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.
Lee Ohanian is Associate Professor at the Department of Economics, University of California, Los Angeles. He specializes in macroeconomic theory, the study of business cycles and growth. He has published in the best journals on monetary policy, war finance, VARs, and other topics. Ohnahian’s RePEc/IDEAS entry.
EconomicDynamics: With Hal Cole, you show in your Minneapolis Fed Quarterly Review article that the major peculiarity of the Great Depression was not so much the sharp decline in 1929-33 but rather the extremely slow recovery until 1939. You argue that the key fact to explain is stagnant hours. Why?
Lee Ohanian: Productivity grew rapidly after 1933. Theory predicts that the economy should have recovered to trend by 1936, with above-trend labor input supporting higher consumption and investment. But hours worked remained 20-25 % below trend until World War II. So why was labor input so low given rapid productivity growth? It wasn’t because other shocks were negative – banking panics and deflation ended in 1933, and real interest rates were low. Hours per adult should have been a lot higher after 1933.
EL: In current work with Hal Cole, you argue that New Deal policies encouraging cartelization linked to high wages are responsible for the slow recovery. Why? How could the government be so wrong?
LO: There must have been a major negative shock to offset the recovery of economic fundamentals and keep the economy depressed. The government adopted some extreme labor and industrial policies (the National Industrial Recovery Act) in 1933 that really distorted markets. These policies suspended the antitrust laws and permitted collusion, provided that the rents were shared with labor. This was accomplished through immediate wage increases and collective bargaining.Our new paper, “New Deal Policies and the Persistence of the Great Depression”, quantitatively analyzes these policies. We build a model of the policies, and embed that model within a dynamic GE business cycle model. In contrast to the fast recovery predicted by standard theory, our model predicts economic activity remains far below trend after 1933. We concluded that these policies were a key factor behind the persistence of the Depression – they can account for about 60 % of the deviation between the predicted trend levels and the actual data.
Ironically, President Roosevelt thought that “excessive” competition was responsible for the Depression, and that these policies would bring recovery. He was wrong. My colleague Armen Alchian was a student at Stanford at the time, and told me that his professors thought the policies were crazy – they couldn’t understand how promoting monopoly could raise employment. It is unfortunate that Roosevelt didn’t listen to these mainstream economists – if he had, the recovery would have been much stronger. These policies were finally weakened during World War II – and employment rose substantially.
There also seemed to be a sentiment to redistribute income during the 1930s. But this policy was a really inefficient method of redistribution. It created a lot of inequality by shutting down employment.
ED: Why do you think it is necessary to use a dynamic general equilibrium model to study the Great Depression? We have all been taught that the economy was not in equilibrium during that period.
LO: Theory has changed a lot since the Depression. I believe that economists took the disequilibrium route because the general equilibrium language of Arrow, Debreu, and McKenzie wasn’t well known at the time. We now know that disequilibrium models should be used very reluctantly, because there are an infinite number of ways an economy can be out of equilibrium. The model Hal and I developed for 1933-1939 is a dynamic general equilibrium model – but with a cartel policy arrangement that generates very low labor input, consumption, and investment.GE theory is important for understanding the Depression. There are a lot of stories about the Depression, but without an explicit GE model you don’t know if the stories hold water. One of the benefits of GE theory is that it forces you to look beyond the direct effects of shocks, and assess the indirect effects. Hal and I are writing a paper for the NBER Macro Annual that uses GE models to study the two most popular shocks for 1929-33: the money stock decline and bank failures. Using GE models, we found that many of the indirect effects of these shocks offset the direct effects, or were at variance with the data.
For example, several economists think money shocks depressed the economy through imperfectly flexible wages. Nominal wages were high in manufacturing because President Hoover told the Fortune 500 C.E.O.’s not to cut wages. But wages did fall in other sectors, so a multi-sector GE model is needed to evaluate this story. We found that high manufacturing wages reduced aggregate output only about 3 % between 1929-33. This is because the direct effect of the wage shock is pretty small, and because the indirect, general equilibrium effects offset some of the direct effect.
The paper also develops a GE model with a banking sector. The model predicts that bank failures should lead firms to substantially increase retained earnings as a substitute for bank finance. However, firms cut retained earnings like crazy during the Depression – In 1930, dividend payments fell by 4% while profits fell by 63%. The model also predicts that regions with more bank failures should have had deeper depressions. But we found little correlation between state-level economic activity and state-level bank failures.
Hal and I thought monetary shocks were the key factor for 1929-33 when we wrote our Minneapolis Fed QR paper. Our view has changed – either we need alternative theories to revive the money and banking hypothesis, or some other shock was responsible for 1929-33.
ED: Were Keynes, and Friedman and Schwartz all wrong?
LO: Keynes didn’t have the benefit of modern theory to help understand the Depression. He was wrong about “animal spirits” driving down investment, employment, and output. Ed Prescott argues in his review of our Minneapolis Fed Quarterly Review paper that the investment decline of the 1930s is not a mystery – it is exactly what theory predicts, given the policy shock that kept labor input so low.Friedman and Schwartz suggest that government policies contributed to the post-1933 depression, which is consistent with our view. But we suspect their emphasis on monetary shocks as a cause for 1929-33 may be misplaced. Believe it or not, productivity (TFP) fell about 15 % relative to trend between 1929-33. This drop isn’t technological regress, and it doesn’t seem to be input measurement error. Hoover intervened in the private economy significantly during this period. Perhaps his interference went beyond wage policies, and affected work practices and expectations about future returns to investment. We don’t know the source of this TFP drop, or all the consequences of Hoover’s actions, but these factors might be important for 1929-33.
ED: Would you make a parallel between the slow recovery of the Great Depression and the “jobless recovery” in the early 1990s?
LO: There are some key differences between these two recoveries. Employment growth was stagnant in the 1930s because of government policies that raised wages and reduced competition. These types of policies weren’t in place during the 1990s. My guess is that a mismatch between new technologies and the existing stock of labor may have contributed to the more recent “jobless recovery”. It is interesting that employment growth has been rapid the last few years as the pool of workers able to use these technologies has increased.
ED: Would you say, like Ed Prescott, that the conclusions of your work with Hal Cole could be applied to contemporaneous France, Spain, and Japan?
LO: Economies normally recover rapidly from downturns. It is pathological for a country to enjoy normal productivity growth but remain depressed for many years. Japan is one of these pathologies. Many economists think that Japan’s problems could be solved if their banks could make more loans and if fiscal and monetary policies stimulated aggregate demand. Some economists even argue that higher inflation expectations would bring recovery.We believe that Japan has more fundamental problems than finding the right mix of fiscal and monetary policy. Japan has tried all sorts of Keynesian stimuli, and it hasn’t worked. When an economy stagnates year after year, you have to ask: “What is preventing people from working and producing more?” Banking problems affected their economy, but we don’t think it is the whole story. 15 years ago, 80 % of Ireland’s banks shut down for six months because of a strike. Their economy did not falter – people found substitutes for closed banks.
If banking is the key to Japan’s Depression, why haven’t the Japanese found substitutes after all these years? The persistence of their depression and the failure of Keynesian policies suggest some other shock is responsible for Japan’s stagnation. My work with Hal suggests we should look for policies that keep employment low. Ed, Hal, and I are planning to start research along these lines soon.
Society for Economic Dynamics: 2000 Meetings in Costa Rica
The 2000 Meetings of the Society for Economic Dynamics will be held June 29-July 2 (Thursday-Sunday), 2000 in San José, Costa Rica. The conference is hosted by INCAE with the co-sponsorship of the Central Bank of Costa Rica. Registration and welcome reception will be held on Wednesday, June 28, 2000. The conference director is Alberto Trejos and the program organizer is Per Krusell. As in previous editions, the program will consist of several rounds of simultaneous sessions (both with submitted and invited papers) throughout the day for each day, followed by a plenary session at the end of each day. We are pleased to inform that Thomas Sargent, Guillermo Calvo and Patrick Kehoe have accepted to be our plenary speakers for the year 2000 edition of the conference.
For those wishing to attend the conference, online registration forms are available both for the hotel and the meetings at the web site. We will also soon post information about vacation packages for those wishing to extend their stay in Costa Rica. The web site already gives much details about attractions and other visitor information.
As the coordinating editor of the Review of Economics Dynamics, I want to update you on recent developments at RED. First, in order to get even faster turn around times for submitted papers, we are now prepared to receive electronic submissions that will be handled fully through email. We require, however, that authors send properly formatted PDF files. The RED web site has all necessary details. Of course, the traditional postal submission on paper is still accepted.
We also have started to put technical appendices of published papers on the web site. We encourage past and future authors to send us such material that should be a useful complement to the space constrained articles. While on the web site, you may also have a peek at the great articles that are forthcoming. Our Web site will soon have a new look. Be sure to check it out.
Finally, I wish to encourage your library to subscribe to RED. The Review is very reasonably priced for institutions, and given the high quality of the Review, no research library should be without it. So please check with your library.
Software: NEOS: Network-Enabled Optimization System
Economics is mostly about constrained optimization. With the increasing complexity or size of models, computers have been gradually taking over the task of finding optima. For such numerical analyses, techniques developed in operations research have helped a lot to solve problems that grew faster in complexity than the power of the computers. But one has to be aware that not all models can be solved efficiently with the same algorithm. Many solution methods have been developed, each geared to a particular set of problems. In fact, there are now so many methods that it becomes a complex task in itself to select the appropriate algorithm.
This richness in algorithms has also the drawback that one has to learn new tools for each model. To help reduce this type of cost to the researcher, the Optimization Technology Center of the Argonne National Laboratory has put together a web site introducing the many techniques, the Network-Enabled Optimization System (NEOS) at http://www-fp.mcs.anl.gov/otc/. This very complete site guides the reader through the various types of problems, showing for each several solution algorithms that are explained and that can even be tried online. In fact, it is possible to have very large and complex problems solved on the NEOS server, a great opportunity for those who do not have the necessary resources available to them directly.
Most of the algorithms on NEOS deal with rather simple objective functions with numerous but simple constraints. In economic dynamics, problems have usually more complex functions and fewer constraints. Yet, the latter can be converted to the former, as shown for example by Trick and Zin (1997) who demonstrate that a standard dynamic problem is equivalent to a linear programming problem with one constraint for each state. Such problems can be solved easily now, even if they have millions of constraints.
Trick Michael, and Stanley Zin, 1997, “Spline approximations to value functions: A linear programming approach”, Macroeconomic Dynamics, v.1, p. 255-277.
EconomicDynamics Links: QM&RBC: Quantitative Macroeconomics and Real Business Cycles
Economic Dynamics covers many fields, but Real Business Cycle theory is certainly one of the more prominent ones. Yet, his theory has difficulties getting mainstream economics because of its technical difficulty. For example, very few undergraduates study it, and for a long time only PhDs coming from some schools could master its techniques. This is changing now, and this is partly due to the Quantitative Macroeconomics & Real Business Cycle home page. The web has democratized the access to information, and this includes getting access to latest research, data and solution techniques.
Indeed, since early 1995, QM&RBC has provided sample codes for solving some of the standard problems in the field. Also, it features links to selected home pages of researchers who provide their latest results online. QM&RBC has provided complementary reading material for many classes, in fact it has even lead many to discover the RBC models they would not have covered in the regular training.
QM&RBC is still being maintained to continue to serve the research community and the aspiring researchers. But the field has evolved and RBC is now synonymous with much more that “real” shocks or “business cycles”. This is why a poll is currently being conducted to choose for a new acronym for RBC that would more appropriately describe it. We encourage you to provide your input.