Self Investment in Human Capital: A Quadratic Model with an. Implication for the Skills Gap

Self Investment in Human Capital: A Quadratic Model with an Implication for the Skills Gap Anthony M. Marino Marshall School of Business University of Southern California Los Angeles, CA 90089-1422 E-mail: amarino@usc.edu October 3, 2018 Abstract This paper considers a hidden action agency model in which an agent can be incentivized to simultaneously work and self train to increase productivity. We determine conditions under which the principal wants to incentivize concurrent working and training and conditions under which the agent would want to participate. Next we consider outside hiring as an alternative to internal working and training. We discuss the conditions under which the agent and the principal would be better off with internal and concurrent work and training as opposed to a pre-work training program, and we provide a reason for the skills gap. JEL Code: L20, L21, L22, and L23 Key Words: Human Capital, Agency. I thank.my colleagues Odilon Camara and Joao Ramos. I also thank the Marshall School of Business for generous research support. 1

1. Introduction Investment in human capital has been and continues to be an important topic in the economics literature. Since the classic papers of Becker (1962) and Hashimoto (1981), the question of whether firm specific versus general human capital is advantageous for a firm has been extensively discussed. While general human capital in their environment is not profitable to the organization, firm specific investment is advantageous with the costs and the benefits being shared by the employee and the firm. Sharing is incentivized by the fact that it decreases the chance that either party will terminate the relationship and impose a loss on the other actor. This early literature studied the human capital investment problem in a perfectly competitive environment in the absence of an agency problem in the firm. Theoretical work has extended this early study to consider the case of imperfectly competitive labor markets, and the results show that general training can be optimal in the imperfect setting (See Kessler and Lulfesmann (2006) or Acemoglu and Pischke (1999).). In a recent paper, Fudenberg and Rayo (2017) present a dynamic model of an apprentice who can work on skill enhancing and non-skill enhancing tasks while working for a principal. The principal optimally incentivizes the apprentice to pay for general training by working for low wages and by working inefficiently hard. This paper has interesting implications for the mix of the agent s efforts, the length of the apprenticeship, and regulatory measures on apprenticeships, but it does not consider hidden action or information. Krakel (2016) has extended the study of human capital investment in the firm to the situation where there is a moral hazard agency problem and where the agent elects whether or not to engage in training in a zero-one manner. He shows that firms will want to invest in general human capital and, in some cases, specific human capital, depending on the assumptions. This paper also extends the study of human capital investment to a hidden action agency setting, but with a focus on an agent learning while working, as in Fudenberg and Rayo (2017). We 2

depart from existing agency literature by assuming that the agent can self invest in human capital through the exertion of self improvement effort. This learning or study effort stochastically leads to possible higher productivity in the firm. During the same period in which the agent attempts to improve skills in the firm, the agent exerts normal work effort to increase cash flow. Both types of effort enter the agent s convex cost of effort function, each on the continuum, with the effect of raising the total and marginal cost of total effort. The agent then becomes a multitasker with self improvement effort and cash flow effort transpiring during the same period. This characterization departs from existing literature in two ways. First, there is no pure training period in which the principal attempts to increase productivity of the agent. Instead, the agent attempts to self improve concurrently during the work period. Second, the agent must exert explicit continuous effort in an attempt to increase productivity, and such additional effort increases both the total and marginal cost of overall effort in the firm. In the cost of effort, we include a parameter which can make the marginal cost of learning effort greater or less than that of working or cash flow effort. We assume that both types of effort cannot be observed by the firm which must contract with the agent. Given these two types of unobservable actions, the principal must devise an optimal contract for the firm which, due to verifiability and observability problems, is based solely on observable cash flow. In this setting, several interesting questions arise. Under what conditions would the principal want to incentivize such learning while working, and under what conditions would the agent want to participate? How does the principal s contract change when learning is incentivized? How does the agent prioritize learning and work efforts? How does a second best equilibrium compare to the efficient equilibrium? Finally, under what conditions would it be more efficient for the firm to set up a separate pre-work training program as opposed to encouraging concurrent learning and working? We will address each of these questions. We employ a simple quadratic cost model in which work and learning efforts are complements, 3

the agent has limited liability, and all actors are risk neutral. This allows for tractability and closed form solutions which, otherwise, would not be possible. We show that there are two types of equilibria depending on the magnitude of a parameter representing the ratio of the marginal benefit of learning to a measure of the marginal cost of learning. If this marginal benefit tocostratiois below a cutoff value, the principal writes an optimal contract which incentivizes the agent not to learn. We call this the non-learning equilibrium. The principal optimally incentivizes learning effort only in situations where the benefit to cost ratio of learning is above that cutoff value. We call this a learning equilibrium. We show that a learning contract is always more high powered than anon-learningone,weranklearningandworkeffort at a learning equilibrium and at the social optimum, and we compare all equilibria to the social optimum. We show that the agent would prefer that the principal s threshold benefit-cost ratio be lower for the learning equilibrium, but the principal incentivizes learning at the higher cutoff. In the interval of the benefit tocostratios between the agent s lower cutoff and the principal s higher one, the agent would gain more surplus than the principal would lose, if the principal were to implement a learning equilibrium as opposed to a non-learning equilibrium. However, due to the fact that the agent has limited liability, it is not possible to create a transfer from the agent to the principal in order to implement the more efficient learning equilibrium in this region. This implication of the model can explain why many employers see a skills gap in hiring workers. 1 The agent who attempts to learn and enhance productivity has a base level of productivity equal to and seeks to increase that base productivity to the level + by exerting learning effort. In the upper range of benefit-cost ratios for which there is a learning equilibrium offered by the principal, we ask hypothetically whether the principal would prefer to hire an outside agent with guaranteed ability + with the endogenously optimal non-learning contract or to optimally 1 See, for example, Bessen (2014) for a discussion of the skills gap. 4

incentivize learning with the possibility that the internal agent might attain the ability level + It turns out that it is more profitable for the principal to hire the outside agent and issue a nonlearning contract. 2 However, we show that the outside agent would receive an equilibrium utility which could be greater than or less than the equilibrium utility of the internal agent in a learning equilibrium. The outside hire receives less utility than the internal aspiring agent in situations where the ratio is low, meaning that the internal agent has just a little to learn. On the other hand, the outside agent receives greater utility if the ratio is sufficiently large, indicating that the aspiring internal agent has a lot to learn. While expected profit is greater if the firm can recruit an agent with productivity + such recruiting may not be possible or it may be that learning can be achieved only within the firm. If so, then the firm might consider a separate training program to take place before the agent works. If the profit from the separate program is greater than that of the concurrent one and is sufficiently high (the agent has a lot to learn), then the separate training program has the advantage of raising both firm profit and employee welfare. If the converse is true and profit is less with the separate program and is sufficiently low (the agent has a little to learn), then the concurrent training-work program creates both higher profit forthefirm and higher employee welfare. Section 2 presents the model, and Section 3 characterizes the principal s optimal contracts. Section 4 analyzes outside hiring, and Section 5 concludes. All proofs are placed in the Appendix. 2. The Basic Model Consider a principal agent situation where the agent has hidden effort. The agent s effort can be directed towards the creation of cash flow for the firm and the agent can exert effort to attempt to increase his or her productivity in the production of cash flow. The agent then has the classic 2 This result is not obvious because the learning contract with base ability carries its own optimal wage which is different from that of the non-learning one. 5

multitasking problem of deciding whether to devote all effort to the production of cash flow or to devote some effort to cash flow production and some effort to the building of skill which leads to a greater possibility of increased productivity in producing cash flow. This problem is akin to the problem of the fishermanwhocanjustfish with current productivity or devote some effort to skill improvement in fishing so as to enhance productivity and also do some fishing during the same period. Let cash flow be denoted ˆ and assume that it can be high or low (zero), ˆ {0} where 0 The probability that cash flow will be high is given by Prob ( =ˆ ) =ˆ where is effort of the agent in attempting to create cash flow and ˆ is a random variable representing a productivity parameter for cash flow effort. The productivity parameter can take on two values given by ˆ { + } The probability that the agent achieves the higher productivity parameter ( + ) is Prob (ˆ = + ) = where represents the effort exerted by the agent to raise his/her productivity level above the base level We assume that 0 and + 1 Given these assumptions, the firm s expected cash flow is (ˆ ) = [( (1 ) +( + ) ]= ( + ) At a later point, we will present feasibility restrictions, denoted (F.i), on parameters which guarantee that, in equilibrium, the two probabilities are in the unit interval, ˆ [0 1) The firm cannot observe either type of effort. Both the firm and the agent know the default 6

productivity level and they know the probability distribution of ˆ at the time of contracting. However, we assume that because of verifiability problems, the firm cannot contract on the agent s productivity level. At time zero, the principal and the agent contract. After contracting, the agent exerts nonnegative learning effort and nonnegative working effort based on the expectation of productivity. Possibly a new productivity level is realized, if learning effort took place. At the end of period one, cash flow is realized. During the work period, the agent can engage in periodic working and learning without fully knowing productivity. For example, an academic researcher can develop a research idea and periodically learn new analytical techniques during the same period without perfect knowledge of productivity. The same would go for a worker creating a physical commodity in that periodic technique development and practice may be intertwined with production, the result of which is an expectation of incremental productivity. 3 For tractability, the cost of effort is assumed to be quadratic and given by ( 2 )( + ) 2 The parameter measures the extent to which the marginal cost of learning effort differs from that of normal cash flow effort, in that it is equal to the ratio of the marginal cost of learning effort to that of cash flow effort. If 1 then it is more costly on the margin for the agent to learn than it is to produce cash flowandconverselyif1 In the former case, learning is more of a chore than normal work effort and in the latter it is a bit of a diversion from everyday work effort. Let us think of as a relative difficulty of learning parameter. The constant represents a cost of effort parameter. We will not consider situations where there is a capacity constraint on total effort, or, equivalently, we look at cases where such a constraint is non-binding. 4 However, there is 3 The notion that an agent does not know his or her productivity with certainty is not new in the agency literature. See Holmstrom (1999). 4 Effort is typically not measured in people hours, and if it were, few if any work 24-7. 7

an opportunity cost of learning effort, because any positive amount raises the marginal and total cost of working effort in the cost of effort function. Both the firm and the agent are assumed to be risk neutral, and the agent has an outside option utility of zero. The principal s contract consists of a possible flat salary and an incentive percentage of cash flow. We will assume that the agent is subject to limited liability in the sense that no promised payment can be negative, = 0 The agent s optimization problem is to max + ( + ) ( { } 2 )( + ) 2 (1) The first order conditions for learning and cash flow efforts are : ( + ) =0 and (2) : ( + ) ( + ) =0 (3) It is interesting to note that condition (2) implies that ( ) 2 =0 Thus, we will have a zero corner solution in learning effort if ( ) 0 for [0 1] This is the case where the ratio of the marginal benefit of learning to the relative difficulty of learning ( ) is lower than thecostofeffort parameter, given the optimal incentive share. We will return to this interesting corner solution when we consider the principal s problem. Assuming that a solution exists, we can solve (2) and (3) for the equilibrium values of and as functions of We have = ( ) (2 ) ( ) and (4) 8

= 2 ( ) (5) (2 ) At this point we note that the principal will choose (0 1) Negative are ruled out by limited liability, and =0would, by (2) and (3), imply that both efforts and profit are zero. A = 1 would imply that the firm s profit is non-positive, under limited liability. Thus, we focus on (0 1) From (4) and (5), for either effort level to be well defined and for to be non-negative, it is necessary that 2 for (0 1) Further, given 2 we see that = 0 if and only if ( ) = 1 Hence, it is necessary that 1 if there is to be a positive solution ( ) 0 In what follows, we will assume A.1 The benefit to cost ratio of learning satisfies 2 1 where It turns out that the condition 2 in A.1 implies that the agent s objective function is strictly concave in ( ) for [0 1] 5 The term measures the total (organizational) marginal benefit oflearningeffort per unit of work effort,. Theterm is the marginal cost of learning effortperunitoftotaleffective effort, ( + ) We will henceforth refer to as the marginal benefit to cost ratio of learning. Next consider the principal s problem. We are interested in characterizing two types of solutions. The first is a solution in which both work effort and learning effort are positive. The second is a corner solution where work effort is positive, but learning effort is zero. We will begin by characterizing the conditional second best contract ( ) for the principal which would generate an interior solution ( )( ) 0 By conditional, we mean conditional on generating an interior solution in both efforts. We will be interested in characterizing the set of parameters for which 5 The Hessian of the agent s objective function will be denoted and the first independent variable will be taken as with taken as independent variable 2. We have that 11 = 2 0 22 = 0 12 = and 11 22 ( 12) 2 0 iff 2 for [0 1] For the latter to be true it suffices that 2 Under this condition, all of the conditions for strict concavity are met 9

this conditional "learning" solution exists. Keep in mind that this solution is profit maximizing, conditional on obtaining an interior solution in efforts (best in the set of interior solutions). It can be that a corner solution in learning effort may dominate the conditional learning solution for the principal, for some parameter values. In Section 3, we will show that in a subset of parameters for which the conditional learning solution exists, the principal prefers a contract which generates a corner solution in learning effort, =0. Following our characterization of the conditional learning equilibrium for positive efforts, we will characterize the set of parameters for which the agent sets learning effort at zero, given the principal s optimal contract for such. Givensuch the agent s new incentive compatibility constraint for alone generates a new profit function for the principal. For this case, we will characterize the principal s optimal contract. The principal will choose and so as to maximize the firm s residual profit subject to the agent s limited liability constraints and the agent s participation constraint. 6 The principal s problem is to max { } +(1 ) ( + ( )) ( ) (6) subject to limited liability and participation: 0 and (LL) + ( + ( )) ( ) ( 2 )( ( )+ ( )) 2 = 0 (P) The following result allows us to simplify the principal s problem. Lemma 1. At a solution to the principal s problem, =0and the participation constraint is 6 Limited liability here means that the principal does not stipulate any negative payments for the agent to pay as part of the contract. All payments to the agent must be nonnegative. 10

non-binding for any (0 1] and any solution with 0. Using the result of Lemma 1, we can employ (4) and (5) to rewrite the principal s problem as max ( ) 2 3 (1 ) { } =2 (2 ) 2 (7) The which solves (7) is =2(1 1 ) (8) The optimal is a positive fraction under A.1. Note that the power of the contract,, isincreasing in the benefit-cost ratio of learning, In the following result, we address the satisfaction of the second order condition to the principal s problem and the question of whether the equilibrium is unique. Lemma 2. There is a (0 1) for which (8) holds, the second derivative of the objective function of problem (7) is negative at,and is a unique interior maximizer of ( ) The condition that 2 is the condition which guarantees that the agent s objective function is strictly concave, and it also implies that 1 The condition A.1, 2 1 implies that there is an interior optimizer in. Next, we can substitute the optimal into the expressions for and given by (4) and (5). We have = ( (3 2)) and = (2 ) 2 2 (2 ) 9 Both and are well defined under A.1. Further, for an interior solution in we require that (3 2) 0 Under this condition, we have that 1 0 so that, from (8), 0 We then have that an interior equilibrium ( ) 0 is guaranteed if 2 3 2 (10) 11

Condition (10) and the solution for imply that (2 3 1) Define = with = For the learning equilibrium to be feasible, the two equilibrium probabilities, and ˆ must be in the unit interval. Given our assumptions, it suffices that 1 From (9), 1 if the feasibility condition ( ) (2 ) 3 2 (F1) holds. Likewise, 1 if ( ) 4 (F2) If 3 2 0 then will be zero, and the agent will not invest in learning. In this corner solution, the first order condition for the agent s optimal choice of work effort, is =( ) ( ) The principal s optimal choice of is 0 and the profit function is ( ) = 2 2 (1 ) (12) where the optimal choice of can be shown to be =1 2 The equilibrium = 2 (11) is in the unit interval if 2 0 12 (F3)

We have Proposition 1. For both learning effort and cash flow effort to be positive and described by (9) and for the conditional second best to be characterized as =2(1 1 ) (2 3 1) it suffices that 2 3 2 At such an interior solution in efforts, the participation constraint is non-binding and =0. If 1 3 2 then learning effort will be zero, work effort is given by = 2 and the optimal =1 2 At such a corner solution, the participation constraint is non-binding and =0 In the conditional learning equilibrium, the principal sets the incentive share in the range where a majority of the cash flow goes to the agent. The learning equilibrium is, therefore, implemented with a high powered contract. If the marginal benefit tocostratio falls below a lower bound, the agent ceases human capital investment and only exerts work effort. In this non-learning equilibrium, the principal s optimal incentive share for the agent is lower or less high powered than it is in the learning equilibrium. The comparative static results at the two solutions are straightforward and are as expected, with cost parameters decreasing effort and productivity parameters increasing efforts. The comparative static results on the incentive share at the interior solution point out that in firms where learning is less difficult and/or where learning is more productive, the sensitivity of the optimal contract will be greater in that agents will be promised greater contingent shares of the firm. The incentive share becomes less when efforts are more costly. Whether the agent spends more time working as opposed to learning depends on the difficulty of learning parameter. We have Proposition 2. If = 1 then There exists a ˆ (0 1) such that if ˆ. The threshold value ˆ is decreasing in and increasing in The second best then admits solutions where there is more learning effort than work effort, if 13

the difficulty of learning is sufficiently lower than that of working, ˆ 1. Otherwiseifˆ, work effort will exceed learning effort. A smaller productivity increase as the result of successful learning or a greater marginal cost of effort will reduce that threshold. It is also of interest to examine how the agent s choice of efforts differ from those that would be chosen by a social planner at a first best solution. The social planner s firstbestproblemisto choose and so as to maximize ( ) ( + ) 2 ( + ) 2 Under A.1 s restriction 2 it can be shown that is strictly concave in ( ) If a an interior solution exists, the first order conditions define a maximum of welfare. They are given by : ( + ) =0 and (13) : ( + ) ( + ) =0 (14) Solving (13) and (14), we obtain first best effort levels = 2 (2 ) and = ( ) (2 ) (15) We note that the planner would want to set human capital investment to zero if the condition 1 5 0 is met. However, this is ruled out by A.1. This is a weaker condition than the condition for zero human capital investment at the second best. That is, the upper threshold for the marginal benefit to cost ratio which causes the planner to cut off human capital investment ( 5 1) islower than that of the firm at the second best ( 5 3 2). Let an superscript denote the first best. If 14

=0 then, at a non-learning equilibrium, = (16) Each of the firstbestefforts at the interior and corner solutions ((15) and (16)) are greater than their counterparts at the second best solutions ((9) and (11)). This is expected since with 1 the agent does not internalize the full marginal benefit of his or her actions. At the first best interior solution, as in the second best above, the planner would want more effort to be devoted to training than to work only if the marginal cost of learning effort is significantly less than that of working. We have Proposition 3. If = 1 then There exists a ˆ (0 1) such that if ˆ. The threshold value ˆ is decreasing in and increasing in 3. The Equilibria In this section, we will investigate globally optimal contracting by the principal and the welfare of the agent under different parametric configurations. From the basic model, we know that there are two possible equilibria. One equilibrium is such that the agent does not learn. At this equilibrium, we have =0 0 and =1 2 This will be called the non-learning equilibrium and it occurs if (1 3 2]. The other equilibrium is the conditional learning equilibrium where 0 and =2(1 1 ) (2 3 1) and it obtains when (3 2 2) If (3 2 2) we have not ruled out the possibility that the principal might prefer a corner solution, with =0 to the conditional learning equilibrium. The following proposition addresses this issue. Proposition 4. The principal s optimal contract and the possible equilibria are described as in the following. 15

(i) If (0 1 2 (1 + 51 2 )) the principal sets =0and =1 2. The agent sets learning effort at zero and work effort is = 2 The principal and the agent have positive expected payoffs. In the intermediate region (3 2 1 2 (1 + 51 2 )) the agent prefers a learning contract, but the principal prefers and invokes a non-learning contract. (ii) If [ 1 2 (1+51 2 ) 2) the principal sets =0and =2(1 1) (2(1 1 ( 1 2 (1+51 2 ))) 1) The agent sets learning effort and work effort as in (9). The principal and the agent have positive expected payoffs. Marginal benefit to cost ratios which are too small result in an equilibrium involving no learning on the part of the agent. At these small benefit tocostratios,theagentdoesnotwishtoexert positive learning effort, given optimal actions by the principal. There is an intermediate region of marginal benefit to cost ratios over which the agent would prefer the learning equilibrium, but the principal does not. In this region, although the agent would like to supply positive learning effort at the principal s conditional learning incentive share, the principal would make greater profit if the non-learning equilibrium is invoked. If marginal to benefit costratiosareaboveathreshold, then both the principal and the agent benefit from a learning type equilibrium, and it is invoked by the principal. Let and denote the principal s expected profit in the learning and non-learning equilibria respectively, and let and denote the agent s expected utility at the same equilibria. In the intermediate region of benefit tocostratios, (3 2 1 2 (1 + 51 2 )) the agent gains expected utility 0 by learning, and the principal loses profit with learning, 0 Define the difference = If this were positive, then it would be more efficient or total surplus would be greater, if the learning equilibrium were implemented. Indeed we have Lemma 3 If (3 2 1 2 (1 + 51 2 )) then 0 While it is Pareto efficient for the firm to move to the learning equilibrium from the non-learning 16

equilibrium in the intermediate region of marginal benefit to cost ratios, the firm, ofcourse, would not do this because it does not maximize profit. Limited liability of the agent prevents the agent from offering the principal a lump sum transfer of as an inducement to implement the learning equilibrium. In the low state of the world with ˆ =0, the agent could not pay the transfer. A transfer contingent on the good state would of course change the calculus of the agent s optimization problem, affect effort and then unravel a transfer scheme from the agent to the principal. 4. Outside Hiring and the Skills Gap Let us consider the learning equilibrium with [ 1 2 (1 + 51 2 ) 2) In this equilibrium, both the principal and the agent are aspiring to see the agent gain the incremental productivity ( + ) and both actors benefit from this endeavor. Would the principal benefit more if an agent of type ( + ) could be hired from the outside as opposed to having the internal agent attempt to learn? The outside agent with productivity ( + ) would be optimally paid =1 2 if the participation constraint is nonbinding. The principal s profit wouldbe + = ( + )2 2 4 and the principal s profit with the internal learning agent is given by = 2 2 3 4 2 (2 ).Wehave + T 0 ifandonlyif ( ) (2 ) 1 2 T (17) Moreover, would the outside hire with ability + enjoyagreaterorlesserutilitythanwould the internal learning agent? The equilibrium utility of this agent is + = ( + )2 2 8 who learns has utility given in = 2 2 ( ) 2 2 (2 ). Rewriting the difference, we have and the agent + T 0 ifandonlyif ( ) 2( 1) 1 2 (2 ) 1 2 T (18) Proposition 5. Assume that feasibility conditions (F.1) and (F.2) are met. In the feasible 17

region [ 1 2 (1 + 51 2 ) 2) the principal is better off hiring an outside agent with ability ( + ) as opposed to training an internal agent, + Further, an internal learning agent can enjoy a greater or lesser utility than an external agent with ability ( + ) + T It is not too surprising that the principal always benefits if the firm can hire an outside agent with guaranteed productivity + The firm would not train this agent, would be guaranteed high productivity, and would pay =1 2 as opposed to the higher powered incentive share (2(1 1 ( 1 2 (1 + 51 2 ))) 1) 1 2 An agent from the outside may or may not be better off than an agent who internally trains and seeks to have high productivity. From (18), we can fix and the ratio = and ask what would make the right side of this condition greater than the left side, so as to make the internal agent better off than the external one. The answer is that would have to be high relative to This condition says that is low, and it indicates that the base agent with ability has just a little to learn in the learning process. A high is indicative of the internal agent having a lot to learn, and, in this case, the external high productivity agent would be better off than the aspiring internal agent. Given that the firm prefers the learned outside agent, if the skill required to attain productivity level + could be acquired outside the firm, the firm would be better off attempting to recruit agents with the higher ability at the start. If this skill level cannot be acquired outside the firm, then the firm might consider a dedicated training program which would precede employment. As in the internal concurrent training equilibrium, there would be no guarantee that the agent would attain productivity + and there would be training costs which would have to be accounted for when comparing expected profits with the internal program outlined above. If the costs of such a program are such that expected profit is greater than that of the concurrent program, then in cases where is high, the agent is also better off and total surplus is gained. If the converse is true and the firm prefers the internal program, then total surplus is greater with the concurrent program, if 18

is low. Finally consider the case where skills are firm specific. If is thought of as a skills gap between what skill level the principal desires and that which is available on the market, then the reticence of the principal to train in the intermediate interval where the employee is willing and the firm is unwilling may help explain the persistence of a skills gap. In the above model, it is caused by the high standards on the benefit to cost ratio of concurrent training optimally maintained by the firm. It is an unwillingness to train problem in the face of low benefit to cost ratios of learning. 5. Conclusion The main takeaways of this paper are succinctly described in the following. We find that concurrent work and training is profitable for the firm only if the marginal benefit to cost ratio associated with training while working exceeds a threshold, and we show that the employee would like to see that threshold lowered. We show that it is not possible for the agent to issue a lump sum transfer to the principal in an intermediate region of benefit-cost ratios so as to induce the principal to implement a learning contract below the principal s threshold but above the agent s lower threshold. This inability for a Pareto efficient transfer is due to the agent s limited liability. A skills gap may arise in cases where skills are firm specific due to the higher pre-conditions the firm optimally places on incentivizing concurrent training. The learning contract is always higher powered than the non-learning contract, learning effort can be less or greater than work effort, and both of these efforts are less than their socially efficient levels. Work effort is always greater than learning effort if learning effortisatleastasdifficult as work effort. We show that the firm is always better off if it can recruit a higher productivity employee and abandon training altogether. Such recruitingmaynotbepossibleespeciallywhenthe higher productivity being sought after involves firm specific skills which must be learned inside the 19

firm. In this case, concurrent training may be optimal, but, when accounting for cost, its expected profit must be compared to that of a pre-work training program. From a total surplus standpoint, if the concurrent training program is more profitable, then it also tends to improve total surplus when the employee has a little to learn as a result of training. If the pre-work training program is more profitable, then total surplus increases when the employee has a lot to learn through training. Appendix ProofofLemma1:The relevant Lagrangian is L = +(1 ) ( + ( )) ( )+ [ + ( + ( )) ( ) ( 2 )( ( )+ ( )) 2 ]+ + The first order condition for is 1 + =0 By 0 we have that 0 Thus, =0 With =0 the agent s objective function, ( )=( + ) ( 2 )( + ) 2 is strictly concave in ( ) under 2 Further, (0 0) = 0 The gradient condition for a strictly concave function then tells us that ( ) ( )+ ( ) =0 in equilibrium, with 0 and 0 It follows that (P) is non-binding. Proof of Lemma 2: Taking the second derivative of the objective function of (7), we obtain 2 2 = 2 2 2 3 2 [ 4 ( 3+ ) ]. The sign of this expression is that of 4 +(3 ) If we ( 2+ ) 4 substitute for the optimal from (8), we obtain 2 + which is assumed to be negative. At any critical point, ( ) is locally strictly concave. Given that the function is twice differentiable and, thus, has continuous first derivative, the existence of more than one interior critical point would imply that the negative second derivative property at an interior critical point would 20

be violated at some point on the convex combination of the two alleged maxima. Thus, uniqueness holds. Finally, existence of is guaranteed by the conditions 0 (0) 0 and 0 (1) 0 By continuity there is a critical point, Firstnotethat 0 (0) = (4( + ) 0, by1 Next, 0 (1) = 2 2 3 (2 ) 2 0 Proof of Proposition 1: The text proves all but the statements regarding the solution with =0 For this case, the principal s Lagrangian is L = +(1 ) ( )+ [ + ( ) 2 ( )2 ]+ + The first order condition for is again 1 + =0 By 0 we have that 0 Thus, =0 From (13), = so that participation constraint is now given by ( ) 2 ( )2 0 The principal solves max { } 2 2 (1 ) It follows that =1 2 ProofofProposition2:We have = (2 +3 2 ) 2 (2 ) and the sign of this expression is that of +3 2 However, takes on a supremum value of 2 so that +3 2 0 and work effort is greater than learning effort if = 1. If 1 then we need more information to determine the sign of (15). The threshold level of ˆ which sets +3 2 =0is defined by ˆ = (9 2 + 8 ) 1 2 3 2 It is clear that ˆ 0 from the 21

fact that the numerator of ˆ is positive, For ˆ 1, we would require 2 Under A.1, we have that 2 so that indeed 2 if 1 Thus, under A.1, there would be a ( 2 ˆ ) such that 0 Note that 2 ˆ by A.1, so that the latter interval is non-empty. We have that ˆ is clearly increasing in and ˆ = 0 (9 +8 ) ProofofProposition3:From (15), ( )= (2 + ) =(1 + ) with (1 2) For = 1 the maintained condition 2 implies that 1+ 0 so that it is socially optimal for the agent to devote more time to working than to learning. For 1 the critical ˆ which sets ( 2 + ) =0is ˆ = ( 2 +4 ) 1 2 2 Note that ˆ 0 Also ˆ 1 if andonlyif 2 A.1 For ( 2 1 By A.1, 2 1 Moreover, ˆ 2 2 ( 2 +4 ) 1 2 2 if 2 1 which is again implied by ) it is socially optimal for the agent to devote more time to learning than to working. Clearly, ˆ is increasing in and ˆ = 0 ( +4 ) ProofofProposition4:In the non-learning equilibrium, the principal s expected profit and the agent s expected utility are = 2 2 4 and = 2 2 8 respectively. In a learning equilibrium, the principal s expected profit and the agent s expected utility become = 2 2 3 4 2 (2 ) and = 2 2 ( ) 2 2 (2 ) The principal would prefer that the agent learn if assuming that the sufficiency conditions for the learning equilibrium are met. We can write this condition as ifandonlyif1 2 2 + 3 1 for (3 2 2) The function ( ) 1 2 2 + 3 is strictly increasing in the marginal to benefit cost ratio with (3 2) = 0 125 and ( 1 2 (1 + 51 2 )) = 0 Clearly, the principal prefers the learning equilibrium only in the interval ( 1 2 (1 + 51 2 ) 2) Next, consider the agent s welfare. If a learning contract is offered and the sufficiency conditions for 0 are met, (3 2 2) then the agent would prefer the learning equilibrium if 22

We can write this condition as ifandonlyif 3 +4 2 2 4 0 for (3 2 2) The function ( ) 3 +4 2 2 4 is strictly increasing in 3 +4 2 2 4 with (3 2) = 0 875 0 Thus, the agent prefers learning in the entire feasible region for the marginal benefit-cost ratio That is, unlike the principal, the agent prefers the learning equilibrium in the intermediate region of benefit costratiosgivenby (3 2 1 2 (1 + 51 2 )) Finally consider the case where (0 1 2 (1 + 51 2 )) For in this region, the principal prefers the non-learning equilibrium. If we can show that the agent selects this equilibrium if =1 2 then the result obtains. If the principal sets =1 2 the agent optimizes over a selection of accordingtothefoc(2)with =1 2 This condition is ( 2 ) 2 =0 Thus, for a positive solution in it is necessary that ( 2 ) 0 or 2 The latter is impossible since 2 Proof of Lemma 3: We have that = 2 (2 3 3 4 2 2 +6 2 2 3 3 3 ) so that, given our 8 2 (2 ) assumptions, the sign of is given by the sign of 2 3 3 +4 2 2 6 2 2 +3 3 3 Dividing this expression by 3 3 0 we obtain 2+4 6 2 +3 3 This expression is positive for (3 2 1 2 (1 + 51 2 )) ProofofProposition5: Because = and appears on both sides of conditions (17)-(18), 23

we will think of as fixed and variations in as being generated by variations in = We will begin the proof by showing that it cannot be true that = + Suppose to the contrary that the latter is true. Then, ( ) 0 However, if we can show that ( ) ( ) then feasibility condition (F.1) is violated and we have a contradiction. We have ( ) ( ) if (2 ) 1 2 (2 ) 3 2 or rewriting 2 4(2 ) Theleftsideisincreasingin so that it has a minimum value of 2.62 while the right side is decreasing in with a maximum value of 8 4(1 62) = 1 52 2 62 Thus, ( ) ( ) and + if the latter exists in the feasible region. To see existence, let =1 75 = 53 = 455 =1and = 32 For this example, we can set 53 1 =1 75 so that =3 3 Here, ( ) =7455 = ( ) =1 02 455 ( ) =1 02 0 and ( )= 698 0 In this example, we have that + and + All that remains to be shown is that there exists a case where + Take =1 71 = 467 =1 28 = 21 and = 6 For this case, ( ) =1 2 1 28 = ( ) =11 64 1 288 = ( ) =1 38 0 and ( )=1 61 0 Thus we have that + and, of course, + References [1] Acemoglu, D. and J.S. Pischke. 1999. "The Structure of Wages and Investment in General Training." Journal of Political Economy 107: 539 572. [2] Becker, G. 1962. "Investment in Human Capital: A Theoretical Analysis." The Journal of Political Economy 70: 9-49. [3] Bessen. J. 2104. "Employers Aren t Just Whining, the "Skills Gap is Real" is Real." Harvard Business Review https://hbr.org/2014/08/employers-arent-just-whining-the-skills-gap-is-real. [4] Fudenberg D. and L. Rayo. 2017. "Training and Effort Dynamics in Apprenticeship." Working Paper, August 9, 2017. 24

[5] Hashimoto, M. 1981. "Firm-Specific Human Capital as a Shared Investment," American Economic Review 71: 475 482. [6] Holmstrom, B. 1999. "Managerial Incentive Probelms: A Dynamic Perspective," Review of Economic Studies 66: 169-182 [7] Kessler, A.S. and C. Lulfesmann. 2006. "The Theory of Human Capital Revisited: On the Interaction of General and Specific Investments," Economic Journal 116: 903 923. [8] Krakel. M. 2016. "Human Capital Investment and Work Incentives. "Journal of Economics and Management Strategy 25, 627-651. 25