A Theoretical Model of On-the-Job Training with Imperfect Competition

A Theoretical Model of On-the-Job Training with Imperfect CompetitionAuthor(s): Margaret StevensSource: Oxford Economic Papers, New Series, Vol. 46, No. 4 (Oct., 1994), pp. 537-562Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2663510 .

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Oxford Economic Papers 46 (1994), 537-562

A THEORETICAL MODEL OF ON-THE-JOB TRAINING WITH IMPERFECT COMPETITION

By MARGARET STEVENS

Trinity College, Oxford OX1 3BH and

Institute of Economics and Statistics, St Cross Building, Oxford OX1 3UL

1. Introduction

IN HIS book Human Capital, Gary Becker (1964, 1975) classified on-the-job training as 'general' if it raises the worker's productivity equally in many firms, and 'specific' if it is of value only in the training firm. Becker suggested that most training was some combination of the two types. His most important conclusion was that workers, and not firms, receive the whole of the return to general training, and will therefore bear the costs. Moreover, since firms do not, and need not, invest in general training (although they may supply it) there is no problem of under-investment due to poaching of trained workers by other firms.

Becker's argument was convincing and has been influential not only in the economics literature-in which almost any discussion of training uses the general/specific classification as a starting point but also on training policy. For example, it was used by Lees and Chiplin (1970) to support their claim that the grant-levy system of the 1964 Industrial Training Act had 'no basis in economic logic'.

Yet some unresolved puzzles remain. Much effort has been expended in trying to explain the empirical finding that firms do, apparently, invest in general training.' Furthermore, it is widely believed that there is under-investment in training, and in spite of the theoretical argument, many authors seem reluctant to abandon the notion that this is in part due to a poaching externality (for example, Finegold and Soskice, 1988).

The contention of this paper is that there is a simple and plausible theoretical explanation for these problems. The general/specific classification does not encompass all types of training in particular that which takes place in the context of imperfect competition between firms in the labour market. When firms have labour market power, a firm may obtain some return to an investment in training, in spite of the fact that the skills are transferable to some other firms; in addition, since those other firms can also benefit from the investment there is an externality which may lead to under-investment.

The paper is organised as follows. In Sections 2 and 3 the classical theory of training is discussed, and extended to allow for imperfect competition. The conditions under which an externality exists are identified. In the remainder

1 See, for example, the collection of papers edited by Stern and Ritzen (1991).

?) Oxford University Press 1994



538 A THEORETICAL MODEL OF ON-THE-JOB TRAINING

of the paper a formal model is developed, to demonstrate some of the implications for training decisions by workers and firms.

2. General, specific and transferable training

Although the terms 'general' and 'specific' are widely used in discussion of training, there are some surprising differences in interpretation by different authors. But Becker is unambiguous about the meaning of 'perfectly general' and 'completely specific' training:2

Definition G. A training programme is 'perfectly general' if it increases the worker's marginal product by exactly the same amount in many firms (p. 20). Examples: clerical training; brick-laying.

Definition S. A training programme is 'completely specific' if it increases the worker's productivity in the firm providing the training, but has no effect on his productivity in any other firm (p. 26). Example: familiarisation programme for new employees.

However, it is when training is neither perfectly general nor completely specific that ambiguities arise. For example, Becker asserts at one point that training which raises productivity more at the firm providing it falls within the definition of specific training (p. 26). Others have tended to expand the definition of general training: Jones (1988), Ritzen (1991), and Hyman (1992) (and many others) define it as of use to some other firms, and Shackleton (1992) says that it is training of use to at least one other firm. Such definitions may be reasonable, and suggest that these authors believe that, in practice, some types of training are of use to small numbers of firms. But the inconsistencies become important if results obtained under one definition are assumed to apply when a different definition is used.

Becker's famous and influential result for general training can be summarised as:

Becker's Result for General Training. If a training programme is perfectly general, and if the labour market for trained workers is perfectly competitive, then the worker's post-training wage is equal to his marginal product, so the whole of the return to the investment in the training programme accrues to the worker.

The implication of this result (p. 25) is that there is no positive externality between firms which would lead to under-investment in general training. Neither the training firm nor any other firm obtains any part of the return to the training investment. The training firm will not, therefore, be pre- pared to incur training costs, but the investment can be financed by the worker since he captures the full return, and hence an optimal level of investment will occur.

2 In what follows, page numbers refer to the 2nd (1975) edition of Human Capital.



M. STEVENS 539

The result is often quoted more briefly as 'the return to general training accrues to the worker' on the assumption, perhaps, that if the training is equally useful to many firms the labour market must be competitive. But the definitional inconsistencies noted above have lead to misuse of this result: those authors who define general training as of use to some other firms frequently make the implicit assumption that Becker's result extends to such training. Yet to do so without considering the characteristics of the corresponding labour market is highly questionable. Becker does not make this claim: indeed, he states that the effect on training when firms possess monopsony power (but are not pure monopsonists) is 'difficult to assess' (p. 36).

Discussing completely specific training, Becker used an informal argument to hypothesise that the wage of a specifically trained worker would lie somewhere between his marginal product and the market wage available to him elsewhere. Thus, the return to an investment in specific training would be shared between worker and firm, and the costs of that investment would also be shared. Also (p. 30) since such training is of no value to other firms, there is no associated externality which could lead to sub-optimal investment.

The final step in Becker's analysis is the recognition that in practice, training is often neither perfectly general, nor completely specific, raising the worker's productivity in other firms but by less than in the training firm. He claims (p. 30) that such training can be regarded as 'the sum of two components, one completely general, the other completely specific'. This claim is important, because it gives the impression that, having understood the economic characteristics of these two polar cases, we can, almost trivially, derive results for any other type of training. So, for example, we know that there is no positive externality between firms associated with either general or specific training. If we believe that all types of training are a 'sum' of general and specific components, we may conclude that the much-discussed 'poaching externality' does not exist-for any type of training.3 However, it is argued below that both this belief and this conclusion would be false.

As a first step, we will formalise the analysis as follows. Suppose that there are many firms in the economy, and consider a single worker who undertakes a training programme in firm 0. Assume, for simplicity, that before undertaking training, the worker's productivity is zero in every firm. Let vi be the productivity of the worker after training if he works in firm i. Then the value of the training programme can be described by the vector

V = (VOw V1, V2, * * * , Vill .. *)

where the length of the vector is large, and corresponds to the total number of firms in the economy. So, using Becker's definitions, a perfectly general training programme can be described by

vg=(g,g,g,...,g,...) forsomeg>O

3Becker did not make this assertion.




and completely specific training by

Vs = (s, O, O.. .., O.. .) for some s > O

A sum of general and specific training would have a vector of the form

Vsg = (s + g,g,g, I . .,g, * ..)

This formulation immediately indicates that there are many other possibilities which cannot be regarded as the sum of a general and a specific component. For example

Vti = (t, t, t, 0,, . .0.. I . .) where t > O. or

vt2 = (VO, Vo, 2VO, . ,Vo, where vo > 0 and O < < 1

According to the alternative definition discussed above, vt1 would correspond to general training. But we will continue to use Becker's definition, reserving the term general for vectors of the form vg above, and use the term transferable (often used interchangeably with general) for training corresponding to vectors such as vt1 and vt2.

Definition T. A training programme is transferable if it is of some value to at least one firm in addition to the training firm.

Having recognised that not all transferable training can be regarded strictly as a sum of general and specific components, we should ask whether this matters. If, in fact, transferable training programmes such as vt1 above have economic properties which are 'intermediate' between the general and specific polar cases, then the distinction would not be of much value. But, as will be shown, in one important respect this is not the case.

In order to analyse the return to any training programme, we require to know not only its value v, but also how the wage of a trained worker will be determined that is, we need to know how the labour market for this type of worker operates. If training is general, it is usual to suppose that the labour market is perfectly competitive. Specific training, on the other hand, corresponds to a situation of pure monopsony: specific skills are those for which there is no competition between firms in the labour market. But for training programmes such as vt1 and vt2 which are transferable but not general, we should consider the possibility that the labour market is imperfectly competitive: vt1 corresponds to oligopsony; vt2 models a situation in which firms have differentiated skill requirements.

It is natural to consider imperfect competition in the context of training, since the acquisition of specialised skills by workers, and the requirement for particular skills by firms, can be regarded as reducing competition in the labour market. In the classical perfectly competitive labour market there are many identical workers and many identical firms, but as workers acquire different bundles of skills, they differentiate themselves from each other. Similarly, firms who use different combinations of specialised technology, or different patterns



M. STEVENS 541

of work organisation, require workers with particular sets of skills and job experience. Specific skills represent the extreme example of the association between training and imperfect competition, although as pointed out by Chapman (1991) it is difficult to think of many examples of purely specific skills. That some skills are required by small numbers of employers is a rather more plausible hypothesis. Furthermore, as in the product market, if firms can gain labour market power by differentiating themselves from others, they may choose to do so.

The inherent competition-reducing effects of training may be further com- pounded by lack of worker mobility. If workers face high moving-costs to other areas, or are prevented from moving by family commitments, their set of potential employers may be further reduced. Even if skills are very general, lack of mobility may mean that their value is effectively represented by a vector such as vt1, rather than vg. The importance of mobility in the analysis of training was emphasized by Oatey (1970). We could incorporate this concept formally by defining the value vector as representing the total value of the training to firms and worker, inclusive of all switching costs. When defined in this way, purely specific human capital may become relatively more important: it can include (as Becker pointed out) recruitment costs, and the costs to a firm of discovering the ability and potential of new employees.

3. The implications of imperfect competition for training

It has been argued firstly that the analysis of the returns to training depends on labour market conditions, and secondly that there is a natural link between training and an imperfectly competitive labour market. When we consider the effects of imperfect competition an important result emerges immediately:

3.1. Basic Result for Transferable Training

If, when an investment in transferable skills is made, the training firm and worker are uncertain about whether the worker will remain with the firm after training or move to an alternative firm which values the skills, and if the labour market is such that alternative firms are able to pay a wage less than marginal product, then part of the total expected return to the investment is captured by alternative firms.

That is, the total private return (the joint return to the worker and training firm) is smaller than the social return which includes that obtained by alternative firms: there is an externality associated with transferable training, which may lead to underinvestment.

This result will be demonstrated in this paper using a particular model, but it is a very general one: any source of imperfect competition leading to wages below marginal product, combined with any source of uncertainty about labour turnover, gives rise to this externality. Becker (p. 29) discusses




the importance of labour turnover, arguing that although it can be ignored in traditional competitive analysis, it is an essential element of the analysis of specific training. Here, we simply extend this argument: labour turnover has important implications whenever the labour market is less than perfectly competitive.

In the special cases in which training is perfectly general or perfectly specific the externality disappears: for general training this is because we then assume a perfectly competitive labour market, and for specific training because there is no possibility that the worker can move to an alternative firm which values these skills. Nor will the externality exist for training which is strictly a sum of general and specific components, represented by the vector vg s above. With many external firms for which the worker has identical value, g , the external wage will be equal to g so these firms cannot obtain any part of the return. It is when training is transferable but not simply a combination of general and specific that the externality problem can arise.

4. A Model of Training with Imperfect Competition Between Firms

We will now develop a model which allows investigation of how the returns to a transferable training programme are shared, and demonstrates the existence and implications of the externality identified above. Two features are therefore required: some form of uncertainty generating labour turnover, and imperfect competition between firms in the labour market. In the following model these features are generated by supposing that there are a small number of firms, which are subject to independent productivity or demand shocks, leading to short-run heterogeneity of firms and hence imperfect labour market competition. In other respects the labour market in this model is competitive: firms set wages in a 'Bertrand' manner, and workers are perfectly sensitive to any difference in wages.4

We will consider an economy in which there may be many workers and firms, but make a simplifying assumption of constant returns to labour which allows us to focus on the training taking place at a single firm, firm 0. That is, the productivity of any worker in any firm is unaffected by the number of other workers employed in that firm. Since workers do not then compete for jobs, the training decisions made at an individual firm can be made independently of those at other firms.

4.1. The Workers

Suppose that, initially, all workers are untrained and a random group of workers is attached to firm 0. Untrained workers have constant productivity, initialised to zero, in all firms. Workers differ from each other only in their

'Stevens (1994) demonstrates how similar results can be obtained when it is the propensity of workers to change jobs which is uncertain, and their lack of perfect sensitivity to wage differences which gives market power to the firms.



M. STEVENS 543

'trainability', which affects the cost of training them, and is represented by a continuously distributed parameter a.

4.2. Timing

There are two periods: a training period and a work period. At the start of the first period each worker and firm 0 make a joint decision on whether the worker should be trained, and if so, choose the amount of training and decide how to share the costs. At the start of the second period the worker enters the labour market; he then works either in the training firm, or for some other firm.

4.3. Training programmes

Firm 0 offers a type of training which is of potential positive value in n other firms in the economy; in all other firms its value is identically zero. So the post-training productivity of a worker is described by the vector

V = (VOw V1, V2, * * * 9 V119 09 09 ... *)

During training the values vi are uncertain, since the firms are subject to random independent and identically distributed productivity or demand shocks. So v is a random vector, whose actual value is realised at the start of the second period.5

Suppose that the expected value of the training is equal in all external firms, but may be higher in the training firm

E(v) = (m + a,m,m,m, . .I. ,mOO, ...) for some m > O, a > O

Let the independent and identically distributed shocks be represented by random variables ei, for i = O,. . . , n, with mean zero, support [- 1, 1] and continuous distribution and density functions F(.) and f(.). Then:

v = (m + a +80, M + 81, M + 82, .-- , M + 8119 0 .)

So, for a given distribution F, the training programme is represented by a set of parameters, {m, a, n}. Looking at the expected value vector, it would be natural to describe m as the level of a transferable element, a as the level of a specific element, and n as a parameter of transferability the higher it is, the more transferable is the training in the sense that there is a larger external market for the resultant skills. When n = 0 the programme is purely specific; when n is very large the transferable element becomes general, and only then might the programme be regarded as a combination of general and specific elements.

In this analysis, it will be assumed that the size of the external market, n, is

'The independence assumption is made for convenience; it is only necessary that the vi's are not perfectly correlated. In interpreting the variation between firms as arising from demand or productivity shocks, it can be noted that these firms are not necessarily operating in the same product market. For example, consider a training programme for the use of a particular computer package which is used by a variety of firms in one locality.




exogenous, given the type of training,6 but that the levels of training, m and a, are chosen optimally when training decisions are made. The cost of the training programme depends on the levels chosen, and the worker's trainability, and is represented by a function C(m, a; o), which is assumed to be increasing and convex in m and a, and to decrease with Lc.

The specific element, a, may include, in addition to specific skills, switching costs for both the worker and the firms. Some such costs may not be under the control of the agents, in which case they can be regarded as exogenous when the programme is chosen, but it is possible that the worker and firm will choose to increase a through some means not classified as training. For example, the firm might encourage the worker to take out a mortgage on favourable terms, thus increasing his future moving costs.

4.4. The labour market

At the start of the second period the worker enters the labour market, then works for one of the n + 1 firms. Although the external firms have identical distributions of value for the worker, the actual values vi will be different when labour market competition takes place, and this short-run heterogeneity gives the firms some market power. We will assume that the labour market operates as follows. The firms observe the worker's actual value, v. There are no information asymmetries-firm i observes not only the value, vi, of the worker to itself, but also his value to its competitors. Then each of the n + 1 firms simultaneously makes a wage offer, wi(i = O,. . . , n), and the worker chooses to work for the one offering the highest wage.

4.5. The training decision

Consider the decisions made in respect of a worker of type a. Figure 1 summarises the timing of events. Both worker and firm are assumed to be risk-neutral, and to make training desisions to optimise their joint net return. It they decide to undertake a training programme, they will bargain as to how the costs should be shared, and this will depend on their individual returns, but the training outcome will not be affected by the sharing of costs provided that neither party is credit-constrained. In the present paper, we will assume perfect capital markets (although the model could easily be adapted to allow for credit constraints) and so will not analyse the cost bargaining stage.

Clearly untrained workers work at a zero wage in both periods. If training occurs, the levels m and a will be chosen to maximise the joint net return on the investment

max Rp(m, a, n) - C(m, a; cx) m, a

6 It may in practice be possible to vary transferability: for example, inclusion in the training of tests leading to some form of certification could be a way of increasing n.



M. STEVENS 545

Choose Bargain Firms Worker levels over Make accepts

of sharing w hi hest training of cost offer ofer w

Ys . Training in firm 0 Work in firm j Yes/ wage = wj

Worker and Worker's firm 0 value v decide PERIOD 1 realized PERIOD 2 whether to train

No\

Work in firm 0 Work in some firm

wage=0 wage=0

FIG. 1. Decisions and timing

where Rp is the joint expected return in the second period and there is no discounting between periods. So training will be undertaken if and only if the optimal net return is positive. Note that when the training externality exists the private return Rp will differ from the social return, so private training decisions will not be socially optimal.

The analysis proceeds as follows: in Section 5 we evaluate the outcome of the labour market stage, given a particular realisation of v, and use this to calculate the expected returns to all parties for a training programme of given parameters. Then in Section 6 we examine how these returns vary with the parameters. Finally, in Section 7 we analyse the maximisation problem above, to evaluate the training investment decisions and the consequences of the training externality.

5. Calculation of expected returns

At the labour market stage, the n + 1 firms observe the realised value vector v of the worker and make simultaneous wage offers. Let v' and v2 be the order statistics representing the highest and second highest of the values vi. Since the firms know all the vi's before making their wage offers, this is a standard full information price game. The following two propositions hold for any continuously distributed random vector v they do not require the particular parameterisation chosen in the previous section.

Proposition 1. The outcome of the second-period labour market competition is that the worker works for the firm for which he has highest value, v', at a wage equal to the second highest value v2.




Proof. The following strategy for firm i(i = O,. . . , n) constitutes a Nash equilibrium of this game, and leads to an outcome with the above properties:

If vi is the highest value (vi = v') offer wi = v2 + e (e small and positive); otherwise Wi = Vi.

This is not the only equilibrium strategy, since when vi is less than the second highest value firm i can offer any wi < vi without affecting the outcome. But it can be shown, following the usual argument for asymmetric Bertrand competition and abstracting from some technical problems (see, for example, Tirole, 1989) that the equilibrium outcome is unique. ED

So, ex-post, the total value of the training programme is v', which is shared between the worker, who receives v2, and the firm that has the highest value, which receives v'-v2. Taking expectations we have:

Proposition 2. Any firm which has positive probability of having the highest value for the worker receives a positive share of the expected return to the training programme. The expected returns to each party are given by:

Total expected return: R = E[vl] Return to worker: RW=E [v2] Return to firm i: RI= E[vl-v2lvi = vl]Pr[vi = v']

Proposition 2 demonstrates the condition for the existence of a training externality in this model. If there is any probability that the training firm will lose the worker to one of its competitors in the labour market, part of the expected return to training accrues to alternative firms: the externality exists. If, on the other hand, the training firm will have the highest value with probability 1 (perhaps due to a large element of specificity) then it is known with certainty when the investment is undertaken that the worker will not leave, and there is no externality.7

To calculate expected returns for the programme with parameters {m, a, n}, let F(y) = F(y)'. This function is the distribution function of y = max{81, 82, . E}. The corresponding density is fn (y) = nf(y)F(y)n-. Then, using the algebraic manipulation given in the appendix, the expected returns can be shown to be (for n > 1)

r1 + a Total Expected Return: R = m + 1 + a-f F,(y)F(y -a) dy

1 +Fa

Return to Training Firm: Ro = Fn(y)( - F(y -a)) dy

'This is an example of the association between externalities and uncertainty identified by Rasmusen (1989) in a variety of other contexts. The common feature is that in the absence of uncertainty there is a critical investment level at which private incentives are aligned with social incentives, which is smoothed away if uncertainty is introduced.



M. STEVENS 547

Sum of Returns to External Firms: X = nF,1(y)(1 - F(y))F(y - a) dy

,al-

Return to worker: RW = m + { yfl,(y) dy - X

The probability that the worker moves to another firm in the second period is given by:

P ={a fnl(y)F(y-a) dy

and the joint private return to the worker and training firm is:

Rp Ro + Rv R -X

6. Dependence of the returns on the parameters of the training programme

6.1. The level of transferable training, m

The parameter m represents the expected value of the transferable part of the training. In this model it simply shifts the distributions of value vi upwards in all (n + 1) firms, and the following result holds:

Proposition 3. The marginal benefit of the transferable element of training is equal to one, accrues to the worker only (provided n > 1) and has no effect on the returns to any of the firms

OR _ RIV ORaO aX __ - AR =-1 ? = =0 am am am am

Hence the existence of the externality, represented by the term X, will not affect the choice of the level of m. To some extent, this result it an artefact of the model, since with many training programmes it may not be possible to vary m without affecting other characteristics of the training. For example, higher levels of transferable training may be associated with greater uncertainty, or with either a larger or smaller external market, both of which would affect the return to external firms. However, within the limitations of this model, it is clear that the level of m has no effect on the degree of competition between firms, and is not, therefore, related to the externality problem.

6.2. The specific element of training, a

An additional unit of specific training raises the value of the worker by one unit, if he remains in the training firm; otherwise its value is lost. This is the direct effect of specific training. But it also has an indirect effect, since the higher is the parameter a, the higher is the chance that, ex-post, the worker will have highest value in the training firm. Thus, the specific element also affects the




probability of movement. The following proposition summarises the impact of specific training on the social return, R.

Proposition 4. (i) The marginal benefit of specific training is equal to the probability that the worker remains in the training firm. (ii) The probability of the worker leaving decreases with the level of specific training. (iii) There are increasing returns to specific training.

Proof (i) Differentiating the total return, R, with respect to a gives

OR +a

a-= J_ Ft.(y)f(y-a) dy Oa a-

= f - An(y)F(y -a) dy (integrating by parts) ,a -1

= 1 - P (where P is the probability of movement, defined above)

= the probability of remaining in the training firm

(ii) -= - f(y)f(y-a) dy -<O Oa .aa-R

(iii) Follows immediately since a2 =-- ? E Oa 2 a

The first part of this proposition formalises the statement by Becker (p. 32) that there are external diseconomies associated with specific training part of its potential value may be lost due to turnover. The 'increasing returns' effect happens because an additional unit of specific training not only increases the direct value of the worker, but also increases the return to earlier units, because there is a lower probability that their value will be lost through turnover.

The results of Proposition 4 do not depend on the existence of an externality: it will be shown below that they hold even in the limiting case as n approaches infinity. But the specific element is also important in relation to the externality problem:

Proposition 5. (i) The total return to external firms, X, falls with the level of specific training. Hence the marginal private benefit of specific training is greater than the marginal social benefit. (ii) There are increasing private returns to specific training.

Proof. (i) Differentiating the expression for X gives

ax F' Aa -f Ja -'n(F(y))y-1 (- F(y))f(y - a) dy < 0 Oa a -l



M. STEVENS 549

and since Rp = R - X we have

8RP OR AX OR

aa aa Oa aa that is, the marginal private benefit is greater than the marginal social benefit.

(ii) It is proved in the appendix that a2Rp/aa2 _ 0. D

Next, we can ask how the private return to specific training is shared between the firm and the worker. Since they optimise their joint net return when choosing a training programme, this division does not affect the investment decision in the present model, but it is interesting to ask whether Becker's sharing hypothesis holds here.

Consider what happens if the worker has higher value in the training firm than elsewhere which is more likely if there is a large element of specific training. Then the training firm pays a wage equal to the best available elsewhere, and so does not appear to share any of the return to the specific part of the training with the worker. This is consistent with later work on the sharing hypothesis by Hashimoto (1981), who showed that further assumptions about asymmetry of information and transaction costs are required to justify the use of a long-term contract which results in sharing. Here, with the assumption of symmetric information, there are no inefficiencies requiring a long-term contract. The firm makes a single take-it-or-leave-it offer so is effectively assumed to possess all the bargaining power, and need not share the worker's specific value with him.

The following proposition follows immediately from differentiation of the expressions for the returns.

Proposition 6. The direct marginal benefit of specific training accrues to the firm: aR0/aa = AR/da. The worker benefits indirectly, from the reduction in return to external firms:

aRwaa = O-X/aa ) 0.

Thus, although in one sense the assumption that the firm sets the wage does result in non-sharing, the worker is not indifferent to the level of specific training when an externality exists.

Finally, consider the case when a is so large that the range of the distribution of value in the training firm does not overlap with the distribution in other firms: that is, a ) 2. Then, the probability of a move to another firm is zero and the marginal benefit of specific training is equal to one. Both the externality and the increasing returns effect disappear. In this case, the benefit of specific training accrues entirely to the training firm.

6.3. The size of the external market, n

The parameter n represents the transferability of the training the higher it is, the more potential employers there are who value the training. It might also




be regarded as an index of competition, since as n increases, we would expect the difference between wage and marginal product the margin between v1 and v2 to fall. In fact, this does not necessarily happen, but it is true for many well-behaved distributions.8 In what follows, n is initially treated as a continuous parameter, to obtain some comparative statics results.

Proposition 7. As the transferability, n, of the training increases: (i) the total return to the training programme increases: aR/dn > 0;

(ii) the return to any individual firm falls: aR0/an < 0 and a(X/n)/an < 0; (iii) the probability of the worker moving to another firm increases: aP/an > 0; (iv) the return to the worker increases (aR,/an ) 0) if the distribution F is

log-concave; (v) the marginal benefit of specific training falls: a2R/laan < 0.

Proof

(i) A J_ log(F (y)) F (y)F (y - a) dy > 0 On La-l (ii) =O log(F(y))F (y)(1 - F(y - a)) dy < 0

Firm i(i = 1, ... n) obtains expected return X/n and

a(X/n) _

an - a-i log(F(y))(F(y))n - 1(I-F(y))F(y - a) dy < 0

(iii) Integrating by parts, the moving probability, P, can be written

P = F(1 -a) - Fn(y)f(y - a) dy , a -1

and hence

an = X log(F(y)) Fn (y)f(y -a) dy > 0 On La - (iv) Proved in appendix.

aa2R J_ log(F (y)) Fn (y)f(y -a) dy < 0 O Oa On a -1 These results are intuitive: the greater the number of firms for which the training has value, the more socially valuable it is, and the more likely it is that some firm other than the training firm has highest value for the worker. Since the worker is more likely to leave, specific training is less

8It can be shown that, for a set of n i.i.d. random variables, a sufficient condition for the expectation of the difference between the first two order statistics to decrease with 71 is that the distribution be log-concave. This is a wide class which includes the normal, truncated normal, and uniform distributions. See Caplin and Nalebuff (1991) and Stevens (1993).



M. STEVENS 551

valuable. The return to any individual firm falls because of increased competition, and this does not require any condition on the distribution because even if the margin between wage and marginal product increases, the probability of that firm obtaining the worker falls, and the latter effect always dominates. The worker does not necessarily benefit from increased competition, but if the distribution is such that the difference between the wage and marginal product falls he will certainly do so.

Note that there is a divergence here (which does not occur with respect to the levels m and a) between the interests of the worker and training firm: the worker prefers training with a large external market, while the firm prefers less transferable training. This divergence does not matter when training is chosen to optimise the joint return and capital markets are perfect, but would affect the training chosen if, for example, there were asymmetries of information between the investing parties.

Although the return to individual firms falls with increasing transferability, it does not follow that the externality problem, represented by the total return to external firms, X, is necessarily alleviated, since there are a larger number of firms obtaining a share of the expected return. The following proposition shows that increasing competition can worsen the problem.

Proposition 8. The total return to external firms, X, achieves a maximum value for some number of firms N E [1, co). There is no upper bound on N: that is, for any No e [1, Ao) there exists some a such that the maximum of X occurs at N > No.

Proof Consider X as a function of n and a M I

X = { n(F(y))'-'(1 - F(y))F(y - a) dy

X is a continuous function of n on [0, so), zero at n = 0 and approaching zero as n tends to infinity. For all intermediate values of n, X is positive. Hence the maximum occurs at some finite positive N, which depends on the parameter a (and the shape of the distribution F).

ax _ r' an-= Ja -' (1 + n log F(y))F(y) n(I - F(y))F(y - a) dy An a -l

Consider a particular No. The term (1 + No log F(y)) is monotonically increasing on (-1, 1], positive when y is close to 1, and negative at low y. So, there exists z e(-1, 1) such that 1 + No log F(y) > 0 iffy > z. Let a - 1 = z. Then the integrand is positive throughout the range, so aX/an > 0 at No, and furthermore, aX/an > 0 for all n < No. Hence the maximum of X occurs at N > No. Since this is true for any finite No, there is no upper bound on the maximand NC

Figure 2 shows the function X for the uniform distribution. The return to




0.4 -

a=0 x xx

0.3 - X _ x

C x

, 0.2 _ a=0.3 x

0.0 x E 0 00

0 5 1 1 2 E0) 0 t

. a =1 Qo0 0 00 X X X X a: 0. 1 ~~~~+ + + + + + 0 000 0 0 +

+ + + + ~~+ + ++ ++0

0.0 a =21 1 I 0 5 10 15 20

Number of external firms, n

FIG. 2. Total return to external firms, X, as a function of firms, n, for the uniform distribution

external firms is highest when there is no specific element in the training, and the number of competing firms is 1 or 2. When the specific element, a, is positive, increasing competition initially increases X: for example, when a = 1, the maximum occurs when n = 4. But (as we know from proposition 5(i)) for given n, X is smaller at higher values of a and reduces to zero at a = 2.

6.4. Two special cases

6.4.1. Fully transferable training, a = 0. If the training contains no specific element, it might be described as 'fully transferable', in the sense that its value has identical distribution in the training firm and the n external firms. The terminology causes some problems here-it may not be 'highly transferable', and certainly not general, if n is small.

The point to be emphasised is that the training firm still obtains a positive expected return-the incentive for a firm to invest in training does not depend on there being some purely specific element.

6.4.2. The limiting case, n -+ oo. The case of a large number of firms might be thought of as approximating a competitive labour market. This is only an approximation: what happens in the limit is that the highest external wage offer will be (with certainty) at the upper limit of the range of values, and equal to the marginal product, m + 1. However, the heterogeneity of firms does not disappear in the limit, so this is not really a classical labour market.

When n is large, any uncertainty about the external labour market disappears: the worker will definitely leave if the wage is less than m + 1, and stay otherwise.

Proposition 9. In the limit, as n approaches infinity, the total return to external firms tends to zero. The return to the training programme is the sum of a



M. STEVENS 553

general component (the competitive wage) which accrues to the worker, and a specific component which accrues to the firm.

Proof.

X n(F(y))n-(1 (- F(y))F(y - a) dy

As n -+ oo, nF(y)n O V y < 1; hence X --O. Integrating the expression for the worker's return by parts gives

'1

RW=m+ 1 ffl(y)dy X

So lim RW =m+ 1 w n -oo

That is, the worker's return is exactly equal to the external wage and is unaffected by the specific element. He receives the same wage whether he stays in the training firm or not. The training firm's return is

(1+a

Ro = Fn(y)(l -F(y -a)) dy

('1 +a

limRo== (1-F(y-a))dy n -o 1

r +a (y 1)f(y -a) dy

= E[vo - w~vo > w]Pr[vo > w]

So, the return to the training firm is the expectation of the worker's additional value in the training firm, given that this is positive so that he will stay, multiplied by the probability that he will stay. The firm's return arises only from the specific element of the training; in particular, if a = 0, RW = 0. D

This result is not very surprising: the parameterisation of the model was chosen so that the expected value would have the form v, g in the limit, so could then be regarded as the sum of two components. But it is important to stress that it is only in the limiting case that the returns can be decomposed in this way: otherwise, part of the return accrues to other firms, the training firm obtains some return even if there is no specific element, and the worker's return is affected by the specific element.

Finally, note that Proposition 4 still holds in the limit as n tends to infinity since the marginal benefit of specific training is then

-= 1 - F(1 - a) = Pr[vo > w] = Probability that the worker stays Oa




and this increases with a, as for finite n. This means that, in the limit, the marginal benefit of specific training is zero at a = 0. The implications of this result will be discussed below.

7. The choice of a training programme

Now consider the choice of training programme for a worker of trainability oc. Suppose that, given the type of training programme, the transferability n is fixed, and there is some minimum level of specific human capital ao, perhaps due to unavoidable switching costs. The training firm and worker are able to choose the levels of transferable training m and additional specific human capital a. The first step in the decision is to find the optimal levels of training for this worker; then they can decide whether it is worthwhile to undertake such a programme.

7.1. Choice of training levels

The optimisation problem is

max Rp(m, a + ao, n) - C(m, a, n; oc) subject to a > 0 in, a

Since Rp = R - X the first order conditions for an interior private optimum are

a = _ aC

am am

OR OX AC

Oa Oa Oa

The social optimum satisfies the same conditions but without the term OX10a. Recall from Proposition 5 that there are increasing private and social returns to specific training (O'RpI@a2 >- 0 and 0Rla2 __ O) so that a unique solution is not guaranteed. An optimum will exist, since if a >, 2 (in which case there is no probability of the worker leaving) the increasing returns effect disappears: aRlaa = aRplaa = I if a > 2. But it may either occur at a = O. or have the property that a is so large as to prevent any labour turnover.

First consider the socially optimal levels (m*, a*). The following result is proved in the appendix but follows directly from the earlier comparative statics results.

Proposition 10. The socially optimal amount of specific training, a*, increases with exogenous switching costs, ao, and decreases with the size of the external market, n.

This proposition tells us, in particular, that when the conditions for a highly competitive labour market prevail-many firms and low exogenous switching costs-there should be little investment in specific human capital. And in such conditions the social optimum will be achieved since the externality disappears.



M. STEVENS 555

In fact, it is likely that the optimum level of specific investment will then be zero; as shown in Section 6.4 the marginal benefit of specific training is zero at a = 0. It is only when the external labour market is less competitive, so turnover is low, or when there is a need to reduce turnover because of switching costs, that investment in specific human capital becomes worthwhile.

Now consider how the existence of the externality affects the choice of a and m.

Proposition 11. If there is a unique solution to the first order conditions for the socially optimal (m*, a*), then (for finite n) the private choice (mp, ap) will be such that ap > a*. Thus, there will be over-investment in specific training. mp may be greater or less than m*.

Proof If the solution for a* is unique, then

V (a, m) such that a < a* and 8C/8m = 1 we must have - > 0 Oa Oa

OR 8C OX But (mp, ap) satisfies 8C/8m = 1 and - -- = _ --

Oa Oa Oa

Hence ap >? a*. From the condition 8C/8m = 1 it is clear that mp is either greater than or less than m* according to the sign of 82C/8a Om. E

It was shown (Proposition 8) that the total return to external firms is greatest for some intermediate value of n. Following an identical argument it can be shown that magnitude of the distortionary term 8X/8a is also greatest for intermediate n, so that it is possible for increasing competition to worsen the over-investment in specific training, although it disappears as n becomes very large.

Combining the results of Propositions 10 and 11 suggests that privately- chosen training programmes will tend to fall into one of two categories. If the exogenous conditions favour a highly competitive labour market, the training programme will be chosen optimally and will include little or no specific training. Thus, a training programme with a substantial perfectly general component will include little specific training. But in the presence of non- competitive influences, such as switching costs or training of a specialised type, the training programme will also include substantial amounts of specific human capital leading to a further reduction in turnover and competition in the labour market. In particular, when the training is of a type which is of use to a small number of employers, it will be accompanied by specific human capital (which may or may not be training) in order to reduce turnover and protect the investment in transferable training. In such a case the parties to the training investment could be said to be erecting socially costly mobility barriers.

7.2. Decision whether to train

For a worker of type oc, training will be undertaken if, at the levels of training which are privately optimal for this worker, the net private return is positive.




This decision will not be socially optimal, and it is intuitively obvious that the following proposition holds.

Proposition 12. If the training programme is transferable, but not a combination of general and specific (that is, if n is finite), too few workers will be trained.

Proof. Consider the net social return to training, as a function of trainability of

T(cx, n) = R(m*, a*, n) - C(m*, a*; o)

Since m* and a* are chosen optimally for a worker of type or, and since 8C/8cx < 0, the envelope theorem implies that T is a strictly increasing function of of. Hence, training will be socially optimal for any worker with o > oc*, where T(cx*, n) = 0. Similarly the net private return Tp(o, n) is strictly increasing in o, and training will be privately optimal if oc > op where T(ocp, n) = 0.

But Tp(>, n) = R(mp, ap, n) - C(mp, ap; o) - X(ap, n) < T(cx, n)

- X(ap, n) < T(cx, n)

Since both functions are continuous, this proves that os > oX*, and hence that the number of workers trained will be too low. D

Note that, since X may initially increase with n, the problem of too few workers trained may (like the problem of over-investment in specific training) be most serious for intermediate values of transferability, n.

To summarise the results of Section 7, it has been shown that, when training is transferable, but is not a combination of general and specific, the number of workers trained will be too small, and that those who are trained will receive training which has too great a specific element, resulting in a skilled labour market characterised by low turnover. On the other hand, if the transferable element of training is truly general, socially optimal training decisions will be made, training programmes will not include much specific training (unless it happens that exogenous switching costs are also high) and the corresponding labour market will be characterised by high turnover. Since the distortions are most severe for training programmes of intermediate transferability, we would expect in practice to observe training programmes of either low transferability, with a substantial specific component, or very general programmes with little specific training.

There is an interesting link between these ideas and the theory of internal labour markets (ILMs) (Doeringer and Piore, 1971) which relies upon the assumption of a high degree of specificity in training. An ILM is one in which firms recruit and train unskilled workers, higher level jobs are filled by internal promotion, and mobility of workers above entry level is low. The present analysis suggests that the endogenous choices of workers and firms may lead to the development of labour markets characterised by low transferability of training, mobility barriers and low turnover. According to Marsden and Ryan (1990) occupational labour markets (OLMs), characterised by general training and high mobility, 'are unstable, and tend to degenerate into internal markets'.



M. STEVENS 557

Some support is offered here for that view, in the sense that the requirements for an OLM are stringent (perfectly general training, financed by the worker, and low switching costs) and unless all these are fulfilled, private choices will lead to the development of ILMs.

8. Conclusions

The most important result of this paper is the simplest: that not all transferable skills are general, and for some types of on-the-job training for transferable skills, firms-both the training firm and external firms can obtain a positive share of the return to the training investment. This may explain why firms have been found to invest in transferable training. It also means that there is an externality of the type sometimes referred to as a poaching externality- associated with some types of on-the-job training programmes, which may lead to under-investment in training.

It has been shown that the classical analysis using a combination of general and specific cannot describe all types of training-for example, training for skills which are of value to a small number of firms. The term transferable has been used to designate training which is of some use to at least one firm in addition to the training firm. Skills for which there is a larger external market can then be described as 'more transferable' than those which are valued by only two or three employers. Specific skills are those for which there is no external market, and general skills correspond to the other limiting case in which the external market is very large.

The size of the external market is important insofar as it affects the degree of competition between employers. It is imperfect competition between employers in the labour market on which the existence of an externality depends: if there is a possibility that the worker will move after training to a a firm which can pay him less than his marginal product, that firm obtains a positive share of the expected return to training. Furthermore, it was argued in Section 2 that this is not merely a theoretical curiousity. Rather, there is a natural link between training and imperfect competition, in that the acquisition of skills, and skill requirements, differentiate workers and firms: training can be a competition-reducing process.

The relationship of the externality problem to transferability is illustrated by Figure 2, in which the horizontal axis represents the number of firms in the external market for some type of training. This can be regarded as an index of transferability, or of the degree of competition. When this index is zero, the training is purely specific, and there is no externality. As the index increases, the externality problem, represented by the total return to external firms, initially increases, reaching a maximum at some intermediate level of competition, then falling towards zero again as the labour market approaches the limiting case of perfect competition, in which case the training is either perfectly general, or a combination of general and specific.

The arguments presented in this paper do not conflict strongly with those




made by Becker, except to the extent that he implied that all types of training were covered by his analysis in terms of general and specific. Becker did not show, or claim (although it is frequently asserted that he did) that a poaching externality did not exist. In fact, he stated clearly that his conclusions with respect to the non-existence of a poaching externality applied to a competitive labour market. Nor did he deny the possibility that some training would take place in the presence of imperfect competition between firms he conjectured that such training would be more like specific, less like general training. In this paper, his analysis has been extended by focusing directly on imperfect competition, and, in one sense his conjecture has been shown to be correct. For if we consider training which is transferable to a small number of other firms and has no specific element, the training firm is able to capture more of the return than it would if the training were completely general, and less than it would if the training were completely specific. But it has been shown that in another sense the conjecture was misleading: training in the presence of imperfect competition differs from both the limiting cases of monopsony and perfect competition, in that part of the return accrues to alternative firms.

In the second part of this paper a model has been developed in order to explore the properties of transferable training, and the consequences of the externality in particular. Analysis of imperfect competition between firms, which has been more common in the industrial economics literature than in labour economics, frequently requires some apparently drastic simplifying assumptions, and this model is no exception. Despite these, the predictions of the model accord well with intuition. It demonstrates how, as the size of the external market for some type of training increases, firms (individually) capture less of the return to training and the worker captures more. It illustrates the relationship between labour turnover, the level of purely specific training, and the size of the external market, and shows how this relationship may distort private training choices. Other authors have hypothesised that there is an incentive to over-invest in specific training (for example, Hyman, 1992): this model demonstrates it explicitly. Furthermore, it shows that the training firm benefits, at the expense of the worker, from reducing the transferability of training; this result could be used to explore the idea that firms may choose to differentiate their skill requirements in order to obtain market power in the labour market.

The effect of the externality is that too few workers receive training which is valued by small numbers of employers. One important prediction is that it is not necessarily the case that increasing competition reduces distortions. It can happen that the increased probability of turnover dominates the reduction in market power and worsens the problem.

If firms and workers are unable to undertake the socially optimal training programme because of the externality, they may choose instead either training which is completely general (or at least highly transferable) or training of low transferability. If training of low transferability is chosen it may not be completely specific, but if not, there is an incentive to include a high specific



M. STEVENS 559

element, or some other kind of mobility barrier, in order to protect the investment from appropriation by rival employers.

ACKNOWLEDGEMENTS

An earlier version of this paper was presented at the Oxford Economic Papers Conference on Vocational Training, and at the Labour Market Imperfections Group seminar at Birkbeck College. The paper has benefited greatly from the comments of the participants, and from those of two anonymous referees. Thanks are also due to my supervisor, Meg Meyer, for her help and advice.

REFERENCES

BECKER, G. (1964, 1975). Human Capital, Columbia University Press, New York. CAPLIN, A. and NALEBUFF, B. (1991). 'Aggregation and Social Choice: A Mean Voter Theorem',

Econometrica, 59, 1-23. CHAPMAN, P. G. (1991). 'Institutional Aspects of Youth Employment and Training Policy in Britain:

A Comment', British Journal of Industrial Relations, 29, 491-5. DOERINGER, P. and PIORE, M. (1971). Internal Labour Markets and Manpower Analysis, Lexington. FINEGOLD, D. and SOSKICE, D. (1988). 'The Failure of Training in Britain: Analysis and

Prescription', Oxford Review of Economic Policy, 4, 21-53. HASHIMOTO, M. (1981). 'Firm Specific Human Capital as a Shared Investment', American Economic

Review, 71, 475-82. HYMAN, J. (1992). Training at Work, Routledge, London and New York. JONES, I. (1988). 'An Evaluation of YTS', Oxford Review of Economic Policy, 4, 54-71. LEES, D. and CHIPLIN, B. (1970). 'The Economics of Industrial Training', Lloyds Bank Review, 96,

29-41. MARSDEN, D. and RYAN, P. (1990). 'Institutional Aspects of Youth Employment and Training

Policy in Britain', British Journal of Industrial Relations, 28, 351-69. OATEY, M. (1970). 'The Economics of Training with the Firm', British Journal of Industrial

Relations, 8, 1-21. RASMUSEN, E. (1989). Games and Information, Blackwell, Oxford, 172-3. RITZEN, J. (1991). 'Market Failure for General Training, and Remedies', in D. Stern and J. Ritzen

(eds), Market Failure in Training, Springer-Verlag, Berlin Heidelberg. SHACKLETON, J. R. (1992). 'Training Too Much? A Sceptical Look at the Economics of Skill

Provision in the UK', Hobart Paper 118, Institute of Economic Affairs, London. STERN, D. and RITZEN, J. (1991). Market Failure in Training, Springer-Verlag, Berlin Heidelberg. STEVENS, M. (1993). 'Some Issues in the Economics of Training', University of Oxford D. Phil thesis. STEVENS, M. (1994). 'Transferable Training and Poaching Externalities', in A. Booth and D. Snower

(eds). Acquiring Skills. Market Failures, their Symptoms, and Policy Responses, Cambridge University Press.

TIROLE, J. (1989). Thie Theory of Industrial Organization, The MIT Press, Cambridge, MA, 211.

APPENDIX

An outline of proofs is provided here. Further details of the algebraic manipulation can be found in Stevens (1993).

1. Derivation of expressions for returns to training in Section 5

Let u' and u2 be the first two order statistics for the values in the external firms only (v1,. . ., v,,). Then the expressions for the returns obtained in Proposition 2 can be written:

Total return, R = E[max(vo, ul)] Return to worker, R., = Ro0, + Row, the sum of the returns received in the training firm and




elsewhere, where:

Ro,. = E[u'lvo > u1]lPr[vo > u']

Return to training firm, Ro = E[voIvo > u1]Pr[v0 > u'] -Ro

RlW = E[u2lu2 > vo]Pr[u2 > vol + E[volul > Vo > u2]Pr[u' > vo > u2]

Sum of returns to external firms, X = E[u1lu' > vo]Pr[u' > v0] -RI

The density functions for (vo-m-a), (ul-m) and (u2-m) are f(.), fn(.) = nf(.)F(.)n- and JG(.) = n(n - 1)F(.)" -2f(.)(1 - F(.)) respectively. Hence, evaluating these expressions

R0= J= (m + Y)fn(Y)f(X) = { (m + y)fn(y)(l - F(y - a)) dy a+x>y

r o r 1 ~~~~~~+a Ro + Row = {f (m + a + x)fn(y)f(x) dx dy = (m + y)F.(y)f(y - a) dy

a+x>y

Ro = (Ro + Ro,) -Ro

=f (inl + y) d-{-Fn(y)(1-F(y-a))} dy T i Fn(y)(l F(y-a)) dy uy - 1

R.= j (m + y)f (y)f(x)dxdy+ (mn+ a + x)f(x)fn (y)nf(z)dxdydz a+x<y y<a+x<z

= f (m + y)f.(y)F(y - a) dy-f n(l - F(y))F(y - a)F(y) n-I dy

X + Rl,,. = (in + y)f.(y)f(x) dx dy = f (m + y)fn(y)F(y - a) dy a x<y

I X = (X + RI,,) -Ri = { n(l - F(y))F(y- a)F(y)n- I dy

RW = Ro^ + (X + Rw) -X = m + yfn(y) dy- X

R = (Ro + Roe,) + (X + RoM,)

fb+a d rt+a = - (m + Y) - {Fn(y)F(y - a)} dy = in + 1 +. a- Fn(y)F(y - a) dy dy

Finally, the probability that the worker moves is

P = Pr[u1 > vol = fj'n (y)f(x) dx dy fn(y)F(y-a) dy aa x a+x<y



M. STEVENS 561

2. Proof of Proposition 5(ii) 2P a

ax 0 _x =

1 - I n( -F(y))f(y-a)F(y)

n dy aa

Integrating by parts gives

ax 0 Oa = F(y -a){f,2(y) - fJ(y)} dy eaa a-i

2aa L T f(y a){f"(y) - f2(y)} dy

We know from proposition 4(i) that Oa2 = fly- a)f-(y) dy

Hence, =0 R

_ -2 =2X fly -a)f.2(y) dy >O a2 a2 aa2 ,)

3. Proof of Proposition 7(iv) aRW ?0 if F is log-concave On

From the expression for R.,, integrating by parts gives

1 (

a(I - F(y))F(y - a)T R,= m + 1-J F.(y)(l L - j)dy

OR= L log(F(y ))FI(y)1 a - F(y))F(y - a)]) Oil ay ~~~~~~~~fly)

It can be shown (Stevens, 1993) that for log-concave distributions

-[( - F(y))F(y - a)] < 1 Vy, from which the result follows immediately.

En' f(y)

da* da* 4. Proof of Proposition 10 > 0,- < 0

dao dn

Suppressing the other arguments we can write the first order condition for a*

(ao + a*) = (a*) Oa Oa




Differentiating with respect to ao gives:

[ da* 02 R da* 02C 1+ - ____

dao ja2 da0 Oa2

da* [2C a2 R] 2R

dao LOa2 Oa2 j a2

Since the term in square brackets must be positive at a*, and 02R/8a2 > 0, da*/dao 0. Differentiating the first order condition with respect to n

,2R da* d2R da* 02C +=

On Oa dn Oa2 dn Oa2

and since 02R/anaa < 0 (Proposition 7), da*/dn < 0



A Theoretical Model of On-the-Job Training with Imperfect Competition

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