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Labour Economics
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Work practices, incentives for skills, and training☆,☆☆
Suraj Prasad a,⁎, Hien Tran a,b
a School of Economics, University of Sydney, Australiab Treasury, Government of Australia, Australia
H I G H L I G H T S
• We suggest a new way in the literature for firms to induce skills from workers.• Unlike other papers, our approach does not rely on court enforceable contracts.• Firms promote workers only when they acquire both general and specific skills.• Drawing on our mechanism, we also provide a new rationale for general training.• Our results on training are consistent with evidence on innovative work practices.
☆ Wewould like to thank Jed DeVaro, Denise Doiron, RicMalcomson, Kieron Meagher, Hodaka Morita, Bill Schwormand participants at the Economic Theory Workshop, UNSWCalifornia State University, East Bay, University of Adelaide,and University of Oxford.☆☆ The views in this article are those of the authors aAustralian Treasury.
We study a worker's incentives to invest in non-verifiable skills. We do this within the context of recent in-novative work practices, where jobs have become more flexible. When jobs are flexible, standard approachesin the literature for firms to credibly reward skills (i.e. committing ex-ante to a contract that ties wages tojobs) may not be feasible. We suggest an alternative approach: firms promote workers only when they ac-quire both firm specific skills and general skills. This promotion scheme reveals information about the gener-al skill to competing firms, which in turn allows the firm to credibly reward firm specific skills. Firms thushave an incentive to pay for general training to induce firm specific skills and they are more likely to payfor it when jobs are flexible.
The past few decades have seen significant changes in the organiza-tion of work in firms. While the details of these changes differ acrossfirms and industries there are some common features. First, there is agreater emphasis on continuous skill development and training at alllevels of an organization. Some of these skills are firm specific (such asdeveloping a relationship with customers and learning to work with
hardHolden, Clare Leaver, James, Margaret Stevens, Joel Watson, Australian National University,University of New South Wales,
nd not necessarily those of the
rasad),
rights reserved.
co-workers), whereas other skills are general (such as developingknowledge for equipment maintenance and learning to perform diag-nostic tests to solve production problems).1 Second, jobs have becomemore flexible in response to changes in technology, markets andcustomer's needs.2 That is, firms rely less on stable job descriptionsandmore on flexiblework-assignment descriptions. And these dynamicdescriptions change as assignments are completed. Third, workers areoffered challenging jobs that utilize their skills as they progress through
1 With respect to training, Osterman (1994), using an establishment level survey acrossindustries, finds that 32% of establishments train their workers off the job. IchniowskiShaw and Prennushi (1997) find that off the job training is used alongwith a cluster of in-novative practices in steel finishing lines. The examples of firm specific and general skillsare drawn from case studies in Bartel et al. (2003) across three industries: medical equip-ment manufacturers, steel manufacturers and valve manufacturers.
2 See O'Toole and Lawler (2006), pp.58, for more details on how jobs have becomeflexible. Also see Ichniowski et al. (1997), Eriksson and Ortega (2006), and DeVaroand Farnham (2011) for empirical work in this area.
5 Starting with Adam Smith (Smith, 1937), there is a large literature which empha-sizes the benefits of specialization (Roy, 1950; Rosen, 1978). There are also other ex-planations for why firms benefit from multiskilling. These include constraints on theextent of the market (Smith, 1937), coordination benefits (Becker and Murphy,1992; Hart and Moore, 2005; Dessein and Santos, 2006; Kato and Owan, 2010), im-proved incentives to solve problems (Owan, 2011), benefits from continuous improve-
67S. Prasad, H. Tran / Labour Economics 23 (2013) 66–76
their careers.3,4 The purpose of our paper is to understand how firmscan design incentives for workers to invest in skills in light of these re-cent trends in the workplace.
The literature on incentives for skills suggests a few approaches.The first approach is to write a contract that rewards workers forskills. The problem with this approach is that skills, such as develop-ing a relationship with a customer or learning to diagnose a produc-tion problem, are often acquired by workers on the job and thuscannot be verified by a third party. A second approach is for firms towrite an indirect contract that ties wages to a variable related to skills,say jobs (Kahn and Huberman, 1988; Prendergast, 1993). This helps afirm to commit ex-ante to rewarding workers in some jobs, which itcan use ex-post by assigning workers to jobs based on skills. Writingthese ex-ante contracts, however, requires job descriptions to be sta-ble over time. When job descriptions change over time in response tochanges in technology, markets, and customers needs, this approachmay not be feasible. Firms can also commit to a tournament wherea fixed fraction of workers from a pool are promoted and rewarded(Carmichael, 1983; Malcomson, 1984). But tournaments can ruin co-operation between workers. Finally, firms can rely on their reputa-tions to commit to reward workers (MacLeod and Malcomson,1988; Baker et al., 1994; Levin, 2003). This approach, however, re-quires firms to be sufficiently patient and requires workers to coordi-nate their punishments.
We make two contributions in this paper. First, we suggest an al-ternative mechanism to induce skills, where workers are promoted tochallenging jobs (where the returns to skills are high) only when theyacquire both firm specific skills and general skills. Unlike the existingapproaches above, our mechanism does not rely on contracts wherewages are attached to skills or jobs. What our mechanism doesneed, instead, is that competing firms in the same industry (whichhave access to the same technology and are familiar with productmarkets) can observe job assignments (as in Waldman, 1984). Sec-ond, building on our mechanism, we suggest a new rationale forwhy firms pay for general training; it allows workers to invest in gen-eral skills which in turn can be used to commit to reward workers forfirm specific skills. This commitment role of general training yieldsthe following testable implication which we discuss within the con-text of recent human resource practices: firms are more likely topay for general training when jobs are flexible.
To understand how our mechanismworks, it is useful to start witha paper by Prendergast (1993). He considers the role of promotions ininducing firm specific skills and his framework has twomain features.First, though firms cannot contract on skills, they can write a contractwhere wages are attached to jobs, before skills have been acquired.Second, jobs vary in their returns to skills and a promotion is an as-signment to a job where the returns to skills are higher. Firms canthen split the promise to reward a worker for skills into two parts.The first promise is to promote the worker for skills (promotionpromise) and the second promise is to pay a higher wage for a pro-motion that covers the cost of skills (wage promise). Because wagecontracts are tied to jobs, firms can always set the reward for a pro-motion high enough and keep their wage promise. And by settingthe reward for a promotion low enough, relative to the output gainfrom promoting a skilled worker, a firm can keep its promotion prom-ise. So, wage contracts tied to jobs help a firm to commit to rewardfirm specific skills.
Our mechanism builds on the framework by Prendergast (1993).Like in Prendergast (1993), a promise to reward workers for skills is
3 As O'Toole and Lawler (2006) point out, a crucial feature of high involvement com-panies is that “employees are offered challenging and enriched jobs and a say in themanagement of their own tasks.”
4 Other studies on innovative work practices include Cappelli and Neumark (2001)and Black and Lynch (2004). For a review of the empirical literature in this area, seeLindbeck and Snower (2000) and Ichniowski and Shaw (2003).
split into two parts in our framework: a promise to promote theworker for skills and a promise to pay a higher wage for a promotionthat covers the cost of skills. But, in contrast to Prendergast (1993),we assume that jobs are flexible so that a firm cannot contractuallycommit to attach wages to jobs, ex-ante. Thus the firm cannot directlyset rewards for a promotion. Instead, rewards for a promotion are de-termined by the information about skills that a promotion schemeconveys to competing firms in a labor market. Promoting the workerbased on a general skill that is sufficiently valuable ensures that thereward for a promotion covers the cost of skills. So the firm cankeep its wage promise. Promoting the worker based on a firm specificskill that is sufficiently valuable, on the other hand, helps the firmkeep its promotion promise. This is because the firm specific skill, asin Waldman (1984), increases the output gains from promotion with-out getting reflected in wage offers by the market. To keep bothpromises, a worker is promoted only when he successfully acquiresboth skills. Thus non-verifiable general skills play exactly the samerole as wage contracts tied to jobs in the Prendergast framework:they help a firm to commit to reward firm specific skills.
We fix this commitment role of general skills in inducing firm spe-cific skills and derive some additional results. First, promotions de-pend on both the firm specific and the general skill. Second, thispromotion scheme induces workers to invest in both firm specificand general skills, even when the skills are not complements in theproduction function. This result stands in contrast to other paperswhich emphasize the role of complementarities in multiskilling(Lindbeck and Snower, 2000; Gibbs and Levenson, 2002).5 Third,this promotion scheme sometimes leads to over-investment in thefirm specific skill.
We also offer a new explanation for why firms pay for generaltraining. To make this point, we allow firms to incur a fixed costand offer a publicly observable off the job training program toworkers upfront. This program in turn allows workers to make in-vestments in the general skill, which only the current employer ob-serves. A promotion that is contingent on both types of skillsselectively reveals information about the general skill to a labor mar-ket only when the worker acquires the firm specific skill. This helpsthe firm commit to reward the firm specific skill and thus it earns areturn on the general training it provides. As a result, firms are willingto pay for general training. Also, because general training plays exact-ly the role (in helping a firm commit to reward firm specific skills) ascontracts that tie wages to jobs, firms are more likely to pay for thetraining as jobs get more flexible. This implication for training,which relies on the inability of firms to contractually commit to tiewages to jobs, is distinct from explanations that rely on labor marketimperfections (Acemoglu and Pischke, 1999 for example).
Two key features of our model, promotions and asymmetric infor-mation between current employers and the labor market also play arole in Waldman (1984).6 In his framework, firms face the followingtradeoff when deciding on a promotion. Promotions increase outputbut they also signal ability to a labor market which increases wages.As a result, there is always some under-promotion relative to the ef-ficient level. Waldman (1984) also shows that firm specific skills,
ments (Aoki, 1986; Koike, 1988; Morita, 2005), better employer and employeelearning (Eriksson and Ortega, 2006), and flexibility associated with multiskilling(Garg et al., 2002; DeVaro and Farnham, 2011).
6 A number of other papers assume asymmetric information about a worker's pro-ductivity. Examples areWaldman (1990), Gibbons and Katz (1991), Milgrom and Oster(1987), Bernhardt (1995), Owan (2004) and Chang and Wang (1995). Also see DeVaroand Waldman (2012) for empirical evidence on the signaling role of promotions andSchonberg (2007) for empirical evidence on asymmetric employer learning.
68 S. Prasad, H. Tran / Labour Economics 23 (2013) 66–76
which are exogenous in his framework, increase output from a pro-motion without altering wages, and thus make promotions morelikely.
A few other papers build on Waldman's “market based approach”,where wages are determined by a labor market, to study investmentin skills.7 Scoones and Bernhardt (1998) focus on skills that are verifi-able by a labor market whereas our focus is on non-verifiable skills.8
We believe that non-verifiability of skills is more likely to hold inmore complex production settings (e.g. when workers have to solvecomplex problems). For simpler jobs where tasks are routine, onthe other hand, verifying skills may be easier. This assumption ofnon-verifiable skills also gives us a distinct testable implication:promotions only depend on the firm specific skill in Waldman(1984) and Scoones and Bernhardt (1998), whereas the likelihoodof a promotion is weakly increasing in both types of skills in ourpaper.9 Zabojnik and Bernhardt (2001) study how firm characteris-tics such as size and profitability influence wages and investmentsin skills. In their paper, however, workers can only acquire bothtypes of skills in fixed proportions. Another difference is that theyhave a fixed number of higher level job slots in their paper. Ghoshand Waldman (2010) study a worker's incentive to exert effort (asopposed to investing in skills) under standard promotion schemesand up or out schemes. Finally, DeVaro et al. (2008) study promo-tion patterns within the context of labor market discriminationwhere general and firm specific skills are acquired in fixed propor-tions. Our key contribution to this “market based approach” litera-ture, is to show that firms have an incentive to pay for generaltraining when skills are not verifiable and wages are determinedby a labor market. Or in other words, the market based approachprovides a new rationale for why firms pay for general training.
Our paper is also related to work on multitasking and multiskilling (Holmstrom and Milgrom, 1991; Baker, 1992). This literaturesuggests that the provision of incentives in multitask settings is diffi-cult because incentives have to be balanced across tasks which differin the preciseness of their performance measures. Our model, in con-trast, suggests that jobs with different types of skills help in the pro-vision of incentives. In this sense, our paper is related to Carmichaeland MacLeod (1993) where multiskilling allows firms to commit toretain workers after workers suggest a labor saving innovation. Aoki(1986) and Morita (2005) point out the benefits of multi skillingworkers when making continuous improvements to the productionprocess is important. Another paper where multiskilling arises inequilibrium without the skills being complements is Owan (2011).In his setting, multi skilling takes the form of workers overlappingwith each other to solve problems. This overlap affects ex-postbargaining power which in turn influences ex-ante investments infirm specific skills. DeVaro and Farnham (2011) study the tradeoff be-tween training workers in skills required to produce different goods(multi skilling) versus training workers to produce one good (special-ization). They find, both theoretically and empirically, that as productmarket volatility increases, innovation becomes more important inmulti-product firms leading to less multiskilling. Finally, our resultwhere workers are rewarded if and only if they are successful atboth skills has a similar flavor to results in MacDonald and Marx(2001).
7 An alternative approach is the “classic” approach of Lazear and Rosen (1981)where there are fixed job slots and where firms can commit to reward workers for apromotion.
8 Promotions do not convey information about skills in their framework though theydo convey information about exogenous ability. So promotions, as in Waldman (1984),only depend on the firm specific skill. Firms, also do not have an incentive to pay forgeneral training.
9 In Scoones and Bernhardt (1998), the general skill increases output from a promo-tion but this increase is exactly offset by an increase in wages. See Lemma 1 in Scoonesand Bernhardt (1998).
2. Model
We build on a model by Prendergast (1993), with a firmwhere theworker is currently working. In addition, there is an outside labormarket with at least two competing firms. All of the players are riskneutral. In the analysis that follows, we often refer to the current em-ployer as the firm and competing firms as the labor market. The modelhas four parts. The first part deals with the skill acquisition process.The second part relates skills with output. The third part specifiescontracts that are feasible for the firm. And the final part describesthe timing of the game and the information that players have at var-ious stages of the game.
Consider the acquisition of skills first. There are two skills that theworker can acquire, a firm specific skill and a general skill, with eachskill having two outcomes. The outcome for the firm specific skill isdenoted by iwith i ∈ {0,f}where 0 denotes failure and f denotes successat the firm specific skill. Similarly, the outcome for the general skill isdenoted by j with j ∈ {0,g} where 0 denotes failure and g denotes suc-cess for the general skill. Thus, there are four possible outcomes, failureat both skills (0,0), success only at the firm specific skill (f,0), successonly at the general skill (0,g) and success at both skills (f,g).
Acquiring skills involves investments on the part of firms andworkers. But to begin with, we assume that firms have alreadytrained workers and that it is the turn of workers to choose their in-vestments. Later in Section 5, we specify how training interacts withworker's investments in the production function and study the firm'sincentives to pay for training. A worker can invest in skills by exertingeffort that he alone privately observes, which in turn influences theprobability of success on each skill. To simplify notation, we assumethat the worker chooses these probabilities directly. The probability of
acquiring the firm specific skill is pf∈ p� ; p�h i
and the probability of
acquiring the general skill is pg∈ p� ;p�h i
and these probabilities are
statistically independent of one another. To ensure that beliefs of thelabor market can be determined by Baye's rule, we assume that p� >
0 and pb1. In the analysis that follows we refer to pf as the level of in-vestment in the firm specific skill and pg as the level of investment inthe general skill.10We assume that theworker has a linear cost function
given by C(pf,pg) = cfpf + cgpg defined over p� ; p�h i
� p� ;p�h i
, where cf
and cg are strictly positive constants. This linear cost function leads tocorner solutions in investments which makes the distinction betweenspecialization and multiskilling stark. As we discuss later, allowing fora more general convex cost function does not qualitatively change anyof the results of our paper.
We also assume that theminimum level of investment in a skill is suf-ficiently small so that the following condition holds: p� yHbmin cf ; cg
� �.
This assumption which allows the worker to choose a very small invest-ment level deters specialization when a worker is rewarded only for ac-quiring both skills.
The following definition is useful in stating the main results of thepaper.
Definition 1. A worker specializes in a skill if max pf ; pgn o
> p� and
min pf ; pgn o
¼ p� .
We say that a worker multiskills if max pf ;pgn o
> p� and he doesnot specialize in a skill.11
10 Notice that skill acquisition is a risky activity. Also notice that the worker can ac-quire skills with positive probability even if he chooses the minimal level of invest-ment. One way to interpret p� , is that these are investments that a worker starts outwith (say through social interactions with other co-workers) before deciding to active-ly invest in skills.11 An alternative definition that we could use for specialization is that |pf − pg| isabove a threshold that is sufficiently close to �p− p� . All of our results go through, ifwe use this alternative definition.
69S. Prasad, H. Tran / Labour Economics 23 (2013) 66–76
Next, consider how skills are related to output. The relationshipbetween skills and output depends on the job assignment. There aretwo jobs, an easy or a low skilled job, job L, where the returns to skillsare low and a challenging or a high skilled job, job H, where thereturns to skills are high. Output for the four possible outcomes is asfollows. When the outcome is (0,0), output is 0 for both the firmand the labor market. When the worker is only successful at thefirm specific skill, output for the firm is λyL for job L and λyH for jobH with 0 b yL b yH and where λ ∈ (0,1) measures the relative valueof firm specific skills to general skills, whereas the labor market getsan output of 0.12 When the worker is only successful at the generalskill, output for both the firm and the labor market is (1 − λ)yL forjob L and (1 − λ)yH for job H. Finally for the outcome (f,g), outputfor the firm is given by yL + Δ for job L and yH + Δ for job H withΔ ≥ 0, whereas output for the labor market is given by (1 − λ)yJwhere J ∈ {L,H}. The parameter Δ measures the extent to which in-vestments in both the skills are complementary. In other words,when we fix a job, Δ is the cross partial of expected output withrespect to pf and pg.13 The table below summarizes the relationshipbetween skills and output for the firm.
1
si1
cotaeq1
h
Outcomes
2 Restricting λ to lie in thes. All of our results go throu3 We could allow for sommpeting firm in the labor mrities in investments wouluilibrium.4 There are various ways tiring a new worker, or firm
(0,0)
open rather thgh if we conse of the gainsarket. If we d
d make it eas
o interpret thespecific capita
(f,0)
an the closedider the end pin complemeid make this aier for a firm
cost F. It coul that is lost.
(0,g)
interval simplifiesoints of the intervantarities to be captssumption, then coto keep its wage p
ld be a firing cost, t
(f,g)
Output from Job L
0 λyL (1 − λ)yL yL + Δ Output from Job H 0 λyH (1 − λ)yH yH + Δ
As in Prendergast (1993), we assume that the firm cannot write acontract where wages are attached to skills. But unlike his model, ourfocus is on settings where jobs are flexible so that job descriptions donot stay the same over time. As a consequence, we assume that thefirm cannot write a contract ex-ante that attaches wages to jobs.
Finally, the timing and the information structure of the game areas follows. The worker chooses the probabilities of success for bothskills which induces a probability distribution over outcomes. Theoutcomes for both of these skills are then realized. The firm observesthese outcomes and decides on the job assignment of the worker.Competing firms in the labor market then observe the job assign-ment, update their beliefs and offer a wage that is contingent on thejob assignment. We focus on symmetric wage offers by competingfirms. The firm then decides whether to match the labor market'soffer for a job. If the firm matches an offer made by the labor market,we assume that the worker stays with the firm. We also assume, thatthe firm incurs a cost F > 0 if it loses the worker.14 This assumptionensures that it is optimal for firms to match offers even when theoffers equal the value of output at a given outcome.
In this setting a strategy for a worker is given by the vector (pf,pg).The firm's strategy is denoted by (σ(x),m) where x = (x0,0,xf,0,x0,g,xf,g)with xi,j ∈ {L,H}, where σ is a probability distribution over x, andwherem is a function which specifies whether the firmmatches offersfor a given outcome, job assignment and wage offer for the job.The labor market observes job assignments and for each job assignsbeliefs to each of the four outcomes. The beliefs associated with eachof the outcomes for job L are given by the vector of probabilitiesμL = (μ0,0L ,μf,0L ,μ0,gL ,μf,gL ) and the beliefs associated with each outcomefor job H are given by the vector of probabilities μH = (μ0,0H ,μf,0H ,μ0,gH,μf,gH ). Wages offered by the labor market are denoted by wL whenjob L is observed and wH when job H is observed.
the analy-l.ured by amplemen-romise in
he cost of
3. Efficiency
We start our analysis by characterizing efficient levels of invest-ment for both skills. This characterization serves as a benchmark.Let (pfeff,pgeff) denote the efficient level of investment. Then (pfeff,pgeff)is the optimal solution to the following problem.
Maxpf∈ p
�;p
h i;pg∈ p
�;p
h ipf 1−pg� �
λyH þ 1−pf� �
pg 1−λð ÞyH þ pf pg yH þ Δð Þ−C pf ; pg� �
We also make the following assumption to ensure that the effi-cient solution always sets at least one of the skill levels above theminimum level.
Assumption 1.
yH > cf þ cg
The following proposition characterizes the efficient skill invest-ment levels. The proofs of all of the propositions and lemmas are inthe Appendix.
Proposition 1. The efficient levels of investment in skills are given by
pefff ¼
p� if Δ≤cfp
and λ∈ 0;cf−pΔ
yH
� �
p�;p
� if Δ≤
cfp
and λ ¼ cf−pΔyH
p if Δ≤cfp
and λ∈ cf−pΔyH
;1� �
p if Δ >cfp
8>>>>>>>>>>><>>>>>>>>>>>:
and
peffg ¼
p if Δ >cgp
p if Δ≤cgp
and λ∈ 0;1− cg−pΔyH
� �
p�;p
� if Δ≤
cgp
and λ ¼ 1−cg−pΔyH
p� if Δ≤cgp
and λ∈ 1− cg−pΔyH
;1� �
:
8>>>>>>>>>><>>>>>>>>>>:
Proposition 1 highlights two factors that determine efficient in-vestment levels: complementarities between skills and the relativeimportance of both skills. When complementarities are high, invest-ment levels are at their maximum for both skills. When complemen-tarities are low, the relative value of each skill plays an important role.If both skills are important (for intermediate values of λ), efficiencyrequires a worker to choose the highest investment level for bothskills. If one skill is more important relative to another (λ close to0 or 1), efficiency requires an extreme form of specialization, thehighest investment level for the more valuable skill and the lowestinvestment level for the less valuable skill. Notice that these cornersolutions arise because of the linear specification for the costfunction. Fig. 1 depicts the efficient level of investment with λ onthe horizontal axis and pf
eff and pgeff on the vertical axis when comple-
mentarities are not too high.
4. Equilibrium
The objective in this section is to compare equilibrium investmentlevels for skills with the efficient counterparts above. The equilibriumconcept we use is a Perfect Bayesian Equilibrium and focus on equilib-ria where investments for at least one skill are strictly above the
Fig. 1. Efficient investment levels.
70 S. Prasad, H. Tran / Labour Economics 23 (2013) 66–76
minimum level.15 To start solving the equilibrium, let us start at thelast stage where the firm decides whether it wants to match offersor not. Because the firm is better informed than competing firmsabout skill outcomes and because of the presence of firm specificskills, a competing firm in the labor market can only successfullyraid workers with a wage that exceeds its value of the worker. Thatis, competing firms face a winners curse and the best they can do isto earn an expected profit of zero by ensuring that the worker stayswith his current employer. The problem is that there are severalwage offers by competing firms which satisfy this condition. To pindown wage offers in equilibrium, we follow Ghosh and Waldman(2010) and allow for the possibility of a firm trembling and notmatching offers when it should. In particular, given a job assignmentand given an offer by a competing firm we assume that the firm trem-bles at the outcome where it has the lowest output.16 Given the pos-sibility of these trembles, a firm in the labormarket thenmakes an offerthat equals the output that it gets from job H for that particular out-come.17 An example helps us better understand how wages are deter-mined. Suppose the firm's promotion strategy is (L,L,H,H). Then givenjob L, the firm can tremble and not match offers at the outcome (0,0)and thus wage offers by competing firms equal 0. For the job H, thefirm can tremble at the outcome (0,g) and thus wage offers by compet-ing firms equal (1 − λ)yH. The following lemma says that rewards arealways associated with promotions in a skill acquisition equilibrium.
Lemma 1. In a skill acquisition equilibrium, wH ≥ wL.The reasoning for this Lemma is the following. To induce skills
there must be some outcome for which a worker is assigned to thelower skilled job (a pooling strategy where a worker is alwaysassigned to the higher skilled job cannot induce skills). But if wL
was strictly greater than wH, the firm could always promote theworker, get at least as much output, pay a lower wage and do strictlybetter.
Proposition 2. In a skill acquisition equilibrium, the firm assigns aworkerto job Hwith positive probability if and only if the outcome (f,g) is realized.Furthermore the worker never specializes in a skill.
Proposition 2 says that in a skill acquisition equilibrium, a workeris promoted with positive probability if and only if he successfully
15 However, when computing the equilibrium we do check for deviations where aworker chooses the minimum investment level for both skills.16 This is similar to the notion of a proper equilibrium (Myerson, 1978). An alterna-tive way to pin down offers is to assume that firm's cost of losing the worker is suffi-ciently large. This way, when a competing firm bids its expected value conditional onthe job it observes, the offer is always matched. The results using this alternative ap-proach are qualitatively similar when we consider pure strategies only. For the caseof mixed strategies, it is possible that the worker specializes in the general skill.17 Notice that when λ ¼ 1
2, it is possible that there are two outcomes which yield thelowest output for the firm, given a job assignment. We assume for this knife edge casethat the firm trembles at the outcome where the value of the worker to competingfirms is the lowest.
acquires both skills. It also says that specialization can never be anequilibrium outcome. The interesting feature of this result is that itholds even if the skills are not complements (Δ = 0).
A simple way to understand Proposition 2 is to restrict attention topure strategies for the firm and eliminate strategies that cannot bepart of a skill acquisition equilibrium. Pooling strategies cannot bepart of a skill acquisition equilibrium. Also, promotion strategiesthat are more likely to promote a worker for outcomes with loweroutput gains cannot be part of an equilibrium. This leaves us withfour possible strategies to check: (L,H,L,H), (L,L,H,H), (L,H,H,H) and(L,L,L,H). Consider (L,H,L,H) which induces specialization in the firmspecific skill. This strategy cannot be part of a skill acquisition equilib-rium because the reward for a promotion equals 0 and the firm can-not keep its wage promise. Next, consider (L,L,H,H) which inducesspecialization in the general skill. This strategy cannot be part of askill acquisition equilibrium because the reward for a promotion,(1 − λ)yH, is too high relative to the output gain (1 − λ)(yH − yL),and the firm cannot keep its promotion promise at (0,g). Similarly(L,H,H,H) can be ruled out using the arguments above. Thus the onlypossible pure strategy equilibrium with skills is (L,L,L,H). When p� yHis sufficiently small relative to the marginal cost of investing in eitherskill, the worker prefers the minimal investment level in both skills tospecializing in one, and thus specialization cannot be part of a skill ac-quisition equilibrium.
The following proposition characterizes necessary and sufficientconditions for a skill acquisition equilibrium to exist. To make it easierto compare our results with Prendergast (1993), we also define the
following cost parameter, c ¼ max cfþcgp�þp ;
cfp ;
cgp
�. This cost parameter,
which accounts for the fact that we have both general and firms spe-cific skills (Prendergast (1993) only considers firms specific skills),yields conditions for existence of an equilibrium that can be com-pared to Prendergast (1993).
Proposition 3. There exists a skill acquisition equilibrium if and only ifthe following conditions hold
c≤yH−yL ð1Þ
λ∈ 1− yH−yLð ÞyH
; min 1− cyH
;yH
yH−yLð Þ þ yH
�� : ð2Þ
Furthermore, this equilibrium, with investment levels p;pð Þ and thepromotion scheme (L,L,L,H), is unique in pure strategies.
The best way to understand Proposition 3 is to compare it withPrendergast (1993). Recall that in his paper there has to be a gap be-tween the output gains from promotion and the cost of investing inskills, which is exactly what the inequality in Eq. (1) states. The firmthen, as in Prendergast's framework, can set the reward for a promo-tion somewhere in between these two values to keep its promotionpromise and to induce skills. The key difference in our framework isthat without a court the firm loses its flexibility in setting the rewardfor a promotion. Instead, as condition (2) shows, the reward is deter-mined by information about skills that a promotion conveys to thelabor market. If only the firm specific skill is valuable, the reward of-fered by the labor market is too small relative to the cost of investingin both skills and if only the general skill is valuable, the firm will nothave an incentive to keep its promotion promise. The main point totake away from Proposition 3 is that general skills, that cannot be ver-ified by a third party, play a similar role as contracts where wages areattached to jobs in Prendergast's framework. That is they help a firmto commit to reward firm specific skills.
We can also compare equilibrium promotions and investment levelswith their efficient counterparts in Section 3. Consider promotions first.As in Waldman (1984) and Scoones and Bernhardt (1998), there issome under-promotion. But because skills cannot be verified by
Fig. 2. Equilibrium investment levels.
19 We focus just on general training and not firm specific training because our prima-
71S. Prasad, H. Tran / Labour Economics 23 (2013) 66–76
competing firms, the pattern is different. Workers are under-promotedin our settingwhen they do not acquire both skills together, whereas inWaldman (1984) the under-promotion depends only on the firm spe-cific skill.
Next, consider investment levels. Because the worker bears all thecosts of investing in skills (given that training by the firm is alreadysunk) and the firm gets all of the benefits we would typically expectunder-investment in skills relative to the efficient level in equilibrium.For λ sufficiently close to 0 or 1, where a skill acquisition equilibriumdoes not exist, this is exactly the case. What is unusual aboutProposition 3 is that there can be an over-investment in skills. Thiscan be seen by comparing the smallest λ for which a skill acquisitionequilibriumexists in Fig. 2with the cutoffλ for specialization in the gen-eral skill in Fig. 1. In fact, when cf−pΔ > yL, there is over-investment inthe firm specific skill. This condition has a simple interpretation. Ascf−pΔ gets larger, firm specific skills are more costly, and the cutofflevel of λ for specialization in the general skill in Fig. 1 moves to theright. As yL gets smaller, on the other hand, firms havemore of an incen-tive to keep their promotion promise, and the cutoff level of λ in Fig. 2for firms to keep their promotion promise moves to the left. Thus,when the condition cf−pΔ > yL holds, there is an interval of λ'swhere there is over-investment in the firm specific skill.
Finally, we end this section by discussing the role that the linearcost function plays in all of the propositions so far. Suppose we useda cost function that is strictly increasing, strictly convex and differen-tiable, then it affects our main propositions in the following way. Inthe case of Proposition 1, the results are qualitatively similar exceptwith interior solutions for intermediate values of λ. Proposition 2continues to hold provided the marginal cost at p
�; p�
� �is sufficiently
high relative to p� yH . And in the case of Proposition 3, we could have
multiskilling with interior rather than corner solutions in equilibriumfor intermediate levels of λ.
5. Training
A key point in Proposition 3, is that general skills that are not ver-ifiable by a third party (courts or labor markets) play exactly the samerole as ex-ante contracts where wages are tied to jobs in Prendergast(1993): they help a firm to commit to reward specific skills. Firms canthus benefit from providing general training to workers in contrast toarguments made in Becker (1964).18 To make this point, we expandour model and allow the firm to incur a one off fixed cost CT > 0and provide a general training program (that is publicly observable)before workers invest in skills. We think of this program as off the
18 Becker argues that in a competitive labor market, wages get bid up to equal all ofthe gains in productivity from training. Thus ex-post, the firm gets none of the returnsfrom training. The firm anticipates this ex-ante and does not incur the training cost.Becker also notes that because workers get all of the benefits from training ex-postthrough higher wage offers, they have an incentive ex-ante to pay for training. Takentogether, his observations imply that if there is training in general skills, the workershould pay for it.
job training that allows workers to make larger investments in skills.If workers are trained, then they can make investments in both the
firm specific and general skills from the interval p� ; p�h i
. If not, they
can make investments in the interval above only for the firm specificskill whereas they are forced to choose p� for the general skill.19 So
the general training program gives a worker access to make invest-ments in the general skill.20
In our analysis in this section, we restrict our attention to purestrategies. We also assume that conditions (1) and (2) inProposition 3 hold. Thus we assume that a skill acquisition equilibri-um exists in pure strategies and that it is unique. Also define thelower and upper bounds of λ in condition (2) as λ� ¼ 1− yH−yLð Þ
yHand
�λ ¼ min 1− cyH; yH
yH−yLð ÞþyH
n o.
Now suppose the firm does not provide training. Then fromProposition 2, a skill acquisition equilibrium does not exist. The bestpossible equilibrium for the firm is then a pooling equilibriumwhere the worker is assigned to job H for every outcome. In thiscase, the worker's investment levels are p
�;p�
� �and the firm's
expected profit is given by
p�
1− p�
� �yH þ p
�2 yH þ Δð Þ
On the other hand, when the firm provides general training, thereis a unique equilibrium in pure strategies given by the promotionscheme (L,L,L,H) and investment levels p�; p�ð Þ. In this case, the firm'sexpected profit is given by
p 1−pð ÞyL þ p2 λyH þ Δð Þ
Comparing the expressions above, we notice two things. First, pro-viding general training helps a firm to commit to reward skills, whichin turn induces a worker to invest in firm specific skills. General train-ing thus plays the same role as contracts where wages are tied to jobsin Prendergast (1993) and helps to induce firm specific skills. Second,notice that promotions selectively reveal information about the gen-eral skill only when a worker acquires firm specific skills. Thus,firms earn a return from general training. In fact, when p
�goes to 0
and p goes to 1 in the limit, a firm's return from training (net of thetraining cost) is λyH + Δ.
Because general training and ex-ante contracts where wages aretied to jobs are substitutes from the point of view of inducing firmspecific skills, the returns from training should be higher whenthese contracts are not feasible. To examine this point, we consideran alternative case where firms can commit to a contract wherewages are tied to jobs. In terms of the model, the only change isthat the firm can credibly commit to a wage (through a contract)for a given job before the worker invests in skills. Competing firms,after observing the job assignment, can make an offer to the worker.And finally the current employer can decide to match this offer if itis strictly above the contractual wage. Given a contractual wage, wesolve for an equilibrium in this alternative case.
As in the analysis without contracts, the best a competing firm cando is to make an expected profit of zero and there are multiple wageoffers that satisfy this condition.21 We pin down wage offers in the
ry interest is to point out that firms do benefit from training workers in general skills.20 It is worth noting that there are other ways to model access to the general skill.First, training could reduce the cost of the general skill for the worker. This approachyields qualitatively similar results. Second, training may constrain the investment setof a worker (rather than expanding it) by ensuring that a worker always makes a min-imal level of investments.21 For a job assignment, fix the outcome where the current employer has the lowestoutput. Then any offer which is less than or equal to the maximum of the contractedwage and the current employer's output at the given outcome is optimal.
72 S. Prasad, H. Tran / Labour Economics 23 (2013) 66–76
same way as in our analysis without contracts. Given a job and a pro-motion strategy, consider the outcome where the current employerhas the smallest output. The offer by a competing firm then equalsits own value of skills at that particular outcome. If this offer is lessthan or equal to the contracted wage, the competing firm is neversuccessful at raiding the worker and makes an expected profit ofzero. If this offer is strictly above the contracted wage, once againthe competing firm makes an expected profit of zero, given that thecurrent employer can tremble at the outcome where it has the lowestoutput. Also, because of the presence of firm specific skills, it must bethe case in equilibrium that wages for a job are at least as large as thewage offers from competing firms.22
The following Proposition compares returns across the case wherefirms cannot commit to a contract where wages are attached to jobsand the case where they can.
Proposition 4. Let p�
be sufficiently close to 0 and let p be sufficiently
close to 1. Also let λ∈ max λ� ;12
n o;λ
� iand max yL
2 þ cf ; c� �
b yH2 . Then
the returns from training are strictly larger when firms cannot committo a contract where wages are tied to jobs.
Proposition 4 says that when firm specific skills are sufficientlyvaluable, then the return from training a worker is higher whenjobs cannot be contracted on. The intuition for this proposition isthe following. Contracts where wages are tied to jobs and generaltraining are substitutes from the point of view of inducing firm specif-ic skills. Thus when these contracts are not feasible, the only way toextract surplus from firm specific skills is by providing generaltraining. The assumptions that λ > 1
2 and yL2 þ cf b
yH2 ensure that the
promotion strategy (L,H,L,H) along with investment levels p;p�
� �is an
equilibrium when wage contracts cannot be tied to jobs and workersare not trained. The assumption that yH
2 > c ensures that λ > 12 so that
the interval λ∈ max λ� ;12
n o;λ
� iis non-empty.
23 Complementarities between firm specific and general skills also increase thereturns to general training in our framework. That is, firms are more likely to providetraining when Δ is higher.24 Using survey data on British establishments, DeVaro and Farnham (2011) considerthe relationship between volatility in the market for a product and firms trainingworkers in jobs other than their own (possibly jobs related to another product) and
6. Testable implications
In this section, we state the main testable implications that arisefrom our analysis where skills cannot be verified by outside firms.As mentioned earlier in the introduction, we believe that skills aremore likely to be non-verifiable in more complex production settings.For example, if workers have to solve complex production problemsthen it should be difficult for outside firms to observe their skill levels.For simpler jobs where tasks are routine, on the other hand, verifyingskills may be easier. There are three main testable implications. Thefirst two implications arise from Propositions 2 and 3 in Section 4and the third implication is from Section 5.
The first implication is that the likelihood of a promotion is weaklyincreasing in both firm specific and general skills. This implication isin contrast to Waldman (1984) and Scoones and Bernhardt (1998),where promotions only depend on the firm specific skill. This is be-cause skills are verifiable to competing firms in their framework.
Second, workers have an incentive to invest in firm specific skillseven though firms cannot commit to a contract that attaches wagesto jobs (i.e. even though jobs are flexible). This implication is in con-trast to Prendergast (1993) and is one of the main points of our paper.
Third, firms are more likely to pay for general training off the jobwhen they cannot commit to a contract that attaches wages to jobs(i.e. when jobs are flexible). This implication is distinct from several ex-planations for general training which rely on different labor market im-perfections. For example, Acemoglu and Pischke (1998), Chang andWang (1996) and Katz and Ziderman (1990) focus on informationalasymmetries between the training firm and other firms in the labor
22 They can be strictly larger as well because firms can contractually commit to awage for a job.
market. With asymmetric information, labor markets do not bidwages up enough, giving firms some returns from general training.Malcomson et al. (2003) showhowapprentice contracts, where retainedworkers are paid a higher wage after an apprenticeship can increase theprofit of the training firm. Acemoglu and Pischke (1999) show how thepresence of an exogenous and complementary firm specific skill alsogives incentives for a firm to pay for general training.23 Firm specificskills are endogenously acquired in our framework and our results donot depend on the skills being complements. Stevens (1994) focusseson labor market imperfections in a setting where skills are not entirelygeneral or firm specific. Balmaceda (2005) and Kessler and Lulfesmann(2006) consider both general and specific investments. Their focus, how-ever, is on the bargaining process between a firm and worker.
A key variable required to test the implications above is job flexibil-ity so that firms cannot commit to attach wages to jobs. One way toproxy for this variable is to see how broad job classifications are.Firms presumably have broader classifications when they want to fre-quently update job descriptions. Ichniowski et al. (1997), in theirstudy of steel finishing lines, find a positive correlation between firmsproviding off the job training and broad job classifications. Some exam-ples of these training programs are training in mechanical operations,the chemistry of steel, statistical process control, documenting produc-tion problems and running problem solving meetings (Gant et al.,2002). These programs help workers become better problem solvers(which fits well with the way we have modeled training) and appearto have a large general component. Furthermore, if we assume thatfirms that provide off the job training also pay for it, then this evidenceis consistent with our model. An alternative way to proxy for job flexi-bility is to consider the volatility of a firm's product market (as inDeVaro and Farnham, 2011). More volatile product markets presum-ably requiremore frequent changes in job descriptionsmaking commit-ment to attach wages to jobs difficult.24
We must point out, however, that there are a few other paperswhich yield a positive relationship between training and job flexibility.In Carmichael and MacLeod (1993), training workers across jobs givesthem incentives to comeupwith labor saving innovationswithoutwor-rying about losing work. In Morita (2005), training the worker acrossmultiple jobs increases the benefit ofmaking continuous improvementsto the production process. Because workers are cross-trained acrossmultiple jobs in both of these papers, training and job flexibility havea positive relationship. The difference between our paper and thesetwo papers is in the interpretation of training. In our case, training isoff the job and allows workers to make investments whereas theirfocus is on cross-training across jobs. We believe the examples of train-ing in the preceding paragraph fit better with our view of trainingwhere workers make subsequent investments.
7. Conclusion
In this paper, we study a worker's incentives to invest innon-verifiable skills. We do this within the context of recent workpractices where jobs are flexible so that firms cannot contractuallycommit ex-ante to tie wages to jobs. We show that firms can induceskills by promoting workers only when they acquire both firm specif-ic and general skills. This mechanism does not rely on court enforce-able contracts. Furthermore, it suggests a new reason in the literature
they find a negative relationship for multi-product establishments. This finding is con-sistent with their theoretical model where volatility increases the innovation gainsfrom training a worker in one product (specialization) versus training a worker in mul-tiple products (multiskilling).
73S. Prasad, H. Tran / Labour Economics 23 (2013) 66–76
for why firms pay for general training: general training helps a firm tocommit to reward firm specific skills. We also show that firms aremore willing to pay for general training when they cannot committo tie wages to jobs.
To emphasize the role of general skills, we abstract from workerheterogeneity in our model. However, high involvement organiza-tions spend substantial resources on carefully screening workersand thus worker heterogeneity should be less of a concern in thesesettings. We must point out, though, that adding worker heterogene-ity to our model does not change our results qualitatively, providedwe assume that the size of the interval from which ability is drawnis not too large relative to the cost of acquiring skills.
Appendix A
Proof of Proposition 1. The first order necessary conditions for theoptimal solution are
λyH þ pgΔ−cf1−λð ÞyH þ pfΔ−cg
� �¼ −μ f
10
� �þ μ
� f−10
� �þ μ g
01
� �
þ μ� g
0−1
� �
where μ f is the non negative multiplier associated with the constraintpf ≤ p and μ
� f is the non negative multiplier associated with the con-straint pf ≥ p. Likewise, μ g and μ� g are non negative multipliers asso-ciated with the inequality constraints for pg.
In the following claim we establish that the efficient investmentlevel is at the maximum level for at least one of the skills.
Claim. The efficient level of investment has either pf ¼ p or pg ¼ p orboth.
Proof. Suppose the optimal solution had pf b p and pg b p. Then thefirst order conditions imply λyH + pgΔ − cf ≤ 0 and (1 − λ)yH + pfΔ − cg ≤ 0. Adding both inequalities and rearranging we getyH + (pg + pf)Δ ≤ cf + cg. But this contradicts Assumption 1. ■
Now suppose Δ >cfp and suppose to the contrary that pf b p at the
optimum. From the claim above it follows that pg ¼ p at the optimum.Since Δ >
cfp the first order condition with respect to pf can be written
as λyH þ pΔ−cf ¼ −μ f− μ� f > 0. From the complementary slacknessconditions we have pf ¼ p at the optimum which is a contradiction.Similarly when Δ >
cgp we must have pg ¼ p at the optimum.
Next suppose Δ≤ cfp and Δ≤ cg
p and consider the following parti-tion of the interval (0,1) into five cases.
First, let 0 b λ bcf−pΔ
yH. Rearranging the second inequality we get
λyH þ pΔ−cf b 0. Combining this inequality with the constraintpg≤p, we can write the first order condition with respect to pf asλyH þ pgΔ−cf ¼ μ f− μ
� f b 0. Since the multipliers are non-negative,
it follows that μ�f > 0. The complementary slackness conditions
imply that pf ¼ p�
at the optimum. It follows from the claim above
that pg ¼ p at the optimum.Second, let λ ¼ cf−pΔ
yH. Rearranging, we get−λyH ¼ pΔ−cf . Adding
yH to both sides of the equality we get
1−λð ÞyH ¼ yH þ pΔ−cf ð3Þ
Using Eq. (3) and Assumption 1, the first order condition with re-spect to pg is 1−λð ÞyH þ pfΔ−cg ¼ μ g− μ� g
> 0. Since the multipliersare non negative it follows that μ g > 0. Thus pg ¼ p at the optimum.Likewise the first order conditions with respect to pf yield
λyH þ pgΔ−cf ¼ μ f− μ� fð4Þ
Substituting λyH ¼ cf−pΔ and pg ¼ p into Eq. (4) we get
λyH þ pgΔ−cf ¼ cf−pΔþ pΔ−cf ¼ 0: ð5Þ
Since the objective function is linear with respect to pf when pg is
held fixed it follows from Eq. (5) that any pf in the interval p�;p�
h iis
an optimal solution.
Third, let cf−pΔyH
b λ b 1− cg−pΔyH
. Rearranging the first inequality we
get λyH > cf−pΔ. Substituting this inequality into the first order con-dition with respect to pf we get λyH þ pgΔ−cf ¼ μ f− μ� f
> 0. Since
the multipliers are non-negative, it follows that μ f > 0. The comple-mentary slackness conditions imply that pf ¼ p at the optimum. Sim-ilar reasoning can be used to show that pg ¼ p at the optimum.
Fourth, let λ ¼ 1− cg−pΔyH
. Using reasoning similar to the secondcase we can show that pf ¼ p at the optimum along with any pg
from the interval p� ;ph i
.
Finally for the fifth case suppose 1− cg−pΔyH
b λ b 1. Using reasoningsimilar to the first case we can show that pf ¼ p and pg ¼ p
�at the
optimum. ■
Proof of Lemma 1. Notice that (H,H,H,H) cannot be part of a skill ac-quisition equilibrium because the worker's wage does not depend onthe outcome. Thus in a skill acquisition equilibrium, the firm must as-sign the worker to a low job for at least one of the outcomes. Supposeto the contrary thatwL > wH, then the firm can deviate from L to H forthat particular outcome, get at least the same level of output and paya lower wage, which is a contradiction. ■
Proof of Proposition 2. We prove this proposition using a more gen-eral cost function C(pf,pg) that is strictly positive, strictly increasing,convex and differentiable in the interior of the domain and suchthat such that for all pf and pg in the interior of the domain,
min∂C pf ;pgð Þ
∂pf;∂C pf ;pgð Þ
∂pg
�≥b. This is to show that the results in this
proposition do not depend on the linearity of the cost function.The proof is divided into a series of claims.
Claim 1. Suppose the firm promotes a worker for some outcome (i,j)with positive probability and suppose the output difference acrossjobs is strictly higher for some other outcome (i′,j′). Then in equilibri-um, the uniquely optimal action for the firm at (i′,j′) is to promote theworker.
Proof. In equilibrium, the firm at a given outcome, assigns the workerwith a probability of one to jobH ifwH − wL is strictly less than theoutputdifference across jobs and is indifferent between jobs L and H ifwH − wL
equals the output difference across jobs. Since the output difference injobs at (i′,j′) strictly exceeds the output difference at (i,j) and since theworker is assigned to H with positive probability at (i,j), in equilibriumthe firm must assign the worker to Hwith probability one at (i′,j′). ■
Claim 2. In a skill acquisition equilibrium, the firm never promotes aworker for the outcome (0,0). Furthermore, wL = 0.
Proof. Suppose the firm promotes the worker with a probability ofone at the outcome (0,0) in a skill acquisition equilibrium. Thenfrom Claim 1, the only possible strategy in equilibrium is the poolingstrategy (H,H,H,H). But in this case the worker has no incentive to in-vest in skills because his wage does not vary with the outcome, whichis a contradiction. Next suppose the firm promotes the worker with apositive probability less than one. Then it must be the case that thefirm is indifferent between the jobs L and H at the outcome (0,0).This implies that wH − wL = 0. Once again the worker has no incen-tive to acquire skills. ■
74 S. Prasad, H. Tran / Labour Economics 23 (2013) 66–76
Claim 3. In a skill acquisition equilibrium, the firm never promotes aworker for the outcome (f,0).
Proof. Suppose the firm promotes the worker with positive probabil-ity at the outcome (f,0) in a skill acquisition equilibrium. There arethen two cases to consider. First, suppose λ≤ 1
2. Then wH = 0 andsince wL = 0 from Claim 2, the worker has no incentive to invest inskills which is a contradiction. Next, suppose λ > 1
2. Then there aretwo possibilities. If the firm does not promote the worker at (0,g)then wH = 0 and the worker has no incentive to invest in skills. Ifthe firm does promote the worker with some positive probability at(0,g) then wH = (1 − λ)yH. In this case the firm deviates at (0,g)and chooses L for sure once again leading to a contradiction. ■
Claim 4. In a skill acquisition equilibrium, the firm never promotes aworker for the outcome (0,g).
Proof. Suppose the firm promotes theworker with positive probabilityat the outcome (0,g) in a skill acquisition equilibrium. There are thentwo cases to consider. First, suppose λ > 1
2. Then wH = (1 − λ)yH andsincewL = 0 fromClaim2, thefirm always deviates at (0,g) and choosesLwhich is a contradiction. Next, suppose λ≤ 1
2. If the firm does not pro-mote theworker at (f,0), then once againwH = (1 − λ)yH and the firmhas an incentive to deviate and choose L at (0,g). If the firm does pro-mote the worker at (f,0) with positive probability then wH = 0 andthe worker has no incentive to invest in skills. ■
Claim 5. In a skill acquisition equilibrium, the firm must promote aworker with positive probability for the outcome (f,g).
Proof. Suppose the firm does not promote the worker at (f,g) in askill acquisition equilibrium. The previous claims then imply thatthe firm's strategy is (L,L,L,L). But in this case, the worker's wagedoes not depend on the outcomes and he has no incentive to investin skills which is a contradiction. ■
Claim 6. In a skill acquisition equilibrium, when p�yH b b, the worker
never specializes in a skill.
Proof. Let q be the probability of the strategy (L,L,L,H) and (1 − q) bethe probability of (L,L,L,L). Then the only possible strategy in a skillacquisition equilibrium is one where q > 0. The worker's expectedpayoff for a given q is
wL þ qpf pg wH−wLð Þ−C pf ;pg� �
:
First suppose pf ¼ p� and pg > p�in equilibrium. If the worker de-
viates and chooses p�;p�
� �the difference in theworker's expected payoff
is
−q p�
pg− p�
� �wH−wLð Þ þ C p
�; pg
� �−C p
�;p�
� �:
Let p�
be sufficiently small so that p�yH b b. Then
C p�; pg
� �−C p
�; p�
� �≥ pg− p
�
� �b > pg− p
�
� �pyH
> q pg− p�
� �p�
wH−wLð Þ
where the first inequality follows from the fact that C is convex withpartial derivatives bounded below by b and the last inequality followsfrom the fact that wH − wL is bounded above by yH. Thus the workerdoes strictly better from the deviation leading to a contradiction.
Similar reasoning can be used to rule out pf > p�and pg ¼ p
�. ■
Proof of Proposition 3. Consider (L,L,L,H) and p;pð Þ. These are partof a skill acquisition equilibrium if and only if the strategies are se-quentially rational for both the firm and the worker.
Notice that a worker chooses p; pð Þ if and only if for all (pf,pg) thefollowing condition holds
p2 wH−wLð Þ− cf þ cg� �
p ≥ pf pg wH−wLð Þ−cf pf−cgpg :
Since the condition above must hold for all (pf,pg), it must hold forthe following cases, i) pf ¼ p, pg ¼ p� , ii) pf b p, pg ¼ p, iii) pf ¼ p,pg b p, iv) pf b p, pg b p. Substituting the values of investment inthe first three cases gives us the following conditions respectively
wH−wL≥cf þ cg
�p þ pð6Þ
wH−wL≥cfp
ð7Þ
wH−wL≥cgp: ð8Þ
For case iv) notice that because the agents expected payoff is lin-ear in one skill when the other is fixed it follows that condition (6)is sufficient for p; pð Þ to be optimal.
Next, notice that wages in equilibrium must be
wH ¼ 1−λð ÞyH
wL ¼ 0:Also, for (L,L,L,H) to be an equilibrium strategy it must be the case
that the firm does not have any incentive to deviate at each of thefour outcomes (0,0), (f,0), (0,g) and (f,g). These conditions aregiven by
0≤wH−wL ð9Þλ yH−yLð Þ≤wH−wL ð10Þ
1−λð Þ yH−yLð Þ≤wH−wL ð11Þ
and
yH þ Δð Þ− yL þ Δð Þ≥wH−wL: ð12Þ
Substituting wH − wL from above notice that Eqs. (9) and (11)always hold. Also we can rewrite Eq. (10) as
λ≤ yHyH−yLð Þ þ yH
ð13Þ
and Eq. (12) as
λ≥1− yH−yLyH
: ð14Þ
Combining Eqs. (6), (7), (8), (13) and (14) gives us
1− yH−yLyH
≤ λ ≤ min 1− cyH
;yH
yH−yLð Þ þ yH
�:
To show that the set of λ's satisfying the inequality is non emptynotice that c ≤ yH−yL implies 1−yH−yL
yH≤ 1− c
yH. Also notice that
1− yH−yLð ÞyH
≤ yHyHþ yH−yLð Þ if and only if (yH − yL)2 ≥ 0 which always
holds.Now to show that these strategies are unique, consider the firm first.
For thefirm, the uniqueness of (L,L,L,H) follows fromProposition 2. Nextconsider the worker.
We know from Proposition 2 that when p� yH b b there cannot beskill acquisition equilibria with the minimal investment level for
75S. Prasad, H. Tran / Labour Economics 23 (2013) 66–76
any of the skills. We still need to rule out other cases besides p; pð Þ aspossible equilibrium investment levels.
First, suppose pf ¼ p and p�b pg≠p in equilibrium. Since the
worker's expected utility is linear with respect to pg with pf fixed atp, it follows that p ; p
�
� �yields exactly the same expected payoff for
the worker and this is given by
p p�
wH−wLð Þ−cf p−cg p�
If the worker deviates and chooses p�; p�
� �the difference in the
worker's expected payoff is
− p�
p− p�
� �wH−wLð Þ þ cf p− p
�
� �:
Let p�be sufficiently small so that p
�yHbb. Then the worker can do
strictly better from the deviation leading to a contradiction.Using the same reasoning above we can rule out pg ¼ p and
p�
b pf ≠ p as an optimal choice for the worker.Next suppose the equilibrium strategy (pf,pg) was interior. Since
the expected payoff of the worker is linear in pf holding pg fixed it fol-lows that by choosingpf ¼ p
�and pg the worker gets exactly the same
payoff. But by the reasoning above the worker can do strictly better
by deviating and choosing p�; p�
� �leading to a contradiction. ■
Proof of Proposition 4. For all of the cases belowwe take limits as p�
tends to 0 andp tends to 1 when computing profits. First, consider thecase where wage contracts cannot be tied to jobs. Because yH
2 > c andcondition (1) in Proposition 3 holds, it follows that the intervalλ∈ max λ� ;
12
n o;λ
� iis non empty. The expected profit for a firm
when a worker is trained in this interval is λyH + Δ and the expectedprofit when a worker is not trained is 0. So the returns from trainingare λyH + Δ when λ∈ max λ� ;
12
n o;λ
� i.
Now consider the case where firms can commit to a contractwhere wages are tied to jobs and suppose the worker is nottrained in general skills so that pg ¼ p
�. Then the promotion ,strat-
egy (L,H,L,H), along with investment levels p ; p�
� �, contracted
wages of wH = max{cf, (1 − λ)(yH − yL)} for job H and wL = 0for job L, and wage offers of 0 for either job are part of an equilib-rium when λ > 1
2. To see this, note that because λ > 12 and
yH2 > yL
2 þ cf , the firm's promotion strategy is optimal at all the fouroutcomes. Also, the workers marginal expected utility is at leastas large as max{cf, (1 − λ)(yH − yL)} − cf which is non-negative.Thus it is optimal for the worker to choose pf ¼ p. The expectedprofit for this equilibrium is λyH − max{cf, (1 − λ)(yH − yL)}.
Finally, consider the case where firms can contractually commit totie wages to jobs and workers are trained in general skills. FromClaim 1 in the proof of Proposition 2, there are only four possible strat-egies to consider for a skill acquisition equilibrium: (L,L,L,H), (L,L,H,H),(L,H,L,H) and (L,H,H,H). We will show that for any of these strategies,expected profits never exceed λyH + Δ. As a result, the return from
training in the interval λ∈ max λ� ;12
n o;λ
� iis strictly larger when jobs
cannot be contracted on. First, consider the strategy (L,L,L,H). Becausewages in equilibrium must be at least as large as competing offers of 0for job L, and (1 − λ)yH for job H, and because λyH + Δ > λyL >(1 − λ)yL, expected profits can never exceed λyH + Δ for this strategy.Second, consider the strategy (L,L,H,H). This strategy induces specializa-tion in the general skill and the expected profit (as p
�tends to 0 in the
limit) is− (1 − pg)wL + pg((1 − λ)yH − wH). Becausewages in equi-librium must be at least as large as competing offers of 0 for job L, and(1 − λ)yH for job H, expected profits do not exceed λyH + Δ. Third,consider the strategy (L,H,L,H). This strategy induces specialization inthe firm specific skill and the expected profit (as p
�tends to 0 in the
limit) is − (1 − pf)wL + pf(λyH − wH). Because wages are non-negative, expected profits can never exceed λyH + Δ. Finally, consider
the strategy (L,H,H,H). Notice that when p is sufficiently close to 1,workers will only specialize in one skill. To see this, suppose to the con-trary that pf > p
�and pg > p
�. Consider two possible cases. First sup-
pose either of the investments is at the maximum level (say withoutloss of generality that pf ¼ p). Then the first order necessary conditionfor the worker with respect to pg is 1−pð Þ wH−wLð Þ−cg ≥ 0. Noticethat since wH ≤ yH for the firm to make a positive profit and since p issufficiently close to 1, the first order conditions do not hold, resultingin a contradiction. Alternatively, suppose (pf,pg) is interior. Because aworker's marginal utility is linear in either variable, it must be thecase that p;pg
� �is also optimal for the worker. But once again, the
worker can do strictly better by setting pg ¼ p�which results in a con-
tradiction. So the only possible investments in equilibrium for the strat-egy (L,H,H,H) are those where the worker specializes in a skill. Onceagain, because wages in equilibrium are non-negative expected profitscannot exceed λyH + Δ. ■
References
Acemoglu, D., Pischke, J.-S., 1998. Why do firms train? Theory and evidence. QuarterlyJournal of Economics 113, 79–119.
Acemoglu, D., Pischke, J.-S., 1999. The structure of wages and investment in generaltraining. Journal of Political Economy 107 (3), 539–571.
Aoki, M., 1986. Horizontal vs vertical information structure of the firm. American EconomicReview 76 (5), 971–983.
Baker, G., 1992. Incentive contracts and performance measurement. Journal of PoliticalEconomy 100, 598–614.
Baker, G., Gibbons, R., Murphy, K., 1994. Subjective performance measures in optimalincentive contracts. Quarterly Journal of Economics 109 (4), 1125–1156.
Balmaceda, F., 2005. Firm sponsored general training. Journal of Labor Economics 23(1), 115–133.
Bartel, A., Ichniowski, C., Shaw, K., 2003. Low wage America: how employers arereshaping opportunity in the workplace. chap. New Technology and its Impact onthe Jobs of High School Educated Workers: A Look Deep Inside Three ManufacturingIndustries.Russell Sage Foundation, New York.
Becker, G., 1964. Human Capital. National Bureau of Economic Research, New York.Becker, G., Murphy, K., 1992. The division of labor, coordination costs and knowledge.
Quarterly Journal of Economics 107 (4), 1137–1160.Bernhardt, D., 1995. Strategic promotion and compensation. Review of Economic Stud-
ies 62, 315–339.Black, S., Lynch, L., 2004. What's driving the new economy?: The benefits of workplace
innovation. The Economic Journal 114 (493), F97–F116.Cappelli, P., Neumark, D., 2001. Do ‘high performance’ work practices improve
establishment-level outcomes? Industrial & Labor Relations Review 54 (4), 737–775.Carmichael, L., 1983. Firm-specific human capital and promotion ladders. Bell Journal
of Economics 14 (1), 251–258.Carmichael, L., MacLeod, W., 1993. Multiskilling, technical change and the Japanese
firm. The Economic Journal 103 (416), 142–160.Chang, C., Wang, Y., 1995. A framework for understanding differences in labor turnover
and human capital investment. Journal of Economic Behavior and Organization 28(1), 91–105.
Chang, C., Wang, Y., 1996. Human capital investment under asymmetric information:the Pigovian conjecture revisited. Journal of Labor Economics 14 (3), 505–519.
Dessein, W., Santos, T., 2006. Adaptive organizations. Journal of Political Economy 114(5), 956–995.
DeVaro, J., Farnham, M., 2011. Two perspectives on multiskilling and product-marketvolatility. Labour Economics 18 (6), 862–871.
DeVaro, J., Waldman, M., 2012. The signaling role of promotions: Further theory andempirical evidence. Journal of Labor Economics 30 (1), 91–147.
DeVaro, J., Ghosh, S., Zoghi, C., 2008. Job characteristics and labor market discrimina-tion in promotions: new theory and empirical evidence. Working Paper, pp. 1–54.
Eriksson, T., Ortega, J., 2006. The adoption of job rotation: testing the theories. Industrial &Labor Relations Review 59 (4), 653–666.
Gant, J., Ichniowski, C., Shaw, K., 2002. Social capital and organizational change in high-involvement and traditional work organizations. Journal of Economics and Manage-ment Strategy 11 (2), 289–328.
Garg, H., Vrat, P., Kanda, A., 2002. Trade-offs between multitasking and inventory in as-sembly line operations under demand variability. International Journal of Opera-tions and Production Management 22 (5), 565–583.
Ghosh, S., Waldman, M., 2010. Standard promotion practices versus up-or-out con-tracts. The Rand Journal of Economics 41, 301–325.
Gibbons, R., Katz, L., 1991. Layoffs and lemons. Journal of Labor Economics 9 (4), 351–380.Gibbs, M., Levenson, A., 2002. The expansion of economics. chap. In: Grossbard-
Schechtman, S., Clague, C. (Eds.), The Economic Approach to Personnel Research.M.E Sharpe, Inc., New York.
Hart, O., Moore, J., 2005. On the design of hierarchies: coordination versus specialization.Journal of Political Economy 113 (4), 675–702.
Holmstrom, B., Milgrom, P., 1991. Multitask principal-agent analysis: incentive con-tracts, job ownership and asset design. Journal of Law, Economics and Organization7, 24–52.
76 S. Prasad, H. Tran / Labour Economics 23 (2013) 66–76
Ichniowski, C., Shaw, K., 2003. Beyond incentive pay: insiders' estimates of the value ofcomplementary human resource management practices. Journal of Economic Per-spectives 17, 155–178.
Ichniowski, C., Shaw, K., Prennushi, G., 1997. The effects of human resource manage-ment practices on productivity. American Economic Review 86, 291–313.
Kahn, C., Huberman, G., 1988. Two-sided uncertainty and “up-or-out contracts”. Jour-nal of Labor Economics 6 (4), 423–444.
Kato, T., Owan, H., 2010. Market characteristics, intra-firm coordination, and the choiceof human resource management systems: theory and evidence. Working Paper.
Katz, E., Ziderman, A., 1990. Investment in general training: the role of information andlabour mobility. The Economic Journal 100 (403), 1147–1158.
Kessler, A., Lulfesmann, C., 2006. The theory of human capital revisited: on the interactionof specific and general investments. The Economic Journal 116 (514), 903–923.
Koike, K., 1988. Understanding Industrial Relations in Modern Japan. St. Martin's Press,New York.
Lazear, E., Rosen, S., 1981. Rank-order tournaments as optimum labor contracts. Journalof Political Economy 89, 841–864.
Levin, J., 2003. Relational incentive contracts. American Economic Review 93 (3), 835–857.Lindbeck, A., Snower, D., 2000. Multitask learning and reorganization of work: from
Tayloristic to holistic organization. Journal of Labor Economics 18 (3), 353–376.MacDonald, G., Marx, L., 2001. Adverse specialization. Journal of Political Economy 109 (4),
864–899.MacLeod, W., Malcomson, J., 1988. Reputation and hierarchy in dynamic models of
employment. Journal of Political Economy 96 (4), 832–854.Malcomson, J., 1984. Work incentives, hierarchy, and internal labor markets. Journal of
Political Economy 92 (3), 486–507.Malcomson, J., Maw, J., McCormick, B., 2003. General training by firms, apprentice
contracts, and public policy. European Economic Review 47 (2), 197–227.Milgrom, P., Oster, S., 1987. Job discrimination, market forces, and the invisibility
hypothesis. Quarterly Journal of Economics 102 (3), 453–476.
Morita, H., 2005. Multi-skilling, delegation and continuous process improvement: acomparative analysis of U.S.–Japanese work differences. Economica 72, 69–93.
Myerson, R., 1978. Refinement of the Nash equilibrium concept. International Journalof Game Theory 7, 73–80.
O'Toole, J., Lawler, E.E., 2006. The New American Workplace. Palgrave-Macmillan.Osterman, P., 1994. How common is workplace transformation and who adopts it?
Industrial & Labor Relations Review 47 (2), 173–188.Owan, H., 2004. Promotion, turnover, earnings and firm-sponsored training. Journal of
Labor Economics 22 (4), 955–978.Owan, H., 2011. Specialization, multiskilling and allocation of decision rights. Advances
in the Economic Analysis of Participatory and Labor-Managed Firms 12.Prendergast, C., 1993. The role of promotions in inducing specific human capital acquisi-
tion. Quarterly Journal of Economics 108 (2), 523–534.Rosen, S., 1978. Substitution and division of labor. Economica 45 (179), 235–250.Roy, A., 1950. The distribution of earnings and of individual output. The Economic Journal
60, 489–505.Schonberg, U., 2007. Testing for asymmetric employer learning. Journal of Labor Econom-
ics 25 (4), 651–691.Scoones, D., Bernhardt, D., 1998. Promotion, turnover and discretionary human capital.
Journal of Labor Economics 16 (1), 122–141.Smith, A., 1937. An Inquiry into the Nature and Causes of the Wealth of Nations. Modern
Library, New York.Stevens, M., 1994. A theoretical model of on-the-job training with imperfect competition.
Oxford Economic Papers 46, 537–562.Waldman, M., 1984. Job assignments, signalling, and efficiency. The RAND Journal of
Economics 15 (2), 255–267.Waldman, M., 1990. Up-or-out contracts: a signaling perspective. Journal of Labor
Economics 8 (2), 230–250.Zabojnik, J., Bernhardt, D., 2001. Corporate tournaments, human capital acquisition and
the firm size-wage relation. Review of Economic Studies 68, 693–716.
