Search Theory beyond the Matching Function, by Shouyong Shi
Shouyong Shi is Associate Professor at the Department of Economics at Queen’s University (Kingston, Canada). He has published extensively on search models, especially applied to monetary economics. His interests also include capital accumulation, specialization, and financial intermediation. Shi’s RePEc/IDEAS entry.
The predominant theory for analyzing a frictional market is the search theory developed by Diamond (1982), Mortensen (1982) and Pissarides (1990). This theory has two distinctive elements. One is an exogenous matching function that captures a time-consuming matching process and generates unemployment in equilibrium. The other is an ex post (after-match) wage determination scheme, often the Nash bargaining formula, which splits the match surplus between the two sides of the match. This theory has been used to organize a wide range of facts related to unemployment, both over the business cycles and along the growth trend, and to make policy recommendations. In contrast to other unemployment models (e.g., efficiency-wage models), the search theory can be easily integrated into an intertemporal framework.Yet the exogenous matching function and the exogenous surplus-sharing rule remain unsatisfactory. First, these exogenous features critically affect the model’s predictions on efficiency (see Hosios (1990)) and on the effects of labor market policies (Shi and Wen (1999)). Second, and more fundamentally, they eliminate any role for wages to direct matches ex ante (before matches occur) and deprive agents of the ability to actively influence their matches. In my current research I develop search models that do not rely on these exogenous elements and apply them to analyze wage inequality.
A simple way to allow agents to actively organize their matches is to replace the matching function by a two-stage, wage-posting game, where firms simultaneously post wages first and then workers apply to jobs after observing the wages. Such a price/wage-posting model, developed by Peters (1991) and Montgomery (1991), preserves the time-consuming feature of the search theory by assuming that a worker can only apply to a small fraction of the job openings in each period. In contrast to the standard search model, wages are determined before, not after, matches occur and so wages “direct” workers’ search. The matching process and the surplus division are both endogenous outcomes of agents’ actions. Each firm can deliberately change the posted wage to affect the number of applicants it receives. Firms and workers maximize the expected gains from a match, making a trade-off between the matching probability and the ex post gains from a match.
I further develop this model and use it to examine the following issues.
1. “Pricing with Frictions”. In this paper with Kenneth Burdett and Randall Wright, we first show that the price-posting equilibrium in a market with finite numbers of sellers and buyers converges to the equilibrium with infinitely many buyers and sellers. Since the latter is considerably easier to characterize, this result greatly simplifies the price/wage-posting game in large markets.
Then we allow firms to differ in capacity. The main finding is that the equilibrium price and the endogenous matching function both depend on not only the number of buyers and the number of goods for sale in the market, but also on how those goods are distributed across sellers. This result suggests that the standard matching function adopted in the literature is mis-specified. That is, the number of new matches should depend on whether there are many firms, each with a few vacancies, or a few firms, each with many vacancies.
2. “Product Market and the Size-Wage Differential”. In this paper I examine whether the wage-posting model can be useful for explaining the size-wage differential, i.e., the fact that employers with more workers pay higher wages than smaller employers do to workers with the same observable skills. This size-wage differential is a significant fraction of the overall wage inequality but has not been well explained by traditional theories.
For this task, I integrate the product market and the labor market into a price/wage-posting framework. In the product market the price-posting game generates the outcome that buyers pay a higher price to a larger seller than to a smaller seller for the higher service probability the larger seller provides. Thus, a large firm obtains a higher expected revenue per worker than a small firm. To capture this revenue differential, a large firm posts a higher wage to fill the vacancy than a small firm does. High and low wages generate the same expected wage to a searching worker because a high wage attracts more applicants and hence is more difficult to obtain. Thus, large firms not only pay a higher wage than small firms but also have higher expected profit, although the workers are identical and the firms are identical (except for size).
An increase in the product demand changes the distribution of employment across firms with different sizes and has ambiguous effects on the size-wage differential. In particular, trade liberalization increases wage inequality when the product demand is initially low but decreases wage inequality when the product demand is already high.
3. “Unskilled Workers in an Economy with Skill-Biased Technology”. In this paper I extend the wage-posting model to incorporate skill differences and skill-biased technology. The purpose is to check whether search frictions are important for explaining the following facts in the US data: In the 1970s, the skill premium fell but the within-group wage differential rose; in 1980s, the skill premium and the within-group wage differential both rose.
In this model workers are either skilled or unskilled, while firms use either a high technology or a low technology. The high technology is biased toward skilled workers. High-tech firms prefer skilled workers to unskilled workers and pay a skill premium, but they also post wages for unskilled workers in case they do not receive any skilled applicants. There is a wage differential among unskilled workers, i.e., unskilled workers in high-tech firms are paid more than those in low-tech firms. This within-group wage differential arises not from match-specific productivity or the complementarity between skilled and unskilled workers, but rather from the trade-off between a wage and the matching probability. A high-tech firm’s high wage comes with a low matching probability for an unskilled worker while a low-tech firm’s low wage comes with a high matching probability.
In this framework an increase in the skill-biased productivity increases the skill premium and the wage differential among unskilled workers simultaneously. In contrast, an increase in the general productivity of all workers increases the skill premium but reduces the wage differential among unskilled workers. These results indicate that search frictions can be important for accounting for both the skill premium and the within-group wage differential.
4. “Frictional Assignment”. In this paper I examine the assignment problem in a frictional market, i.e., the two-sided matching problem in a market where agents on each side are heterogeneous. In a frictionless market, Becker (1973) has shown that the market assignment is efficient and positively assortative (i.e., it matches high attributes on one side with high attributes on the other side of the market). Neither feature holds in a frictional matching market modeled in the standard search theory. I re-examine these issues with a wage-posting framework and focus on the assignment between skills and machines.
Two results emerge. First, the efficient assignment in this frictional world may not be positively assortative even when skills and machine qualities are complementary with each other in production. This is because skills and machines are not fully utilized and so, by matching high skills with low-quality machines and high-quality machines with low skills, efficiency may be improved if such a matching scheme increases the utilization of both high skills and high-quality machines. Second, the efficient assignment can be decentralized as follows. Each firm chooses three things before matches occur: a machine quality, a desired skill to be matched with, and a wage for the skill. After observing these choices, workers apply to the firms. The ex ante competition between firms and the endogenous division of the match surplus encourage the right number of firms to enter the market to target each skill and induce them to select the efficient machine quality for each skill.
The price/wage-posting framework is tractable and useful for modeling large, frictional labor markets. It allows search theory to go beyond exogenous matching functions and exogenous surplus-splitting rules, to make predictions and policy recommendations that are not vulnerable to these exogenous elements, and to explain some well-known facts about inequality. More fundamentally, the framework reinstates prices the ex ante role of allocating resources.
Becker, Gary S., 1973, “A theory of marriage: part I
,” Journal of Political Economy
Burdett, Kenneth, Shouyong Shi and Randall Wright, 1998, “Pricing with Frictions”
, Federal Reserve Bank of Philadelphia Working Paper 98-9.
Diamond, Peter, 1982, “Wage determination and efficiency in search equilibrium,” Review of Economic Studies
Hosios, Arthur, 1990, “On the efficiency of matching and related models of search unemployment,” Review of Economic Studies
Montgomery, James D., 1991, “Equilibrium wage dispersion and interindustry wage differentials
,” Quarterly Journal of Economics
Mortensen, Dale T., 1982, “The matching process as a non-cooperative/bargaining game,” in J.J. McCall (ed.) The Economics of Information and Uncertainty
, Chicago: University of Chicago Press.
Peters, Michael, 1991, “Ex ante price offers in matching games: non-steady states
Pissarides, Christopher A., 1990, Equilibrium Unemployment Theory
, Oxford: Blackwell.
Shi, Shouyong, 1997, “Product Market and the Size-Wage Differential
Shi, Shouyong, 1998, “Unskilled Workers in an Economy with Skill-Biased Technology”
, CREFE working paper 73.
Shi, Shouyong, 1998 “Frictional Assignment”
, CREFE working paper 74.
Shi, Shouyong and Quan Wen, 1999, “Labor market search and the dynamic effects of taxes and subsidies,” Journal of Monetary Economics