A note on the name 'O-ring'. The O-ring production function was introduced by Kremer (1993). The name comes from the fact that it was an O-ring failure that caused the space shuttle Challenger disaster. The basic idea is that the failure of a small component can have large adverse consequences. Here one part of the production process failing causes the whole process to fail.
Rauh assumes a production process that can be divided into a number of distinct tasks. This makes it possible for the tasks to be allocated across workers (the division of labour) and for workers to make investments in task-specific human capital (specialisation). This is the kind of situation just discussed in the Becker and Murphy paper. We saw that an increase in employment gave rise to a greater division of labour, that is, fewer tasks assigned to each worker, and greater specialisation and thus higher productivity. Importantly Rauh postulates an additional feature of the production process: a breakdown at any point in production, which could be due to shirking, poor decision-making or a negative shock, will have serious adverse consequences for the successful manufacturing of the product--this is the 'O-ring' type production function.
This second condition has important implications for the moral hazard problems that arise within a firm. In the first best case, the principal can directly monitor individual worker effort and thus will be able to identify and respond to any shirking by workers with probability one. In the second best case, individual output can be monitored and again shirking can be punished with probability one. Note that in this case a worker who experiences a negative shock will also be punished. In the third best case all workers will be punished, with probability one, if any single worker shirks. In each of the three cases there will be no free rider issues since shirkers cannot hide behind the efforts of their co-workers.
Rauh considers a production process where the set of tasks is the unit interval. The principal chooses the number of workers, and the set of tasks to be performed is divided equally across all workers. Each of the workers is able to choose their production effort and their level of investment in task-specific human capital for each task they are assigned. To produce one unit of output requires one unit of output of each task. This means that you get zero output if any of the workers shirks or suffers an adverse shock in any of their assigned tasks. In line with Becker and Murphy (1992) greater levels of employment implies fewer tasks being assigned to each worker, which in turn means the workers can increase their investments in human capital for each of their reduced set of assigned tasks. This results in greater productivity and thus increasing returns to employment.
The stochastic (O-ring) nature of the production function is thought about in the following way.
"In addition to production effort and investments in human capital, each agent monitors his assigned tasks and makes decisions about whether or not a problem has arisen, whether or not to halt production to fix it, whether he can fix it himself, and which potential solution is appropriate. When there is only one agent, there is a high probability that at least some of these decisions will be faulty because he has limited cognitive resources and performs all the tasks himself. When there are two agents, the probability that either one will make a mistake should be lower because each performs only half the set of tasks and can therefore devote more care and attention to each of them. On the other hand, we now have two probabilities instead of one, so the effect of an increase in employment is ambiguous" (Rauh forthcoming: 2).More formally, the probability that a worker suffers a negative shock to at least one of the tasks they have been allocated is an increasing function of the proportion of tasks being performed by that worker. Under an assumption of independence, the probability of a product defect is the product of the individual probabilities. If the number of workers is increased this results in two effects. First, it will decrease the probability that each worker will suffer a negative shock. Secondly, it will increase the number of points in the production process at which a negative shock can occur. Rauh then defines a production process as satisfying the O-ring property if the probability of a defect occurring is increasing in the number of workers and converges to one as the number of workers goes to infinity.
Given this background, the main question for the paper is then considered: What limits the size of a firm? For Rauh the answer has to do with the effects (or lack of effects) of moral hazard. Since there is a one-to-one relationship between the division of labour and the level of employment in the paper, the question can be rephrased as, What limits the division of labour? As has been noted above Becker and Murphy (1992) see this limit as be determined not by the extent of the market, as Adam Smith argued, but rather by coordination costs, including agency costs.
When determining the relationship between moral hazard and the size of the firm, "[ ... ] the optimal employment level balances the following considerations: (i) the increasing returns to employment due to specialization and division of labor, (ii) the O-ring property of the production technology, where the probability of team failure increases with the size of the team, and (iii) the marginal cost of employment (the cost of hiring another agent)" (Rauh forthcoming: 2).
In the first best case of no moral hazard Rauh shows that the standard zero incentive, full insurance contract is employed. Effectively the firm is behaving as if it were a perfectly competitive wage-taker despite it being a monopolist. Since, in this case, each worker's payment is fixed, the firm's labour costs (the number of workers times the expected payment to each worker) are linear in workers and the marginal cost of a worker is constant. Importantly, however, given increasing returns to employment, which arises from specialisation and the division of labour, but only linearly increasing costs to employment, these costs cannot limit the extent of employment. Thus, in this case, the extent of the market for labour or the O-ring property must be limiting employment and thus the size of the firm. If it wasn't for these constraints the first best firm would be of infinite size since there are increasing returns to employment.
Next Rauh considers the second best contract. Here effort cannot be observed but individual output can. Rauh shows that the optimal (second best) contract involves awarding a bonus to a worker when their individual output is high, i.e., when the worker's effort is first best and there is a positive shock, and replacing the worker otherwise. Rauh shows that the worker’s bonus is decreasing in employment. This follows from the fact that as employment increases the proportion of tasks carried out by each worker falls which increases the likelihood of a positive shock. This increases the expected value of the worker’s payment if the worker selects the first best effect level. This means the principal can reduce the bonus paid to the worker. It is also shown that this reduction in the bonus reduces the expected payment to the worker and this implies that the payment is decreasing in employment as well. If this type of effect is large enough then the marginal cost of an extra worker can decline with employment and could even be negative. In this situation the second best cost of employment could be less than the first best (constant) marginal cost of employment. This would mean the second best firm could be larger than the first best firm. Thus, the second best firm would have weak incentives (low bonus), low expected pay (small worker payment) and an excessive division of labour (and an excessive amount of specialisation). Motivation is provided by the fact that shirking workers will be identified and fired, rather than through the use of incentive schemes. As before, as the level of employment increases fewer tasks are carried out by each worker and the probability of a positive shock converges to one. This means that the second best expected payment to a worker converges to the first best payment. In turn, this means that the second best cost function tends towards the (linear) first best cost function. Thus as with the first best case the increasing returns resulting to employment resulting from the division of labour and specialisation cannot be contained by an asymptotically linear cost of employment. Rauh concludes from this that when the principal can monitor individual output, even if not effort, the size of the firm under moral hazard is again limited by either the total number of workers available or the O-ring property .
Lastly, Rauh looks at the third best situation where the where the principal can observe only team output. Here the results are the opposite of the second best case. This is because the third best incentive relies on the probability that all workers experience a positive shock rather than depending on the probabilities that individual workers experience a positive shock. Given the O-ring property, an increase in workers increases the probability that an individual worker experiences a positive shock but reduces the probability that all workers experience a positive shock. In this case increasing the number of workers decreases the team probability of success and this decreases the expected payments made to workers when they put in the first best level of effort. This means that the principal will increase the third best bonus, which increases the third best expected payment to workers and the marginal cost of employment. From this it is clear that all of the third best bonus, expected payments and the marginal cost of a worker are increasing in the number of workers. This is the opposite of the second best case above.
As the number of workers employed continues to increase, the third best bonus, expected payments and the marginal cost of employment all explode. This is contrary to the second best case where all these variables tended to their first best levels. The third best marginal cost of employment is shown to always exceeds the first and second best marginal costs of employment. This means the third best firm is usually smaller than either the first or second best firms.
Thus for Rauh's model moral hazard concerns only limit the division of labour, and the size of the firm, when the principal can monitor just the output of the whole team. When either worker's effort or individual output can be observed either the extent of the labour market or the O-ring property limit the extent of the division of labour or the size of the firm.
Rauh's paper is interesting in part because it combines, in some ways, the older division of labour approach to the firm with the more modern principal agent approach to the firm. The more modern mainstream approaches to the firm don't emphasise the division of labour with their emphasis being more on incomplete contracts and agency problems. The division of labour approach has largely fallen out of favour.
Well worth a read if you are into the theory of the firm.
- Becker, Gary S. and Kevin M. Murphy (1992). 'The Division of Labor, Coordination Costs, and Knowledge', Quarterly Journal of Economics, 107 (4) November: 1137-60.
- Kremer, M. (1993). 'The O-ring theory of economic development', Quarterly Journal of Economics, 108(3) August: 551-75.
- Rauh, Michael T. (forthcoming). 'The O-ring theory of the firm', Journal of Economics & Management Strategy.