In Production and Learning in Teams, Herkenhoff, Lise, Menzio and Phillips build a very sophisticated labor-search model with on-the-job learning. While the model is motivated by a single empirical fact, it is extremely flexible, allowing for many potential causal mechanisms and capturing a panoply of equilibrium outcomes in the labor market.
The motivating empirical fact is the following: When workers go through a spell of unemployment before finding a new job, a 10% increase in the coworkers’ average wage in the first job forecasts a 1.5% higher wage in the second job only for workers who are paid less than their coworkers in the first job. There is no such effect for workers who are paid more than their coworkers in the first job.
What could be happening here? Due to search frictions, a worker’s wage often lags behind her true human capital. Workers may be able to leverage an offer by an outside firm to negotiate a wage increase, but this does not happen every day and so wages are “sticky.” Therefore, on the day you leave your first job, you are likely to have a higher level of human capital than your wage indicates. The empirical fact suggests that human capital might be higher for those whose first job involved interacting with more skilled coworkers. It also suggests that human capital is not depreciated by colleagues with lower skill. In essence, workers learn from more skilled colleagues, but they do not “unlearn” from those with less skill.
An employment-to-employment transition suffers from the possibility that workers are leveraging connections at their current job in order to vault into a better paying position in the next job. That’s why the authors restrict their attention to employment-unemployment-employment transitions. They want to write a model where the starting wage reflects underlying human capital, rather than political or business connections, so they must restrict their attention to new jobs that are preceded directly by a spell of unemployment.
The model features a continuum of firms and a continuum of workers. The workers are indexed by levels of human capital, 1-7 with 7 being the highest. Firms can employ one or two workers. The production function is designed so that the output of a firm with two workers always exceeds the combined outputs of two one-person firms employing workers with the same level of human capital. This is logical : otherwise firms would never employ multiple workers. Depending on the parameterization, the production function can either be supermodular or submodular. Supermodularity means that, for firms employing workers of human capital x and y, f(x,x) + f(y,y) > 2*f(x,y) if x ≠ y.
There are four value functions, one for an unemployed worker, and three for workers in various states of employment. The value functions take into account the current value of a worker’s production as well as the probabilities of transitioning to different states in the next period. As with most search models, there is no curvature in the utility function, so consumption = output = utility. Like many search models, a key variable is the “surplus,” which captures the combined value to a worker, a coworker, and a firm of employment. Wages are just a constant fraction of this surplus.
The model is able to capture a number of important features of the labor market:
· Loss of skills: human capital stays the same or depreciates with some exogenous probability during a spell of unemployment.
· Learning by doing: human capital stays the same or appreciates with some exogenous probability even in the absence of having a coworker.
· Learning from coworkers: the probability of increasing your human capital depends positively on the difference between your coworker’s human capital and your own. The model is flexible enough to allow for non-linear rates of learning. You may learn faster in the presence of a more skilled co-worker than you would “unlearn” in the presence of a less-skilled coworker.
· Divergent values for leisure: the value of home production (what you can do with your free time) depends on the level of human capital.
· Poaching of workers and negotiated salary increases.
Features of the equilibrium:
· Firms slowly upgrade their workforce through letting on-the-job learning take place or by replacing current employees with new ones.
· Firms lose employees for a variety of reasons: poaching by other firms, optimally choosing to replace a current employee with a new one, and exogenous transitions by the worker into unemployment or exiting the labor force altogether.
· Due to search frictions the human capital of a worker may increase without the wage immediately reflecting that. The wage of a worker only reflects his human capital at the time of an initial hire or the last time he received a sufficiently attractive offer from a poaching firm.
· After a spell of unemployment, the wage reflects the underlying human capital.
· The production function is supermodular: firms like to match workers with similar levels of human capital. This results in positive assortative matching, which is inefficient relative to the social outcome. There is too little opportunity for workers to learn from each other when there is positive assortative matching, so the aggregate human capital accumulation in the economy is too low, and total output is too low.
The process by which the authors calibrate the model parameters is discussed in great detail. Some of the calibrations, such as targeting an average of 35 years in the labor force, are standard in the search-theoretic literature. Interesting moments they target in order to parameterize the learning process include: the relation of between-firm to within-firm wage variance (which reflects equilibrium sorting of workers in firms according to human capital), the rate at which workers take a new job if they are employed with more-skilled coworkers vs less-skilled coworkers (which reflects the potential for on-the-job learning from coworkers depending on their skill level), and life cycle wage growth (which reflects the underlying human capital accumulation).
A key question to ask any structural model is the extent to which the model’s predictions are “baked in” as a result of its assumptions. The model developed by Herkenhoff, Lise, Menzio and Phillips allows for a great deal of flexibility. Learning from colleagues can be non-linear. There can also be no learning from colleagues. The model allows for the existence or lack thereof of learning-by-doing. While a parametric form for the production function must be assumed, the production process can either be super- or sub-modular. The production function is explicitly designed for the “knowledge economy” in that it depends only on human capital.
It would be interesting to see if a production function that included capital and allowed for some complementarities between capital and workers of various skills would generate the same conclusions. The assumption that firms employ no more than two workers does not seem overly restrictive. Most models that feature learning in teams still have all team members learning from only the most skilled person on the team (see Akcigit: Dancing with the Stars).
Some of the calibrated parameters don’t look that great to me. A discount rate of 15% a year is quite high. Many labor-search models suffer from this because of the lack of curvature in the utility function. Also, some of the moments that authors target do not seem to make a great fit. See below and judge for yourself.
The model does, however, make a couple of curious predictions and I suspect that they may be generated by an assumption about the number of human capital types. The model predicts that firms currently employing two very high-skilled workers will fire the more highly-skilled of the two and replace him with a medium-skilled worker. The model also predicts that if a firm encounters a highly skilled unemployed worker, it will fire its highest-skilled employee in order to hire the unemployed worker. The authors provide the following explanation: “The firm finds this optimal because the worker can teach more to the worst employee than to the best employee.” While this is true, I wonder if the results would still hold if there were no upper bound on human capital. In the model, a firm employing two workers with human capital equal to 7 will not experience any growth. These workers will neither be able to learn from each other nor learn by doing, because the model constrains human capital to top out at 7. If the model were changed to allow for 100 values for human capital rather than just 7, a firm currently employing two type-7 workers would have two additional options that are not available to it in the existing model: it could either hire a worker with human capital greater than 7 if it encounters one, or it could let its current employees “learn by doing.” Therefore, the firm might not want to fire the type-7 worker in order to hire a type 4-worker as the current model predicts. In real life, the potential for human capital is unbounded, so you would not want the boundedness assumption to be driving some of the model’s predictions.
Nevertheless, the paper is extremely impressive. The model is able to generate lots of interesting life-cycle patterns in wages as well as cross-sectional distributions in skills and assortative matching such as the one below:
There are a lot of fascinating potential extensions of the model, some of which the authors allude to in their conclusion. I have an additional interesting thought that they do not mention: How would equilibrium outcomes differ if workers had heterogenous bargaining power in the wage determination process? In the United States, some workers can negotiate a higher salary based on their unique and hard-to-replace skills. I suspect that allowing for heterogenous bargaining power among workers would reduce the value to a firm of employing a high-skilled worker and therefore would result in more negative assorting matching relative to the current model.