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Jarque, Arantxa. "Hidden effort, learning by doing, and wage dynamics." Economic Quarterly. Federal Reserve Bank of Richmond. 2010. HighBeam Research. 18 Jul. 2018 <https://www.highbeam.com>.
Jarque, Arantxa. "Hidden effort, learning by doing, and wage dynamics." Economic Quarterly. 2010. HighBeam Research. (July 18, 2018). https://www.highbeam.com/doc/1G1-263455217.html
Jarque, Arantxa. "Hidden effort, learning by doing, and wage dynamics." Economic Quarterly. Federal Reserve Bank of Richmond. 2010. Retrieved July 18, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-263455217.html
Many occupations are subject to learning by doing: Effort at the workplace early in the career of a worker results in higher productivity later on. (1) In such occupations, if effort at work is unobservable, a moral hazard problem arises as well. The combination of these two characteristics of effort implies that employers need to provide incentives for the employee to work hard, possibly in the form of pay-for-performance, (2) while taking into account at the same time the optimal path of human capital accumulation over the duration of the contract.
The recent crisis had a big impact on the labor market with high job-destruction rates. If firm-specific human capital accumulation is important, the effect of these separations on welfare may come from several channels. A direct channel is through the loss of human capital prompted by the exogenous separation, as well as the loss in welfare from the decrease in wealth because of unemployment spells of workers. A less direct channel, but potentially an important one, is the change in the cost of providing incentives when the (exogenous to the incentive provision) separation rate increases. However, we are far from being able to understand and measure the importance of this cost, since little is known so far about the structure of incentive provision in the presence of learning by doing. (3) This article constitutes a modest first step in this direction: Abstracting from separations and in a partial equilibrium setting, this article studies the time allocation of incentives and human capital accumulation in the optimal contract. This simplified analysis should be a helpful benchmark in future studies of the fully fledged model with separations and general equilibrium.
We modify the standard repeated moral hazard (RMH) framework from Rogerson (1985a) to include learning by doing. In the standard framework, a risk-neutral employer, the principal, designs a contract to provide incentives for a risk-averse employee, the agent, to exert effort in running the technology of the firm. Both the principal and the agent commit to a long-term contract. The agent's effort is private information and it affects the results of the firm stochastically: The probability distribution over the results of the firm (the agent's "productivity") in a given period is determined by the effort choice of the agent in that same period only. We introduce the following modification to this standard framework: We specify learning by doing by assuming that the probability distribution over the results of the firm in each period is determined by the sum of past undepreciated efforts of the agent, as opposed to his current effort only. In other words, the agent's productivity is determined by his "accumulated human capital." More human capital implies higher expected output, although all possible output levels may realize under any level of human capital. In this specification, the agent determines his human capital deterministically by choosing effort each period. Lower depreciation of past effort is interpreted as "more persistence" of effort.
We present a model of two periods. The first period represents the junior years, when the worker has just been hired and has little experience. The second period represents the mature worker years, when human capital has been potentially accumulated and there are no more years ahead in which to exploit the productivity of the worker. A contract contingent on the observed performance of the agent is designed by the principal to implement the path of human capital accumulation that maximizes the principal's expected profit (expected output minus expected payments to the agent).
In our analysis, we find the following two main implications of the presence of learning by doing. First, the principal does not find it optimal to require a high level of human capital in the last period of the contract, since there is not much time left to exploit the productivity of the worker. Hence, the more experienced workers are not the most productive ones, since they optimally are asked to let their human capital depreciate. This implies that workers exert the most effort in their junior years, and the least in their pre-retirement years. In a comparison with the standard RMH problem, we find that the frontloading of effort, as well as the low requirement at the end of the worker's career, differ markedly from the optimal path of effort in a context without learning by doing. Second, and in spite of this difference in effort requirements over the contract length, we find that learning by doing does not imply a change in the properties of consumption paths; hence, the properties of consumption paths found by previous studies, such as Phelan (1994), remain true in this context (see also Ales and Maziero [2009]).
It is worth noting that in our analysis we assume perfect commitment to the contract both from the employer and the employee, and we do not allow for separations to be part of the contract. This means we need to abstract from the usual career concerns that have been explored in the literature (see Gibbons and Murphy [1992]). The implications of the hidden human capital accumulation that we model here should be viewed as complementary to the implications of career concerns.
As pointed out above, the problem studied here differs from the standard RMH in that the contingent contract needs to take into account the persistent effects of effort on productivity. On the technical side, this highly complicates solving for the optimal contract. The fact that both past and current effort choices are not observable means that, at the start of every period, the principal does not know the preferences of the agent over continuation contracts (that is, the principal does not know the true productivity of the agent for a given choice of effort today). Jarque (2010) deals with this difficulty and presents a class of problems with persistence for which a simple solution can be found. The article studies a general framework in which past effort choices affect current output, as opposed to other forms of persistence that one may consider, such as through output autocorrelation (see, for example, Kapicka [2008]). The learning-by-doing problem that we are interested in, hence, constitutes a fitting application of the results in Jarque (2010). We adapt the assumptions in Jarque (2010) to a finite horizon and we show how this specification of learning by doing greatly simplifies the analysis of the optimal contract.
In Section 1 we introduce the common assumptions throughout the article. Section 2 presents, as a benchmark, the case in which the principal can directly observe the level of effort chosen by the agent every period, and hence can control his human capital at all times. For reference, we also discuss the case in which the effort of the agent does not have a persistent effect in time. The analytical properties of the problem are discussed in both cases. Then we analyze the main case of interest of this article, in which effort is unobservable and contracts that specify payments contingent on the observable performance of the agent are needed to implement the desired sequence of human capital accumulation. In Section 3, we discuss the case without persistence--a standard two-period repeated moral hazard problem. In Section 4 we discuss the technical difficulties of allowing for effort persistence in problems of repeated moral hazard, and the solutions provided in the literature. Section 5 presents the framework of hidden human capital accumulation, a particular case of effort persistence. As the main result, we provide conditions under which the problem with hidden human capital can be analyzed by studying a related auxiliary problem that is formally a standard repeated moral hazard problem. Hence, the discussion of the properties of the standard case in Section 3 becomes useful when deriving the properties of the case with persistence. The numerical solution to an example is presented in Section 6, together with a comparison to the standard RMH without learning by doing, and a discussion of the main lessons about the effects of hidden human capital accumulation on wage dynamics. Section 7 concludes.
1. Description of the Environment
The results in this article apply to contracts of finite length T; however, in order to keep the exposition and the notation as simple as possible, we discuss here the case of a two-period contract, T =2. We assume that both parties commit to staying in the contract for the two periods. For tractability, we assume that the principal has perfect control over the savings of the agent. They both discount the future at a rate ft. We assume that the principal is risk neutral and the agent is risk averse, with additively separable utility that is linear in effort.
Assumption 1 The agent's utility is given by U ([c.sub.t], [e.sub.t]) = u ([c.sub.t]) = v([e.sub.t]), where u is twice continuously differentiable and strictly concave and c, and e, denote consumption and effort at time t, respectively.
There is a finite set of possible outcomes in each period, Y = {[y.sub.L], [y.sub.H]} Histories of outcomes are assumed to be observable to both the principal and the agent. We assume both consumption and effort lie in a compact set: [c.sub.t] [member of] [0, [y.sub.t] and [e.sub.t], [member of] E = [e.bar],[bar.e] for all t.
We model the hidden accumulation of human capital by assuming that the effect of effort is "persistent" over time, in a learning-by-doing fashion. That is, we depart from the standard RMH framework, which assumes that the probability distribution over possible outcomes realizations at t depends only on [e.sub.t],. In our human capital accumulation framework, the probability distribution at t depends on all past efforts up to time t. Assumption 2 states this formally for the two-period problem.
Assumption 2 The agent affects the probability distribution over outcomes according to the following function:
Pr([y.sub.t]=[y.sub.H]|[s.sub.t]) [equivalent to] [pi]([s.sub.t]),
where
[s.sub.1]=[e.sub.1], (1)
[s.sub.2]=[[rho][s.sub.1]+[e.sub.2]] (2)
and [pi] (s) is continuous, differrentiable, concave, and [rho] [member of] (0,1) In the human capital accumulation language, we could equivalently write the law of motion for human capital as
[s.sub.1]=[e.sub.1],
[s.sub.2]=(1-[delta])[s.sub.1]+[e.sub.2,]
where [delta] = 1 - [rho] would represent the depreciation rate. Then,
f([S.sub.t])={[y.sub.H] with probability [pi] ([S.sub.t]),[y.sub.L] with probability 1-[pi] ([S.sub.t])
could be interpreted as the production function or technology of the firm.
In the rest of the article, we loosely refer to Assumption 2 as effort being "persistent," we refer to [s.sub.t], as the accumulated human capital at time t, and we refer to [rho] as the persistence rate.
The strategy of the principal consists of a sequence of consumption transfers to the agent contingent on the history of outcome realizations, c = {[c.sub.i], [c.sub.i,j] sub.i,j=1.,H']}. to which the principal commits when offering the contract at time 0. The agent's strategy is a sequence of period best-response effort choices that maximize his expected utility from t on, given the past history of output: e ={[e. sub.1],[e. sub. 2i]}.sub.i = L.H]} At the beginning of each period, the agent chooses the level of current effort, [e.sub.t]. Then output [y.sub.t], is realized according to the distribution determined by all effort choices up to time t. Finally, the corresponding amount of consumption is given to the agent.
A contract is a pair of contingent sequences c and e. For the analysis in the rest of the article, it will be useful to follow Grossman and Hart (1983) in using utility levels [u.sub.i] = u[(c.sub.i)] and u = u [(c.sub.ij)] as choice variables. (4) To denote the domain for this new choice variable, we need to introduce the following set notation:
[U.sub.i]={u|u=u([c.sub.i]) for some ([c.sub.i]) [member of] [0,[y.sub.i]],i=L,H}
[U.sub.ij]={u|u=u([c.sub.ij]) for some ([c.sub.ij]) [member of] [0,[y.sub.j]]i,j=L,H}
The contingent sequence of utility is then denoted u = [{[u.sub.i],[u.sub i,j]}i,j=L,H'] and we assume that [u.sub.i] [member of] [U.sub.i],[u.sub.ij] [member of] [U.sub.ij]
In order to keep the expressions in the article as simple as possible, and abusing notation slightly, we also introduce some notation shortcuts. We denote [c.sub.i] = [u.sup.-1] for all i. We also write Pr ([y.sub.t] = [y.sub.H] | [S.sub.t] as [[pi].sub.H] and Pr([y.sub.t] = [y.sub.L] || [[pi].sub.L] ([s.sub.t]).
The expected profit of the principal, denoted by V (u, e), depends on the contract as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [s.sub.t], changes with [e.sub.t], as detailed in (1). In the
same way, we can write the agent's expected utility of accepting to participate in the contract as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Within this environment we are now ready to set up the problem of finding the optimal contract that will provide the right incentives for human capital accumulation at the least expected cost. …
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