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Jan Eeckhout and Xi Weng provided a model that predicts the matching pattern of the firms and the workers as well as various wage variables such as the change of wage at turnover and the distribution of wages.In their model, both firms and workers are heterogeneous and the productivity of a firm-worker pairis high in association with the worker’s skill, which is initially not known. Depending on the characteristics of the firms, workerslearn at different rates about productivity and the knowledge of discovering new wage levels or what they deserve depending on what they produce. The Bayesian learning model is used toupdate the belief of unknown skill in productivity, andlearning might be faster in low-type firms.

The aim of this paper is to achieve a model that solves the matching of the workers together with their respective spot wages depending on the belief of skill in productivity. The equilibrium achieved by the model turns out to be dynamically efficient. This summary is based onprofessor Xi Weng and his co-author JanEeckhout’s paper,Assortative Learning, published in Economica(2022).

BACKGROUND

Heterogeneity and sorting seemed to lack in past models that looked at the effects of turnover on wages patterns and productivity. Therefore, this paper looks athow skill sorting can be done when there exists a trade-off between wages and learning speeds. For example, working in a top law firm will expose the individual to more information than working in a local family business. However, the worker sometimes faces a trade-off as they have to choose to either work at a lower wage in a more productive firm and have a chance to learn more about techniques of production, or work in a low productive firm at a higher wage and learn more slowly. The worker, therefore, can choose to sacrifice wages for dynamic gains from learning or vice versa. This paper differs substantially from previous research by showing that, under super-modularity, positive assortative matching always obtains in a market equilibrium with several learning opportunities.

RESEARCH DESIGN AND THE MODEL

The population of firms and workers

Both workers and firms are heterogeneous. The firm type  represents its productivity, which is observable to all players in the economy. The fraction of H-type firms is fixed. And all firms are infinitely lived. The worker ability is not visible both to the firms and the workers. Only a belief is held by both workers and the firms denoted by p∈(0, 1). The spot wages vary with the belief, i.e., the worker with the highest belief is paid highest and vice versa. When a particular worker is born, the worker is type H with probability  or type L with probability (1 -  The workers die with an exogenous probability, and new workers are born at the same rate. Information is symmetric because the worker's ability is not exclusively known, but the firm's wage patterns are unpredictable as the change of the belief of the worker's ability is random. Strict supermodularity is assumed throughout the paper ( > −  ). Consider a situation where the learning is closed so that the belief p∈ (0, 1) is invariant and hence the problem is essentially static. Based on standard results, it is claimed that positive assortative matching is the equilibrium allocation under strict supermodularity. In equilibrium, the H-type firms will always match with workers with higher beliefs p. Interestingly, this result also holds when there is learning and the firm’s level of learning speeds does not matter. The impact of learning rates is compensated by the changing wage rates, which are solely determined at the turnover.

Dynamic analysis

The analysis of the learning case involves dynamic programming. The workers rely on their equilibrium value functions when choosing which firm to work with. There are several technical properties about the functions. For example, it is shown thatthe equilibrium value functions are strictly convex and strictly increasing for p∈ (0, 1). The key claim states that under strict supermodularity, it is impossible to have p1 < p2 and equilibrium value functions WL (for p∈ [p1, p2]), WH 1 (for p < p1), WH 2 (for p > p2) such that the equilibrium boundary conditions are satisfied simultaneously. This result rules out complicated matching patterns which are not assortative.

OTHER RESEARCH FINDINGS

Under Bayesian learning, the belief updating follows a martingale process. This turns out to be a crucial assumption. Under the non-Bayesian learning process, assortative matching may not hold even if the strict supermodularity condition is satisfied. In a situation where the learning speed is sufficiently higher in low productivity firms, workers whose beliefs are close to 0 or 1 will get a match with a high productivity firm. In contrast, those workers whose belief is intermediate will get a match with a low productivity firm. One such example is the human capital accumulation. The above result implies that when the accumulation is sufficiently faster in low productivity firms, workers with extremely high or low human capital will probably stay in high productivity firms, enjoying the high pay but having a slow accumulation of human capital, while workers with intermediate level of human capital will probably sacrifice their wages for a faster accumulation of human capital. Therefore, assortative matching does not hold any longer.

ABOUT THE AUTHOR

Weng Xi received his PhD degree in economics from University of Pennsylvania in 2011. Prior to that, he received his bachelor and master's degrees from Peking University, in 2004 and 2006, respectively. He is currently a tenured professor at Guanghua School of Management, Peking University. His research focused on microeconomic theory, in particular, game theory, information economics and organizational economics. His research has been published in top journals such asJournal of Finance, Management Science, Economic Journal, Journal of Economic Theory, International Economic Review, and American Economic Journal: Microeconomics.