utility for two reasons. First, when the minimum wage allows the winning principal to lower the wage payment, which arises whenever the minimum wage eliminates the principal’s best competitor(s). See Example 4.2 for an illustration. Second, when the minimum wage causes the agent to relocate to another principal and condition (UD) holds, which is generically true. This is illustrated in Example 4.1, where the minimum wage lowers the agent’s utility from 6 to 4.23 Intuitively, the agent’s utility generically suffers if he relocates because the higher cost overcompensates the higher wage payment. Summarizing, even if the agent stays employed after the introduction or increase of a minimum wage, his utility may suffer—no matter whether he switches to another principal or not. If he switches, the agent’s utility generically decreases, and this holds even if the minimum wage exceeds the previously paid wage. See Figure 2for the overview. 23 In this example, for all minimum wages between 9 and 12, condition (UD) is only violated in the non-generic case where θ1−c1=θ2−c2. 123 Analyzing the effects of minimum wages 961 Fig. 2 Effects on the agent’s utility 4.4 Effects on the principals’profits By comparing Propositions 3.2 and 3.4, we directly obtain the effects on the profits. Theorem 4.4 Suppose there is a minimum wage w≤θmax. (I) Principal i’s profit decreases if and only if either (a) i =i∗= i∗and |Nmax|= 1,or(b)i=i∗=i∗and w>w i∗. (II) Principal i’s profit increases if and only if either (a) i =i∗= i∗,w= θi∗, and |Nmax|=1,or(b)i=i∗=i∗,w<w i∗, and (EBC)holds. Theorem 4.4 shows that a principal can benefit from a minimum wage for two reasons. First, if the minimum wage forces an agent to relocate, the principal who is able to attract the agent now generically earns a positive profit. See Example 4.1. Second, if the minimum wage shrinks the set of competing principals when no relocation occurs and the minimum wage falls short of the initial wage. See Example 4.2. Interestingly, a minimum wage can simultaneously lower the agent’s utility and the aggregate profits, as Corollary 4.4 and Theorem 4.4 reveal. See the second example in the introduction for an illustration. 5 Alternatives to the basic bargaining procedure In Appendix B, we examine alternative bargaining procedures. First, we let the agent propose wages to the principals. Second, we abstract from how the parties bargain 123 962 C. Thielen, P. Weinschenk and what they know by examining stable outcomes. We show that the set of stable outcomes is given by the convex combinations of the outcome obtained when the principals make offers and the outcome when the agent makes offers. Remarkably, for any fixed bargaining power of the agent except for the boundary case corresponding to the situation where the agent makes offers, the effects of minimum wages for stable outcomes are always qualitatively identical to the effects observed when the principals make offers. That is, if a minimum wage increases [decreases, does not change] the surplus, the wage, the agent’s utility, or the principals’ profits for stable outcomes, the same holds true when the principals make offers (and vice versa). This implies that all effects of minimum wages we have identified before for the procedure where the principals make offers are the consequence of the stability requirement, and not the consequence of the specific bargaining procedure or information structure. 6 Multiple agents In Appendix C, we show that the model is readily generalized to the case where multiple agents interact with multiple principals. An important insight we obtain is that, in the situation where the principals have no (or no binding) capacity constraints that limit the numbers of agents they can employ, the principals compete for each of the magents independently of the presence of the other agents. Accordingly, the situation decomposes into msingle-agent problems, to which the analysis from the previous sections applies. In the absence of binding capacity constraints, all results shown for the single-agent case thus fully carry over to case with multiple agents. The same holds true if each principal has a capacity of one and the multi-agent model is generated by a duplication of a single-agent model. For the general case of capacity constraints, we prove the existence of stable outcomes via an algorithmic approach and show that the previously obtained effects of minimum wages still arise. 7 Conclusion This paper uses a microeconomic approach to study the effects of minimum wages. We allow that agents can have different productivities at different principals as well as different costs of working. We identify a rich set of effects. We inter alia show that minimum wages may also lower efficiency when leaving employment unaffected, and destroy jobs that generate high levels of surplus when affecting employment. Minimum wages could harm agents even if these stay employed, while principals could benefit from minimum wages. Our analysis further reveals that the productivity gains caused by minimum wages are generically accompanied by efficiency losses. We provide a complete characterization of the effects of minimum wages and show that all effects are robust. The effects are the consequence of stability, and not the consequence of specific bargaining procedures or information structures. The results are policy-relevant. Suppose, for instance, that a government executes a minimum wage reform, and the unemployment rates stay unchanged. Our insights imply that we then cannot conclude that the reform necessarily helps workers and 123 Analyzing the effects of minimum wages 963 leaves efficiency unchanged. Another important issue is that we have to distinguish between the productivity effects and the efficiency effects caused by minimum wages. If we observe that a minimum wage reform increases workers’ productivities, these gains should not be taken as evidence of a positive efficiency effect. The model also provides testable implications. First, an important mechanism in the model is that minimum wages may cause the relocation of workers to other employers where they suffer from higher costs. One could test whether minimum wages affect key componentsofworkers’costs,e.g., increasecommutingcosts/distances.24 Second, the model predicts that minimum wages reduce employment to a lesser extent (a) in environments where parties have more information,25 (b) in more diverse economic environments,26 and (c) when better transportation technologies are available.27 Since the information and transportation technologies have improved substantially over the last decades, the model predicts that the employment effect of minimum wages should be weaker nowadays than in the past. This is precisely the pattern meta studies find.28 Finally, the model might also help us to understand the variety of the empirical findings in given time periods, which is a notable puzzle of the empirical literature on minimum wages (cf. the brief overview in the introduction). Some of the variety of the empirical findings could be due to differences in the afore-described factors (a)–(c) among the different studies. Appendices A Discussion of the bargaining procedure and equilibrium concept In this section, we discuss and substantiate the basic bargaining procedure and equilibrium concept used in Sects. 3and 4. In Sect. A.1, we justify the tie-breaking rules used in this bargaining procedure. Sect. A.2 shows the effects of allowing weakly dominated strategies to be played by the principals, while Sect. A.3 considers the generalization of the equilibrium concept to mixed-strategy Nash equilibria. Section A.4 shows that the effects observed when introducing a minimum wage carry over to the case where an existing minimum wage is increased. Finally, Sect. A.5 analyzes an 24 Dustmann et al. (2022) document that the introduction of a minimum wage in Germany has increased the commuting distances of low-wage workers by 1.5 km (or 8%) relative to high-wage workers. Moreover, as predicted by our model, the minimum wage has led to a relocation of low-wage workers to more productive firms. These observations can also be explained by the model of Bhaskar and To (1999), who also note that some workers might have to accept less preferred jobs after the introduction of a minimum wage. 25 Formally, more information leads to more employment opportunities, which expands the set Nand, thus, causes an agent to remain employed for a larger set of minimum wages. 26 A more diverse economic environment can be captured by a mean-preserving spread of the productivities, which causes an agent to remain employed for a larger set of minimum wages. 27 Lower transportation costs lead to an expansion of the set of principals for whom an agent can profitably work and, thus, to a larger set of minimum wages for which the agent stays employed. 28 In their review, Brown et al. (1982) establish a much-cited consensus that the employment elasticity of minimum wages was between −0.3and−0.1. Wolfson and Belman’s meta-analysis (Wolfson and Belman 2019) uses more recent data and establishes that the range has shifted to −0.13 and −0.07. 123 964 C. Thielen, P. Weinschenk alternative bidding procedure in which the principals make ascending offers to the agent. A.1 Justification of the tie-breaking rules We now discuss the tie-breaking rules used in the basic bargaining procedure by showing that, in the natural two-stage game where the principals first make their offers and the agent then either chooses one of these offers or rejects all offers, he always behaves according to these rules in any subgame-perfect Nash equilibrium. While we concentrate on the case of unrestricted wages in the following discussion, the arguments readily carry over to the case of restricted wages. The first proposition shows that, in any subgame-perfect Nash equilibrium of the two-stage game, the agent always chooses an offer maximizing the surplus if several offers maximize his utility: Proposition A.1 Suppose that the principals’ offers are such that i,i ∈argmaxi∈Nwi −ciwith si>si . Then the agent choosing principal i does not constitute a subgame -perfect Nash equilibrium. Proof If wi >θ i , then πi =θi −wi <0, so principal i can increase her profit to zero by reducing her offer to θi .Ifwi ≤θi , then ui =wi −ci ≤θi −ci = si <si=θi−ci. Consequently, principal ican increase her profit by offering θi−for some 0 <<θ i−ci−si (which makes the agent accept the offer of i).  The next proposition shows that, in any subgame-perfect Nash equilibrium, the agent always accepts an offer instead of no offer in case of indifference: Proposition A.2 Suppose that the principals’ offers satisfy maxi∈Nwi−ci=0. Then the agent choosing to reject all offers does not constitute a subgame-perfect Nash equilibrium. Proof If the agent rejects all offers, the assumption that θi>cifor all iimplies that any principal i∈Ncan increase her profit by offering wi=ci+for some 0<<θ i−ci(which makes the agent accept i’s offer).  Our last assumption on the agent’s behavior states that he always chooses an offer according to an arbitrary deterministic tie-breaking rule (e.g., choosing the offer of the principal with the lowest index) in case that several offers simultaneously maximize both his utility and the surplus. While using a stochastic tie-breaking rule in this case would make the winning principal and (possibly) the paid wage stochastic, it would not influence the agent’s utility, the generated surplus, or the principals’ profits. A.2 Allowing for weakly dominated strategies In the analysis of the basic bargaining procedure, we concentrate on equilibrium strategies that are not weakly dominated. If weakly dominated strategies are allowed, Part (I) 123 Analyzing the effects of minimum wages 965 of Proposition 3.2 and Parts (I) and (II), (1) of Proposition 3.4 still hold by the same argumentation as in the original proofs. Consequently, the winning principal and the equilibrium surplus remain unchanged and are still uniquely determined. However, the wage and, hence, the distribution of the surplus between the winning principal and the agent are, in general, not unique anymore even for a fixed winning principal i∗∈Nmax. We now demonstrate this for the case of unrestricted wages – the case of restricted wages in similar. Specifically, we now show that, for unrestricted wages, any wage wi∗∈[max{0,θj−cj:j∈N\{i∗}} + ci∗,θ i∗]can be obtained in a Nash equilibrium if weakly dominated strategies are allowed. This holds since the following strategy profile constitutes a Nash equilibrium: principal i∗∈Nmax offers wi∗∈[max{0,θj−cj:j∈N\{i∗}} + ci∗,θ i∗], at least one other principal j∈N\{i∗}offers wj=wi∗+cj−ci∗, and each of the remaining principals k∈N\{i∗,j}offers a wage wkweakly below wi∗+ck−ci∗. Note that agent j’s wage offer wjexceeds the productivity θj—and is, thus, weakly dominated—unless j∈argmax{θj−cj:j∈N\{i∗}} and wi∗=max{0,θj−cj:j∈N\{i∗}} + ci∗. Two remarks are in order. First, an equilibrium in which weakly dominated strategies are played requires an implausibly high degree of coordination among players. At least one principal jwho plays a weakly dominated strategy must exactly match the utility offered to the agent by the winning principal i∗∈Nmax, despite the fact that there is a continuum of possible utilities the agent could be offered by i∗.Simultaneously, principal jtakes the chance of a loss if coordination fails, i.e., if she offers a slightly higher utility. This is why equilibria in weakly dominated strategies are usually ruled out. Second, it is interesting to recognize that, although Nash equilibria in weakly dominated strategies are implausible in the basic bargaining procedure where the principals design the offers, the set of possible equilibrium wages that can result from these equilibria is identical to the set of wages in stable outcomes; cf. the characterization of stable outcomes in Sect. B.2. A.3 Mixed-strategy nash equilibria We next show that, for both unrestricted and restricted wages, the generated surplus, the agent’s utility, and the principals’ profits in any (mixed-strategy) Nash equilibrium are as in every pure-strategy Nash equilibrium with probability 1. Consequently, the effects of a minimum wage on the surplus, utility, and profits do not depend on whether pureor mixed-strategy equilibria are considered. With respect to the paid wage, this also holds if either we are in the generic case where |Nmax|=|Nmax|=1, or |Nmax|,|Nmax|≤2 and a deterministic tie-breaking rule is used in case that several offers simultaneously maximize both the agent’s utility and the surplus. If |Nmax|≥3or|Nmax|≥3, the equilibrium wage can be stochastic in some mixed-strategy equilibria. Theorem A.1 Suppose that the wages are unrestricted and |Nmax|=1, i.e., there is a unique principal imax ∈Nmax. Then every (mixed-strategy) Nash equilibrium satisfies: (I) With probability 1, principal imax offers a wage of ˇwimax :=max{0,θj−cj:j∈ N\{imax}} + cimax . (II) With probability 1, the winning principal is i∗=imax and wi∗=ˇwimax . 123 966 C. Thielen, P. Weinschenk Consequently, with probability 1, the agent’s utility, the principals’ profits, and the generated surplus are as in every pure-strategy Nash equilibrium (cf. Proposition 3.2). Proof (I) Since no principal iever offers a wage wi>θ i, principal imax wins with probability 1 when offering wimax =ˇwimax , so offering a higher wage could only decrease her expected profit. Consequently, we must have wimax ≤ˇwimax with probability 1 and it only remains to show that also wimax ≥ˇwimax with probability 1. Suppose for the sake of a contradiction that principal imax offers a wage lower than ˇwimax with positive probability. If |N|=1, i.e., if imax is the only principal, we have ˇwimax =cimax , meaning that imax offers a wage lower than the agent’s cost with positive probability. This is a contradiction since offering a wage below the agent’s cost yields profit zero for principal imax whereas she can obtain positive profit by offering wimax :=ˇwimax =cimax <θ imax . For the case where |N|≥2, we let suppidenote the support of i’s mixed strategy for i∈N. Then, since imax offers a wage below ˇwimax with positive probability, we have inf(suppimax )< ˇwimax , and we claim that the offers of the other principals satisfy max{wi−ci:i∈N\{imax}} >inf(suppimax )−cimax (A.1) with probability 1. Note that if this was not the case, we would in particular have inf(suppi)−ci≤inf(suppimax )−cimax for any principal i∈argmaxi=imax si,so shehas offers inhersupport thatmakeherlose with probability1and, consequently, yield expected profit zero. For any 0 << ˇwimax −inf(suppimax ), however, an offer of wi:=θi−would make principal iwin with positive probability and, thus, yield a positive expected profit, which is a contradiction. Consequently, (A.1) holds with probability 1, which means that there exist offers in suppimax for which principal imax loses (and obtains profit zero) for sure. This yields a contradiction since principal imax can obtain positive profit by offering wimax :=ˇwimax . (II) Since no principal iever offers a wage wi>θ iand principal imax offers ˇwimax with probability 1, it follows directly that i∗=imax and wi∗=ˇwimax with probability 1.  Theorem A.2 Suppose that the wages are unrestricted and |Nmax|≥2. Then every (mixed-strategy) Nash equilibrium satisfies: (I) At least two principals i ∈Nmax offer wages of wi=θiwith probability 1. (II) With probability 1, the winning principal i∗belongs to Nmax and wi∗=θi∗. Consequently, with probability 1, the agent’s utility, the principals’ profits, and the generated surplus are as in every pure-strategy Nash equilibrium (cf. Proposition 3.2). Proof (I) We first show that at least one principal i∈Nmax offers wi=θiwith probability 1. Suppose for the sake of a contradiction that no principal i∈Nmax offers a wage of wi=θiwith probability 1. Then, since no principal i∈Nmax ever offers a wage wi>θ iin equilibrium, we must have inf(suppi)<θ ifor all i∈Nmax,somin{θi−inf(suppi):i∈Nmax}= :>0. 123 Analyzing the effects of minimum wages 967 We now distinguish two cases: Case 1: argmax{θi−inf(suppi):i∈Nmax}Nmax. Consider any principal i0∈argmax{θi−inf(suppi):i∈Nmax}. Because argmax{θi−inf(suppi):i∈Nmax}Nmax, there exists a subset of suppi0 with positive probability mass such that all offers in this set make principal i0lose with probability 1 and, thus, yield expected profit zero for i0. However, denoting the difference between the maximum surplus and the second-highest surplus by :=smax −max{0,si:i∈N\Nmax}>0, principal i0can win with positive probability and obtain a positive expected profit by offering wi0=θi0−min{,} 2, which is a contradiction. Case 2: argmax{θi−inf(suppi):i∈Nmax}=Nmax. We distinguish two subcases: Case 2.1: For some principal i1∈Nmax, the offer wi1=inf(suppi1)has probability 0 (i.e., there is no mass point at inf(suppi1)). Let i0∈Nmax \{i1}be arbitrary. Then, as in Case 1, offering wi0=θi0−min{,} 2 yields a positive expected profit for principal i0, which we denote by ˜πi0. Since the probability of principal i1offering wi1=inf(suppi1)is zero, however, principal i0’s probability of winning tends to zero as wi0approaches inf(suppi0). Thus, since principal i0’s profit obtained from any offer is upper bounded by si0,also principal i0’s expected profit tends to zero as her offer approaches inf(suppi0). Consequently, there exists a subset of suppi0with positive probability mass such that all offers in this subset yield an expected profit lower than ˜πi0for principal i0, which is a contradiction. Case 2.2: Every principal i∈Nmax offers wi=inf(suppi)with positive probability (i.e., there is mass point at inf(suppi)for every i∈Nmax). Then, with positive probability, all principals in Nmax offer wi=inf(suppi) at the same time. If some principal i0∈Nmax had expected profit of zero in this situation (and hence, also unconditional expected profit zero from the offer wi0=inf(suppi0)), this principal could again obtain positive expected profit by offering wi0=θi0−min{,} 2(where is as in Case 1), which is a contradiction. Consequently, all principals in Nmax must have positive expected profit in this situation and, in particular, each one must have a positive probability of winning. Thus, each principal in Nmax must also have a positive probability of losing when all principals i∈Nmax offer wi=inf(suppi). This yields a contradiction since each principal in Nmax could then improve her (unconditional) expected profit by slightly raising her offer. Hence, in all cases, at least one principal i∈Nmax offers a wage of wi=θi with probability 1. Now suppose that only one principal i0∈Nmax offers a wage of wi0=θi0with probability 1. Then, inf(suppi)<θ ifor all i∈Nmax \{i0} and, thus, ¯:=min{θi−inf(suppi):i∈Nmax \{i0}} >0. Then, offering wi0=θi0−min{¯,} 2for as in Case 1 above yields positive expected profit for principal i0whereas i0’s current offer of wi0=θi0yields expected profit zero, which is a contradiction. Consequently, at least two principals i∈Nmax must offer wages of wi=θiwith probability 1 as claimed. 123 968 C. Thielen, P. Weinschenk (II) Since no principal iever offers a wage wi>θ iand at least two principals i∈Nmax offer wages of wi=θiwith probability 1, it follows directly that i∗∈Nmax and wi∗=θi∗with probability 1.  Theorem A.3 When a minimum wage wis imposed, every (mixed-strategy) Nash equilibrium satisfies: (I) If N =∅, then, with probability 1, no principal makes an offer. (II) If N =∅: (1) If |Nmax|=1, i.e., there is a unique principal imax ∈Nmax, then, with probability 1, principal imax offers a wage of ˇwimax :=maxw,max{0,θj−cj: j∈N\{imax}} + cimax and i∗=imax. (2) If |Nmax|≥2, then, with probability 1, at least two principals i ∈Nmax offer wages of wi=ˇwi=θiand the winning principal i∗belongs to Nmax. Consequently, with probability 1, the agent’s utility, the principals’ profits, and the generated surplus are as in every pure-strategy Nash equilibrium (cf. Proposition 3.4). Proof (I) If N=∅and some principals offer wages with positive probability, then at least one principal’s expected profit will be negative, so she could improve by never making an offer. (II) For any principal i∈N\N, all mixed strategies in which imakes an offer with positive probability are weakly dominated by the strategy in which inever makes an offer. Hence, the principals in N\Nnever make offers with positive probability and the game reduces to a game between the principals in N. Then, Claim (1) follows since either |N|=1, in which case the only remaining principal imax offers a wage of max{w, cimax }= ˇwimax ,or|N|≥2, in which case one can argue as in the proof of Theorem A.1 except that the offered wages cannot fall short of w. Similarly, for Claim (2), one can argue as in the proof of Theorem A.2 (again taking into account that the offered wages cannot fall short of w).  A.4 Increasing an existing minimum wage Anothernatural question is whether theeffectsobservedin ourmodel when introducing a minimum wage carry over to the case where an existing minimum wage is increased. To this end, consider an instance in which an initial minimum wage winitial already exists and is increased to wnew >w initial.LetNinitial :={i∈N:θi≥winitial}and Nnew :={i∈N:θi≥wnew}denote the set of principals for which the productivity weakly exceeds the initial/new minimum wage, respectively. Moreover, let Ninitial max := argmaxi∈Ninitial si. We claim that the instance can be transformed into an equivalent instance in which no minimum wage exists initially. For the construction of the transformed instance, we distinguish two cases: Case 1: |Ninitial max |≥2, or |Ninitial max |=1 and cimax ≥winitial for imax ∈Ninitial max . 123 Analyzing the effects of minimum wages 975 If |Nmax|=1, the winning principal is again unique and wNE i∗<ˆwNE i∗. Consequently, the unique value αcan again be interpreted as the agent’s bargaining power, and we can express the agent’s utility and the winning principal’s profit as ui∗=α·ˆuNE i∗+(1−α) ·uNE i∗=: uα i∗and (B.5) πi∗=α·ˆπNE i∗+(1−α) ·πNE i∗=: πα i∗,(B.6) which implies that ui∗∈[uNE i∗,ˆuNE i∗]and πi∗∈[ˆπNE i∗,πNE i∗]for any stable outcome. Similar to the case of unrestricted wages, the agent’s bargaining power αthus determines how the parties share the excess surplus created by their relationship. Formally, ui∗∈[uNE i∗,ˆuNE i∗]=[σ, smax]and πi∗∈[ˆπNE i∗,πNE i∗]=[0,smax −σ], where σ:=max{w−ci∗,ssecond}with ssecond :=max{0,θj−cj:j∈N\{i∗}} denoting the second-highest surplus among the principals in N. B.2.3 Effects of minimum wages In order to explore the effects of a minimum wage, we consider the case where the agent does not get unemployed, i.e., where w≤θmax. Moreover, in order to avoid tedious case distinctions, we concentrate on the generic case where |Nmax|=1 and |Nmax|=1, in which case there exists a unique winning principal i∗∈Nmax without a minimum wage and a unique winning principal i∗∈Nmax with a minimum wage. We additionally assume that the bargaining power of the agent does not change when introducing the minimum wage, i.e., α=α.31 As we can see from Equations (B.1)to(B.6), the effects of a minimum wage w on the wage, utility, and winning principal’s profit for any fixed value of α∈(0,1) are given by the weighted effects arising in the extreme cases α=0 and α=1. The following theorem shows that, actually, the qualitative effects on each of these variables for any α∈(0,1)are the same as for α=0. Theorem B.2 Suppose that |Nmax|=|Nmax|=1. The following holds for each of the variables generated surplus, paid wage, agent’s utility, and winning principal’s profit: If the minimum wage wdecreases the variable / leaves the variable unchanged / increases the variable for α=0, then the same applies for all α∈[0,1). Proof We consider each of the variables separately. For the generated surplus, we note that, by Propositions B.3 and B.4, without a minimum wage, every stable outcome generates a surplus of smax and, with a minimum wage w, every stable outcome generates a surplus of smax. Thus, the change in the generated surplus is identical for all α∈[0,1]. 31 Note that, as described before, the agent’s bargaining power determines how the parties share the excess surplus smax −ssecond (or smax −σin case of restricted wages) created by their relationship. Hence, while both the generated surplus and the excess surplus—and, thus, the amount of surplus over which the parties bargain—may change when introducing a minimum wage, it is natural to assume that the agent’s bargaining power remains unchanged. 123 976 C. Thielen, P. Weinschenk For the paid wage, we note that, with α=αand the unique winning principals i∗ and i∗without and with a minimum wage, respectively, Eqs. (B.1) and (B.4)show that wα i∗−wα i∗=α·wα=1 i∗−wα=1 i∗+(1−α) ·wα=0 i∗−wα=0 i∗.(B.7) If w≤wα=1 i∗, then wα=1 i∗−wα=1 i∗=0 by Theorem B.1,sowα i∗−wα i∗=(1−α) · (wα=0 i∗−wα=0 i∗), which shows that the change in the paid wage for any α∈(0,1) is qualitatively the same as for α=0. If w>w α=1 i∗, then also w>w α=0 i∗since wα=1 i∗=θi∗≥wα=0 i∗by Proposition B.1. Consequently, the minimum wage causes all wages to increase since wα=1 i∗−wα=1 i∗>0 and wα=0 i∗−wα=0 i∗>0in(B.7), so also wα i∗−wα i∗>0. Hence, the change in the paid wage for any α∈(0,1)is again qualitatively the same as for α=0. For the agent’s utility, we similarly have uα i∗−uα i∗=α·uα=1 i∗−uα=1 i∗+(1−α) ·uα=0 i∗−uα=0 i∗(B.8) by Equations (B.2) and (B.5). If w≤maxj∈Nmax θj=θi∗, then uα=1 i∗−uα=1 i∗=0by Theorem B.1,so uα i∗−uα i∗=(1−α) ·uα=0 i∗−uα=0 i∗, which shows that the change in the agent’s utility for any α∈(0,1)is qualitatively the same as for α=0. If w>maxj∈Nmax θj=θi∗, then i∗= i∗and the agent’s utility decreases in all cases since uα=1 i∗−uα=1 i∗<0 by Theorem B.1 and uα=0 i∗−uα=0 i∗<0 by Corollary 4.4. Thus, the change for any α∈(0,1)is again qualitatively the same as for α=0. For the winning principal’s profit, we have πα i∗−πα i∗=α·πα=1 i∗−πα=1 i∗+(1−α) ·πα=0 i∗−πα=0 i∗(B.9) by Equations (B.3) and (B.6). Exploiting that πα=1 i∗=πα=1 i∗=0 by Theorem B.1, this shows the desired result.  Theorem B.2 reveals that the qualitative effects of minimum wages on each single variable of the model obtained for the basic bargaining procedure (where the principals have all bargaining power, i.e., α=0) extend to all cases where the principals have nonzero bargaining power (i.e., α<1).32 32 Even though the theorem focuses on the generic case where |Nmax|=|Nmax|=1, an even stronger result holds in the nongeneric case where |Nmax|,|Nmax|≥2. Then, we have si∗=ui∗=smax,πi∗=0, 123 Analyzing the effects of minimum wages 977 Concerning wage overshooting (which is formally defined for an arbitrary value of αin the following definition), however, the quantitative effect of the minimum wage on the paid wage is also important, as we now demonstrate. Definition B.2 For a given bargaining power α∈[0,1]of the agent, a minimum wage w>w α i∗causes wage overshooting if the agent stays employed and the paid wage increases to wα i∗>w. The following theorem shows that, if wage overshooting occurs for a minimum wage win one of the extreme cases α=0orα=1, then it also occurs for all intermediate cases. Theorem B.3 Suppose that |Nmax|=|Nmax|=1. If a minimum wage wcauses wage overshooting for either α=0or for α=1, then the minimum wage walso causes wage overshooting for all α∈(0,1). Proof Note that |Nmax|=1 implies that the agent stays employed in all cases, so this prerequisite for wage overshooting does not have to be considered in the rest of the proof. First assume that wcauses wage overshooting for α=0. Fix some α∈(0,1). Then by Definition 4.1 and Corollary 4.2: wNE i∗>w>w NE i∗,(B.10) i∗= i∗.(B.11) Together with (B.4) and ˆwNE i∗≥w, this yields wα i∗=α·ˆwNE i∗  ≥w +(1−α) ·wNE i∗  >w >w. It remains to show that wα i∗<w. To this end, first note that i∗∈Nwould imply i∗∈Nmax,soi∗=i∗, which contradicts (B.11). Thus, we must have i∗/∈N, which implies that θi∗<wand also ˆwNE i∗<w. Hence, by (B.1) and (B.10) wα i∗=α·ˆwNE i∗  <w +(1−α) ·wNE i∗  <w <w.(B.12) Now assume that wcauses wage overshooting for α=1. Again fix some α∈(0,1). Then by the definition of wage overshooting given in Corollary B.1: ˆwNE i∗>w>ˆwNE i∗.(B.13) and wi∗=θi∗in any stable outcome without a minimum wage and, similarly, si∗=ui∗=smax, πi∗=0, and wi∗=θi∗in any stable outcome with a minimum wage. Consequently, the qualitative and quantitative effects on the generated surplus, the agent’s utility, and the winning principal’s profit are completely independent of which stable outcomes are considered, while the effect on the paid wage only depends on the choice of the winning principal without and with the minimum wage from the sets Nmax and Nmax, respectively. 123 978 C. Thielen, P. Weinschenk Together with (B.4), this yields wα i∗=α·ˆwNE i∗  >w +(1−α) ·wNE i∗  ≥w >w. In addition, since ˆwNE i∗=θi∗by Proposition B.1 and wNE i∗≤θi∗, the second inequality in (B.13) yields that also wNE i∗<w. Hence, by (B.1), we obtain wα i∗=α·ˆwNE i∗  <w +(1−α) ·wNE i∗  <w <w.  While wage overshooting in one of the extreme cases α=0orα=1 is sufficient for wage overshooting in all intermediate cases α∈(0,1), it is not necessary. Indeed, as the following example demonstrates, wage overshooting can occur for intermediate cases when it does not occur at the extremes: Example B.1 Consider a situation with two principals, where θ1=12, θ2=10 and c1=2, c2=2. Without a minimum wage, we have i∗=1, wi∗=10, and ˆwi∗=12. Now fix some α∈(0,1)and consider a minimum wage wwith 10 +2α<w<12. Then Nmax =Nmax ={1}and the Nash equilibrium wages at the winning principal i∗=1arewNE i∗=wand ˆwNE i∗=12. In particular, no wage overshooting occurs if the agent has no or all bargaining power. However, when the agent’s bargaining power is α, wage overshooting does occur since we have wα i∗=α·12 +(1−α) ·10 =10 +2α<w and wα i∗=α·12 +(1−α) ·w>w. C Analysis of the multi-agent model We previously examined how a minimum wage affects the interaction between an agent and a set of principals. In this section, we generalize the model to the case of m≥2 agents. We denote the set of agents by M={1,...,m}, and let ci,jand θi,j denote agent j’s cost and productivity, respectively, when working for principal i∈N. Similarly, ui,j:=wi,j−ci,jdenotes the utility of agent jif he works for principal i at wage wi,j, and πi,j:=θi,j−wi,jdenotes the profit resulting for principal ifrom this employment. Each agent can work for at most one principal, but a principal may employ several agents. The total profit πiof principal iamounts to the sum of the profits resulting from all these employments, where πi:=0 if she does not employ any agent. Similarly, we let ujdenote agent j’s utility at the principal he works for, where uj:=0 if agent jdoes not work for any principal. The surplus generated when agent jworks for principal iis denoted by si,j:=θi,j−ci,j. Note that, while we assumed (without loss of generality) that θi>cifor all i∈N in the single-agent case, we now allow that θi,j≤ci,jfor some i,j(which means that 123 Analyzing the effects of minimum wages 979 agent jcan never be profitably employed by principal i). This allows us to consider situations where each agent is only productive (i.e., has productivity exceeding cost) at a subset of the principals. In the following, it will be useful to summarize the costs and productivities in two (n×m) matrices C=(ci,j)i,jand =(θi,j)i,j, respectively. Here, each row corresponds to a principal i∈Nand each column to an agent j∈Mand the corresponding entries ci,jof Cand θi,jof are the cost and productivity, respectively, resulting when principal iemploys agent j. Similarly, the matrices W=(wi,j)i,j,U=(ui,j)i,j, and =(πi,j)i,jsummarize the (offered) wages, utilities, and profits, respectively. We set wi,j:=NO if principal imakes no offer to agent j. The assignment stating which principals employ which agents is summarized in the binary matrix A=(ai,j)i,j, where ai,j=1 if principal iemploys agent jand ai,j=0, otherwise. A pair (A,W) consisting of the assignment Aand the wages Wwill be referred to as an outcome. In the situation in which the principals have no capacity constraints limiting the numbers of agents they can employ, the principals compete for each agent independently of the presence of the other agents. Consequently, the situation decomposes into msingle-agent problems, to which the analysis from the previous sections applies. Hence, in the absence of capacity constraints, all results shown for the single-agent case carry over to case of multiple agents. In the situation where each principal has a capacity of one, i.e., a principal can only employ a single agent, it is easy to see that the results from the single-agent case carry over to the multi-agent case if the economy is simply duplicated a given number of times, i.e., there are kidentical agents and kcopies of each principal for some k≥2. We now seek to explore the general multi-agent case with capacity constraints. Here, each principal i∈Nhas a positive integer capacity κi∈N>0that specifies the maximum number of agents she can employ and we summarize the capacities in a vector κ=(κ1,...,κ n). We start by considering the canonical extension of our basic bargaining procedure in which each principal can offer wages to at most as many agents as her capacity allows her to employ. The following example, however, demonstrates that, in this case, pure-strategy Nash equilibria may fail to exist and mixed-strategy Nash equilibria may not be stable – even when no minimum wage is imposed: Example C.1 Consider the situation with two principals with unit capacities κ1= κ2=1 and three agents shown in Figure 3, where no minimum wage is imposed and all costs are zero. Obviously, there cannot be any pure-strategy Nash equilibrium in which either no principal makes an offer to agent 2 or both principals make their offers to agent 2. Hence, in a pure-strategy equilibrium, exactly one principal—say, principal 1—would have to make an offer to agent 2. Given that principal 2 does not make an offer to agent 2, principal 1’s optimal offer to agent 2 is w1,2=0. But principal 2 could then increase her profit by offering w2,2=1 to agent 2. There does, however, exist a (unique) symmetric, mixed-strategy Nash equilibrium in which each principal makes an offer to agent 2 with probability 1/6, in which case she distributes the offered wage on [0,2], and offers wage zero to the agent who yields productivity 10 for her, otherwise. Then, however, agent 2 receives no offer with probability (5/6)2=25/36, in which case each principal could improve her profit ex post by employing agent 2. 123 980 C. Thielen, P. Weinschenk Fig. 3 Example C.1. Connections that are omitted yield negative surplus We now show that stable outcomes do always exist in the setting with multiple agents and capacity constraints both with and without a minimum wage. Here, stability is formally defined as follows: Definition C.1 If a minimum wage wis imposed, an outcome (A,W)is called stable if it respects the capacities and the minimum wage, i.e., j∈Mai,j≤κifor all i∈N and wi,j≥wfor all i,jwith ai,j=1, is individually rational, i.e., πi,j,ui,j≥0for all i,jwith ai,j=1, and, for all i,jwith ai,j=0, we have: (a) If j∈Mai,j<κ i, then uj≥si,jor θi,j≤w. (b) If j∈Mai,j=κi, then uj+min{πi,j:ai,j=1}≥si,jor θi,j≤w+ min{πi,j:ai,j=1}. For unrestricted wages, stability is defined analogously, except that the conditions involving the minimum wage ware omitted. Note that, even though an outcome (A,W)contains wage offers wi,jfor all pairs (i,j), only those offers where ai,j=1 are relevant for the stability of the outcome. An outcome respecting both the capacities and the minimum wage is stable if and only if all employments in the current assignment yield nonnegative profit and utility for the corresponding principals and agents, respectively, and no principal ican offer a wage to an agent jcurrently not working for her in a way that both iand jwould improve and the capacities as well as the minimum wage are still respected. In Case (a), where principal i’s capacity κiis not exhausted in the current assignment, the condition that principal icannot offer such a wage to agent jmeans that either j’s current utility ujalready weakly exceeds the surplus si,jhe would generate when working for i, or the minimum wage weakly exceeds j’s productivity θi,jat principal i. If both of these conditions are violated, there exist wage offers improving j’s utility, while at the same time respecting i’s capacity and the minimum wage and yielding a positive profit πi,jfor principal i, for instance w i,j:=θi,j−1 2min{si,j−uj,θ i,j−w}. In Case (b), where principal i’s capacity κiis already exhausted in the current assignment, the conditions are similar except that principal ithen has to get rid of one 123 Analyzing the effects of minimum wages 981 (least profitable) agent she currently employs, which yields a profit loss of min{πi,j: ai,j=1}.33 In the following, we refer to a wage offer of a principal ito an agent jthat would increase both principal i’s profit and agent j’s utility by at least a given amount >0 as an -improving offer: Definition C.2 Given an outcome (A,W), a minimum wage w, and >0, a wage offer w i,j≥wof principal ito agent jis called an -improving offer if w i,j−ci,j≥ uj+and either j∈Mai,j<κ iand θi,j−w i,j≥,orj∈Mai,j=κiand θi,j−w i,j≥min{πi,j:ai,j=1}+. For unrestricted wages, the definition is the same except that the condition w i,j≥wis omitted. The outcome (A,W)is called -stable if it respects the capacities and the minimum wage if one exists, is individually rational, and no principal ican make an -improving offer to any agent j. The following observation is obtained directly from the definition: Observation C.1 For any outcome (A,W), the following holds: (I) If principal i can make an -improving offer to agent j, then ai,j=0, i.e., j does not work for i in the current assignment. (II) If (A,W)is -stable for some >0, then it is also -stable for all >. The next proposition provides the connection between stability and -stability: Proposition C.1 An outcome (A,W)is stable if and only if it is -stable for all >0, i.e., if and only if there does not exist an -improving offer for any >0. Proof If there exists an -improving offer of some principal ito some agent jfor some >0, then ai,j=0 and the conditions in a) and b) of Definition C.1 are clearly violated for i,j. Conversely, if the outcome (A,W)respects the capacities and the minimum wage if one exists but is not stable, then some principal ican make an offer to some agent j such that both i’s profit and j’s utility improve. This offer is then an -improving offer when choosing as the minimum of the two improvements.  We now algorithmically prove the existence of -stable outcomes for every >0, and then use Proposition C.1 in order to derive also the existence of a stable outcome. While Algorithm 1is formulated for the case with a minimum wage, it straightforwardly applies also to the case without a minimum wage by setting ˘wi,j:=ci,j+uj+in line 3. As long as there exists a principal who can make an -improving offer to some agent, Algorithm 1chooses such a principal iwho then makes an -improving offer to an agent jin a way that maximizes her profit. To do so, principal icould have to replace a least profitable agent lshe has previously employed by agent jin case that her capacity κiis already exhausted. Such a replacement, however, never actually occurs in any iteration of the algorithm, as the proof of the following proposition shows. 33 If both conditions in b) are violated, a possible wage offer improving j’s utility and i’s profit and respecting the minimum wage is given by w i,j:=max w,θ i,j−min{πi,j:ai,j=1}− δ 2,where δ:=si,j−uj−min{πi,j:ai,j=1}. 123 982 C. Thielen, P. Weinschenk Algorithm 1 -stable outcome Input: Set Nof principals, set Mof agents, minimum wage w, productivities ,costsC, capacities κ,and>0. Output: An -stable outcome (A,W). 1 Initialize ai,j:=0, wi,j:=NO, πi,j:=0, uj:=0 for all i∈N,j∈M. 2while there exists a principal iwho can make an -improving offer to some agent do 3 Choose j∈argmax j∈M {θi,j−˘wi,j},where ˘wi,j:=max{w,ci,j+uj+}. 4if  j∈M ai,j=κithen 5 Choose l∈argmin j∈M:ai,j=1 {πi,j}. 6Setai,l:=0, πi,l:=0, and ul:=0. 7end if 8Setai,j:=1andai,j:=0 for all i∈N\{i}. 9Setwi,j:=˘wi,jand wi,j:=NO for all i∈N\{i}. 10 Set πi,j:=θi,j−˘wi,jand πi,j:=0 for all i∈N\{i}. 11 Set uj:=˘wi,j−ci,j. 12 end while 13 return (A,W) Proposition C.2 Algorithm 1terminates after a finite number of iterations with an -stable outcome (A,W). Proof Since the outcome returned by the algorithm at termination is clearly -stable, we only have to show that the algorithm terminates after a finite number of iterations of the while loop. To this end, we now show that the utility ujof each agent jis monotonously increasing during the while loop. Since the utility of some agent jincreases by at least >0 in each iteration of the while loop and the sum of all utilities of the agents (which is zero before the first iteration) is bounded from above (e.g., by the sum of all si,j), this will directly imply that the number of iterations of the while loop is finite. In order to show the desired monotonicity of the utilities, we first observe that an agent k’s utility ukcan never decrease as long as the agent stays employed (i.e., as long as i∈Nai,k=1 during the while loop). This holds since the agent then stays employed at the same principal for the same wage until he receives an -improving offer, which increases his utility by at least . Moreover, observe that the only point in the algorithm where a previously employed agent can become unemployed is in lines 4–6, where a principal ireplaces an agent lby another agent jbecause her capacity κi is already exhausted when making an -improving offer to agent j. Consequently, the claim follows if we show that the condition of the if statement in line 4 is never satisfied, i.e., that no principal ever replaces an agent by another agent because her capacity is already exhausted. Suppose that iteration tis the first of the while loop in which some principal i replaces an agent lby another agent j.Lett<tdenote the latest previous iteration in which principal ioffered a new wage ˘wi,l(t)to agent l. Then, since no agent was ever replaced by another agent during the iterations 1,...,t−1, agent j’s utilities uj(t) and uj(t)at the start of iterations tand tmust satisfy uj(t)≤uj(t). Thus, the 123 Analyzing the effects of minimum wages 983 values ˘wi,j(t)and ˘wi,j(t)in iterations tand talso satisfy ˘wi,j(t)≤˘wi,j(t). Hence, denoting the values πi,kat the beginning of iteration tby πi,k(t),wehave θi,j−˘wi,j(t)≥θi,j−˘wi,j(t)≥πi,l(t)+=θi,l−˘wi,l(t)+>θ i,l−˘wi,l(t), where the second inequality follows since l∈argmin j∈M:ai,j=1{πi,j(t)}and ˘wi,j(t) is an -improving offer in iteration t. This yields the desired contradiction since agent i should have made an -improving offer to agent jinstead of agent lin iteration t according to line 3.  In particular, Proposition C.2 yields the following result: Corollary C.1 For every >0, there exists an -stable outcome. We now use Proposition C.1 and Corollary C.1 to establish the existence of a stable outcome. To this end, for any fixed assignment Arespecting the capacities, consider the following mixed integer linear program (MILP), which computes the smallest value ≥0 such that the assignment Atogether with suitable wages Wforms an outcome (A,W)that is -stable for all > : min s.t. (C.1) uj= i:ai,j=1 (wi,j−ci,j)∀j∈M(a) πi,j=θi,j−wi,j∀(i,j)with ai,j=1(b) max{w, ci,j}≤wi,j≤θi,j∀(i,j)with ai,j=1(c) si,j·xi,j≤uj+2∀(i,j)with ai,j=0, j∈M ai,j<κ i(d) si,j·xi,j≤uj+πi,˜ j+2∀(i,j)with ai,j=0, j∈M ai,j=κi,∀˜ jwith ai,˜ j=1(e) (θi,j−w) ·yi,j≤∀(i,j)with ai,j=0, j∈M ai,j<κ i(f) (θi,j−w) ·yi,j≤θi,˜ j−wi,˜ j+∀(i,j)with ai,j=0, j∈M ai,j=κi,∀˜ jwith ai,˜ j=1 (g) xi,j+yi,j≥1∀(i,j)with ai,j=0(h) xi,j,yi,j∈{0,1}∀(i,j)(i) ≥0 (j) Note that the variable wi,jrepresenting the wage paid by principal ito agent j only exists in case that ai,j=1, i.e., if agent jworks for principal i.IntheMILP,(a) and (b) link the utilities ujand the profits πi,j, respectively, to the paid wages wi,j. The constraints (c) ensure individual rationality and that all paid wages adhere to 123 984 C. Thielen, P. Weinschenk the minimum wage. The constraints (d)–(i) ensure that no principal ican make an -improving over to any agent jfor any > : Either xi,j=1 and the surplus si,j allows a joint improvement of at most 2(by (d), or (e)) or yi,j=1 and principal i can improve by at most due to the minimum wage (by (f)or(g)). For each assignment Arespecting the capacities, we denote the optimum objective value of MILP (C.1)by˜(A). Note that an optimal solution always exists as long as (C.1) is feasible (cf. Schrijver (1998)). If no feasible solution exists for a given assignment A,welet˜(A):=+∞. We then have ˜(A)≥0 for every assignment A due to constraint (j). Moreover, we let ˜:=minA˜(A)≥0 denote the minimum of the values ˜(A)over all the (finitely many) assignments A. Note that ˜is finite since (C.1) has a feasible solution for at least one assignment Arespecting the capacities: If ai,j=0 for all i,j, then setting xi,j:=1 and yi,j,π i,j,uj:=0 for all i,j, and choosing :=maxi,j si,j 2is clearly feasible. We now use MILP (C.1) to establish the existence of stable outcomes: Theorem C.1 There exists a stable outcome (A,W)for any productivities , costs C, capacities κ, and any minimum wage w. Proof We show that ˜=0 for any productivities , costs C, capacities κ, and any minimum wage w. This will prove the claim since a corresponding assignment A for which ˜(A)=0 together with the wages wi,jfrom an optimal solution of (C.1) for this Aconstitute an outcome (A,W)that is -stable for all >0. Thus, this outcome (A,W)is stable by Proposition C.1. Suppose for the sake of a contradiction that ˜>0. Then, by Corollary C.1, there exists an outcome (A,W)that is ˜ 2-stable, and by Observation C.1, this outcome is also -stable for all > ˜ 2. Thus, the wages given by Winduce a feasible solution of (C.1) with objective value at most ˜ 2,so˜(A)≤˜ 2, which yields a contradiction since ˜(A)≥˜by definition of ˜. Theorem C.1 also holds in case no minimum wage is imposed, since a minimum wage w≤mini,jci,jhas no effect.34 Further, recognize that the result in Theorem C.1 is constructive in the sense that, for any assignment Arespecting the capacities, we can use MILP (C.1) in order to compute wages Wsuch that the outcome (A,W)is stable (these wages are given by the variables wi,jin an optimal solution if the optimum objective value ˜(A)is zero) or decide that no such wages exist for assignment A (which is the case if ˜(A)>0). The following example demonstrates that the effects observed in the single-agent case still arise in the case of multiple agents: Example C.2 Consider the situation with six principals with unit capacities κi=1for all iand four agents shown in Figure 4. When no minimum wage is imposed, the only possible assignment in a stable outcome is a1,1=a3,2=a4,3=a6,4=1 and 34 For the case where no minimum wage is imposed, the existence of a stable outcome can also be shown by using linear programming duality in order to calculate the utilities and profits in a stable outcome; see Shapley and Shubik (1971). This technique, however, does not easily generalize to the case in which a (nontrivial) minimum wage exists, since the minimum wage may prevent certain combinations of utilities and profits for each principal-agent pair. 123 Analyzing the effects of minimum wages 991 Schrijver, A.: Theory of Linear and Integer Programming. John Wiley & Sons, Chichester (1998) Shapley, L.S., Shubik, M.: The assignment game I: the core. 1, 111–130 (1971) Smith, A.: The wealth of nations. New York Modern Library, reprint of 1776 original (1937) Sorkin, I.: What does the changing sectoral composition of the economy mean for workers? 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