Adverse Selection in an Efficiency Wage Model with Heterogeneous Agents

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1 Adverse Selection in an Efficiency Wage Model with Heterogeneous Agents Ricardo Azevedo Araujo Deartment of Economics, University of Brasilia (UnB), Brazil Adolfo Sachsida Brazilian Institute for Alied Economic Research (IPEA) and IBMEC-DF, Brazil Abstract This aer studies efficiency wages in the resence of heterogeneous workers and asymmetric information. It includes an incentive comatibility constraint (ICC) in the efficiency wage model with heterogeneous workers to show that the imlementation of efficiency wages in the resence of heterogeneity faces the roblem of adverse selection. Emloyees with a smaller effort aversion suly a smaller level of effort than what is otimal under erfect information due to hidden information. In this vein only a second best solution is obtained. Keywords: Efficiency Wages, Adverse Selection, Asymmetric Information JEL Classification: J41, M59 Resumo Este artigo analisa a utilização de salários de eficiência na resença de trabalhadores heterogêneos e assimetria de informação. Através da inclusão de uma restrição de comatibilidade de incentivos nesse modelo com agentes heterogêneos, é ossível mostrar que ocorre o roblema de seleção adversa. Trabalhadores com menor aversão ao esforço ofertam um menor nível de esforço do que aquele que é ótimo sob informação erfeita. Assim aenas uma solução de segundo ótima é ossível. Submitted in June 2010, acceted in August We would like to thank without imlicating Emilson Silva and an anonymous referee for helful suggestions. The usual disclaimer alies. Financial suort from the CNPq is acknowledged. address: rsaaraujo@unb.br Revista EconomiA Setember/December 2010

2 Ricardo Azevedo Araujo and Adolfo Sachsida 1. Introduction Efficiency wages models redict that homogeneous workers may receive different wages relying on the degree of shirking aversion of firms. In their classical article, Shairo and Stiglitz (1984) roosed a non-shirking condition which once satisfied avoid workers to incur in shirking. More recently Charness and Kuhn (2005) roose an efficiency wage model with heterogeneous workers (high roductivity and low roductivity workers) to exlain why firms can comress wages. The main idea of efficiency wage models is that emloyers are able to set wages u its cometitive level to avoid shirking. That is, emloyer offers a contract to an emloyee that is able to avoid shirking, this is the non-shirking condition. Since the worker accets the contract this imlies that her utility to work hard is bigger than the utility of incurring in shirking. In other words, the non-shirking condition may be viewed as a articiation constraint. Acceting the non-shirking condition (NSC) as a articiation constraint raises the question about the incentive comatibility constraint. While NSC warrants articiation, it does not imly that the worker will reveal his true tye. In models with homogeneous workers it is not a roblem. However, it is hardly defensible that the work force is homogeneous. Difference in abilities, skills, and motivation are simle examles which can be mentioned to refute homogeneity hyothesis. While skills can be observed, motivation and abilities can not be easily observed. However, all of them have imact over roductivity. In the literature of efficiency wages with heterogeneous workers it is usual to assume different tyes of individuals with articular roductivities. Furthermore, it is imlicit assumed that the identification of the roductivity level of the worker is easy. Since the emloyer can always verify the worker roductivity and classify him/her as low or high roductive the incentive comatibility constraint is not necessary. Facing a heterogeneous suly of labor, emloyers have some difficulties to learn about emloyee s roductivity, mainly during the hiring rocess. In this scenario, the incentive comatibility constraint (ICC) becomes imortant and a device to force workers to reveal their true tyes becomes necessary. In order to tackle this oint this aer extends the Shairo-Stiglitz model to allow for heterogeneity between workers. 1 Assuming that the heterogeneity comes from the disutility to work, we introduce an ICC condition in the efficiency wage model. Our main result shows that the imlementation of efficiency wages in the resence of heterogeneous workers gives rise only to a second best solution due to the issue of hidden information: workers with a smaller disutility of effort act as if they had a higher aversion to effort in order to obtain a better contract. This aer is structured as follows: Section 3 resents the formal model and Section 4 concludes. 1 Shairo-Stiglitz original aer suggests that heterogeneous workers could be a source of stigma from loosing a job. This stigma could serve as a disciline device. However, they do not deal with the imlications of the heterogeneity of workers over wages which is the objective of this aer. 496 EconomiA, Brasília(DF), v.11, n.3, , Se-Dec 2010

3 Adverse Selection in an Efficiency Wage Model with Heterogeneous Agents 2. The Model Let us assume an one eriod model in which the utility function is given by: u(w,e) = w Re, where w is the wage aid by the firm, e is the effort, and R is a ositive constant that measures the disutility of effort (inverse of motivation). 2 The emloyee has outside oortunities that rovide him with a reservation salary w with no effort (e = 0). There is a robability to lose the job attached to shirking behavior. The exected utility is given by: V N = w Re (1) V S = w+(1 )w (2) where V N is the utility associated with no shirking and V s the utility associated with shirking behavior. The efficiency wage is chosen to satisfy the inequality: V N V S (3) Due to the maximization behavior of the firm the equality holds, that is: Imlicit derivation of w as a function of R yields: w Re = w+(1 )w (4) w = w+ Re (5) This seems that wage may be an increasing function of R. Let us secify the contract under erfect information, which is assumed to be given by: MaxF [e] w (6) w,e s.t. w Re w +(1 )w (NSC) where the rice of the roduct is normalized to one, and F(.) is a roduction function, with F (.) > 0 and F (.) < 0. From the constraint we obtain: w = w+ Re. This exression may be rewritten as e = (w w) R. By inserting this exression into the objective function and deriving with resect to w we obtain: F (.) = R. This exression shows that the marginal roductivity of effort is an increasing function of the disutility of effort(r). By assuming that:f [e] = e α then after some algebraic maniulation the exression above yields the following efficiency wage: ( w = w+α 1 1 α α 1 α R) (7) 2 Katz (1986), Akerlof and Yellen (1990), and Charness and Kuhn (2005) also used an effort that is a function of wages. EconomiA, Brasília(DF), v.11, n.3, , Se-Dec

4 Ricardo Azevedo Araujo and Adolfo Sachsida By taking the artial derivative of w with resect to R we conclude that: w R = α 1 α 1 ( α 1 α R R) < 0. This result shows that the otimal strategy for the firm is to set wages to make it a decreasing function of R. Let us assume now that there are only two tyes of workers: 3 low aversion to effort, R 1, and high aversion to effort, R 2, with R 2 >R 1. The incentive comatibility constraint for worker tye 1 requires that: w 1 R 1 e 1 w 2 R 1 e 2 (IC1) where w 1 and w 2 stand for the wages for the workers of tye 1 and 2, resectively and IC1 stands for the incentive comatibility constraint for worker of tye 1. Exression (IC1) requires that the utility to behave like tye 1, when the worker is tye 1, should be larger (or at least equal) than the utility to behave like tye 2. In the same way, there is an incentive comatibility restriction to worker tye 2, namely (IC2), which is written as: w 2 R 2 e 2 w 1 R 2 e 1 (IC2) The non-shirking conditions for workers 1 and 2, namely (NS1) and (NS2) resectively, require that: w 1 R 1 e 1 w+(1 )w (NS1) w 2 R 2 e 2 w+(1 )w (NS2) They state that each worker refers the contract that is designed for her. When the firm is hiring the worker or roosing the contract it does not know the tye of the emloyee. Let us assume that the only information the firm has is the distribution of workers of tye 1 and 2 in the oulation and that the robability of hiring a worker of tye 1 is σ, and the remainder, 1 σ, is the robability of hiring a worker of tye 2. The contract under imerfect information that satisfies the revelation rincile is given by: Max σ[f(e 1 ) w 1 ]+(1 σ)[f(e 2 ) w 2 ] (8) w 1,w 2,e 1,e 2 s.t. w 1 R 1 e 1 w 2 R 1 e 2 w 2 R 2 e 2 w 1 R 2 e 1 w 1 R 1 e 1 w+(1 )w (IC1) (IC2) (NS1) 3 The number of tyes could be extended to consider $n$ different tyes but the case in which there are two tyes in enough to convey the main imlications of the model without cumbersome algebraic maniulations. 498 EconomiA, Brasília(DF), v.11, n.3, , Se-Dec 2010

5 Adverse Selection in an Efficiency Wage Model with Heterogeneous Agents w 2 R 2 e 2 w+(1 )w (NS2) A ossible way of tackling the above roblem is to adot the Kuhn-Tucker Theorem. Before doing this let us show that one of the restrictions of the roblem, namely restriction (NS1), may be ignored. This is the content of the following: Lemma 1: The constraint (NS1) may be ignored in roblem (8). Proof. From (IC1) from the fact thatr 1 <R 2 one obtains:w 1 R 1 e 1 w 2 R 1 e 2 > w 2 R 2 e 2. From (NS2) it is ossible to conclude that: w 1 R 1 e 1 w+(1 )w. Another useful Lemma is the one that shows that (NS2) holds with equality. Lemma 2: (NS2) holds with equality. Proof. Assume for the sake of contradiction that the otimal contract is the one in which w 2 R 2 e 2 > w +(1 )w. Then the firm may choose another contract by choosing w 2 ǫ that still satisfies the above equations and yields smaller costs. But this is a contradiction to fact that the first contract was otimal. This result may also be roven by solving the maximization roblem (8) subject to (IC1), (IC2) and (NS2) through the Kuhn-Tucker method. By using Lemma 1, the roblem may be rewritten as: Max σ[f(e 1 ) w 1 ]+(1 σ)[f(e 2 ) w 2 ] (9) w 1,w 2,e 1,e 2 s.t. w 1 R 1 e 1 w 2 R 1 e 2 w 2 R 2 e 2 w 1 R 2 e 1 w 2 R 2 e 2 w+(1 )w The Lagrangean function related to this roblem may be written as: (IC1) (IC2) (NS2) L=σ[F(e 1 ) w 1 ]+(1 σ)[f(e 2 ) w 2 ]+λ 1 [w 1 R 1 e 1 w 2 +R 1 e 2 ]+λ 2 [w 2 R 2 e 2 w 1 +R 2 e 1 ]+µ[w 2 R 2 e 2 w (1 )w] where λ 1, λ 2 and ì are the Kuhn-Tucker multiliers related to constraints (IC1), (IC2) and (NS2) resectively. The first order conditions are given by: w 1 = σ +λ 1 λ 2 = 0 w 2 = (1 σ) λ 1 +λ 2 +µ = 0 e 1 = σf (e 1 )+λ 1 R 1 λ 2 R 2 = 0 e 2 = (1 σ)f (e 2 ) λ 1 R 1 +λ 2 R 2 +µr 2 = 0 (A1) (A2) (A3) (A4) EconomiA, Brasília(DF), v.11, n.3, , Se-Dec

6 Ricardo Azevedo Araujo and Adolfo Sachsida The Kuhn-Tucker conditions are given by: λ 1 λ 1 = λ 1 [w 1 R 1 e 1 w 2 +R 1 e 2 ] = 0,λ 1 0, λ 1 = w 1 R 1 e 1 w 2 +R 1 e 2 0 (A5) λ 2 λ 2 = λ 2 [w 2 R 2 e 2 w 1 +R 2 e 1 ] = 0,λ 2 0, λ 2 = w 2 R 2 e 2 w 1 +R 2 e 1 0 (A6) µ µ = µ[w 2 R 2 e 2 w (1 )w] = 0,µ 0, µ = w 2 R 2 e 2 w (1 )w 0 (A7) It is ossible to show see the Aendix that from the eight ossible cases just one is feasible, namely case (v), in which λ 1 = 0, λ 2 = σ > 0 and µ = 1 > 0. In this context the following roositions hold. Proosition 1: The utility of a worker of tye 1 is larger than the utility of a worker of tye 2, that is: w 1 R 1 e 1 > w 2 R 2 e 2 Proof: From (IC1) we know that:w 1 R 1 e 1 w 2 R 1 e 2. SinceR 1 <R 2 we conclude that: w 1 R 1 e 1 w 2 R 1 e 2 > w 2 R 2 e 2. This roosition shows that that the firm makes the utility of an emloyee of tye 1, with higher willingness to work to be larger than the utility of an emloyee of tye 2. At a first glance the contract rovides right incentives since it gives more rewards in terms of utility to workers with higher willingness to work. Proosition 2 shows that the effort of a worker of tye 1 is chosen to be larger or at least equal than the effort of a worker of tye 2. Proosition 2: The effort of a worker of tye 1 is larger or equal than the wage of a worker of tye 2. That is: e 1 e 2. Proof: By summing (IC1) and (IC2) and after some algebraic maniulation we obtain: (R 1 R 2 )[e 2 e 1 ] 0. Since R 1 <R 2 we conclude that: e 2 e 1. Hence the effort of a worker of tye 1 is larger or equal than the salary of a worker of tye 2. From Proositions 1 and 2 it is ossible to conclude that in general a worker of tye 1 has better reward than a worker of tye 2. This result is according to what was established as the wage chosen by the firm as the outcome of the rofit maximization behavior of the firm under erfect information and it shows that the conventional wisdom that workers with higher willingness to work have to receive larger wages than those which higher effort aversion revails. However since the worker of tye 1 has to rovide an effort level higher or at least equal to the one rovided by worker or tye 2 it may find wrong incentives to announce that she is a worker of tye 2 in order to rovide less effort than what it is otimal under erfect information. Proosition 3: An emloyee of tye 1 chooses a contract designed for an emloyee of tye EconomiA, Brasília(DF), v.11, n.3, , Se-Dec 2010

7 Adverse Selection in an Efficiency Wage Model with Heterogeneous Agents Proof: From Proosition 2, we know that: e 2 e 1. From (IC2) we know that w 1 R 1 e 1 w 2 R 1 e 2. Note that it is not ossible to exclude the ossibility that: w 1 R 1 e 1 = w 2 R 1 e 2. In fact this result holds as it is shown in case (vi) of the Aendix. In this vein a worker of tye 1 by announcing of being a worker of tye 2 may rovide smaller effort and receives the same utility as if he admitted his own tye. By announcing to be a worker of tye 2 allows the worker of tye 1 to rovide less effort while keeing her utility of being a worker of tye 1. The consequence of this strategy is that the salary earned by her will be smaller than if she admitted her true tye. Of course this fall in wages will be comensated in the utility function by the smaller effort rovided. But a further analysis on this outcome shows that it can roduce only a second best result for the firm: rofits that the firm obtains hiring an emloyee of tye 1 is smaller than what it would obtain under erfect information. This is the content of the next roosition. Proosition 4: Profits of the firm hiring worker of tye 1 are smaller than rofits it obtains under erfect information. Proof: From the rofit maximization roblem of the firm it is ossible to conclude that the rofit of the firm under erfect information by hiring a worker of tye 1 is: Π 1 = F [e 1 ] w 1 where w 1 = w + R1e1. This yields: Π 1 = F [e 1 ] w R1e1. The same result holds for the rofit of the firm by hiring a worker of tye 2 under erfect information, that is: Π 2 = F [e 2 ] w R2e2. It is ossible to show that the rofit of the firm by hiring a worker of tye 1 is large or at least equal to the rofit of the firm hiring a worker of tye 2: as R 1 <R 2 then F [e 2 ] w R1e2 > F [e 2 ] w R2e2 = Π 2. But we know that Π 1 = F [e 1 ] w R1e1 F [e 2 ] w R1e2 since e 1 is the solution for the rofit maximization under erfect information if the effort aversion is R 1. Hence we conclude that Π 1 > Π 2. Note that the common term in both inequalities is nothing but the rofit of the firm by hiring a worker of tye 1, acting as worker of tye 2, which is smaller than the rofit of a worker of tye 1 under erfect information and higher than the rofit of a worker of tye 2 under erfect information. This result is not the first best since the effort rovided by a worker of tye 1 is smaller than her effort under erfect information which leads to smaller rofits. Hence the contract is not otimal since it roduces only a second best solution. The model shows that after the contract is signed and roductivity is revealed it is ossible to identify workers of tye 1 and 2 by the efforts they rovide. Hence a ossible way to avoid the second best solution due to hidden information is to establish a robation eriod in which the roductivity of workers is screened. In this case, workers of tye 1, by knowing that they are identified during the robation eriod find right incentives to announce his true roductivity when the contract is signed. But it is imortant to bear in mind that this device works only in the case in which the robation eriod does not create incentives to workers of tye 1 to remain only during this eriod and quit the job after that and that they are aware of their own roductivity. EconomiA, Brasília(DF), v.11, n.3, , Se-Dec

8 Ricardo Azevedo Araujo and Adolfo Sachsida Pouyet et al. (2008) assuming that rincials comete for attracting heterogeneous agents by offering contracts have found that when the agents tyes are ublicly observed then cometitive equilibria are efficient, a result similar to the one obtained here. When tyes are rivately observed these authors continue to obtain the result that efficiency holds rovided that rincials do not directly care about the agents rivate information. Here we have shown that only a second best solution is ossible in the case of hidden information. 3. Concluding Remarks In this aer we get aart from homogeneous work hyothesis and derive an incentive comatibility constraint (ICC) for the efficiency wage aroach. Suosing that the heterogeneity arises from different disutility to work (motivation) between workers, we show that the efficiency wage contract rovides only second best solution: emloyee of tye 1 sulies a smaller level of effort than what is otimal under erfect information due to hidden information. References Akerlof, G. & Yellen, J. (1990). The fair-wage effort hyothesis and unemloyment. Quarterly Journal of Economics, 105(2): Charness, G. & Kuhn, P. (2005). Pay inequality, ay secrecy and effort: Theory and evidence. NBER Working Paer Katz, L. (1986). Efficiency wage theories: A artial evaluation. In NBER Macroeconomics Annual, ages NBER. Pouyet, J., Salanié, B., & Salanié, F. (2008). On cometitive equilibria with asymmetric information. The B.E. Journal of Theoretical Economics, 8(1). Available at: htt:// Shairo, C. & Stiglitz, J. E. (1984). Equilibrium unemloyment as a worker disciline device. American Economic Review, 74(3): EconomiA, Brasília(DF), v.11, n.3, , Se-Dec 2010

9 Adverse Selection in an Efficiency Wage Model with Heterogeneous Agents Aendix Let us analyze the eight ossible cases, which arise from the ossible combination of the Kuhn-Tucker multiliers: (i) λ 1 = λ 2 = µ = 0. From exression (A1) it yields that σ = 0, a contradiction. (ii) λ 1 = λ 2 = 0 and µ > 0. For the same reason of case (i) we exclude this ossibility. (iii) λ 1 = µ = 0 and λ 2 > 0. From exression (A1) it yields λ 2 = σ < 0, a contradiction. (iv) λ 2 = µ = 0 and λ 1 > 0. From exression (A1) it yields λ 1 = σ. By substituting this result into exression (A2) we conclude that µ = 1, a contradiction. (v) λ 1 = 0, λ 2 > 0and µ > 0. From exression (A1) it yields λ 2 = σ < 0, a contradiction. (vi) λ 1 > 0, λ 2 = 0 and µ > 0. From exression (A1) it yields λ 1 = σ > 0. By substituting this result into exression (A2) we conclude that µ = 1. Note that in this case, from exression (A6) it is ossible to conclude that w 2 R 2 e(w 2 ) = w 1 R 2 e(w 1 ), which is the content of Proosition 3, meaning that a worker of tye 1 is indifferent between her contract and a contract for the tye 1. Besides as µ = 1 > 0 then (NS2) holds with equality which is the content of Lemma 2. (vii) λ 1 > 0, λ 2 > 0 and µ = 0. By substituting (A1) into (A2) we conclude that µ = 1, a contradiction to the fact that µ = 0. (viii) λ 1 > 0, λ 2 > 0 and µ > 0. From (A1) λ 1 λ 2 = σ. By inserting this exression into exression (A2) one obtains µ = 1. As all the multiliers are different from zero then the constraints are binding. By summing u (A3) and (A4) and after some algebraic maniulation we conclude that: σf (e 1 )+ (1 σ)f (e 2 ) + R 2 = 0. All the terms in the left hand side of this equality are larger than zero then it is not ossible to hold. EconomiA, Brasília(DF), v.11, n.3, , Se-Dec