Job Design: Task-Specific Human Capital, Workers Discretion and Incentives in a Competitive Labor Market

Transcription

1 Job Design: Task-Specific Human Capital, Workers Discretion and Incentives in a Competitive Labor Market Felipe Balmaceda 1 Centro de Economía Aplicada (CEA) Universidad de Chile 2 July 5, I would like to express my gratitude to Mike Waldman for his valuable comments to an earlier version and to participants in the regular seminar at CEA-University of Chile and the Econometric Society Meeting, Mexico. Financial support was provided by Fondecyt through research grant # Rep ublica 701, Santiago , Chile. fbalmace@dii.uchile.cl

2 Contents 1 Introduction 1 2 The Model The Basic Set-Up The Wage Determination Procedure The First-Best Outcome The Efficient Time Allocation First-Best Efficient Job Design First-best Efficient Training The Spot Market Equilibrium: The no Pay-for-Performance Case The Optimal Time Allocation Optimal Job Design Spot Market Task-Specific Training The Spot Market Equilibrium: The Pay-for-Performance Case The Optimal Time Allocation Optimal Job Design Spot Market Task-Specific Training Job Design, Technical Change and the Skill Premium Technical Change The Skill Premium Final Comments 40 1

3 Abstract This paper provides a theory of job design driven by task-specific human capital and hold-up. I provide a rationale for the adoption of job designs that make an extensive use of multi-tasking, multi-skilling and workers discretion. The model s predictions are broadly consistent with the empirical evidence with regard to the relationship between technical change, changes in the quality of the labor force, job design and wage evolution. For instance, it is show that technical changes that increases skill complementarity lead to a greater wage inequality and wage growth for able workers. Keywords: Task-Specific Training, Hold-up, Discretion, Technical Change, Wage Evolution JEL-Classification: J41, J24, D21.

4 1 Introduction In the last decades most corporations have experienced a significant transformation in terms production and management. The traditional "fordistic" or "tayloristic" firm, characterized by mass production, high specialization of the workforce and centralization of decision making authority, was the predominant (and generally successful) model in the industrial economy of the past century. However, from the end of the 1980s, its rigid vertical structure gave way to new organizational structure, known as "Post tayloristic" firm, better equipped to operate in more competitive and customer-oriented markets, able to fully exploit the potential of new technologies (such as computers, more sophisticated tools, multi-task machines, etc.) and with a requirement for a more flexible and skilled workforce. This trend went along with a significant increase in the average education and skills of the workforce, an increase of labor market flexibility, the development of new information and communication technologies and a rise in wage inequality. 1 I propose an equilibrium theory of job design that has four main elements: firm-provided task-specific human capital; individuals are heterogeneous in their ability to perform in different tasks; jobs can be set up to be accomplished with a single task or multiple tasks; and wages are determined by bargaining with outside options. This theory provides an economic rationale for the rise of the "Post tayloristic" organization. Based on the model results, I argue that theories that seek to guide empirical research on the link between wage inequality and technical progress must consider job design at the center of the analysis of organizations and link the organizational structure with aggregate wage variables via equilibrium frameworks. Task-specific human capital is the part of on-the-job training that is specific to a task being performed on the job, as opposed to being specific to the firm. 2 Hence, when task-specific capital is accumulated, multiple firms value the capital, so a share of the value of this capital will be reflected in the worker s wage. The idea of task-specific human capital is closely related to the idea of occupation-specific training. However, task-specific training differs from occupationspecific training in that it does not fully depreciate if an individual leaves his occupation, and the applicability of this concept is different from that for task-specific capital. The relevance of task-specific human capital is warranted by its empirical importance. Gath- 1 For evidence on new work practices see, Hammer and Champy (1993), Osterman (2000, 1994), Black and Lynch (2001), Autor, Levy, and Murnane (2003), and Lindbeck and Snower (2000, 2001). This evidence and other is presented in detail in section 6. 2 In what follows I use human capital and training as synonyms. 1

5 mann and Schonberg (2007) show that task-specific human capital is an important source of individual wage growth, in particular for university graduates. For them, at least 40 percent of overall wage growth over a ten year period can be attributed to task-specific human capital. For the low- and medium-skilled, task-specific human capital accounts for at least 35 and 25 percent of overall wage growth respectively. This shows that workers are more likely to move to occupations with skills requirements similar to their current occupation. It also implies that task-specific human capital will be an important source of individual wage growth compared to general or more specific labor market skills. Kambourov and Manovski (2007) document the existence of substantial returns to occupational tenure. They find that everything else being constant, ten years of occupational tenure are likely to increase wages by at least 19% and, when occupational experience is taken into account, tenure with an industry or an employer have little importance in explaining the wage a worker receives. This paper considers a two-period competitive labor market model with risk neutral firms and workers that do not discount future income. A firm s output is the sum of the output on each job and each job requires one worker. Jobs can be set up to be accomplished with a single task or multiple tasks. The output of a job depends on the tasks involved in the design of the job and whether the worker succeeds or fails in the task. Task can be complements or substitutes. The probability of success in each task is independent of that in the other tasks and it is equal to the time the worker allocates to the corresponding task. Each worker has one unit of time that must be allocated across tasks as he sees fit. For the sake of simplicity, I assume that there are two tasks: a routinized and a discretionary task. For any given worker, time costs in each task depends on his ability to perform in that task and on his task-specific training. As either of this rises, time costs falls, and as one rises, the marginal decrease in time costs due to the increase in the other skill rises. Thus, ability and task-specific training are complements. There is a contractible variable that is an imperfect proxy of the time spend on the routinized task. When the contractible variable is set to a given level, the worker must allocate at a given amount of time to the routinized task that depends as on a random variable privately learned by the worker before choosing his time allocation. I assume that output is non-contractible and study the case in which pay-for-performance contracts are not feasible and then assume following Baker (1992) and Baker, Gibbons, and Murphy (1994) that there is a contractible, but imperfect performance measure, on which pay-for-performance contracts can be based on. The timing is as follows. At the beginning of period 1, firms compete for workers of known 2

6 ability in a Bertrand-like fashion offering a one period wage contract. Upon acceptance, and during the first period, each firm decides how much training to provide to its workers. Simultaneously, workers acquire costless on-the-job specific training. Before period 2 begins, an idiosyncratic productivity shock that determines the worker s firm-specific training productivity is publicly realized. Thus, symmetric learning is assumed. After the productivity shock is realized, firms in the market make offers consisting on a fixed wage and a job assignment. After observing the market offer, the incumbent firm chooses a job assignment and then, the firm and the worker either negotiate the wage for the period, or alternatively, they may either refuse to trade, or agree to trade with a third party instead. The wage determination procedure within the relationship is based on the outside option principle found, for example, in Sutton (1986). 3 The first-best outcome entails training able workers in both tasks and assigning them to a multi-task job, while training the less able workers in the easy task only and assigning them to a specialized job comprising the easy task only. Under spot contracting, the first-best outcome cannot be implemented because of the hold-up problem due to the fact that task-specific training is transferable across firms. More specifically, when the idiosyncratic productivity shock is such that the worker s outside option does not bind, the firm gets only a share of the return to task-specific human capital, while when the shock is such that the outside option binds, the firm gets no return to training. The reason is that training is equally productive across firm-tasks pairs, the labor market is competitive, and in equilibrium all firms adopt the same job design. Thus, in no state firms are full residual claimants on their investment in human capital. 4 With regard to the investment decisions. The firms trade-off the productivity gains from their investments against the increase in the severity of the hold-up problem and the cost of their investments. The former effect arises because task-specific training increases the worker s outside option and thus, ceteris-paribus, the worker is more likely to appropriate the return to 3 The outside option principle assumes that bargaining and employment on the spot market are mutually exclusive. In this case, taking a job outside the firm or hiring a replacement worker terminates the bargaining process. Therefore, the no-trade payoff would be an outside, rather than an inside option in bargaining terminology. In the case where only the responder can take the outside option after receiving a counteroffer, which is the case focused on this paper, the equilibrium is unique. 4 If training were to be continuous, there will always be under-investment in task-specific training. See Balmaceda (2005) for a detailed discussion on the relationship between the hold-up problem and human capital investments. 3

7 training, and the latter depends on whether training investments are substitutes or complements. With regard to task choices. Firms trade-off the productivity gain from multi-tasking against the direct cost of designing multiple task jobs and the greater wage cost implied by labor market competition. The direct cost of multiple-task jobs depends on how much costly is to accommodate the second task at the margin, when the first task has been chosen. In other words, it depends on the degree of complementarity across tasks. In equilibrium, firms train less able workers in the easy task and assign them to an easy-task job when training costs are small, while they provide them with no training and assign them to the easy-task job when costs are high. Able workers are trained in both tasks and allocated to a multi-task job when training and job design costs are small, while they are either trained in the difficult task only and assigned to a difficult-task job, or they receive no training and are assigned to an easy-task job when costs are high. Thus, multi-skilling and multi-tasking are observed among able workers in industries where training and task complementarities are sufficiently large to compensate for the impossibility to appropriate the full return to training due to the hold-up problem. In contrast, specialization as an equilibrium outcome is the result of under-training and lack of skill and task complementarities. The paper also deals with two other important issues: (i) employees discretion and (ii) long-term contracting. Mainly, I show that workers are more likely to be granted discretion to make decisions that affect their productivity at the task level when multi-skilling and multitasking are optimal, and that long-term contracts ameliorate the hold-up problem. This implies that multi-skilling, multi-tasking and worker s discretion are more likely to be found in jobs where long-term contracts are in placed such as Japanese firms. For example, an MIT study of automobile production found that Japanese plants in Japan gave their newly hired workers 9 times more hours of training that their US counterparts, make their workers to undertake more tasks and provide them with greater discretion. 5 Since the seminal work of Katz and Murphy (1992), the study of wage inequality has taken the organization of production as fixed and determined by a particular specification of technology, and it ignores both the source of the interaction between workers and the organizational aspects of this interaction. These shortcoming is particularly important in light of growing empirical 5 For evidence on the difference between Japan and the US along the dimensions considered in this paper see Morita (2001, 2005). 4

8 evidence that points to the complementarity between organizational change and changes in the distribution of wages (e.g., Bresnahan and Hill (2002) and Autor, Levy, and Murnane (2002). Using the results of this model I argue that theories that seek to guide empirical research on these areas must put task-specific training and complementarities at the center of the analysis that links the organizational structure with wage evolution via equilibrium frameworks. The model predicts that any technical change that rises task and training complementarity increases the likelihood of multi-skilling, multi-tasking and discretion. The same holds for any skilled biased technical change under reasonable conditions. These predictions provide a foundation for the observed empirical relationship between job design, wage inequality and wage growth. In particular, the model shows that any technical change that induces job enrichment increases wage inequality and wage growth among able workers, reduces the tenure effect and increases equilibrium turnover. There is a small but growing literature that studies the issue of job design in a multitasking principal-agent setting. Much of that literature focuses on how incentive contracts together with the quality of performance measures affect the allocation of tasks across jobs. Holmström and Milgrom (1991) consider an assignment problem with two identical workers, and show that the measurability of task output is an important determinant of job assignments. Itoh (1994) also considers a multi-task assignment problem with identical agents where jobs differ by the number of tasks. In his model, specialization has benefits because tasks are substitutes, but it has costs as well because more agents have to be paid a risk premium. MacDonald and Marx (2001) also study a multi-task setting, but they focus on inducing the appropriate time allocation through complementarities and incentives, whereas my focus is on firms incentives to provide task-specific skills, the assignment of heterogeneous workers to different jobs, and the design of jobs. Meyer (1994) analyze optimal task assignment when a firm needs to learn the abilities of employees. She shows that when projects require collaboration between juniors and seniors and only team outputs are observable, having juniors divide their time between two projects is less informative about their abilities, but more informative about their senior teammates abilities, than having juniors devote all their time to a single project. These papers while speaking to job design issues, they deal with different issues related informational problems, which are fully absent in my model. In addition, none of them are concerned with investment in human capital and the difference between multi-tasking and multi- 5

9 skilling. Thus, I see them as complementary to mine. In addition, I do not see them as being able to explain the trend from tayloristic firms to firms that make an extensive use of multi-skilling, multi-tasking and discretion. Following Adam Smith, on the determinants of specialization, there is also some literature that looks at the issue of job design by focusing mainly on the returns to specialization vis-a-vis the costs of coordinating the activities of different workers (Yang and Borland (1991), Becker and Murphy (1992), and Bolton and Dewatripont (1994)). These papers predict that as coordination costs decrease (due to for instance lower communications costs), the degree of specialization among workers within firms raises. While this is an important component of job design, the paper here does not deal with coordination problem as theirs do not deal with issues I deal here. The papers closest to this one are Lindbeck and Snower (2000), Gibbons and Waldman (2005) and Carmichael and MacLeod (1993). Lindbeck and Snower (2000) study job design in a setting in which work organization is modeled through the time allocation of workers among two tasks. Specialization arises when firms allocate a worker s time to perform one task, while multi-tasking does when they allocate his time between the two tasks. When deciding time allocation, firms trade-off the return on intratask learning the idea that the more time a worker spends on a task, the higher his productivity in that task and the return from inter-task learning the idea that the more time spent in one task, the higher the productivity in the other task. Their paper is similar to mine in that complementarities play an important role in job design. The difference is that they do not study how job assignments vary with worker heterogeneity, assume exogenous skills, and labor market competition plays no role. They do not consider discretion and differentiate between multi-skilling and multi-tasking. Gibbons and Waldman (2005) is the only one that I am aware off discusses job design and task specific training. In particular, they argue that jobs should be designed to minimize the under-utilization of task-specific training. However, they do not study firms incentives to provide task-specific training nor they provide a formal model of task allocation across jobs. Furthermore, discretion plays no role in their model and there is no link between job design, technical change and wage evolution as considered here. Carmichael and MacLeod (1993) argues that by training workers in more than one job, a firm assures its workers that they will not be made worse-off by labor-saving innovations and hence workers have incentives to participate actively in the creation 6

10 of technological progress. 6 The rest of the paper is structured as follows. The next section, Section 2, presents the model and presents the wage determination procedure. Section discusses the first-best efficient investment policy and job design. The next section derives the optimal investment in taskspecific training and optimal job design in a competitive labor market. In Section??, I extend the model to deal with workers discretion and long-term contracting. Section 6 discusses the link between job design, technical change and wage inequality. And finally, in Section 7, I offer some concluding remarks. 2 The Model 2.1 The Basic Set-Up I consider a two-period model between workers (l) and firms (f ), both of whom are risk neutral and do not discount future income. There is a continuum of workers with mass 1, who supply labor inelastically and labor is the only input. Total output is the sum of the output on each job and each job requires one worker. Jobs can be set up to be accomplished with a single task or multiple tasks. The output of a job is a function of the outputs on the tasks involved in the design of the job in a manner that will be explained below. I view period 1 as the early career of workers. During the first period, firms decide how much observable, but non-contractible task-specific training to provide to workers. In addition, during the same period, workers acquire costless firm-specific training. Task-Specific Training. Training is assumed to be specific to the tasks being performed in the job, as opposed to being specific to the firm. Hence, when task-specific capital is accumulated, multiple firms value the capital. Task-specific training in task j is denoted by τ j and belongs to [0, ]. Task-specific training costs ψ per-unit, and thus total training costs are ψ j τ j. Workers Discretion. I assume that there are two different types of tasks. Tasks that involve some degree of discretion as to how they are accomplished (time or effort intensive). Examples are creative tasks such as R&D as well as developing marketing and investment strategies. And 6 Morita (2005) discusses similar issues. 7

11 tasks that have to be carried out at a given time intensity such as routine or standardize tasks that do not leave much room for discretion with respect to intensity they are undertaken or if they are undertaken. Examples are administrative work and clerical tasks. But also performing some production tasks as well as the implementation of (predetermined) marketing and investment strategies leaves much less room for discretion. For these tasks the time allocation problem can be solved at low or no costs. To distinguish between the two types of tasks, routinized tasks are called type-r tasks (i.e., j = r), and discretionary tasks are called type-d tasks (i.e., j = d). For the sake of simplicity, a job can have at most two tasks, one of each kind, and therefore there are three different job designs available to the firm. A multi-task job where a type-r and a type-d task are bundled together and two different single-task jobs: a type-d job and a type-r job. When task j, j = r, d, is included in the job, the variable φ j takes the value 1, else φ j = 0. Let denote the vector (φ r, φ d ) by φ and the set of all binary vectors of dimension 2 by Φ. Thus a job design consists in choosing φ Φ. Skills. There are 2 skills that a worker may be born with. Type-r skills which are valuable in routinized tasks and type-d skills which are valuable in discretionary tasks. Workers differ by their skills, denoted by s (s r, s d ), with s distributed across workers with cumulative distributions G( ) with support [0, 1] 2 and marginal densities g h ( ) for h = r, d. Technology The output in task j depends on whether the task succeeds, denoted by y j = 1, or fails, denoted by y j = 0, and on the job design in the manner described below. The probability of success for a type-j task is equal to the time that the worker spends into that particular task θ j. Probabilities of success are independent across tasks. Thus, P rob(y j = 1 θ j ) = θ j. I assume that the employer can observe and contract on the total time (i.e., 1) spent on the job, but not how the worker divides his time across tasks within a job. Thus, the total time allocated to a job must satisfy the following: j φ jθ j = 1. Furthermore there is a contractible variable, denoted by θ u, that it is positively correlated to the time allocated to the routinized task as follows. When the contractible variable is set to θ u, the worker must allocate ɛθ u units of that time to the routinized task in order for the contractible variable to take the value θ u, where ɛ is a random variable privately learned by the worker before 8

12 choosing his time allocation. Thus, θ u is an imperfect proxy for the worker s time spent on the routinized task. I assume that E(ɛ) = 1. Thus, ɛθ u is unbiased estimator of the time spent on the type-r task. I assume also that the distribution function is such that in equilibrium ɛθ u < 1. As in Baker et al. (1994), I interpret ɛ as follows: there are days (i.e. values of ɛ) when long time allocations to the type-u task forces the worker to spend lots of time in the type-r tasks, days when long time allocations do not increase θ r, and days when short time allocations to the type-u task increase θ r ; these days correspond to ɛ around one, ɛ near zero, and ɛ much larger than one, respectively. Consider the following example. The contractible variable consists on writing an economic report about macroeconomic forecast made by experts. In order to write such report, which is irrelevant from the worker s productivity viewpoint, only the knowledge about these forecasts matter, yet the boss may ask the worker to write such a report so that to be sure that the worker spends some time doing the research to become informed about the forecasts. In writing such a report, the worker must know something about the forecasts and thus he spends a share ɛ of θ u doing the research. Time costs (measured in dollars) in any given job are given by: C(φ, θ) = c(s r, τ r )φ r θ r + c(s d, τ d )φ d θ d. The cost of all other uses of time is normalized to zero. As in MacDonald and Marx (2001), I adopt this linear specification to emphasize the conflicting preferences of the firm and worker; that is, the worker sees time allocated to different activities as substitutable whereas, as detailed below, the firm may regard them as complements. To save on notation, I will denote c(τ j, s j ) by c j. Worker i s output when job design (φ d, φ r ) = (1, 1) is chosen is given by Y (φ, θ) = θ d [θ r Y 11 + (1 θ r )Y 10 ] + (1 θ d )[θ r Y 01 + (1 θ r )Y 00 ] where Y 11 is the output when both tasks succeed, Y 10 is the output when one task succeeds and the other fail and Y 00 is the output when both tasks fail, with Y 11 > Y 10 > Y 00. Worker i s output when a single-task job design entailing task j is chosen is given by Y (φ, θ) = θ j Y 1j + (1 θ j )Y 0j where Y 1j is the output when the task succeeds and Y 0j is the output when the task fails, with Y 1j > Y 0j. 9

13 Because worker i also acquires firm-specific training, his output with the first-period employer is different from that with any other employer. More specifically, worker i s total output with his first-period employer when he is assigned to a multi-task job is Y (φ, θ) + δ, and that when he is assigned to a single-task job j is Y (φ, θ) + δ, where δ is worker i s specific-human capital productivity 7. Thus, task- and firm-specific training are neither complements nor substitutes. The specific human capital productivity δ has density f( ) with fixed support [δ L, δ H ], mean E(δ) and constant variance. Furthermore, f is twice-continuously differentiable and log-concave; that is, the hazard function f( ) 1 F ( ) is non-decreasing δ. Assumptions. A1 c(τ j, s j ) is strictly decreasing in (τ j, s j ) and strictly convex in τ j. A2 c(τ j, s j ) is sub-modular in (τ j, s j ) for j {r, d}. A3 lim τj 0 c(τ j,s j ) τ j > ψ and lim τj c(τ j,s j ) τ j 0. A4 Y 0j = 0 for j {r, d} and Y 1r = Y 1d = Y 10 = Y 01. Assumption (A1) implies that there are benefits from specialization in training in the sense that a worker s time cost in a given task falls with the level of task-specific training and taskspecific skills in the corresponding task. Assumption (A2) establishes that training and skills are complements. Assumption (A3) ensures that it is first-best efficient to train workers and that the solution is interior. Thus, training specialization is valuable. Assumption (A4) establishes that, ceteris-paribus, the output in a single-task job is the same for both tasks and the output after a success in a single-task job is the same as that after a success and a failure in a multi-task job. This facilitates the analysis since for any time allocation that assigns a positive time to both tasks, the output in a multi-task job is greater than that in a single task job when tasks are complements and lower than when tasks are substitutes. In addition, the output in a multi-task job when the worker assigns all the time to one task is the same as that in a single-task job. It is also worthwhile to mention that this does not mean that tasks are symmetric since the surplus (output minus time costs) from any task depends on the time costs which indeed depend on the worker s skill and task-specific training level. 7 δ can be thought of as idiosyncratic productivity shock and the results do not depend on this interpretation 10

14 The role of firm-specific training. A remark with regard to role of the productivity of firm-specific training in the model is in order here. The existence of firm-specific training is needed in order to have rents from continuing the relationship. In the presence of rents and bargaining with outside options, Balmaceda (2005) shows that firms have an incentive to invest in general training. Since task-specific training is general in Becker s sense, firm-specific training is needed in order to have firm-provided training. In a model with labor market distortions such as asymmetric information or search costs and bargaining, this is not needed since workers are not paid their productivity outside of the firm and thus the firm gets a share of the return to general training. The surplus created from firm-specific human capital seems of a more general nature than the surplus arising from informational distortions. In particular, in light of the fact that this not need to be complementary to general training to have an impact on the distribution of the surplus between the firm and the worker. Timing. The precise timing is as follows. At the beginning of period 1, firms compete for workers in a Bertrand-like fashion and each firm-worker pair negotiates a one period contract for the supply of one unit of labor. Then, it provides workers with task-specific training and workers costlessly acquire firm-specific training. After that firms choose a job design for the worker second period job that consists in choosing the allocation of tasks to the job. At the end of period 1 and before period 2 begins, the productivity of firm-specific training is publicly revealed and training is observed by everyone else that is, symmetric learning and observed training is assumed. After these are revealed, the market makes a wage offer to the worker. Then the incumbent firm, upon observing the market offer, the firm and the worker negotiate the wage for the period. After that the worker chooses the time allocation in order to maximize total productivity. 2.2 The Wage Determination Procedure The bargaining game between the incumbent firm and worker adopted here is Rubinstein s alternating-offer game with the addition of outside options for both the firm and the worker. Bargaining takes place over a number of periods. At the beginning of the second period, the worker is chosen to be the proposing party with probability 0.5 the worker s bargaining power and the firm with probability 0.5 the firm s bargaining power. If the proposing party is the worker, he proposes a wage w 2. The firm can either accept or reject this offer, if it accepts, then the firm gets Y +δ w 2, while if it rejects, then either the firm and worker gets zero and bargaining 11

15 goes to the next round where the firm makes a proposal or the firm chooses to terminate the bargaining process taking its outside option. If bargaining is terminated, the worker also gets his outside option which is equal to Y m, where m denotes the job in the which worker i is placed in the market; that is, the worker s productivity in his best alternative (the worker s outside option). Note that only the responding party is allowed to terminate bargaining. This ensures a unique solution for the bargaining game. Furthermore, because complete information is assumed, the bargaining process ensures that trade is ex-post efficient; that is, the firm-worker relationship continues whenever continuing the relationship generates more benefit than separating; i.e., Y + δ Y m. It follows from this and the outside option principle that when neither outside option is binding, the surplus from continuing the relationship is divided according to each party s bargaining power (hereinafter, the surplus sharing outcome); that is, the worker gets 1 2 (Y + δ) and the firm gets 1 2 (Y +δ); when only the worker s outside option is binding, the worker gets his outside option and the firm gets the total surplus minus the worker s outside option; that is, Y + δ Y m ; and when only the firm s outside option is binding, the worker gets the total surplus from continuing the relationship minus the firm s outside option, which is, 0, and the firm gets its outside option. Finally, when the worker and the firm s outside options are both binding, they are better-off terminating the relationship and each getting the corresponding outside option because what is generated by continuing the relationship is less than what can be generated if the firm and worker terminate their relationship. Because the firm s outside option is 0 and Y m 0 for all τ, whenever the firm s outside option is binding it is optimal to terminate the relationship. To visualize this, notice that the firm s outside option is binding when 1 2 (Y + δ) < 0. This implies that the surplus within the relationship Y + δ is negative, which means that Y + δ < Y m. That is, the surplus inside the relationship is lower than the surplus when the relationship is terminated. Thus, the firm and worker i s payoffs are as follows: (Y + δ) if δ > 2Y m Y, π 2 Y + δ Y m if Y m Y δ 2Y m Y, 0 if δ < Y m Y 8 For a formal proof of the outside option principle adopted here see Sutton (1986). (1) 12

16 and w (Y + δ) if δ > 2Y m Y, Y m if δ 2Y m Y, (2) where π 2 denotes the firm s second-period profit and w 2 denotes a worker s second period wage. 3 The First-Best Outcome In this section, I derive the first-best efficient outcome. 3.1 The Efficient Time Allocation Given a job design φ the first-best efficient time allocation maximizes expected output minus total costs of time allocated. In a single-task job, it is efficient to allocate all the time available to the corresponding task. In a routinized job the efficient time allocation will be θ r = 1 and in a discretionary job it will be θ d = 1. In a multi-task job, the efficient time allocation is chosen to solve the following problem, max {θ d[θ r Y 11 + (1 θ r )Y 10 ] + (1 θ d )[θ r Y 01 + (1 θ r )Y 00 ] c d θ d c r θ r } (θ d,θ r ) [0,1] 2 subject to θ d + θ r = 1. The first-order conditions are given by θ d : θ r (Y 11 + Y 00 2Y 10 ) + Y 10 Y 00 c d λ 0 θ r : θ d (Y 11 + Y 00 2Y 10 ) + Y 10 Y 00 c r λ 0 where λ is the Langrange multiplier for the restriction that the total time allocated between the two tasks must ad up to 1. Observe that if Y is supermodular, then the time allocated to task d and that allocated to r are strategic complements, otherwise they are substitutes. When d and r are strategic complements, increasing θ d causes increases in θ r to be more attractive to the firm, and vice versa. In what follows, let Y be Y 11 + Y 00 2Y 10 and c be c d c r. Then, a standard analysis of the first-order conditions leads to the following lemma. Proposition 1 (1) Suppose that the job comprises task j only, then it is efficient to allocate all the time to the type-j task; i.e, θ j Y > c, then θ d = Y c 2 Y = 1; and (2) suppose that the job comprises task d and r. If and θ r = 1 θ d. Else, θ d = 1 if c < 0 and θ d = 0 otherwise. 13

17 This lemma establishes that a necessary condition for having an efficient time allocation in which success in both tasks occurs with positive probability (that is, θrθ d > 0), is that the time allocated to task d and that allocated to r are strategic complements. Because tasks are not symmetric, a sufficient condition is that the strategic complementarity, as measured by Y, is greater than the time cost difference, measured by c. 3.2 First-Best Efficient Job Design The first-best efficient job design entails to maximize the total surplus with respect to the tasks that a job should comprise. Let τ j design entails be the efficient task-specific training. Then the efficient job { } max Y (φ, θ ) C(φ, θ ) + δdg. (3) φ Φ 0 Let the solution to the optimization problem be φ. Because the probability of separation is independent of the job design, it is clear that when the efficient time allocation is such that the whole time is allocated to one task, the efficient job design is single-tasking. In particular, the job must comprise the task in which the worker s cost of time is lower. That is, the routinized task if c(τ r, s r ) is lower than c(τ d, s d ) and the discretionary task otherwise. Thus, total output minus time costs in a single task job is given by: Y 10 min j {r,d} c(τ j, s j ). (4) When in a multi-task job, it is efficient to allocate time to both tasks, total output minus time costs is given by ( Y + c) 2 + Y 10 c d. (5) 4 Y Comparing the surplus when a multi-task job design is chosen with one with that of a singletask job design, it can be concluded that it is efficient to adopt a multi-task job design when ( Y + c) 2 4 Y + Y 10 c(τ d, s d) Y 10 min j {r,d} c(τ j, s j ) (6) This leads to the following result. Proposition 2 Suppose that Y > c, then the first-best efficient job design entails a multitask job. Else, the job entails one type-d task if c(τ d, s d) c(τ r, s r ) and one type-r task otherwise. 14

18 This shows that when complementarity is small, the first-best efficient job design entails a single-task job comprising the task in which the worker s time cost is lower, while when complementarity is high, the job entails a multi-task job design. 3.3 First-best Efficient Training When the efficient job design is chosen, the first-best efficient training level maximizes total surplus minus training costs despite the fact that a separation may occur. Thus, the goal is to choose τ to solve the following problem: max τ R 2 + { Y (φ, θ ) C(φ, θ ) + δdg ψ(φ d τ d + φ rτ r ) 0 where Y (φ, θ ) C(φ, θ ) is given by ( Y + c) 2 4 Y + Y 10 c(τd, s d) if Y > c, Y 10 min j {r,d} c(τj, s j) if Y c. }, (7) First, suppose that task complementarity is such that Y c. Then, the optimal job design entails a single task for all task-specific training levels and therefore the efficient taskspecific training level is determined by the following first-order condition c(τ j, s j ) τ j ψ = 0. (8) Because c(τ j, s j ) is strictly convex and sub-modular in (τ j, s j ), the next result directly from the first-order condition for τ j. Lemma 1 Suppose Y c. Then, in a single-task job, there is a unique first-best efficient training level τ j and this rises with the worker s skills in task j. Next, suppose task are complements (that is, Y > 0). Because the objective function is continuously differentiable and globally concave, the first-order conditions are given by: c(τ d, s d ) τ d θ d ψ 0, c(τ r, s r ) τ r (1 θ d ) ψ 0. It is interesting to note that task-specific training in task d and that in task r are substitutes. That is, the marginal return to type-d task-specific training falls as type-r task-specific training rises. The reason is that as one type of training rises, the efficient time allocation in the other 15 (9)

19 tasks falls since the agent sees the time allocated to one task as a substitute of the time allocated to the other task. Assuming that the first-order conditions are sufficient, it follow from them that the first-best efficient training levels must be so that Y + c 2 Y = c τd (τ d, s d ) c τd (τ d, s d ) + c τr (τ r, s r ). (10) It readily follows from this that when the worker is equally skillful in both tasks, he will receive the same training in both tasks and that a worker who is more skillfull in any given task he will receive more training in that task. Observe also that the objective function is not necessarily concave and thus the first-order conditions may be not be sufficient. Thus, in order to have an interior solution, it is necessary to find conditions under which the objective function is concave. Lemma 2 Y (φ, θ ) C(φ, θ ) is strictly concave in τ if and only if Y > c+( c(τ j,s j ) τ j ) 2 / c(τ j,s j ) τ j τ j for j = r, d. Proof. Differentiating the FOC for τ j, one gets that c(τ j, s j ) τ j τ j θ j + ( c(τj, s j ) τ j 1 2 Y ) 2 (11) In order for this to be negative, Y > c + ( c(τ j,s j ) τ j ) 2 / c(τ j,s j ). Noting that the cross partial derivative of the FOC is given by: by: ( c(τ d, s d ) ( 1 Y c(τ d,s d ) τ d 1 2 Y θd τ d τ + c(τ d, s d ) 1 d τ d 2 Y c(τ d, s d ) τ d c(τ r, s r ) τ r ) 2 τ j τ j c(τ r,s r ) τ r, the determinant of the Hessian is given ) ( c(τ r, s r ) θr + c(τ r, s r ) 1 τ r τ r τ r 2 Y After a few steps of simple algebra this can be written as follows ( c(τd, s d ) c(τ r, s r ) Y Y c(τ d, s d ) c(τ r, s r ) c(τ ) d, s d ) c(τ r, s r ) τ d τ d τ r τ r τ d τ r τ r τ d τ d τ r ( c(τd, s d ) c(τ r, s r ) c c c(τ d, s d ) c(τ r, s r ) + c(τ ) d, s d ) c(τ r, s r ) 0. τ d τ d τ r τ r τ d τ r τ r τ d τ d τ r Because the first term in parenthesis is positive, the first term inside the first parenthesis is positive and Y c, the determinant greater than 2 c c(τ d, s d ) τ d τ d c(τ r, s r ) τ r. 16 )

20 This is positive when c > 0, since c(τ r, s r ) is strictly decreasing and convex in τ r. While when c 0, the determinant is positive when Y > 0 is sufficiently large since the determinant rises with Y > 0. Because skills and task-specific training are complements, the worker receives more training in the task in which his ability is higher. Thus, a worker for whom s r > s d, will receive more training in the routinized task, while if s r s d, the opposite will occur. Lemma 3 Suppose that Y (φ, θ ) C(φ, θ ) is strictly concave in τ. Then (i) there exists a unique positive task-specific training level τj for each task; (ii) τ j rises with the worker s skills in the corresponding task and falls with the worker s skills in the other task; and (iii) if s r > s d, then τr > τd and vice-versa. In a multi-task job, an increase in the worker s skills in one task, while it increases the return to training in the corresponding task, it decreases the marginal return to training in the other task. Thus, a worker with unbalanced skills is also trained in an unbalanced way. That is, he receives more training in the task whose skill endowment is biased towards. In addition, when a worker s skill endowment is such that he is equally skillful in both tasks, he is trained with the same intensity in both tasks. Now I am ready to state the main result of this section. Proposition 3 Suppose that Y (φ, θ ) C(φ, θ ) is strictly concave in τ, then the optimal job design entails a multi-task job and multi-skilling. Else, the optimal job design entails a single-task job comprising a type-d task if s d s r and a type-r task otherwise. There are several interesting corollaries that arise from this proposition. Corollary 1 A worker whose skill endowment is more evenly distributed across tasks, it is more likely to be assigned to a multi-task job and to be trained in both tasks. Corollary 2 A worker whose skill endowment is biased towards task j is more likely to be assigned to a single-task job comprising task j only. Corollary 3 Task complementary is necessary for multi-skilling and multi-tasking, and the more biased the skill endowment towards a given skill, the high the complementary between tasks needed for the efficiency of multi-tasking. 17

21 Corollary 4 A worker receives more task-specific training in a type-j task when he is assigned to a type-j job than when he is assigned to a multi-task job. 4 The Spot Market Equilibrium: The no Pay-for-Performance Case In this section, I derive the spot market equilibrium. 4.1 The Optimal Time Allocation Given a job design φ, the worker allocates his time between tasks to maximize his expected utility. Because the wage is fixed, this entails to minimize total costs of effort subject to the contractible requirements made by the employer. In a single-task job, the worker allocates all the time available to the task since the employer can force the worker to do so. Thus, if the job entails a routinized task, then θ m r comprises a discretionary task, then θ m d = 1. = 1, while if it In a multi-task job in which the firm forces the worker to spend θ u units in the unproductive task, the worker chooses to solve the following problem, max (θ d,θ r ) [0,1] 2{w c dθ d c r θ r } subject to θ d + θ r = 1 and θ r ɛθ u It follows from this that if c j c j, then the worker allocates as much time as possible to task j. Because the employer can force the worker to spend θ u units of time in the type-u task, the remaining time is assigned by the worker to minimize his time costs. Proposition 4 (1) Suppose that the job comprises task j only, then the worker allocates all his time to task j; i.e., θj m = 1; and (2) suppose that the job comprises task d and r. If c d c r, then θd m = 1 ɛθ u and θr m = min{ɛθ u, 1}. Else, θd m = 0 and θm r = 1. This shows that when time costs are lower in a type-d task, the worker spends as much time as possible in that task. This means that he spends the required time in the type-r task (i.e., ɛθ u ) and the remaining time in the type-d task. In contrast, when time costs are lower in a type-r task, the worker allocates all his available time to that task. Thus, the contractible variable θ u 18

22 can be useful in terms of modifying the worker s time allocation only when he wishes to allocate some time to the discretionary task. Given the worker s choice of time allocation, the output in a single task job is Y (φ, θ m ) = Y 10 and the output in a multi-task job is Y (φ, θ s (1 ɛθ u )ɛθ u Y + Y 01 if c d c r, ) = Y 01 if c d > c r. Observe that regardless of who is the second-period employer, the worker will allocate his time in the same way within the firm as well as outside of it as long as the job design outside of the firm is the same as that with the first-period employer. 4.2 Optimal Job Design Because the market is competitive, for any given time allocation made by the worker, firms in the market will compete for the worker in a Bertrand-like fashion and thus will choose the job design that maximizes the worker s utility. Because wages are fixe, market firms will offer the worker a wage equal to his productivity. Thus, spot-market firms solve the following problem: max E ɛ{y (φ, θ m ) c(φ, θ m )} φ Φ,θ u [0,1] If c d > c r, the optimal is to set θ u = 0, since the worker will allocate all his time to the routinized task. Else, the first-order condition with respect to θ u is (E(ɛ) 2E(ɛ 2 )θ u ) Y + E(ɛ) c 0. (12) Observe that the objective function is strictly concave in θ u. Thus, the first-order condition is necessary and sufficient. Thus, θ m u { } 1 Y + c = max 0,, 1 + V ɛ 2 Y Observe that the worker s level of discretion (denoted by D), measured by the time he is free to allocate between tasks as he sees fit (i.e., 1 θ m u ), is given by This leads to the following result. D = (1 + 2V ɛ)) Y c. 2(1 + V ɛ ) Y 19

23 Proposition 5 Suppose that Y > c and c 0. Then the worker s discretion rises with time costs in the routinized task and the average quality of the contractible time proxy (i.e., lower V ɛ ) and falls with time costs in the difficult task and the degree of task complementarity. There are several remarks about this result. First, it says that workers with higher skills in the the discretionary task are given more discretion, while workers with higher skills in the routinized task are given less discretion. The reason is that the goal of limiting discretion is to ensure that the worker allocates time to the routinized task. Because the higher the skills in a given task, the more attractive the task from the worker s point of view, as the worker s skill in the type-r task increase, it is harder to induce him to allocate time to the type-d task. Second, as task complementary rises, workers are given less discretion. The reason is that the value of inducing the worker to allocate time to both tasks rises and this can achieved only by limiting his discretion. Third, as the quality of the proxy about the time the worker spends in the routinized task worsen, the worker is given more discretion. The reason is that the cost in terms of a distortion in the ex-post time allocation, due to the worse quality of the contractible measure, decreases the gain from exploiting task complementarity and thus the return to worker s discretion. Lemma 4 (i) Suppose that c(τ m d, s d) c(τ m r, s r ), then if Y > c and c 0, the market assigns the worker to a multi-task job and requires the worker to spend 1 > θ m u > 0 units of time in the type-r task, otherwise it assigns him to a single-task job comprising a type-d task and chooses θ m u = 0; and (ii) suppose that c(τ m d, s d) > c(τ m r, s r ), then the market assigns the worker to a single-task job comprising a type-r task and chooses θ m u = 0. Let denote the worker s outside option by U m l Y (φ m, θ m ) c(φ m, θ m ). Note that U m l is given by 1 ( Y + c) 2 1+V ɛ 4 Y + Y 10 c d if Y > c and c 0 Y 10 min j {r,d} c(τ m j, s j) otherwise Because at the time the incumbent firm chooses the job design, the idiosyncratic productivity shock has been revealed to everyone and the worker s outside option has been already determined, the first-period employer chooses the job design that maximizes the total surplus Y (φ, θ s ) + δ c(φ, θ s ) minus the maximum between the worker s outside option and half of the surplus within the relationship; that is, Y (φ, θ m ) + δ c(φ, θ m ) max { 1 2 (Y (φ, θm ) + δ c(φ, θ m )), U m l 20 (13) }. (14)