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Allocation: Adverse selection Gastón Llanes Francisco Ruiz-Aliseda November 11, 2015 1 Adverse selection and employee hiring In previous lectures we have studied the moral hazard problems that arise when the organization has to contract with agents whose actions will be di�cult to observe or measure. In this lecture, we study the problems that arise when the organization cannot observe the agents’ characteristics before contracting. When this happens, we say there is an adverse selection problem. Adverse selection arises when the principal and the agent contract given that the agent is known to privately hold a piece of information that is relevant for the princi- pal. Such piece of information that is observed by the agent but not by the principal may have to do with (i) factors exogenous to the agent (such as her innate ability for solving problems), or (ii) unobservable actions taken by the agent before contracting with the principal (such as the agent’s investment in developing expertise for execut- ing a task). Together with moral-hazard problems, adverse-selection problems are the most relevant informational frictions in economic exchange. Recall that moral-hazard problems arise when an agent takes unobservable actions after contracting with the principal. One particular instance in which adverse selection arises is when the principal has to hire employees. In this case, the principal may worry that the labor contract she o�ers will be accepted by an applicant with inadequate or undesirable characteristics. For example, she may o�er a contract for a job that requires certain skills, and may be concerned that these skills are not possessed by the applicant she will select. In some cases, the principal may test the applicant before hiring her, to determine if she is indeed a good �t for the job. However, in some cases, it may be di�cult or costly 1 to test an agent’s characteristics before contracting with her. In these cases, we shall see that the optimal way to address this adverse-selection problem is to o�er a menu of contracts to the agent, and allow the agent to choose the contract that is best for her. Such a choice will allow the principal to correctly infer the agent’s characteristics. Because the agent anticipates such information revelation, she will have to be given incentives to do so. As we shall see, this will imply both a transfer of rents by the principal and ine�cient contracting relative to when the agent’s characteristics are observed. The principal must design the menu of contracts appropriately, so that an agent with certain characteristics is not tempted to choose a contract that was designed for an agent with distinct characteristics. For example, the principal may o�er two types of contracts to job candidates whose unobserved sales ability is either high or low: one with a large �xed wage and a small sales commission, and another with a small �xed wage and a large sales commission. If the menu of contracts is designed correctly, an agent with a low sales ability will prefer the contract with the large �xed wage (she knows it is unlikely that she will sell a lot), and an agent with a high sales ability will prefer the contract with the high sales commission (she knows she can obtain a large salary by making lots of sales). Therefore, even if the ability of agents is unobservable to the principal, agents with di�erent abilities will sort themselves out among the contracts that are being o�ered. To study these issues formally, consider a relationship between a risk-neutral prin- cipal and a risk-neutral agent who may be hired by the principal. The agent’s cost of e�ort is c (e ) = γe2, where γ ∈ {γL,γH } is a parameter that represents the agent’s cost of exerting e�ort, and 0 < γL < γH . If γ = γL we say the agent is “productive,” and if γ = γH we say the agent is “unproductive” (since her marginal cost of e�ort is higher). It is common knowledge that the proportion of productive agents in the population is p ∈ (0, 1). The value created by the agent’s e�ort for the principal is π = e . Let w be the payment from the principal to the agent for exerting e�ort e ≥ 0, which for simplicity is assumed to be contractible (i.e., there is no moral hazard problem). Then the princi- pal’s net pro�t is e−w and the agent’s utility isw−γke2 (k ∈ {H ,L}). Regardless of how costly the agent �nds it to exert e�ort, the agent has an outside alternative which gives her utility u (reservation utility) if she does not contract with the principal. Note that 2 the agent’s outside alternative does not depend on her unobserved productivity. That is, we assume for simplicity that the alternative job gives the same wage to productive or unproductive agents (perhaps because the alternative job requires a di�erent set of skills that are unrelated with the productivity at the principal’s �rm). For this reason, suppose that the agent o�ers a menu of contracts so that she can perfectly identify the e�ort cost function of the agent, where a contract speci�es a level of e�ort required and an associated wage. In particular, the principal designs a menu {(eH ,wH ), (eL,wL)}, so that an agent with γ = γk (k ∈ {H ,L}) prefers to accept (ek ,wk ), where ek is the e�ort that the principal requires such an agent to exert andwk is the corresponding �xed wage (paid conditional on the agent exerting the required e�ort). Consider �rst a benchmark case in which the principal can observe γk when deal- ing with the agent. In this case, she would choosewk and ek to maximize ek−wk subject to the individual rationality constraint thatwk−γk e2k ≥ u. The principal would clearly o�er the minimal wage so that the agent accepted the contract, sowk = u+γke2k would hold. The principal would then choose ek to maximize ek−γke2k−u, so she would choose êk = 1/(2γk ) and ŵk = u + 1/(4γk ) to maximize her expected payo�. Suppose now that the principal cannot observe γk when dealing with an agent. If the principal acts naively and o�ers the menu of contracts {(êH , ŵH ), (êL, ŵL)}, a pro- ductive agent would prefer to pretend to be unproductive, and would choose (êH , ŵH ) over (êL, ŵL), since γH > γL implies that ŵH − γLê 2 H = u + γH ê 2 H − γLê 2 H > u = ŵL − γLê 2 L. Killing this incentive of a productive agent to prefer accepting the contract intended for an unproductive agent will be a critical constraint in devising the optimal menu of contracts. Since the proportion of productive agents in the population is p ∈ (0, 1), when γ is not observed by the principal, the probability of interacting with a productive agent is p and the probability of interacting with an unproductive agent is 1 − p. To �nd out the optimal menu of contracts that deals with adverse selection in the hiring of 3 employees, the principal must choose eH , wH , eL and wL to maximize p (eL −wL) + (1 − p) (eH −wH ) subject to the following constraints: wL − γLe 2 L ≥ u, (IRL) wH − γHe 2 H ≥ u, (IRH ) wL − γLe 2 L ≥ wH − γLe 2 H , (ICL) wH − γHe 2 H ≥ wL − γHe 2 L. (ICH ) Note that the principal’s objective function is constructed based on the presump- tion that the menu is designed in such a way that an agent who is productive (some- thing that happens with probability p) chooses contract (eL,wL) and an agent who is unproductive (something that happens with probability 1 − p) chooses contract (eH ,wH ). The �rst and second constraints, called individual rationality or participa- tion constraints, require that an agent prefers to accept the contract designed for her, given her productivity type, over not refusing any contract. The other constraints, called incentive compatibility constraints, require that an agent prefers to accept the contract designed for her, given her productivity type, over the other contract. Instead of solving a complex optimization problem with several inequality con- straints, one can notice that the optimal solution must satisfy the following properties: (1) Constraint (IRH ) mustbe binding. (2) Constraint (ICL) must be binding as well. (3) The optimal solution must be such that eL ≥ eH . (4) Constraint (ICH ) can be safely ignored. (5) Constraint (IRL) can be safely ignored as well. One can prove property (1) using (ICL), γH > γL, and (IRH ), so that wL − γLe 2 L ≥ wH − γLe 2 H ≥ wH − γHe 2 H ≥ u. 4 If we had that wH − γHe2H > u (so that wL − γLe 2 L > u also held), then both wL and wH could be lowered a bit without violating any constraint and the principal’s payo� would increase, which would contradict optimality, so we must have that (IRH ) binds at the optimal solution. To prove property (2), suppose to the contrary that (ICL) did not bind at the optimal solution. Then γH > γL and the fact that (IRH ) is binding would imply that wL − γLe 2 L > wH − γLe 2 H > wH − γHe 2 H = u, so one could lower wL a bit without violating (ICL), (IRL) or (ICH ) and the principal’s payo� would increase, a contradiction with optimality, so (ICL) must indeed bind at the optimal solution. To show that property (3) is true, add up (ICL) and (ICH ). Rearranging yields that (γH − γL) (e 2 L − e 2 H ) ≥ 0, so γH > γL yields that eL ≥ eH . Property (4) then can be shown to hold with some work. Because (ICL) is binding and γH is greater than γL, eL ≥ eH implies that wL −wH = γL (e 2 L − e 2 H ) ≤ γH (e 2 L − e 2 H ), an inequality that can be rewritten as wH − γHe 2 H ≥ wL − γHe 2 L, which is inequality (ICH ). It follows that the fact that (ICL) is binding implies that (ICH ) must necessarily hold, as claimed, and hence it can be ignored. Finally, to prove property (5), use (ICL), γH > γL, and (IRH ), so that wL − γLe 2 L ≥ wH − γLe 2 H ≥ wH − γHe 2 H ≥ u. This shows that that constraint (IRL) can be safely ignored because it is implied by satisfaction of other constraints. 5 We have therefore shown that the principal chooses eH ,wH , eL andwL to maximize p (eL −wL) + (1 − p) (eH −wH ) subject to (IRH ) and (ICL) holding with equality. When (IRH ) holds with equality, we have that wH = u + γHe 2 H , so the equality in (ICL) can be rewritten as wL = u + γLe 2 L + (γH − γL)e 2 H . Using these values in the respective �xed wage components on the principal’s payo� function yields that the principal chooses eH and eL to maximize p (eL − u − γLe 2 L − (γH − γL)e 2 H ) + (1 − p) (eH − u − γHe 2 H ). Once the optimal values of eH and eL are obtained, the optimal values of wH and wL can be obtained from the previous equalities, so their determination will be ignored henceforth. To �nd out the optimal values of eH and eL, note that the �rst-order conditions yield p (1 − 2γLeL) = 0 and (1 − p) (1 − 2γHeH ) − 2p (γH − γL)eH = 0. The �rst condition is similar to the one that obtains for an agent who is observed to be productive, so the principal does not distort the e�ort required from a productive agent. As can be seen from the second �rst-order condition, the principal does distort the e�ort required from the agent with �nds it more costly to exert e�ort (see last term in the equation). This downward distortion arises because the principal wants to make sure that the contract intended to reward such an agent should not be appealing to a productive agent. Such distortion will be greater the bigger p or γH − γL are. 6 From the �rst �rst-order condition, we obtain e∗L = 1 2γL , whereas the second �rst-order condition yields after some rearranging that e∗H = 1 2[γH + p 1−p (γH − γL)] . Note that e∗L = 1 2γL > 1 2γH > 1 2[γH + p 1−p (γH − γL)] = e∗H . In optimally providing incentives for an unproductive agent without making the con- tract attractive for a productive agent, the principal is led to lower e∗H relative to the cases in which it can observe how productive the agent is. This ine�ciency arises be- cause the unobservability of the agent’s characteristics induces the principal to make contract (eH ,wH ) unappealing to the productive agent: by requiring a lower e�ort from an unproductive agent, such an agent receives a lower salary and hence is less tempted to take the contract unintended for her, especially because the salary received by a productive agent is greater than ŵL.1 It is also worth noting that, at the optimal solution, a productive agent is indi�erent between accepting the contract (e∗L,w ∗ L) and accepting the contract (e ∗ H ,w ∗ H ). However, an unproductive agent is indi�erent between accepting the contract (e∗H ,w ∗ H ) and not accepting it, whereas a productive agent is strictly better o� accepting the contract (e∗H ,w ∗ H ) than not accepting any contract. The expected utility that such an agent ob- tains above and beyondu is called her informational rent. Because the principal wants a productive agent to prefer (e∗L,w ∗ L) over (e ∗ H ,w ∗ H ), such an agent must be rewarded to reveal her unobserved productivity when accepting the contract designed for her. 1Formally, w∗L = u + γL (e ∗ L ) 2 + (γH − γL ) (e ∗ H ) 2 > u + γLê 2 L = ŵL . 7 2 Adverse selection and moral hazard In the previous section, we studied a model in which the agent’s characteristics were unobservable to the principal, but her actions were veri�able, which means that they could be included in a contract. Thus, the only source of information asymmetries were the unobservable characteristics of the agent. However, many cases in reality exhibit both adverse selection and moral hazard problems. For example, an agent’s e�ort may be di�cult to observe or contract upon (moral hazard problem), but the principal may also be unsure about the agent’s degree of risk aversion (adverse selec- tion problem). To study this issue in more detail, consider a prospective agent that is to be hired by a principal. The agent’s cost of e�ort is c (e ) = e2/2. If the agent anticipates a random utility θ , then her expected utility equals E (θ ) − λVar (θ )/2. We assume that the value of parameter λ is known by the agent, but not by the principal. It is common knowledge however that the principal believes that the agent’s degree of risk aversion equals λL ≥ 0 with probability p ∈ (0, 1) and λH > λL with probability 1−p. An agent has a reservation utility of u regardless of her degree of risk aversion. The agent’s e�ort generates pro�t π = e + ε for the principal, where ε is a random variable with zero mean and variance equal to σ 2ε . The agent’s e�ort is not veri�able but the principal’s pro�t is veri�able, so the principal o�ers a contract that is linear in pro�t, s (π ) = a + b π , in order to provide incentives to exert e�ort. The principal does not observe how much the agent dislikes variability in her �nal utility and hence is unsure of how to structure the linear contract, so it o�ers a menu of contracts so that it can perfectly identify the degree of risk aversion of the agent she is to hire. In particular, the principal designs a menu {(aH ,bH ), (aL,bL)} so that an agent whose degree of risk aversion is λk (k ∈ {L,H }) prefers to accepts (ak ,bk ). From previous lectures, we know that an agent exerts e�ort e∗ k = bk under contract (ak ,bk ), so the expected utility of an agent with risk-aversion coe�cient λl (l ∈ {H ,L}) under contract (ak ,bk ) (k ∈ {L,H }) is equal to ak + b2 k 2 (1 − λlσ 2ε ). Therefore, if the principal could observe the degree of risk aversion of the agent, she 8 would o�er a contract such that âk = u − 1 − λkσ 2ε 2(1 + λkσ 2ε )2 and b̂k = 1/(1 + λkσ 2ε ) for k ∈ {L,H }. However, naively o�ering this menu of contracts when the principal cannot ob- serve the agent’s type would induce an agent with low degree of risk aversion to choose sH (π ) = aH + bHπ over sH (π ) = aL + bLπ . To see this, note that the expected utility for such an agent of choosing sH (π ) = aH + bHπ equals aH + b2H 2 (1 − λLσ 2ε ) = u + 1 − λLσ 2ε 2(1 + λHσ 2ε )2 − 1 − λHσ 2ε 2(1 + λHσ 2ε )2 , which exceeds aL + b2L 2 (1 − λLσ 2ε ) = u + 1 − λLσ 2ε 2(1 + λLσ 2ε )2 − 1 − λLσ 2ε 2(1 + λLσ 2ε )2 = u because λH > λL. Similarlyto the analysis of the previous section, killing this incentive of an agent with low degree of risk aversion to accept the contract designed for an agent with high degree of risk aversion will be a critical constraint when devising the optimal menu of contracts. To �nd out the optimal menu of contracts that deals with adverse selection and moral hazard, the principal must choose aH , bH , aL and bL to maximize p [bL (1 − bL) − aL] + (1 − p) [bH (1 − bH ) − aH ] 9 subject to a number of constraints: aL + b2L 2 (1 − λLσ 2ε ) ≥ u, (IRL) aH + b2H 2 (1 − λHσ 2ε ) ≥ u, (IRH ) aH + b2H 2 (1 − λHσ 2ε ) ≥ aH + b2H 2 (1 − λLσ 2ε ), (ICL) aH + b2H 2 (1 − λHσ 2ε ) ≥ aL + b2L 2 (1 − λHσ 2ε ), (ICH ) where we have used the result that the optimal e�ort of the agent for a given contract (ak ,bk ) is e∗k = bk . Instead of solving a complex optimization problem with several inequality con- straints, one can again notice that the optimal solution must satisfy the following properties: (1) Constraint (IRH ) must be binding. (2) Constraint (ICL) must be binding as well. (3) The optimal solution must be such that bL ≥ bH . (4) Constraint (ICH ) can be safely ignored. (5) Constraint (IRL) can be safely ignored as well. One can prove property (1) using (ICL), λH > λL, and (IRH ), so that aL + b2L 2 (1 − λLσ 2ε ) ≥ aH + b2H 2 (1 − λLσ 2ε ) ≥ aH + b2H 2 (1 − λHσ 2ε ) ≥ u. If we had that aH + b2H 2 (1−λHσ 2 ε ) > u (so that aL+ b2L 2 (1−λLσ 2 ε ) > u also held), then both aL and aH could be lowered a bit without violating any constraint and the principal’s payo� would increase, so we must have that (IRH ) binds at the optimal solution. To prove property (2), suppose to the contrary that (ICL) did not bind at the optimal solution. Then λH > λL and the fact that (IRH ) is binding would imply that aL + b2L 2 (1 − λLσ 2ε ) > aH + b2H 2 (1 − λLσ 2ε ) > aH + b2H 2 (1 − λHσ 2ε ) = u, 10 so one could lower aL a bit without violating (ICL), (IRL) or (ICH ) and the principal’s payo� would increase, a contradiction with optimality, so (ICL) must indeed bind at the optimal solution. To show that property (3) is true, add up (ICL) and (ICH ). Rearranging then yields that b2L (λH − λL)σ 2 ε 2 ≥ b2H (λH − λL)σ 2 ε 2 , so λH > λL yields that bL ≥ bH . Property (4) then can be shown to hold with some work. Because (ICL) is binding and λH is greater than λL, bL ≥ bH implies that aH − aL = (b2L − b 2 H ) (1 − λLσ 2 ε ) 2 ≥ (b2L − b 2 H ) (1 − λHσ 2 ε ) 2 , an inequality that can be rewritten as aH + b2H 2 (1 − λHσ 2ε ) ≥ aL + b2L 2 (1 − λHσ 2ε ), which is inequality (ICH ). It follows that the fact that (ICL) is binding implies that (ICH ) must necessarily hold, as claimed, and hence it can be ignored. Finally, to prove property (5), use (ICL), λH > λL, and (IRH ), so that aL + b2L 2 (1 − λLσ 2ε ) ≥ aH + b2H 2 (1 − λLσ 2ε ) ≥ aH + b2H 2 (1 − λHσ 2ε ) ≥ u. This shows that that constraint (IRL) can be safely ignored because it is implied by satisfaction of other constraints. We have therefore shown that the principal chooses aH , bH , aL and bL to maximize p[bL (1 − bL) − aL] + (1 − p)[bH (1 − bH ) − aH ] subject to (IRH ) and (ICL) holding with equality. When (IRH ) holds with equality, we have that aH = u − b2H 2 (1 − λHσ 2ε ), 11 so the equality in (ICL) can be rewritten as aL = u + b2H 2 (λH − λL)σ 2 ε − b2L 2 (1 − λLσ 2ε ). Using these values in the respective �xed wage components on the principal’s payo� function yields that the principal chooses bH and bL to maximize p bL (1 − bL) − b2H 2 (λH − λL)σ 2 ε + b2L 2 (1 − λLσ 2ε ) − u + (1 − p) bH (1 − bH ) + b2H 2 (1 − λHσ 2ε ) − u . Once the optimal values of bH and bL are obtained, the optimal values of aH and aL can be obtained from the previous equalities, so their determination will be ignored henceforth, as usual. To �nd out the optimal values of bH and bL, note that the �rst-order conditions yield p [ 1 − 2bL + bL (1 − λLσ 2ε ) ] = 0 and (1 − p) [ 1 − 2bH + bH (1 − λHσ 2ε ) ] − pbH (λH − λL)σ 2 ε = 0. The �rst condition is similar to the one obtains for an agent whose degree of risk aver- sion is known to be λL. The principal does not distort the e�ort-provision incentives given to the agent that needs to be compensated less for the risk taken. As can be seen from the second �rst-order condition, the principal does distort the e�ort-provision incentives given to the agent that needs to be compensated more for the risk taken (see last term in the equation). This downward distortion arises because the principal wants to make sure that the contract intended to reward such an agent should not be appealing to an agent whose degree of risk aversion is lower. Such distortion will be greater the bigger p, λH − λL or σ 2ε are. From the �rst �rst-order condition, we obtain b∗L = 1 1 + λLσ 2ε , 12 whereas the second �rst-order condition yields after some rearranging that b∗H = 1 1 + λHσ 2ε + ( p 1 − p ) (λH − λL)σ 2 ε . Note that b∗L = 1 1 + λLσ 2ε > 1 1 + λHσ 2ε > 1 1 + λHσ 2ε + ( p 1 − p ) (λH − λL)σ 2 ε = b∗H . In optimally providing incentives for a highly risk averse agent without making the contract attractive for an agent who is less risk averse, the principal is led to ine�- ciently lower b∗H relative to the cases in which it can observe how risk averse the agent is. It is worth noting that, at the optimal solution, an agent whose degree of risk aversion is low is indi�erent between accepting the contract sL (π ) = aL + bLπ and accepting the contract sH (π ) = aH + bHπ . However, an agent whose degree of risk aversion is high is indi�erent between accepting the contract sH (π ) = aH + bHπ and not accepting it, whereas an agent whose degree of risk aversion is low is strictly bet- ter o� accepting the contract sH (π ) = aH + bHπ than not accepting any contract. The expected utility that such an agent obtains above and beyond u is called her informa- tional rent. Because the principal wants an agent whose degree of risk aversion is low to prefer sL (π ) over sH (π ), such an agent must be rewarded to reveal her unobserved degree of risk aversion when accepting the contract designed for her. 13 Adverse selection and employee hiring Adverse selection and moral hazard
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