Schooling Choice, Labour Market Matching, and Wages

\@startsection{section}{1} \z@{1.0\linespacing\@plus\linespacing}{.8\linespacing}{Introduction} Since the 1980s, economists have attributed rising wage inequality to a number of sources. One possible source of such inequality is positive assortative matching between workers and firms - the tendency for the quality of workers and firms who match with one another to be positively correlated.\footnote{Recent empirical papers examining the role of sorting on wage inequality include \citet*{Card/Heining/Kline:13:QJE}, \citet*{Barth/Bryson/Davis/Freeman:2016:JL}, and \citet*{Kantenga/Law:2016}.} Unfortunately, studying matching in labour markets presents a serious challenge when the decisions of individual job seekers affect each other's hiring outcomes. This paper develops a methodology to address this challenge. In particular, we show how cross-sections of matched employer-employee data can be used to study the role that a labour market matching technology plays in shaping the equilibrium distributions of education and wages. The structural model we develop can capture assortative matching between workers and firms even in the absence of complementarities between worker and firm types in the match production function.\footnote{The value of a match between any type of worker, $h$, and any type of firm, $k$, can be represented using a positive, increasing function, $f(h,k)$. We say that the types are complements in $f$ when the marginal product of an $h$ type is higher when matched with a higher $k$ type (and vice versa). }

A general overview of the labour market in the model is as follows. Agents from one side of the market sequentially choose agents from the other side according to their preferences. Preference rankings of the choosers depend on a preference parameter, along with the capital of both types of agents. The order in which the choosers pick depends on the chooser's capital and a matching technology parameter. Before matching, the agents who will be chosen are allowed to simultaneously decide their capital given the distribution of the chooser's capital and the underlying parameters of the economy (including the frictions).

This paper contributes to the econometric literature concerned with inference in two-sided matching models.\footnote{See \citet*{Chiappori/Salanie:16:JEL} for a review of this literature. A seminal paper in this literature is \citet*{Choo/Siow:06:JPE}, which considers inference in a transferable utility setup with a continuum of agents. } A key feature of our setup is that the characteristics of agents on one side of the market are endogenous - in particular, arising in equilibrium from a pre-matching investment game. We show how, rather than making the empirical analysis intractable, accommodating such pre-matching investments provides the researcher useful information for inference.\footnote{A popular approach for estimating two-sided matching models builds on the notion that the observed matching is pairwise stable. For example, see \citet*{Fox/Bajari:13:AEJM}, \citet*{Echenique/Lee/Shum:13:SEM}, and \citet*{Menzel:15:Ecta}. Requiring that the observed matching be pairwise stable may be unrealistic in the context of a frictional labour matching market of the sort that is the focus of this paper.} We propose a two-stage approach for inference on the agents' preferences and the matching technology. In the first stage, we fix the matching technology and construct confidence regions for the preference parameter by estimating the Bayesian game associated with the workers' pre-match investment in education decision. We show that this problem can be cast in a discrete choice framework yielding tractable and consistent estimation via maximum likelihood when the workers' educational decision takes one of two values (college, or no college). In the second stage, we construct confidence intervals for the matching technology using a simulation-based inference approach. In the first stage, the presence of the matching function in workers' expected utility function makes estimating workers' equilibrium expectations highly non-trivial. Nevertheless, under reasonable assumptions, we show that workers' equilibrium expectations can be written in a closed form suitable for consistent estimation. The second-stage inference on the matching technology uses the following insight: once the matching process is specified, the finite sample distribution of the observed matching is known up to a parameter.\footnote{This idea of using a structural model to characterize the joint distribution of a discrete matching model that can then be used for inference on the model parameters builds from \citet*{Kim/Schwartz/Song/Whang:19:Econometrics}. Although computationally intractable when the dimension of the parameter is large, this approach is attractive for inference on the matching technology parameter in the second stage of our approach.} We construct a test statistic that measures the distance between the observed joint distribution of worker education and matched firm capital to simulated counterparts. A confidence interval for the matching technology can then be constructed by inverting the test.

This paper builds on the fundamental insights of \citet*{Becker:73:JPE} and \citet*{Gale/Shapley:62:AMM} to illustrate how an economically meaningful notion of sorting can be captured in a model that assumes additive worker and firm effects. Since \citet*{Abowd/Kramarz/Margolis:99:Ecta} (AKM), the availability of matched employer-employee data has allowed researchers to study the role that unobserved worker and firm attributes play in driving wage variation over time. In AKM, the correlation of worker and firm fixed effects from wage regressions is taken to capture a notion of sorting. Although popular for investigating the wage structure, a burgeoning literature has criticised the viability of AKM for detecting sorting on unobservables. In particular, the additive structure of AKM implies that wages are monotone in firm type - an implication that is difficult to reconcile with equilibrium models of sorting with and without frictions (\citet*{Eeckout/Kircher:11:ReStud}, \citet*{LopesDeMelo:18:JPE}).\footnote{\citet*{Gautier/Teulings:06:JEEA} was an early empirical study that detected a concave relationship between wages and firm type.} For example, in \citet*{Eeckout/Kircher:11:ReStud}, a low-type worker can receive a lower wage at a high-type firm since the worker must implicitly compensate the high-type firm in equilibrium for forgoing the opportunity to fill a vacant job with a higher-type worker.\footnote{There are other reasons wages may be non-monotonic in firm type. In \citet*{Postel-Vinay/Robin:2002:Ecta}, workers may be willing willing to accept lower wages at higher type firms when they expect to receive higher wages in the future. }

In his seminal 1973 paper on the marriage market, Becker argued that when the match production function is supermodular\footnote{When $f$ is differentiable, (strict) supermodularity is equivalent to $\partial^{2}f(h,k)/\partial h\partial k>0$. } and utility is transferable between matched agents, high types can outbid low types for the best partners, leading to an equilibrium with positive assortative matching. In the same paper, Becker noted that sorting can also arise in an non-transferable utility (NTU) framework when the payoffs of the agents on both sides of the market are monotonic in the other agent's type. To explain why, Becker invokes the notion of pairwise stability \citet*{Gale/Shapley:62:AMM}.\footnote{This insight comes to us by way of \citet*{Chade/Eeckhout/Smith:17:JEL}'s excellent review of the search and matching literature. } To illustrate, consider an economy with four agents where a low-type firm is paired with a high-type worker and vice-versa. Such a matching is unstable when high types are preferred, because both high-types will agree to abandon their low-type partners for one another. NTU arises naturally in our model from the assumption that wages for any matched pair are determined exogenously (in fact, by a Nash bargaining assumption). In a special case of our model in which the preferences of agents are indeed monotonic, sorting - and some inequality - may emerge. In this case, complementarities are not necessary for sorting but merely amplify the effects of sorting, since interactions between worker and firm types in the wage function lead to more wage dispersion than when such interactions are absent.

Search and matching models have emerged as the leading alternative to the AKM framework for studying sorting in labour markets.\footnote{\citet*{Hagedorn/Law/Manovskii:17:Ecta}, \citet*{Bagger/Lentz:2018:ReStud}, \citet*{Lise/Meghir/Robin:16:RED} and \citet*{LopesDeMelo:18:JPE} all find evidence of positive sorting when an AKM approach finds negligible sorting.} In this literature, the standard matching technology is one that converts aggregates of vacancies and unemployed workers into matches. Although treating matching at the aggregate level simplifies the analysis considerably, any strategic interdependence that may be present in the matching process is assumed away (\citet*{Chade/Eeckhout/Smith:17:JEL}). One contribution of this paper is to develop and estimate a model that takes such strategic interdependence in the matching process seriously. Capturing such interdependence is desirable, since in many labour markets the impact of an individual's decisions on the outcomes of other workers are highly relevant and cannot be ignored.\footnote{For example, a worker's decision to get a master's degree in finance will not only affect the likelihood that he gets a job at an investment bank, but also the likelihood that his competitors get the job.} In the equilibrium of our model, (and in contrast to standard search models), the probability that a worker matches to a given firm typically depends on the decisions of all the other agents in the economy. Another key facet of search models is that workers direct their job search based on the wages that employers set for them. However, recent studies of online job markets have found that it is relatively uncommon for positions to explicitly post wages.\footnote{For example, \citet*{Marinescu/Wolthoff:16:WP} study the role of job titles in directing the search of workers report that only 20% of job the advertisements CareerBuilder.com report a wage. } Another way in which this paper differs from the traditional search literature is that we do not require workers do not observe posted wages directly. Instead, workers know the underlying distributions of job characteristics and the matching process prior to simultaneously investing in education. In this sense, the worker's decision to invest in education is the channel by which workers are able to direct their search.

The framework in this paper supposes that each equilibrium gives rise to a single large matching between workers and firms.\footnote{This contrasts with cases in which the researcher sees many independent copies of games involving few players, such as those studied by \citet*{Bresnahan/Reiss:91:JPE}, \citet*{Ciliberto/Tamer:09:Ecta}, \citet*{Berry:92:Ecta} and many others. See \citet*{Xu:18:IER}, \citet*{Song:14:WP}, \citet*{Menzel:16:ReStud} for more papers discussing the estimation of large Bayesian games. } Under familiar assumptions (e.g., iid and separable private information), we follow similar arguments to \citet*{Aguirregabiria/Mira:19:WP} to prove that an equilibrium exists. The setup here, however, also allows us to provide sufficient conditions for equilibrium uniqueness.

This paper is also part of the literature concerned with estimating cross-sectionally dependent observations. In our setup, the observed matching of workers to firms exhibits cross-sectional dependence of an unknown form due to the matching process. This means that asymptotic inference approaches that appeal to the the law of large numbers and central limit theorems will not work. The approach we pursue builds on \citet*{Kim/Schwartz/Song/Whang:19:Econometrics} which discusses how inference in structural matching models are possible when knowledge of the matching process can be used to characterize the joint distribution of the observed matching. This paper shows how such a simulation-based inference approach, cumbersome when the dimension of the parameter space is high dimensional or complex, is useful for estimating a subset of the parameters in structural models with cross-sectional dependence.\footnote{The simulation-based approach used in the second-stage of the inference procedure is known as a Monte Carlo test. Monte Carlo tests have a history in econometrics dating back at least to the 1950s, as discussed by \citet*{Dufour/Khalaf:01:TE} in their overview of the technique.}

Section (ref) provides intuition on how our model captures the relationship between matching frictions, sorting, and inequality. In particular, we illustrate how a fall in matching frictions can yield two opposing impacts on wage inequality via their effects on sorting and the supply of highly educated workers. In the model parameterization considered, a fall in frictions leads to both an increase in the equilibrium supply of highly educated workers and an increase in positive assortative matching between workers and firms. That is, the latter sorting effect increases wage inequality while the former supply effect acts in the opposite direction. In general, the impact of matching frictions on wage inequality are more pronounced when worker and firm types are complements in the match production function. The section also illustrates how a fall in information frictions can lead to a dramatic rise in the education wage premium through sorting while at the same time, a much more modest increase in the supply of highly educated workers. Thus, changes in informational frictions may be a useful way to explain a puzzling empirical findings concerning the relationship between wage premia and educational attainment.\footnote{See \citet*{Card/Lemieux:01:QJE}.}

Section (ref) introduces the model of two-sided labour market matching with frictions. In the baseline model of Section (ref), workers and firms with exogenous characteristics match with one another and split the match surplus according to a Nash bargaining rule. The rest of the paper is organized as follows. Section (ref) extends the baseline model to allow for endogenous worker characteristics - after observing their type, workers simultaneously invest in education prior to entering the labour market. Section (ref) outlines an approach for inference on the parameters of the structural model of Section (ref). Section (ref) presents a small simulation study illustrating the finite sample size and power performance of the first-stage inference. Section (ref) concludes. Mathematical proofs are confined to two appendices: Appendix A contains results concerning the existence and uniqueness of the equilibrium of the game described in Section (ref) while Appendix B contains supplemental results relating to the first-stage inference on preferences.

Our goal is to study the distribution of education and wages using separate cross-sections of matched employer-employee data. The first subsection introduces the core elements of the model that will serve as the basis for the structural model in the second subsection.

\@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{Baseline model}

Let $N_{h}=\{1,...,n_{h}\}$ be the set of workers and $N_{f}=\{1,...,n_{f}\}$ be the set of firms, where $n_{h}$ and $n_{f}$ are used to denote the total number of workers and firms, respectively. Each worker seeks one job and each firm seeks to hire one worker.

The matching of workers to firms will be determined by the preference rankings of workers and firms. Workers value the capital of firms, $K=(K_{j})_{j\in N_{f}}$, and firms value the human capital of workers $H=(H_{i})_{i\in N_{h}}$, where $K_{j}$ and $H_{i}$ are scalars. Any worker $i$ who is matched with firm $j$ receives wage $w_{ij}\geq0$ while firm $j$ receives profit $\rho_{ji}\geq0$, where both wages and profits may also depend on a parameter, $\theta\in\mathbf{R}^{d}$.\footnote{In this setup, $\theta$ represents the preferences of both workers and firms. As we will see, $w_{ij}$ and $\rho_{ji}$ depend on the output of worker $i$ at firm $j$, and the production function that gives rise to this output will depend on a part of $\theta$. } Since our framework supposes that wages and profits are always non-negative for any worker and firm that could match, we will assume throughout the paper that no agent will ever unilaterally dissolve a match to become unmatched. This requirement that any matching satisfy an individual rationality constraint is embodied in the following condition:\footnote{The current setup is tailored to settings where the researcher has at least one cross-section of matched employer-employee data and the agents who are unmatched are not of primary interest in the analysis. An interesting (and challenging) extension of the current framework would accommodate the possibility of unmatched agents, and hence unemployment.}

Condition IR (Individual rationality of matches): For each $i\in N_{h}$, and $j\in N_{f}$ $w_{ij}\geq0$ and $\rho_{ji}\geq0$.

Based on the values of $\rho_{j}=(\rho_{ji})_{i\in N_{h}}$ each firm $j$ can construct preference rankings over the workers. We suppose that if the firm is ever indifferent between one or more workers, then the firm picks preference rankings over these workers at random. Next, we introduce a condition on the worker's wage function that will grant us a natural economic interpretation of the matching process (along with our notion of information frictions).

Condition H (Homogeneous worker preferences): For each $i\in N_{h}$, the wage of worker $i$ is increasing in the capital of their matched firm.

The condition is tantamount to a notion of worker preference homogeneity, implying that all workers prefer higher capital firms. Supposing that workers accurately observe the capital of firms, the condition implies that a matching algorithm in which the highest capital firm, $j_{1}$, choose his preferred worker, $i_{1}\in N_{h}$, the second highest capital firm, $j_{2}$, choose his preferred worker $i_{2}\in N_{h}\backslash\{i_{1}\}$ and so on is an example of the serial dictatorship mechanism and would produce a stable matching.\footnote{\citet*{Satterthwaite/Sonnenschein:81:ReStud}. See Section 2.2. of \citet*{Roth/Sotomayor:92}. }

In order to build a model that accounts for the possibility of mismatches between workers and firms, we suppose that information frictions are present in the market. Specifically, we suppose that workers do not directly observe realizations of the firm's capital. Instead, each worker sees $v=(v_{j})_{j\in N_{f}}$, where $v_{j}$ is a `noisy' measure of firm $j$'s capital. In particular, suppose that workers see

for each $j$, where $\beta\in B$, $B\subset\mathbf{R}$ is the parameter space of $\beta$, and $\eta_{j}$ is a random variable that is independent across $j$. The size of the variance of $\eta_{j}$ relative to the magnitude $\beta$ represents the magnitude of information frictions in the matching process. It is clear that when $\beta$ is zero and the variance of $\eta_{j}$ is positive, then this setup yields random matching from firm to worker the characteristics, since variation across firm capital plays no role in determining the realizations of $v$. Furthermore, when $\beta\neq0$ and $\text{Var}(\eta_{j})=0$, it will be as if firm capital is observed by the worker, since $v_{j}$ is determined entirely by the firm's capital. In the latter case, when $\beta>0$ workers would favour firms with the largest realizations of $v$, while in the case that $\beta<0$, workers would favour firms with the smallest realizations of $v$. However, even in the case that $\text{Var}(\eta_{j})>0$, $v_{j}$ still conveys some useful information to the worker under certain circumstances. To see this, suppose that $v_{j}$'s follow equation (ref) with $\beta>0$ and let $\eta_{j}$'s be iid\footnote{We will impose such an assumption in a later section.}. Then, any worker who sees $v_{j1}$ exceed $v_{j2}$ will prefer matching with Firm $j_{1}$ over Firm $j_{2}$, since the worker recognizes that the distribution of $K_{j1}$ conditional on $v_{j1}=\tilde{v}_{j1}$ stochastically dominates the distribution of of $K_{j2}$ conditional on $v_{j2}=\tilde{v}_{j2}$ when the worker observes $\tilde{v}_{j1}>\tilde{v}_{j2}$.

The following condition specifies the matching process we will use throughout the paper.

Condition SD (Matching process): The matching of workers to firms in the economy arises as follows. The highest $v$ firm, $j_{1}$, chooses his preferred worker, $i_{1}\in N_{h}$, the second highest $v_{2}$ firm, $j_{2}$, chooses his preferred worker $i_{2}\in N_{h}\backslash\{i_{1}\}$, and so on, until the lowest $v$ firm, $j_{n_{f}}$, chooses his preferred worker among those not chosen by any higher ranked firms.

One way of understanding this matching algorithm in economic terms is to consider the following thought experiment. Imagine a situation in which a group of job-seekers have assembled in a large room on the day of a job fair. Workers do not observe the true quality of any of the firms, (represented by $K$), but they do see each firm's value of $v$. When $\beta>0$ and $\eta_{j}$'s are iid, each worker is happiest to match with the highest $v$ firm, since the distribution of capital associated with the highest $v$ firm stochastically dominates the distribution of capital associated with any of the lower $v$ firms. A procedure in which the highest $v$ firm, $j_{1}$, chooses his preferred worker, $i_{1}\in N_{h}$, the second highest capital firm, $j_{2}$, chooses his preferred worker $i_{2}\in N_{h}\backslash\{i_{1}\}$ and so on, will have no complaints from any of the participants at the job fair \textendash that is, until uncertainty associated with $K$ is revealed. In this world, agents will typically have more regret (and hence a greater desire to rematch) when the frictions in $v$ are large. However, rematching is outside the scope of the model.

Next, we add some further structure to wages and profits. In particular, we will assume that the payoffs for any two matched agents follow a Nash bargaining structure. Let $\tau\in(0,1)$ be the bargaining weight. A worker $i$ who matches with a firm $j$ receives

where $f$ is the worker-firm output function and $g(H_{i})$ is an outside option function, both of which may depend on elements of $\theta$. In a subsequent section, we will allow worker covariates, $X_{i}$, to effect wages through the outside option function, $g.$\footnote{$X_{i}$'s have support $\mathcal{X}\subset\mathbf{R}^{d}$, where $d$ is an integer greater than or equal to one.} The following condition requires $f$ to satisfy some intuitive properties with respect to the worker and firm capital variables.

Condition F (Production function): $f$ is increasing in human capital and firm capital.

Condition F merely requires that more capital leads to more output - it does not impose that the worker and firm attributes be complements in $f$. Section (ref) goes into further detail about the role of $f$ in this model.

\@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{Frictional Matching Model with Worker Investments}

We now introduce a structural model where workers simultaneously invest in education prior to the serial dictatorship matching process as outlined in the previous section. A general overview of the matching process is as follows: i) workers, observing only their type, simultaneously choose a level of education, ii) $v$ is realized, iii) firms, seeing only the education of workers, match according to Condition SD.

Although firms select their preferred workers in the serial dictatorship phase after constructing preference rankings over the workers, firms are not considered strategic agents within the context of the investment game itself.

There are $n_{h}$ players indexed by $i\in N_{h}$. Each player chooses an education level, $h_{i}$, from the discrete set $\mathcal{H}\equiv\{1,...,J\}$ to maximize their expected payoff. Let $\lambda=(\theta',\beta)$, where $\beta$ is the matching frictions parameter and $\theta\in\mathbf{R}^{d}$ is a preference parameter. The payoff function of player $i$ comprises the wage less a cost of education,

where $h_{-i}\in\mathcal{H}_{-i}$ are the choices of the other agents,\footnote{Since the set of pure strategies for each agent is $\mathcal{H}$, it follows that $\mathcal{H}_{-i}=\mathcal{H}^{n_{h}-1}$ for each $i$, where $\mathcal{H}^{n_{h}-1}$ denotes the $(n_{h}-1)$-ary Cartesian power of $\mathcal{H}$. } $x_{i}\in\mathcal{X}$ and $\varepsilon_{i}\in\mathbf{R}^{J}$ are the private information of worker $i$, and $k\in\mathbf{R}^{n_{f}}$ and $\eta\in\mathbf{R}^{n_{f}}$ are vectors of exogenous firm variables that are unobserved by the workers. Although $\varepsilon_{i}$ and $x_{i}$ are private information of the worker, we will assume $x_{i}$ is observed by the econometrician in a subsequent section. The variable $\varepsilon_{i}$ represents the worker's private cost associated with each of the $J$ education levels. In Section (ref) we will supply explicit assumptions on worker and firm information that illustrates why, given the matching process, the components of the payoff function depend on model's underlying variables in the way stipulated by equation ((ref)).

We now provide additional conditions that establish the existence of a Bayesian Nash equilibrium for our game (which we prove in Section (ref)).

The assumptions of separability and independence are common in the structural literature.\footnote{For example, see the discussions in \citet*{Kasahara/Shimotsu:08:JoE} and \citet*{Xu:14:EJ}.} In Section (ref), we show that Assumptions (ref) and (ref) are sufficient for establishing the existence of the Bayesian Nash equilibrium for the game of this section. For now, we will provide some intuition into the worker's education decision problem. First, we define the set of pure strategies as $\sigma=\{\sigma_{i}(x_{i},\varepsilon_{i}):i\in N_{h}\}$ where $\sigma_{i}$ is a function that maps from $\mathcal{X}\times\mathbf{R}^{J-1}$ into $\mathcal{H}$. Assumption (ref) says that we can write the expected utility of agent $i$ with covariates $x_{i}$, who chooses $h_{i}$ under beliefs $\sigma$ as

where the first term in the expected utility is

and

where $\tilde{\omega}_{i}(h_{i},h_{-i},x_{i})$ is given by

and expectation is taken with respect to the distributions of $K$ and $\eta$. By Lemma ((ref))m we can rewrite equation (ref) as

Throughout this paper, we will consider the case in which the wages of workers are determined by Nash bargaining. As in equation (ref), we will suppose that firm capital only enters the worker's payoff through the production function. Denote $\mathcal{M}(i)$ as the identity of the firm that worker $i$ matches to as a result of the matching process, and $K_{\mathcal{M}(i)}$ as the level of capital associated with firm $\mathcal{M}(i)$. Under these assumptions, we may write $\tilde{\omega}_{i}(h_{i},h_{-i},x_{i})$ as

where

the expectation is taken with respect to the distributions of $K$ and $\eta$, and we have allowed the worker's characteristics to enter the payoff function through the outside option function, $g$.\footnote{Here, $\tau_{i}=\tau$ for each $i$. The framework here can be extended to incorporate heterogeneity in worker bargaining positions. \citet*{Bagger/Lentz:2018:ReStud} emphasize the importance of endogenous search intensity and matching variation (e.g., \citet*{Postel-Vinay/Robin:2002:Ecta}) in understanding the causes of wage inequality.}

Education affects the worker's expected utility in a number of ways. The first two are obvious: since $f$ is increasing in $h_{i}$ by Condition F, the worker who invests in a higher level of education obtains a higher wage at any firm he matches to. The worker's choice of education also affects his payoff through the outside option function, $g$. The novel channel in this setup is that $h_{i}$ also determines the expected quality of the firm that $i$ matches to. Even though (as mentioned before) firms in this model are non-strategic agents, the functional form of the production function, $f$, plays a key role in determining whether or not firms with different levels of capital exhibit different preferences for workers of differing levels of education. To see how $f$ determines whether or not firms' preferences are heterogeneous, consider the Nash bargaining preferences of a firm for any worker who chooses education level $h$:

where $\tilde{g}=\mathbf{E}g(h,X_{i})$ and the expectation is taken with respect to the distribution of $X_{i}$.\footnote{Here, we implicitly assume that firms do not observe workers' covariates and rank workers only in terms of their education. We make these assumptions concerning firm information explicit in a subsequent section.} Suppose that $X_{i}$ is iid, $K$ takes two values $k_{1},$ $k_{2}$ and there are two levels of education, $h_{1}$, $h_{2}$ with $h_{2}>h_{1}$. Let us denote the set of firms that prefer high education ($h_{2}$) as

If $f(h,k)$ is of the form $a(h)+b(k)$, where $a$ and $b$ are two functions that map the capital variables to the real numbers, then $M_{2}^{+}(\theta)$ will be either $\{1,2\}$ or $\emptyset$. In this case we say that firms have homogeneous preferences, since both types of firms in the economy prefer the higher educated workers. Alternatively, if $f(h,k)$ is of the form $a(h)b(k)$ then $M_{2}^{+}(\theta)$ will be either $\{1,2\}$, $\emptyset$, or $\{2\}$. This is the case of heterogeneous firm preferences. In this latter case where $f$ exhibits complementarities in worker and firm types, the set of firms types that prefer high to low education is more finely partitioned. Moreover, the presence or absence of complementarities will play a key role in determining the severity of wage inequality. More general than all these points, however, is the following fact about the model: as long as $k$ appears somewhere in $f$, $k$ does not have to interact directly with $h$ in $f$ for the information frictions represented by $\beta$ to matter in worker's investment decision.

\@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{Some Implications of Frictional Matching Model}

In this section, we explore some key features of the model. We will suppose that the functional forms, underlying distributions, and firm preferences are such that firms always strictly prefer higher educated workers. In the following subsection, we will illustrate sorting without any direct interactions between worker and firm types in the production function.

\@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Sorting Without Complementarities}

In Figure 1 and Figure 2, we compare the equilibrium probability of investing in education and the equilibrium Gini coefficient for a range of the friction parameters under two specifications of the production function: Specification 1 allows direct interaction between worker and firm types, $f=\theta_{1}hk$, while such interactions are absent in Specification 2, $f=\theta_{1}(h+k)$. Each point on the plot is the average of 500 simulations of endogenous variable from the equilibrium of the model. The outside option parameter is set to $\theta_{2}=(-.75,.25,.5)$. There are 500 workers and firm positions. In Specification 1, the high value of $\theta_{1}$ is 3, and the low value of $\theta_{1}$ is 1. In Specification 2, the high value of $\theta_{1}$ is 2, and the low value of $\theta_{1}$ is 1. There are two levels of of firm capital: $K=1/2$ and $K=1$. The fraction of each type of firm is .5 in the economy.\footnote{Across all the specifications, we set the outside option function to be $g =\exp(h\cdot x\theta_{2})$.}

A number of implications are straightforward: the equilibrium probability of investing in high education is higher when $\theta_{1}$ is higher and frictions are lower. When $\theta_{1}$ is higher, workers will be compensated more for higher levels of education. When $\beta$ is higher, the probability of matching to a higher type firm when they choose high education is higher.

The effect of increasing $\beta$ (lowering matching frictions) on both the education and wage inequality is typically much more dramatic in Specification 1. A rise in $\beta$ (a lessening in matching frictions) increases sorting in both specifications. In Figure 1, the correlation between worker and firm types rises from approximately zero to 45% when $\theta_{1}$ is high, but from zero to 70% when $\theta_{1}$ is lower; in Figure 2, the correlation between worker and firm types rises from zero to 51% in the high theta case whereas it rises from zero to 68% in the low theta case. The overall level of inequality in Specification 1 is also higher since whatever sorting is present is amplified to a greater extent when the types interact in the wage equation than when they do not.

The high $\theta_{1}$ case in the right hand panel of Figure 1 also illustrates the role that two competing effects of changes in $\beta$ play on the level of wage inequality. When $\beta$ rises from 0 to 1, the level of inequality increases through the sorting channel. However, as $\beta$ continues rises, the equilibrium probability of investing in education also continues to rise. As the fraction of highly educated surpasses 80%, the level of inequality begins to level off (at $\beta=2$) and then begins to fall. This phenomenon is also illustrated to a lesser degree in the high $\theta_{1}$ case of the right hand side panel of Figure 2.

\@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Supply of Highly Educated Workers and Education Premia}

In this section, we show how simulation of our static model can capture a puzzling phenomenon discussed in \citet*{Card/Lemieux:01:QJE}. How can dramatic increases in the education wage premium lead to only modest increases in the supply of highly educated workers? The authors note that, over a roughly 30 year period beginning in the early 1970s, the college-high school wage gap rose considerably in the United States, Canada, and the United Kingdom, and that this rise occurred mostly for younger workers. They argue that an important source of this trend is a stagnation in the rate of educational attainment among workers born in the 1950s and thereafter.

In Figure 3, we show how this pattern can be driven entirely by changes in the matching technology over time. The wage premium is measured as the difference between the average wages of the workers with high education and the average wages of workers with low education. Each point on the plot represents the average of 500 simulations of the model. We use Specification 1, $f=\theta_{1}hk$, under the same setup as before with only one difference; we choose the low value of $\theta_{1}$ to be 0.7 and the high $\theta_{1}$ to be 2.5. In the case that $\theta_{1}$ is very low, the effect of raising $\beta$ is to dramatically increase sorting without inducing a large benefit to the workers from acquiring education in equilibrium.

\@startsection{section}{1} \z@{1.0\linespacing\@plus\linespacing}{.8\linespacing}{Econometric Inference}

In this section, we outline the general empirical strategy for performing inference on the underlying model parameters. In Section (ref), we describe how the main model can be used to characterize the observed distribution of the matching of workers to firm and hence the wages of all the workers in the economy. The goal is to then use these representations to construct confidence regions for the preference and matching technology parameters.

However, if the model is high dimensional, the Monte Carlo inference approach may be cumbersome to apply in practice. For this reason, we propose a two-stage inference approach that relies on the construction of a first-stage confidence interval for a subset of the model parameters. We demonstrate this approach in practice in Section (ref) by estimating the Bayesian game from (ref) for fixed values of $\beta$.

\@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{Two-Stage Inference Accommodating Cross-Sectional Dependence of Observed Matching}

The econometrician observes a matching of workers to firms, $\mathbf{M}=(\mathbf{M}(i))_{i\in N_{h}}$, where for each $i\in N_{h}$, $\mathbf{M}(i)$ takes values in the set of firms.\footnote{Throughout this paper, we will suppose that the matching is one-to-one between workers and firms. In practice, “firms” in this context can be viewed as positions at particular firms. } The main challenge associated with inference is the fact that the distribution of $\mathbf{M}$ exhibits cross-sectional dependence of a complicated form. The matching of workers to firms can be thought of as discrete choice problem on the part of the firm where the choice sets of firms are endogenously constrained by the choices of firms with higher $v$-indices, which depends on $\beta$, $\eta$ and $k$. Hence, the event that worker $i$ matches to firm $j$ cannot be considered independent from the event that a worker $i'\neq i$ matches to firm $j$. Also, the fact that firm preferences may be heterogeneous means we cannot condition on the $v$-index and firm preferences in a way to remove the cross-sectional dependence as was done by \citet*{Diamond/Agarwal:17:QE}.

The econometrician observes the vector $\mathbf{M}\in\mathbf{R}^{n_{h}}$, which represents a matching of workers to firms. Given the serial dictatorship matching process, the joint distribution of $\mathbf{M}$ is known up to a parameter. Let $\mathbf{K}=(\mathbf{K}(i))_{i\in N_{h}}$, where $\mathbf{K}(i)=K_{\mathbf{M}(i)}$; i.e., the capital of the firm matched to by worker $i$.

Our model also implies that the finite sample distribution of wages, $(\mathbf{W}(i))_{i\in N_{h}}$, is known up to a parameter. Under Nash bargaining (and a specification of the post-match wage function based off an equation such as (ref)), we have for each $i\in N_{h}$ \[ \mathbf{W}(i)=w(H_{i},\mathbf{K}(i)). \] We denote all the match-related observables as $\mathbf{Y}=(\mathbf{K},\mathbf{M})$. $\mathbf{M}$ is observed whenever the researcher has matched employer-employee data. $\mathbf{K}$ is observed when the researcher can use the matching data, $\mathbf{M}$, and the firm capital data, $K$, to find the capital of the firm each worker in the sample is employed at. Using $\mathbf{Y}$ and worker observables $H$ and $X$, the econometrician wishes to infer $\lambda_{0}$.

\@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Finite Sample Inference on Parameters}

Next, we consider a test statistic that matches the moments of the distribution of the matched-related observables with their simulated counterparts. To simplify the exposition, we discuss the construction of a confidence interval for $\beta_{0}$ alone, i.e., supposing that we knew the true values of $\theta_{0}$. Denote $R+1$ as the total number of simulations in the Monte Carlo inference procedure. Drawing $\eta_{r}$ from some continuous parametric distribution function,\footnote{We will specify a particular parametric family that this distribution belongs to, along with additional assumptions, in Section (ref). } we simulate a version of the matching for each $\beta\in B$ and each $r=1,...,R+1$, which we write as $\mathbf{M}_{r}(\beta)=\{\mathbf{M}_{r}(i;\beta):i\in N_{h}\}$. The simulated wages are then \[ \mathbf{W}_{r}(i;\beta)=w(H_{i},K_{\mathbf{M}_{r}(i;\beta)})). \] It is convenient to define

Next, we will propose a test statistic that depends on both the observed matching data, $\mathbf{Y}$, and the simulated matching data, (along with simulated versions of this test statistic). That is,

An example of such a test statistic is one that compares the observed joint distribution of worker human capital and matched firm capital with simulated counterparts. For example, we may consider the test statistic\footnote{This test statistic is similar to the one used in \citet*{Kim/Schwartz/Song/Whang:19:Econometrics}. See also Diamond/Agarwal:17:QE.}

where

That is, $\hat{P}$ is an $J\times M$ matrix\footnote{In this example, we are implicitly assuming that the distribution of $K$ is discrete and has $M$ support points. We will make this assumption explicit in a subsequent section.} whose $(j,m)$ element is the estimated probability that a worker of education level $h_{j}$ matches to a firm of capital level $m$. $\hat{P}_{r}(\beta)$ is defined similarly to $\hat{P}$, except we replace the observed matching with the $r$th simulated matching, $\mathbf{M}_{r}(\beta)$.

Using our test statistic, we may compute a confidence region for $\beta$ as \[ C_{\alpha,R}^{\beta}=\{\beta\in B:T(\beta)\leq c_{\alpha,R}(\beta)\}, \] where the critical value is computed as the $(1-\alpha)$ -quantile of the empirical distribution of $\{T_{r}(\beta):r=1,...,R\}$:

Under Assumption (ref), it can easily be shown that finite sample inference on $\beta_{0}$ satisfies $P\{\beta_{0}\in C_{\alpha,R}^{\beta}\}\geq1-\alpha$ when the procedure outlined above involves the true parameter, $\theta_{0}$.

In practice, we do not know the true value of $\theta_{0}$. In situations in which the full parameter vector $\lambda_{0}$ is not very large, it may be feasible to construct a $(1-\alpha)100\%$ confidence region for this parameter that exhibits finite sample validity. That is, we construct

where $T(\lambda)$ and $c_{\alpha,R}(\lambda)$ are defined analogously to $T(\beta)$ and $c_{\alpha,R}(\beta)$. In the case that $\Lambda$ is high-dimensional, the finite sample procedure outlined above may not be practical due to the unreasonable computational cost. In the following subsection, we explore a two-stage inference approach that admits inference on $\beta_{0}$ when the researcher is able to construct a first-stage confidence region for a subset of the parameters, $\theta_{0}$.

Note that plugging in a consistent estimator of $\theta_{0}$, $\hat{\theta}_{n}$, for the true value in inference procedure outlined above will generally not lead to valid inference on $\beta_{0}$. This is because there is no reason to expect that plugging in $\hat{\theta}_{n}$ for $\theta_{0}$ will make the distribution of the simulated matching, $\mathbf{M}_{r}$, equal to the distribution of the observed matching, $\mathbf{M}$. The fact that $\mathbf{M}_{r}$ is not equal in distribution to $\mathbf{M}$, in turn implies that $\mathbf{K}_{r}$ does not follow the same distribution as $\mathbf{K}$. The severe consequences of estimation error in $\hat{\theta}_{n}$ occur because the firm preferences are typically misspecified at all values of $\theta$ other than the true value, $\theta_{0}$. Moreover, this problem is not alleviated by conditioning on $H,K,$ or exogenous variables. In the following section, we discuss a general two-stage inference approach that can be used when the econometrician can construct an (asymptotically) valid confidence first-stage confidence interval for $\theta_{0}$. In Section (ref), we extend our baseline economic model of Section (ref) in a manner that admits the application of this two-stage inference approach to our setup.

\@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Two-Stage Inference on $\beta$ using Test-Inversion Confidence Interval }

Suppose that we wish a ($1-\alpha$)-level asymptotic confidence interval for $\beta_{0}$, and can construct a confidence interval for $\theta_{0}$. Let us denote the test statistic and its simulated counterpart from the previous section, where the $\theta$ arguments make explicit the test statistic's dependence upon a given value of $\theta\in\Theta$:

Note that according to the notation we used in the last section we have $T(\beta;\theta_{0},\theta_{0})=T(\beta)$. Our inference on $\beta$ proceeds in two steps:

Step 1. Using the first stage estimates of $\hat{\theta}(\beta)$, we construct a confidence region for $\theta_{0}$, $\hat{C}_{\alpha/2}(\beta)$, with $(1-(\alpha/2))$ asymptotic coverage.

Step 2. Next, we construct a test statistic that doesn't involve $\theta$. Define

We now construct a confidence set for $\beta$ as

where the critical value $c_{1-(\alpha/2),R}^{*}(\beta)$ is computed as the $(1-(\alpha/2))$ -quantile of the empirical distribution of $\{S_{r}^{*}(\beta):r=1,...,R\}$; that is,

The following lemma establishes the asymptotic validity of the two-stage inference procedure.

In the following section, we provide assumptions under which we can construct a confidence region for $\theta_{0}$ using a maximum likelihood approach. In Section $\ref{sec:identification}$, we argue that $\theta_{0}$ is identified up to $\beta_{0}$, and provide standard conditions under which the maximum likelihood estimator is consistent and asymptotically normal. In Section $\ref{sec:MonteCarlo}$, we then present a small Monte Carlo study that illustrates how this estimator can be used as the basis for the first-stage inference on preferences. In particular, we show how a parametric bootstrap can be used to construct a $\hat{C}_{\alpha,R}$ with reasonable finite sample size and power properties.

\@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{First-Stage Estimation of Preferences }

In this section, we show how $\theta$ can be estimated for a particular fixed value of $\beta$. We will write an estimator of such an object as $\hat{\theta}(\beta)$. The main challenge associated with this problem is that of estimating the worker's expected utility from equation (ref). The problem is difficult because the workers must somehow resolve uncertainty associated with the serial dictatorship matching process in order to compute the expected output under the equilibrium education choices. In spite of these complications, it turns out that, under reasonable assumptions, the parameters are tractably estimable using discrete choice methods with a fixed point constraint when there are only two education choices. We now provide and discuss these assumptions.

Under part (a) of Assumption (ref), firms do not take covariates into account when forming their preference rankings over workers. Thus, workers with the same education level are equally desirable to any given firm. When worker $i$ considers the desirability of choosing education $h_{j}$, he need only consider the capital a generic agent who chooses level $h_{j}$ expects to receive in the matching process. In many contexts, (a) will be reasonable for a host of variables that affects the worker's education decision (e.g., marital status, number of dependent children).\footnote{In some cases in which employers do see these worker characteristics, they are prohibited from discriminating based on them due to state or federal anti-discrimination laws.} Part (b) says that workers know only the distribution of firm capital without knowing the precise realizations of capital. Assumption (ref) (b) also stresses that the worker's knowledge of the distribution of capital is not sufficient for knowledge of the distribution of the number of firms that prefers each education class, which will turn out to be crucial for our results of this section.

Part (a) says the distribution of firm capital has discrete support. In practice, we can let $M$ be as large as our application requires. In concert with (b) and the parametric structure for $v$ stipulated by equation (ref), (a) allows us to express the unconditional distribution of $v_{j}$ as a mixture of normals, $G\equiv\sum_{m=1}^{M}q_{m}F_{m}$, where $F_{m}$ is $N(\beta k_{m},\sigma^{2})$.\footnote{In the simulation sections of the paper we normalize $\sigma^{2}=1$ when we perform inference on the model parameters.} Part (c) is an assumption on the worker's unobserved costs that allows us to estimate the model parameters using conventional discrete choice methods.

We wish to obtain a convenient representation of each worker's conditional expectation of the production function, for each education level that the worker can choose. Under the model of Section (ref) the identity of the firm that worker $i$ matches with, $\mathcal{M}(i)$, depends on $K$, $H$, $\beta$, and, $\theta$. Therefore, for each $i\in N_{h}$ and $h_{j}\in\mathcal{H}$, we wish to estimate

where the expectation is taken with respect to the distribution of $K$, $H_{-i}$ and $\eta$. Under Assumption (ref) (a), we can express the expectation on the preceding line as

where $f_{j}=(f_{j1},...,f_{jm})'$ is an $M\times1$ vector with the $m$-th element of $f_{j}$ given as $f_{jm}=f(h_{j},k_{m})$ and $\pi_{j}^{(i)}=(\pi_{1j}^{(i)},...,\pi_{Mj}^{(i)})'$ is an $M\times1$ vector with the $m$-th element of $\pi_{j}^{(i)}$ given as

This is the probability that worker $i$ matches to a firm of capital level $k_{m}$ when he has chosen education level $h_{j}$.\footnote{Note that although these terms depend on $\theta$ and $\beta$, we will occasionally omit these from our notation for convenience.} Given that there are $M$ education levels, $J$ choices, and $n_{h}$ workers, the dimensionality of the problem appears daunting. However, under our assumptions the problem is simplified considerably, and we can show that for each $j$ and $m$, $\pi_{mj}^{(i)}=\pi_{mj}$, and hence, $\tilde{f}_{ij}=\tilde{f}_{j}$.\footnote{The argument for why this is the case is given in the proof of Proposition (ref).}

Although it is unclear how to represent $\pi_{mj}$'s analytically when the worker faces a choice between a large number of education levels, the problem becomes tractable when there are only two (i.e., $J=2$). Proposition (ref) shows that under our informational assumptions, firms (and workers) cannot distinguish between workers with the same education level during the matching process. As a consequence, we find that a worker is only concerned with the number of other workers who picked one of the two education levels (and not which particular workers chose what). Independence and identical distributions assumptions imply that the probability that $n_{j}$ workers picked education level $h_{j}$ can be represented using the binomial probability mass function. However, the number of workers choosing education level $h_{j}$ is unknown to workers, so they must take expectations. Thus, instead of having to sum over $n_{h}-1$ indices associated with actions of each of the other workers to compute the worker's expectation, we need only sum over one: the number of workers choosing a particular education level.

We will also allow $\theta$ to enter $\pi_{mj}$'s through the distribution of the number of firms that prefer high (or low) education. The following assumption is a natural way to specify this distribution. We use the notation $M_{j}^{+}(\theta)$ to denote the set of firm types that prefer education level $h_{j}$.\footnote{That is, $M_{j}^{+}(\theta) =\{m\in\{1,...,M\}:\rho(k_{m},h_{j};\theta)\geq\rho(k_{m},h_{j'};\theta),j\neq j'\}$. See also the discussion before Proposition (ref).}

The explicit representation of the matching probabilities are given in Propositions (ref), (ref), and Lemma (ref). These results can be used to construct estimates of the $\pi_{mj}$'s - and hence the $\tilde{f}_{j}$'s - for fixed values of $\theta$ and $\beta$. Using a given functional form for the production function, we denote an estimate of the expected production function when the worker chooses education level $h_{j}$ as

where our notation emphasizes the dependence of the objects upon the parameter values. To construct $\hat{\pi}_{mj}$'s we must estimate the terms of equation (ref). $\hat{P}(n_{j})$ is constructed as $B(n_{j};n_{h}-1,\hat{p}_{j})$ where the latter denotes the binomial probability mass function with $\hat{p}_{j}=P(H_{i}=h_{j})$.\footnote{In so doing, we pursue a two-step approach for estimating the choice probabilities, such as \citet*{bajari2013game}. See for example \citet*{Kasahara/Shimotsu:12:Ecta} for an alternative approach.} Similarly, $\hat{P}(n^{(j)};\theta)$ is constructed as $B(n_{j};n_{h}-1,\hat{q}_{j}(\theta))$, where $\hat{q}_{j}(\theta)=\sum_{m\in\hat{M}_{j}^{+}(\theta)}\hat{q}_{m}$, with $\hat{q}_{m}=\hat{P}(K_{j}=m)$, \[ \hat{M}_{j}^{+}(\hat{\theta})=\left\{ m\in\{1,...,M\}:\hat{\rho}(k_{m},h_{j};\theta)\geq\hat{\rho}(k_{m},h_{j'};\theta),j\neq j'\right\} , \] and $\hat{\rho}(k_{m},h_{j};\theta)$ is as in equation ((ref)), except we use $\hat{g}_{j}=\frac{1}{n}\sum_{i=1}^{n}g(h_{j},X_{i})$ in place of $\tilde{g}$.

Lastly, the $P_{h_{j},n_{j},n^{(j)}}(m)$'s, from equation (ref) - that is, the probability that a worker matches to a firm of type $m$ when they choose education level $h_{j}$, $n_{j}$ other workers choose $h_{j}$, and $n^{(j)}$ firms prefer $h_{j}$ - can be simulated for fixed values of $\theta$ and $\beta$. Propositions (ref) and (ref) show how these can be represented using probabilities involving order statistics. Under Assumption (ref) (b), we can construct $\hat{P}_{h_{j},n_{j},n^{(j)}}(m)$'s by averaging functions of simulated draws of beta-distributed random variables.\footnote{In particular, see Corollaries (ref) and (ref), which follow the order statistic result in Lemma (ref).}

Once we have estimated $\hat{f}_{j}(\theta,\beta)$ for each education level, we may use the specification of the wage from equation (ref) to write the expected wage as

When there are two choices ($J=2$), the worker chooses high education ($h_{j}=1$) if and only if \[ U_{1i}^{*}-U_{i0}^{*}>0. \] Under the assumption that $\varepsilon_{i}$'s follow the extreme value distribution (Assumption (ref)), the probability that worker $i$ chooses high education can be written as \[ \hat{p}_{i}(\theta,\beta)=\frac{\exp(\hat{\omega}_{1i}(\theta,\beta)-\hat{\omega}_{0i}(\theta,\beta))}{1+\exp(\hat{\omega}_{1i}(\theta,\beta)-\hat{\omega}_{0i}(\theta,\beta))}. \] Since the covariates $\{X_{i}\}_{i=1}^{n}$ are iid we can write the joint likelihood as the product of the marginal likelihoods. We can then define the estimator of $\theta$ (for a fixed value of $\beta$) as the minimizer of the standard logit likelihood function:\footnote{When $\beta$ is fixed, maximizing the likelihood by computing the $\hat{f}_{j}(\theta,\beta)$'s for each candidate value of $\theta$ can be slow. The following strategy can be used to estimate $\theta$ for fixed $\beta$ more quickly provided that the support of $K$ is not too large. First, note that $\theta$ enters $\hat{f}_{j}(\theta,\beta)$ only through the set of firm types that prefer education level $h_{j}$, $\hat{M}_{j}^{+}(\theta)$. Given our assumptions on the production function and firm preferences, $\hat{M}_{j}^{+}(\theta)$ must take one of $M+1$ possible values. Therefore, for fixed $\beta$ , we can avoid simulating $\hat{f}_{j}(\theta,\beta)$ for each candidate value of $\theta$ by pre-allocating the $\hat{q}_{j}(\theta)$'s and $P_{h_{j},n_{j},n^{(j)}}(m)$'s for each of the $M+1$ cases for $\hat{M}_{j}^{+}(\theta)$. It then suffices to evaluate $\hat{M}_{j}^{+}(\theta)$, select the appropriate dimension of the array of terms, then assemble the terms according to equation (ref).}

\[ \ln L_{n}(\theta,\beta)=-\sum_{i=1}^{n}\left[h_{i}\ln\hat{p}_{i}(\theta,\beta)+(1-h_{i})\ln(1-\hat{p}_{i}(\theta,\beta))\right]. \]

\@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{Matching Probabilities }

In this section, we consider the role of frictions, or the magnitude of $\beta_{0}$ relative to the variance of $\eta$, in shaping matching patterns between workers and firms. Note that these frictions play no role in determing firm preferences, or which firm types prefer high education.\footnote{We discuss the role of firm preferences on matching patterns at the end of Section (ref).} Nevertheless, because the frictions do affect sorting patterns, they are of considerable importance to workers when they decide how much to invest in education.

In the following example, we will suppose that that the set of firms that prefer education level $h_{j}$, $M_{j}^{+}$, contains at least two types of firms, $m$ and $\tilde{m}$ with $k_{m}\neq k_{\tilde{m}}$. Suppose we fix $N_{j}$, the number of workers who chose education level $h_{j}$, at some $n_{j}$ and we fix $N^{(j)}$, the number of firms who prefer highly-educated workers at some $n^{(j)}$ such that $n_{j}+1<n^{(j)}$. In this situation, there are strictly more firms who prefer type $h_{j}$ workers than there are workers of this type. Let $\kappa=n^{(j)}-n_{j}+1$, and denote $p_{m\kappa}\equiv P(v_{m}>v_{(\kappa)})$ for each $m$ in $M_{j}^{+}$. Proposition (ref) says that the difference in the probability of matching to a type $\tilde{m}$ versus a type $m$ firm at these values of $n_{j}$ and $n^{(j)}$ in such a situation is given by

with \[ c_{\kappa}\equiv\sum_{m\in M_{j}^{+}}p_{m\kappa}q_{m}^{+}, \] where $q_{m}^{+}=q_{m}/\sum_{m\in M_{j}^{+}}q_{m}$. Under Assumption (ref), the case of $\beta_{0}=0$ gives us that $p_{mk}=p_{\tilde{m}k}$, implying that the first term in the parentheses of equation (ref) is zero. This means that when matching frictions are highest (i.e., when $\beta_{0}=0$), the difference in the probability of matching to one type of firm that prefers $h_{j}$ over another is captured by the relative prevalence of those types of firms in the economy.

In the case that $\beta_{0}>0$, Assumption (ref) implies that $p_{\tilde{m}\kappa}-p_{m\kappa}$ becomes larger as $k_{\tilde{m}}-k_{m}$ becomes larger. This means that higher capital firms have a better chance of matching with the high education workers when $\beta_{0}>0.$ On the other hand, in the case that $n_{j}+1>n^{(j)}$ (i.e., $h_{j}$ is demanded by fewer firms than there are in the economy), then the above probabilities once again depend solely on the relative prevalence of the each type of firm.

\@startsection{section}{1} \z@{1.0\linespacing\@plus\linespacing}{.8\linespacing}{A Small Monte Carlo Simulation Study}

In this section, we investigate the finite sample size and power properties of the estimator of preferences, $\hat{\theta}_{n}(\beta)$, under a variety of parameters and functional form assumptions. The results in this section are for the case where the matching technology, $\beta$, is known to the econometrician.

We consider the following general structure for the worker's expected utility function: \[ \tilde{U}_{i}=\left(f_{i}+g_{i}\right)/2+d(H_{i})\varepsilon_{i}, \] where $\theta=(\theta_{1},\theta_{2})'\in\mathbf{R}^{2}$, $X_{i}\in\mathbf{R}^{2}$, and $d(H_{i})$ is a $2\times1$ vector with one in the $H_{i}$-th row where $H_{i}$ takes values of one or two. $X_{i}=(X_{1i},X_{2i})'$ are drawn independently across $i$ and one another from $U[0,1].$ Firm capital takes the value of $1/2$ and $1$ with equal probability. We also suppose that $\varepsilon_{i}\in\mathbf{R}^{2}$ follows the extreme value distribution so that the best response probability function has the logit structure. $\eta$ is drawn independently from the standard normal distribution. We use 100 draws of beta random variables to compute the matching probabilities. We also interpolate the supports of $N^{(j)}$ and $N_{(j)}$ so that they have $n/50$ support points rather than $n-1$ support points. In our experiments, we set the true value of preferences to be $\theta_{0}=(1,1)'$.

We consider two functional forms for the production function which we call $f_{i1}$ and $f_{i2}$:

$f_{i1}$ implies production complementarities between the worker and firm variables whereas any complementarities in $f_{i2}$ are forced through the worker's expectation of firm capital $\pi_{i}'k$. We also consider the performance of the inference under two versions of the outside option, $g_{i1}$ and $g_{i2}$:

Note that these choices of the outside option function ensure non-negativity. For each simulation sample, the $H_{i}$'s are generated as follows. First, we solve for fixed point in the best response operator to obtain $\mathbf{P}^{*}$.\footnote{In experiments with different starting values, iterating the best response operator yielded the same fixed point each time.} Then we compute the best response at the simulated covariates

\[ \Psi_{i}(H_{i}|X_{i},\mathbf{P}_{-i}^{*})=\frac{\exp(\tilde{U}_{i}^{*}(H_{i},X_{i}))}{\sum_{j=1}^{2}\exp(\tilde{U}_{i}^{*}(H_{j},X_{i}))}. \] Letting $\Psi_{i}^{*}(X_{i})\equiv\Psi_{i}(H_{i}|X_{i},\mathbf{P}_{-i}^{*})$ we then generate the actions as, \[ H_{i}=1\{\Psi_{i1}^{*}>\omega_{i}\} \] where $\omega_{i}$'s are drawn iid from the uniform distribution on $[0,1]$.

\@startsection{section}{1} \z@{1.0\linespacing\@plus\linespacing}{.8\linespacing}{Conclusion}

This paper presents an empirical strategy for studying wages and education in a labour market where the decisions of workers matter in the matching process. We demonstrate the feasibility of our approach in the case that the worker faces a choice between two education levels.

One limitation of the current approach is its reliance on cross-sectional variation alone for inference. In effect, useful information concerning unemployment and job-to-job transitions by workers is unused in our framework.

This paper has also demonstrated how the decision to invest in education - and wage inequality - is sensitive to the presence of a particular source of matching frictions in the economy. Although firm capital is exogenous in this paper, the role of information frictions on capital accumulation in an extended framework could be a fruitful way to study not only wage inequality, but also economic growth.

\@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}*{Acknowledgements} I thank Kyungchul Song, David Green, Vadim Marmer, and Florian Hoffmann for their advice and kind support throughout the course of this project. I have also benefited from comments from Joris Pinske, Aureo de Paula, Hiro Kasahara, Paul Schrimpf, Michael Peters, Anna Rubinchik, Jonathan Graves, Anujit Chakraborty, and the participants of presentations at the University of British Columbia, the University of Haifa, and the University of California (Davis).

\@startsection{section}{1} \z@{1.0\linespacing\@plus\linespacing}{.8\linespacing}{Appendix A: Equilibrium Existence and Uniqueness}

In this section, we characterize the equilibrium of the incomplete information game of Section (ref). First, we introduce a representation of the worker's expected utility function that proves useful for establishing the existence of the Bayesian Nash equilibrium of the game as a fixed point of a best probability response operator. We begin by defining relevant terms. A profile of strategy functions (or decision rules) is \[ \sigma=\{\sigma_{i}(x_{i},\varepsilon_{i}):i\in N_{h}\}, \] where the functions $\sigma_{i}:\mathcal{X}\times\mathbf{R}^{J-1}\rightarrow\mathcal{H}$. The conditional probability that a worker with covariates $x_{i}$ chooses action $h_{i}$ can be written

\[ P_{i}(h_{i}|x_{i},\sigma_{i})\equiv\int1\{\sigma_{i}(x_{i},\varepsilon_{i})=h_{i}\}dF(\varepsilon_{i}). \] Since $X_{i}$'s are private information in this model, each agent $i$ must take expectations with respect to the distribution of $X_{-i}$. The following result shows that under the independence assumptions embodied by Assumption (ref), the agent's expected utility has a very convenient form - it is only affected by the behaviour of the other agents through the choice probabilities.

We will show the existence of the equilibrium for our model. The solution concept for the game described in Section (ref) is Bayesian Nash Equilibrium (BNE), which we now define.

The notation and arguments in this section follow \citet*{Aguirregabiria/Mira:19:WP}, but we include them here for completeness. Under Assumption (ref), we write the expected utility of $i$ as

By Lemma (ref) we can express the first term on the right hand side of the the preceding equation as

Note that $\tilde{U}_{i}(h_{i},x_{i},\sigma)$ only depends on the choices of other agents through the choice probabilities of the other players that are induced by $\sigma$. We write the choice probabilities of the people other than $i$ as \[ \mathbf{P}_{-i}\equiv\{P_{j}(h_{j}):(j,h_{j})\in N\backslash\{i\}\times\mathcal{H}\backslash\{1\}\}. \] For any $\mathbf{P}_{-i}$, we can define a best response probability function as:

$\tilde{\Psi}_{i}$ tells us the probability that a particular action is optimal for $i$ with covariates $x_{i}$ when others choose according to probabilities $\mathbf{P}_{-i}$.\footnote{Note that when $\varepsilon_{i}$'s have the extreme value distribution (as in Assumption (ref)) then we have

\[ \tilde{\Psi}_{i}(h_{i}|x_{i},\mathbf{P}_{-i})=\frac{\exp(\tilde{U}_{i}(h_{i},x_{i}))}{\sum_{j=1}^{J}\exp(\tilde{U}_{i}(h_{j},x_{i}))}. \] } Let

An equivalent to Definition 1 (ref) is that the equilibrium probabilities, $\mathbf{P}^{*}\equiv\mathbf{P}(\sigma^{*})$, satisfy the fixed point constraint, $\mathbf{P}^{*}=\Psi(\mathbf{P}^{*})$, where $\Psi$ is the best response probability mapping:

We can demonstrate that the Bayesian equilibrium is unique under mild conditions on the derivatives of the best response probability mapping. Define $J_{n}(\boldsymbol{P})\equiv\frac{\partial\Psi(\boldsymbol{P})}{\partial\boldsymbol{P}'}-I_{n}$, where $I_{n}$ is the identity matrix and let $\det(A_{n})$ denote the determinant of an $n$-by-$n$ matrix, $A_{n}$. A result of Kellogg:76:PAMS, as stated in Konovalov/Sandor:10:ET, says that the equilibrium is unique if $\Psi$ has no fixed points on the boundary of $\mathcal{P}$ and provided that $\det(J_{n}(\boldsymbol{P}))$ is non-zero for each $\boldsymbol{P}\in\mathcal{P}$. Note that under the conditions of Lemma (ref), $J_{n}(\boldsymbol{P})$, is a matrix with $-1$'s on the diagonal and $\varphi(\boldsymbol{P})=\partial\Psi_{i}(\boldsymbol{P})/\partial p_{j}$ for all $i\neq j$ (the off-diagonals). Therefore, $J_{n}(\boldsymbol{P})$ is a circulant matrix implying the following explicit formula: \[ \det(J_{n}(\boldsymbol{P}))=\left(\varphi(\boldsymbol{P})\cdot(n-1)-1\right)\left(-(1+\varphi(\boldsymbol{P}))\right)^{n-1}. \]

This determinant is guaranteed to be non-zero provided that $\varphi(\boldsymbol{P})\neq1/(n-1)$ for every $\boldsymbol{P}\in\mathcal{P}$ and $n\geq2$. In our setup, the requirement that $\Psi$ have no fixed points on the boudnary of $\mathcal{P}$ holds under weak conditions on the distribution of $\varepsilon_{i}'$s.

\@startsection{section}{1} \z@{1.0\linespacing\@plus\linespacing}{.8\linespacing}{Appendix B: First-Stage Estimation of $\theta_0$} \@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{Characterization of Matching Probabilities} The remaining results of this section allow us to represent the matching probabilities from equation (ref), hence workers' expectations, in a convenient way. These representations can then be used to estimate $\theta(\beta)$ using maximum likelihood.

When $J=2$ we can partition the types of firms, $m=1,...,M$ into two sets: those who prefer $h_{j}\in\mathcal{H}$ and those who prefer $h_{j'}$ with $j'\neq j$. It is convenient to introduce the following notation:

recalling that firm preferences are given in (ref). The firm classes that prefer $h_{j}$ are pinned down by the functional form for firm preferences, $\rho$, the preference parameter, $\theta$, and the distribution of $X_{i}$. Furthermore let us denote

where

Note that this object depends on both $\beta$ and $\theta$ through the matching function. For each firm type $m=1,...M$ let $F_{m}\equiv N(\beta k_{m},\sigma^{2})$, and define the following for each education choice $h_{j}$: \[ G_{j+}\equiv\sum_{m\in M_{j}^{+}}q_{m}F_{m}\text{ and }G_{j-}\equiv\sum_{m\in M_{j}^{-}}q_{m}F_{m}. \] Furthermore, define the posterior firm types as follows: \[ q_{m}^{+}\equiv q_{m}/\sum_{m\in M_{j}^{+}}q_{m}\text{ and }q_{m}^{-}\equiv q_{m}/\sum_{m\in M_{j}^{-}}q_{m}. \] We also define $v_{(b_{1},b_{2};F)}$ as the $b_{1}$-order statistic of $b_{2}$ random variables independently distributed according to cdf $F$. Propositions (ref) and (ref) are characterizations of $P_{h_{j},n_{j},n^{(j)}}(m)$'s of the model in the case that $J=2$ and $n_{h}=n_{f}=n$.

When considering these results, it is important to recall one core feature of the matching model as we outline it in Section (ref): that there is no unemployment. Therefore, when reading the arguments, the reader should take for granted the fact that the probability that each worker matches to some firm occurs with probability one.

Next we define $G\equiv\sum_{m=1}^{M}F_{m}q_{m}$. Proposition (ref) characterizes the matching probabilities in the case that all firms types prefer one level of education; that is, in the case that firm preferences are homogeneous over worker education types. The arguments are abridged, since they are very similar to those used in the proof of Proposition (ref).

The following result takes for granted a well-known fact that uniform order statistics follow the Beta distribution.\footnote{For example, see Chapter 2 \citet*{Ahsanullah/Nevzorov/Shakil:13:Springer}. }

The following results are a direct application of the previous results. They are useful for constructing the $\pi_{mj}$'s that are used in the structural estimation of this paper. Recall the definitions of $G$, $G_{j+}$, $G_{j-}$, and $v(b_{1},b_{2},F)$ from before. We introduce the following notation: \[ a(\kappa,n,m;G)\equiv\mathbf{E}\Phi\left((G^{-1}(U_{(\kappa;n)})-\beta k_{m})/\sigma_{m}\right), \] where $U_{(\kappa;n)}$ is the $\kappa$-order statistic of $n$ uniform random variables and $\mathbf{E}(\cdot)$ is taken over the distribution of $U_{(\kappa;n)}$.

\@startsection{subsection}{2} \z@{.8\linespacing\@plus.7\linespacing}{.7\linespacing}{Identification of $\theta_0$}

In extremum estimation problems, it is common to argue that the parameter is identified via a set of sufficient conditions requiring continuity (or semi-continuity) of the underlying population criterion function. See for example, \citet*{Newey/McFadden:94:HE}. In our context, assuming continuity of this criterion function is inappropriate, since the probabilities that workers match to certain firm types conditional on their choice of education are discontinuous in $\theta$. The problem arises because these probabilities depend on the sets of firm types that prefer high and low education, which are discontinuous functions of $\theta$.\footnote{Recall the definitions in equation (ref).} Under the model of Section (ref) we provided assumptions under which it was natural to estimate $\theta$ using maximum likelihood. In this section, we show that the discontinuity of the population likelihood function is not an obstacle to identification. As we will see, the required conditions are only slightly stronger than those commonly used to identify the standard logit model.

Before presenting the identification result and the proof, we define terms. The population objective function is

where $p_{i}(\theta)=\Lambda(\Delta_{i}(\theta)),$ $\Lambda(z)=\exp(z)/(1+\exp(z))$, and $\Delta_{i}(\theta)=\omega_{1i}(\theta,\beta_{0})-\omega_{0i}(\theta,\beta_{0})$ are the population version of the objects in equation (ref).\footnote{Note that our notation in this section will occasionally omit $\beta_{0}$ for the purposes of clarity. } Let us also define $\psi(X_{i};\theta)\equiv g(h_{1},X_{i};\theta)-g(h_{0},X_{i};\theta)$ and write the probability of high education as

where

and $\phi_{j}(\theta)'\equiv(f(h_{j},k_{1};\theta),...,f(h_{j},k_{M};\theta))'\in\mathbf{R}^{M\times1},\text{ for }j=0,1.$ We are now ready to present the main result of this section.

The requirement that $\mathbf{E}|\psi(X_{i};\theta)|<\infty$ follows under familiar circumstances. For example, when we take an outside option of $g(h,x;\theta)=h\cdot x'\theta$, the condition follows from the existence of finite second moments. Our condition is more general so as to allow additional flexibility in the choice of $g$. The existence of $|\Phi(\theta)\Pi(\theta)|$ is also satisfied under very mild assumptions. Since $\pi_{1}(\theta,\beta_{0})$ and $\pi_{0}(\theta,\beta_{0})$ are always bounded between zero and one, a sufficient condition for the existence of $|\Phi(\theta)\Pi(\theta)|$ is that $f$ be such that $f^{2}(h,k;\theta)<\infty$ for each $\theta\in\Theta$, $h$, and $k$.

The following result establishes a set of sufficient assumptions for the first condition of Theorem (ref).

For example, it is clear that linear outside option functions satisfy the conditions of the above lemma. An example of a non-linear outside option function satisfying the conditions is $g(h,x'\theta)=\exp(h\cdot x'\theta)$.\footnote{One way in which Theorem (ref) appears strong is that it implicitly assumes that all elements of $\theta$ enter the outside option. However, the simulation study presented in Section (ref) offers evidence that $\hat{\theta}_{n}(\beta_{0})$ remains consistent when this requirement is relaxed somewhat. In particular, the simulation demonstrates that the inference performs well when a single element of $\theta_{0}$ enters the production function but not the outside option.}

Consistency of $\hat{\theta}_{n}(\beta_{0})$ holds when the data are iid and $\Theta$ is compact under Theorem 2.5. of \citet*{Newey/McFadden:94:HE}, under the additional requirement that the likelihood function be continuous at each $\theta\in\Theta$ with probability one. As pointed out by the authors, the latter condition is mild in the sense that it does not require the likelihood be continuous at every $\theta$ for a given realization of the random variables. In our context, this requirement is satisfied if the probability that a firm is indifferent between a worker with high and low education is zero for every value of the preference parameter, $\theta$. Asymptotic normality of $\hat{\theta}_{n}(\beta)$ then follows provided the conditions of Theorem 3.3 of \citet*{Newey/McFadden:94:HE} hold. One of the requirements of this asymptotic normality theorem is that the population density be twice differentiability in $\theta$ within a neighborhood of $\theta_0$. Although the density in our context is not guaranteed to be differentiable for every $\theta\in\Theta$, it may nonetheless satisfy this much weaker requirement of twice differentiability in a neighborhood of the true parameter.