University of Pennsylvania and NBER
We study two wage bargaining games between a firm and multiple workers. We revisit the bargaining game proposed by Stole and Zwiebel (1996a). We show that, in the unique Subgame Perfect Equilibrium, the gains from trade captured by workers who bargain earlier with the firm are larger than those captured by workers who bargain later, as well as larger than those captured by the firm. The resulting equilibrium payoffs are different from those reported in Stole and Zwiebel (1996a) as they are not the Shapley values. We propose a novel bargaining game, the Rolodex game, which follows a simple and realistic protocol. In the unique no-delay Subgame Perfect Equilibrium of this game, the payoffs to the firm and to the workers are their Shapley values.
We introduce menu costs in the search-theoretic model of imperfect competition of Burdett and Judd (1983). When menu costs are not too large, the equilibrium is such that sellers follow a (Q,S,s) pricing rule. According to the rule, a seller lets inflation erode the real value of its nominal price until it reaches some point s. Then, the seller pays the menu cost and resets the real value of its nominal price to a point randomly drawn from a distribution with support [S,Q], where s<S<Q. A (Q,S,s) equilibrium differs with respect to a standard (S,s) equilibrium: (i) In a (Q,S,s) equilibrium, sellers sometimes keep their nominal price constant to avoid paying the menu cost, other times because they are indifferent to changes in the real value of their price. An exploratory calibration reveals that menu costs account less than half of the observed duration of nominal prices. (ii) In a (Q,S,s) equilibrium, higher inflation leads to higher real prices, as sellers pass onto buyers the cost of more frequent price adjustments, and to lower welfare.
“Equilibrium Price Dispersion Across and Within Stores,” with Nicholas Trachter, Review of Economic Dynamics, 2018, 28 (2), 205-220.
We develop a search-theoretic model of the product market that generates price dispersion across and within stores. Buyers differ with respect to their ability to shop around, both at different stores and at different times. The fact that some buyers can shop from only one seller while others can shop from multiple sellers causes price dispersion across stores. The fact that the buyers who can shop from multiple sellers are more likely to be able to shop at multiple times causes price dispersion within stores. Specifically, it causes sellers to post different prices for the same good at different times in order to discriminate between different types of buyers.
“Evidence on the Relationship between Recruiting and Starting Wage,” with Jason Faberman, Labour Economics, 2018, 50 (1), 67-79.
Using data from the Employment Opportunity Pilot Project, we examine the relationship between the starting wage paid to the worker filling a vacancy, the number of applications attracted by the vacancy, the number of candidates interviewed for the vacancy and the duration of the vacancy. We find that the wage is positively related to the duration of a vacancy and negatively related to the number of applications and interviews per week. We show that these surprising findings are consistent with a view of the labor market in which firms post wages and workers direct their search based on these wages if workers and jobs are heterogeneous and the interaction between worker’s type and job’s type in production satisfies some rather natural assumptions.
Are nominal prices sticky because menu costs prevent sellers from continuously adjusting their prices to keep up with inflation or because search frictions make sellers indifferent to any real price over some non-degenerate interval? The paper answers the question by developing and calibrating a model in which both search frictions and menu costs may generate price stickiness and sellers are subject to idiosyncratic shocks. The equilibrium of the calibrated model is such that sellers follow a (Q,S,s) pricing rule: each seller lets inflation erode the effective real value of the nominal prices until it reaches some point s and then pays the menu cost and sets a new nominal price with an effective real value drawn from a distribution with support [S,Q], with s<S<Q. Idiosyncratic shocks short-circuit the repricing cycle and may lead to negative price changes. The calibrated model reproduces closely the properties of the empirical price and price-change distributions. The calibrated model implies that search frictions are the main source of nominal price stickiness.
“Shopping Externalities and Self-Fulfilling Unemployment Fluctuations,” with Greg Kaplan, Journal of Political Economy, 2016, 124 (3), 771-825.
We propose a novel theory of self-fulfilling unemployment fluctuations. When a firm increases its workforce, it increases the demand facing other firms---as employed workers spend more than unemployed workers---and decreases the extent of competition facing other firms---as employed workers have less time to search for low prices than unemployed workers. In turn, the increase in demand and the decline in competition induces other firms to hire more labor in order to scale-up their presence in the product market. The feedback between employment and product market conditions generates multiple equilibria---and the possibility of self-fulfilling fluctuations---if the differences in the shopping behavior of employed and unemployed workers are large enough. Empirical evidence on spending, shopping and prices paid suggests that this is the case.
We develop a life-cycle model of the labor market in which different worker-firm matches have different quality and the assignment of the right workers to the right firms is time consuming because of search and learning frictions. The rate at which workers move between unemployment, employment and across different firms is endogenous because search is directed and, hence, workers can choose whether to seek low-wage jobs that are easy to find or high-wage jobs that are hard to find. We calibrate our theory using data on labor market transitions aggregated across workers of different ages. We validate our theory by showing that it correctly predicts the pattern of labor market transitions for workers of different ages. Finally, we use our theory to decompose the age profiles of transition rates, wages and productivity into the effect of age variation in work-life expectancy, human capital and match quality.
“Equilibrium Price Dispersion with Sequential Search,” with Nicholas Trachter, Journal of Economic Theory, 2015, 160 (6), 188-215.
The paper studies equilibrium pricing in a product market for an indivisible good where buyers search for sellers. Buyers search sequentially for sellers, but do not meet every sellers with the same probability. Specifically, a fraction of the buyers’ meetings lead to one particular large seller, while the remaining meetings lead to one of a continuum of small sellers. In this environment, the small sellers would like to set a price that makes the buyers indifferent between purchasing the good and searching for another seller. The large seller would like to price the small sellers out of the market by posting a price that is low enough to induce buyers not to purchase from the small sellers. These incentives give rise to a game of cat and mouse, whose only equilibrium involves mixed strategies for both the large and the small sellers. The fact that the small sellers play mixed strategies implies that there is price dispersion. The fact that the large seller plays mixed strategies implies that prices and allocations vary over time. We show that the fraction of the gains from trade accruing to the buyers is positive and non-monotonic in the degree of market power of the large seller. As long as the large seller has some positive but incomplete market power, the fraction of the gains from trade accruing to the buyers depends in a natural way on the extent of search frictions.
This paper is a study of the shape and structure of the distribution of prices at which an identical good is sold in a given market and time period. We find that the typical price distribution is symmetric and leptokurtic, with a standard deviation between 19% and 36%. Only 10% of the variance of prices is due to variation in the expensiveness of the stores at which a good is sold, while the remaining 90% is due, in approximately equal parts, to differences in the average price of a good across equally expensive stores and to differences in the price of a good across transactions at the same store. We show that the distribution of prices that households pay for the same bundle of goods is approximately Normal, with a standard deviation between 9% and 14%. Half of this dispersion is due to differences in the expensiveness of the stores where households shop, while the other half is mostly due to differences in households’ choices of which goods to purchase at which stores. We find that households with fewer employed members pay lower prices, and do so by visiting a larger number of stores, rather than by shopping more frequently.
“Taxation and Redistribution of Residual Income Inequality,” with Mikhail Golosov and Pricila Maziero, Journal of Political Economy, 2013, 121 (6), 1160-1204.
This paper studies the optimal redistribution of income inequality caused by the presence of search and matching frictions in the labor market. We study this problem in the context of a directed search model of the labor market populated by homogenous workers and heterogeneous firms. The optimal redistribution can be attained using a positive unemployment benefit and an increasing and regressive labor income tax. The positive unemployment benefit serves the purpose of lowering the search risk faced by workers. The increasing and regressive labor tax serves the purpose of aligning the cost to the firm of attracting an additional applicant with the value of an application to society.
“Monetary Theory with Non-Degenerate Distributions,” with Shouyong Shi and Hongfei Sun, Journal of Economic Theory, 2013, 148 (6), 2266-2312.
We construct and analyze a tractable search model of money with a non-degenerate distribution of money holdings. We assume search to be directed in the sense that buyers know the terms of trade before visiting particular sellers. Directed search makes the monetary steady state block recursive in the sense that individuals' policy functions, value functions and the market tightness function are all independent of the distribution of individuals over money balances, although the distribution affects the aggregate activity by itself. Block recursivity enables us to characterize the equilibrium analytically. By adapting lattice-theoretic techniques, we characterize individuals' policy and value functions, and show that these functions satisfy the standard conditions of optimization. We prove that a unique monetary steady state exists. Moreover, we provide conditions under which the steady-state distribution of buyers over money balances is non-degenerate and analyze the properties of this distribution.
Why do some sellers set prices in nominal terms that do not respond to changes in the aggregate price level? In many models, prices are sticky by assumption. Here it is a result. We use search theory, with two consequences: prices are set in dollars since money is the medium of exchange; and equilibrium implies a nondegenerate price distribution. When money increases, some sellers keep prices constant, earning less per unit but making it up on volume, so profit is unaffected. The model is consistent with the micro data. But, in contrast with other sticky-price models, money is neutral.
“Efficient Search on the Job and the Business Cycle,” with Shouyong Shi, Journal of Political Economy, 2011, 119 (3), 468-510.
The paper develops a model of directed search on the job where transitions of workers between unemployment, employment and across employers are driven by heterogeneity in the quality of firm-worker matches. The equilibrium is such that the agents' value and policy functions are independent of the distribution of workers across employment states. Hence, the model can be solved outside of steady-state and used to measure the effect of cyclical productivity shocks on the labor market. Productivity shocks are found to generate large fluctuations in workers' transitions, unemployment and vacancies when matches are experience good, but not when matches are inspection goods.
This paper revisits the no-recall assumption in job search models with take-it-or-leave-it offers. Workers who can recall previously encountered potential employers in order to engage them in Bertrand bidding have a distinct advantage over workers without such attachments. Firms account for this difference when hiring a worker. When a worker first meets a firm, the firm offers the worker a sufficient share of the match rents to avoid a bidding war in the future. The pair share the gains to trade. In this case, the Diamond paradox no longer holds.
We study the long-run relation between money (inflation or interest rates) and unemployment. We document positive relationships between these variables at low frequencies. We develop a framework where money and unemployment are modeled using explicit microfoundations, providing a unified theory to analyze labor and goods markets. We calibrate the model and ask how monetary factors account for labor market behavior. We can account for a sizable fraction of the increase in unemployment rates during the 1970s. We show how it matters whether one uses monetary theory based on the search-and-bargaining approach or on an ad hoc cash-in-advance constraint.
Consider a labor market in which firms want to insure existing employees against income fluctuations and, simultaneously, want to recruit new employees to fill vacant jobs. Firms can commit to a wage policy, i.e. a policy that specifies the wage paid to their employees as a function of tenure, productivity and other observables. However, firms cannot commit to employ workers. In this environment, the optimal wage policy prescribes not only a rigid wage for senior workers, but also a downward rigid wage for new hires. The downward rigidity in the hiring wage magnifies the response of unemployment to negative shocks.
“Directed Search on the Job, Heterogeneity and Aggregate Fluctuations,” with Shouyong Shi, American Economic Review, 2010, 100 (2), 327-332.
In this paper, we prove the existence of a Block Recursive Equilibrium for a model of directed search on the job in which workers are ex-ante heterogeneous with respect to some observable characteristic such as education.
“Block Recursive Equilibria for Stochastic Models of Search on the Job,” with Shouyong Shi, Journal of Economic Theory, 2010, 145 (4), 1453-1494.
We develop a general stochastic model of directed search on the job. Directed search allows us to focus on a Block Recursive Equilibrium (BRE) where agents’ value functions, policy functions and market tightness do not depend on the distribution of workers over wages and unemployment. We formally prove existence of a BRE under various specifications of workers’ preferences and contractual environments, including dynamic contracts and fixed-wage contracts. Solving a BRE is as easy as solving a representative agent model, in contrast to the analytical and computational difficulties in models of random search on the job.
“A Theory of Partially Directed Search,” Journal of Political Economy, 2007, 115 (5), 748-769.
This paper studies a search model of the labor market where firms have private information about the quality of their vacancies, they can costlessly communicate with unemployed workers before the beginning of the application process, but the content of the communication does not constitute a contractual obligation. At the end of the application process, wages are determined as the outcome of an alternating offer bargaining game. The model is used to show that vague non-contractual announcements about compensation---such as those one is likely to find in help-wanted ads---can be correlated with actual wages and can partially direct the search strategy of workers.
We develop a theory of endogenous and stochastic fluctuations in aggregate economic activity. Individual firms choose to randomize over firing or keeping workers who performed poorly in the past to give them an ex-ante incentive to perform. Different firms choose to correlate the outcome of their randomization to reduce the probability with which they fire non-performing workers. Correlated randomization leads to aggregate fluctuations. Aggregate fluctuations are endogenous---they emerge because firms choose to randomize and they choose to randomize in a correlated fashion---and they are stochastic---they are the manifestation of a randomization process. The hallmark of a theory of endogenous and stochastic fluctuations is that the stochastic process for aggregate "shocks" is an equilibrium object.
“Relative Price Dispersion: Evidence and Theory,” with Greg Kaplan, Leena Rudanko and Nicholas Trachter, NBER Working Paper, Revise and Resubmit at American Economic Journal: Microeconomics (September 2017).
We use a large dataset on retail pricing to document that a sizeable portion of the cross-sectional variation in the price at which the same good trades in the same period and in the same market is due to the fact that stores that are, on average, equally expensive set persistently different prices for the same good. We refer to this phenomenon as relative price dispersion. We show that relative price dispersion might stem from sellers' attempts to discriminate between high-valuation buyers who need to make all of their purchases in the same store and low-valuation buyers who are willing to purchase different items from different stores. We calibrate our theory and show that it is not only consistent with the extent and sources of dispersion in the price that different sellers charge for the same good, but also with the extent and sources of dispersion in the prices that different households pay for the same basket of goods.
“Declining Search Frictions, Unemployment and Growth,” with Paolo Martellini, NBER Working Paper (April 2018).
Over the last century, unemployment, vacancy, job-finding and job-loss rates as well as the Beveridge curve have no trend. Yet, the last century has seen the development and diffusion of many information technologies—such as telephones, fax machines, computers, the Internet—which presumably have increased the efficiency of search in the labor market. We explain this phenomenon using a textbook search-theoretic model of the labor market. We show that there exists an equilibrium in which unemployment, vacancies, job-finding and job-loss rates are constant while the search technology improves over time if and only if firm-worker matches are heterogeneous in quality, the distribution of match qualities is Pareto, and the quality of a match is observed before the start of the employment relationship. Under these conditions, improvements in search lead to an increase in the rate at which workers meet firms and to a proportional decline in the probability that the quality of a firm-worker match is acceptable leading to a constant job-finding rate, unemployment, etc... Interestingly, under the same conditions, unemployment, vacancies, job-finding and job-loss rates are independent of the size of the labor market even in the presence of increasing returns to scale in search. While declining search frictions do not lower unemployment, they contribute to growth. The magnitude of the contribution depends on the thickness of the tail of the Pareto distribution. We present a simple strategy to measure the decline in search frictions and its contribution to growth. A rudimentary implementation of this strategy suggests that the decline in search frictions has been substantial, it has been caused by both improvements in the search technology and increasing returns to scale in the search process, and it has had a non-negligible impact on growth.
We propose a theory of intermediation as rent extraction, and explore its implications for the extent of intermediation, welfare and policy. A frictional asset market is populated by agents who are heterogeneous with respect to their bargaining skills, as some can commit to take-it-or-leave-it offers and others cannot. In equilibrium, agents with commitment power act as intermediaries and those without act as final users. Agents with commitment trade on behalf of agents without commitment to extract more rents from third parties. If agents can invest in a commitment technology, there are multiple equilibria differing in the fraction of intermediaries. Equilibria with more intermediaries have lower welfare and any equilibrium with intermediation is inefficient. Intermediation grows as trading frictions become small and during times when interest rates are low. A simple transaction tax can restore efficiency by eliminating any scope for bargaining.
The paper studies the Rolodex bargaining game in networks. We first revisit the Rolodex game originally proposed in the context of intra firm bargaining, in which a central player bargains sequentially with multiple peripheral players. We show that the unique no-delay SPE of this game yields the Myerson-Shapley value for the star graph in which the central player is linked to all peripheral players. Second, we propose a Rolodex game for a general graph. Links in the graph negotiate sequentially, with one of the linked players making an offer to the other. If the respondent rejects, the link moves to the end of the line and the direction of the offer is reversed for the next negotiation of this link. As in the original Rolodex game, all agreements are renegotiated in the event of a breakdown. We show that the unique no-delay SPE of this game yields the Myerson-Shapley value for the corresponding graph.
“A Search Theory of Rigid Prices,” PIER Working Paper 07-031 (2007).
This paper studies price dynamics in a product markets characterized by: (a) search frictions—in the sense that it takes time for a buyer to find a seller that produces a version of the good he likes; (b) anonymity—in the sense that sellers cannot price discriminate between first-time buyers and returning costumers; (c) asymmetric information—in the sense that sellers are subject to idiosyncratic shocks to their marginal cost of production and privately observe the shocks’ realizations. I find that the joint dynamics of costs and prices may be very different than in a standard Walrasian market. When shocks are i.i.d., the price remains constant in the face of fluctuations in a seller’s marginal cost. When shocks are moderately persistent, the price adjusts slowly and imperfectly in response to changes in a seller’s cost. Finally, when shocks are sufficiently persistent, the price adjusts instantaneously and efficiently as soon as a seller’s production cost varies.
“A Cheap-Talk Theory of Random and Directed Search,” Revised for the Journal of Political Economy as “A Theory of Partially Directed Search” (2006).
This paper studies a search model of the labor market where firms have private information about the gains from trade and post cheap-talk messages to advertise their vacancies before workers decide which location to visit. The surplus of a match is divided ex-post according to the outcome of an alternating-offer bargaining game of asymmetric information. When this bargaining game is fast, I show that the maximum amount of information that can be transmitted through the cheap-talk depends non-monotonically on the tightness of the labor market. In particular, if the ratio of unemployed workers to vacancies is either sufficiently high or sufficiently low, the unique equilibrium has the firms babbling and search is random. If the tightness of the labor market takes on intermediate values, there is also an equilibrium where the cheap-talk is informative, high and low productivity firms post different messages and the search process of the workers is directed.
“High-Frequency Wage Rigidity,” Job Market Paper (2005).
In the context of a frictional model of the labor market with off and on the job search, I advance a novel model of wage determination where contracts are non-binding and firms have private information about the productivity of labor. The characterization of the intra-firm bargaining game leads to a reduced-form model where the firm chooses the wage subject to a non-discrimination and consistency constraints. The fundamental property of the optimal firm-wage policy is high-frequency wage rigidity. While the firm does not respond to productivity shocks whose persistence falls below a critical threshold, the wage is a non-degenerate function of the long-term component of labor productivity. A calibrated version of the model shows that the cyclical behavior of the model is quantitatively consistent with the empirical regularities of the labor market at the business cycle frequency. Among other things, wages are nearly acyclical, the semi-elasticity of the average labor productivity to unemployment is smaller than one, and vacancies are almost perfectly correlated with unemployment.
Work in Progress
“Workers Transitions and the Diffusion of Knowledge,” with Kyle Herkenhoff and Jeremy Lise.
“Exploitation of Labor,” with Claudio Michelacci.
“The Strategic Origin of Search Frictions,” with Pieter Gautier.