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A Note on the Estimation of Job Amenities and Labor Productivity
{ Keywords: Matching, Observed transfers, Structural estimation, Value of Statistical Life.}
{ JEL Classification: C35, C78, J31.\vskip50pt }
Identification and estimation of both agents' value of a match in one-to-one matching models with transferable utility have been the subject of increasing interest in the last decade. Two important applications are in the fields of family economics with the marriage market (where the econometrician observes matching patterns, but not the transfers) and labor economics with the labor market, or more generally the literature on hedonic models (where the econometrician observes both the matching patterns and the transfers), although the former has thus far received most of the attention.
In the case when transfers are not observed, thus in the case of the marriage literature, Choo and Siow (2006) is a seminal reference which allowed to bring theoretical models to the data. Subsequent references such as Chiappori et al. (2015), Galichon and Salani\'{e} (2021) and Dupuy and Galichon (2014) have extended the structure of the model in various dimensions. In particular, Dupuy and Galichon (2014) have provided a framework for estimation of a matching model where agents match on continuous characteristics, which they have applied to marriage market data.
In the case when transfers are observed, however, transfers may potentially provide useful supplementary information about the partners' values of a match. In the analysis of the labor market, for example, wages may be observed. The literature referred to above is not very explicit on how this information may be used. Many authors, such as Ekeland et al. (2004), Heckman et al. (2010) and Galichon and Salani\'{e} (2021), among others, suggest techniques that implicitly or explicitly require to perform nonparametric estimation (\textquotedblleft hedonic regression\textquotedblright ) of the wage curve prior to the analysis. While this works well in the case when the relevant characteristics is single-dimensional, as in Ekeland et al. (2004) and Heckman et al. (2010), or discrete, as in Galichon and Salani\'{e} (2021), this is more involved when the characteristics are continuous and multivariate. In this framework, Salani\'{e} (2015) shows that this structure implies quite strong testable restrictions.
In this paper, we build a flexible and tractable model of equilibrium matching and wages on the labor market, and show how to estimate the model using a maximum likelihood approach. This work therefore extends our previous work, Dupuy and Galichon (2014), to the case when transfers are observed.
We illustrate our method by revisiting the literature on compensating wage differentials (CDW) initiated as an application of Rosen's (1974) hedonic model to the labor market by Lucas (1977) and Thaler and Rosen (1976), and soon followed by many others; see Rosen (1986) for an elegant presentation of the theory and a review of the early empirical literature, and Viscusi and Aldy (2003) for a more recent review of the empirical literature. The approach in this vein consists in performing the reduced form estimation of the risk-wage gradient to uncover workers' marginal willingness to accept certain levels of fatal injury risk at their job and herewith derive an estimate of the Value of Statistical Life. A crucial assumption of this approach is that the data contains rich enough information about a worker's skills to control for wage differentials due to productivity differentials across workers. Departure from this assumption implies an inherent bias in estimates of the compensating wage differentials, and Hwang et al. (1992) have shown this bias can be large in magnitude. Attempts to avoid this bias have consisted in either using panel data to estimate workers' fixed effects and control for unobserved heterogeneity (see for instance Brown, 1980) or an instrumental variable approach (see for instance Garen, 1988). More recently, Kniesner et al. (2007) have argued in favor of adopting a structural hedonic model to identify the “underlying fundamentals (preferences), [...] that would further generalize estimates of Value of Statistical Life”.
To the extent of our knowledge, however, the structural hedonic model approach has been largely ignored in the applied literature. Our method contributes to this discussion by proposing a structural estimation of preferences for risky jobs that explicitly takes into account the matching of workers to jobs while estimating the equilibrium hedonic wage equation. This method comes at no cost on the data since information about who matches with whom is, by definition, already available in the data needed to perform the hedonic wage regression in the first place. Accounting for the matching of workers to jobs results in the likelihood of observing the data given parameters being expressed as a weighted sum of two terms: the contribution of the first term is to equate the predicted moments of the matching distributions to their sample counterparts whereas the contribution of the second is to equate the predicted wages with their sample counterparts.
Following Viscusi (2003, 2007 and 2013), we use US data and merge the Census of Fatal Occupational Injuries (CFOI) by occupation and industry to the 2017 Current Population Survey (CPS). This allows us to have access to data on hourly wages, workers' characteristics and the rate of fatal injuries in their job. Our main results quantify the extent to which US workers dislike risky jobs, their utility dropping by log-points per hour of work as the probability of fatal injury on the job increases by per $100,000$. This amounts to a Value of Statistical Life (VSL hereafter) of \$million (\$2017). This estimate is about \$3 million lower, though not statistically so, than the estimate obtained when applying a hedonic (log)wage regression that does not account explicitly for the sorting of workers into jobs, i.e. \$9.7 million (\$2017), which itself lies in the range of previous estimates using similar data (e.g. Viscusi, 2013).
Our model is also related to a growing empirical literature applying the celebrated estimation technique proposed in Abowd et al. (1999) to decompose workers' wage differentials into differentials due to observed workers' characteristics, unobserved workers' heterogeneity and firms' heterogeneity using matched employer-employee panel data, see among others Abowd et al. (2002), Andrews et al. (2008;2012), Gruetter and Lalive (2009), Woodcock (2010) and Torres et al. (2013). Workers and firms fixed effects capture reduced form notions of workers and firms types that are fixed over time and are identified using the mobility of workers across firms over time.
The outline for the rest of the paper is as follows. Section 2 introduces the model and characterizes equilibrium. Section 3 presents our parametric specification of the model and a maximum likelihood estimator on data about matches and wages. Section 4 presents the empirical application and section 5 summarizes and concludes.
The purpose of this section is to succinctly present our model, which is a bipartite continuous matching model with transferable utility and logit unobserved heterogeneity. In our context, equilibrium transfers (wages) are observed. This is relevant for instance in the labor market, as opposed to the marriage market where transfers are typically unobserved. We limit ourselves to the introduction of the notation needed for the construction of our estimator, emphasize on the additional identification and estimation results obtained when transfers are observed and refer the interested reader to the original paper, i.e. Dupuy and Galichon (2014), for more details about its other main properties. To fix ideas, we use in the remainder of the paper the example of the labor market where transfers (wages) are observed, in line with the application of Section 4.
Populations and matching. We shall assume that workers' characteristics are contained in a vector of attributes $x\in \mathcal{X}= \mathbb{R}^{d_{x}}$, while firms' characteristics are captured by a vector of attributes $y\in \mathcal{Y}=\mathbb{R}^{d_{y}}$.\footnote{In our one-to-one matching model, the terms \textquotedblleft job\textquotedblright\ and \textquotedblleft firm\textquotedblright\ are interchangeable. Our model would continue to work in a one-to-many context where firms offer multiple jobs as long as there is perfect substitutability between jobs (workers) within firms, i.e. as long as the surplus of a firm is the sum of the surplus at each job (of each worker) in the firm. Our model can therefore be seen as a matching model of jobs to workers within firms under perfect substitutability. A recent application of this idea is found in a model of polygamy in the marriage market context by Andr\'{e} and Dupraz (2017).}
Our first main assumption is about the distribution of workers' and firms' types in the economy.
Since workers of type $x$ have a density of probability $f\left( x\right) $, and firms of type $y$ have a density of probability $g\left( y\right) $ and workers and firms are in equal number, a feasible matching between workers and firms will consist in the probability density ${\Greekmath 0119} \left( x,y\right) $ of occurrence of a $\left( x,y\right) $ pair, which should have marginal densities $f$ and $g$. More formally, we define the set of feasible matchings as
Demand and supply. Let $w\left( x,y\right) $ denote the wage of a worker of type $x$ when working for a firm of type $y$. It is assumed that a worker of type $x$ not only values her wage but also the amenities of her job. The value of amenities is further assumed to be decomposed into a systematic value ${\Greekmath 010B} \left( x,y\right) $, which is the same for all workers of type $x$, and a random value ${\Greekmath 0122} \left( y\right) $ that is specific to a particular worker, holds for all firms of a given type $y$ and is known by the worker at the time the matching occurs. This specification therefore contrasts with the literature on search and matching that typically introduces a match-specific shock that is revealed after the matching occurred.
In particular, we assume the value for a worker of type $x$ of working for a firm of type $y$ at wage $w\left( x,y\right) $ is given by ${\Greekmath 010B} \left( x,y\right) +w\left( x,y\right) +{\Greekmath 011B} _{1}{\Greekmath 0122} \left( y\right) $ where ${\Greekmath 010B} \left( x,y\right) +w\left( x,y\right) $ is deterministic, ${\Greekmath 0122} \left( y\right) $ is a worker-specific random process and ${\Greekmath 011B} _{1}$ is a scaling factor. As in Dupuy and Galichon (2014), we choose to model the random process ${\Greekmath 0122} \left( y\right) $ as a Gumbel random process, introduced by Cosslett (1988) and Dagsvik (1988), which is constructed as follows.\footnote{See appendix (ref) for more details about Gumbel random processes.} Assume that, in a first step, workers form their demand by drawing a random pool of observable types of firms, along with the corresponding utility shocks. We model the random pool by a Poisson point process, so that its cardinality does not have to be fixed and finite. More specifically, we assume that this Poisson process is valued in $\mathcal{Y\times \mathbb{R} }$ with intensity $dye^{-{\Greekmath 0122} }d{\Greekmath 0122} $. The random pool sampled by a worker is $\left\{ \left( y_{k},{\Greekmath 0122} _{k}\right) ,k\in \mathbb{N} \right\} $, where $y_{k}$ is the type and $ {\Greekmath 0122} _{k}$ the corresponding utility shock. Define $ {\Greekmath 0122} \left( y\right) =\max_{k}\left\{ {\Greekmath 0122} _{k}:y_{k}=y\right\} $, with the convention that $\max \emptyset =-\infty $.
By construction, the problem of a utility-maximizing worker of type $x$ reads as
Note that workers' preferences only depend on their potential partner's type. Once the desired type has been determined, workers are indifferent between firms of that type.
By symmetry, we assume the value for a firm of type $y$ of hiring a worker of type $x$ at wage $w\left( x,y\right) $ is given by ${\Greekmath 010D} \left( x,y\right) -w\left( x,y\right) +{\Greekmath 011B} _{2}{\Greekmath 0111} \left( x\right) $ where ${\Greekmath 010D} \left( x,y\right) -w\left( x,y\right) $ is deterministic, and ${\Greekmath 0111} \left( x\right) $ is a firm-specific Gumbel random process.
Assuming that ${\Greekmath 0122} \left( y\right)$ and ${\Greekmath 0111} \left( x\right)$ follow Gumbel random processes allows us to get a continuous logit framework. Indeed, Proposition (ref) in the appendix, which was obtained by Cosslett (1988) and Dagsvik (1988), shows that the density of demand for firms of type $y$ originating from workers of type $x$, is proportional to $\exp({\Greekmath 010B}(x,y)+w(x,y))$. This leads us assuming continuous logit demands from workers and firms, which we formalize in the following assumption.
Note that the conditional density demands rewrite as
where $u\left( x\right) $ given by
interprets in the Gumbel framework as the expected indirect utility of a worker of type $x$ and symmetrically,
where $v\left( y\right) $ given by
interprets as the expected indirect profits of a firm of type $y$.
In a second step, agents determine equilibrium by tatonnement over $w(x,y)$ using the demand functions defined in step 1. Note that since workers (resp. firms) are indifferent between firms (workers) of the same type, once the desired type has been determined, workers (resp. firms) can match with any of the firms (workers) of that type.
Note also that agents need not to observe other agents' idiosyncratic shocks $ {\Greekmath 0122} $ and ${\Greekmath 0111} $ to form their demand. Even if they had access to that information, they would not use it.
Equilibrium. At equilibrium, the wage curve $w\left( x,y\right) $ is such that the density ${\Greekmath 0119} \left( x,y\right) $ of pairs $\left( x,y\right) $ emanating from the workers' problem coincides with the density of pairs $\left( x,y\right) $ emanating from the firms' problem, and hence must satisfy
where
Substituting out $w\left( x,y\right) $ in system ((ref)) yields
where ${\Greekmath 011B} :={\Greekmath 011B} _{1}+{\Greekmath 011B} _{2}$, while substituting out ${\Greekmath 0119} \left( x,y\right) $ yields
Note that ${\Greekmath 011B}$ is the amount of heterogeneity in the model and when the scaling factors of the random values ${\Greekmath 011B} _{1}$ and ${\Greekmath 011B} _{2}$ tend to zero, i.e. there is no heterogeneity in the model $ {\Greekmath 011B} \rightarrow 0$, the firm's problem and the worker's problem converge to the deterministic maximization problems
and the equilibrium problem consists in finding $w\left( x,y\right) $ and $ {\Greekmath 0119} \left( x,y\right) $ which are compatible with optimality in these problems. See section (ref) below.
Note also that in our model, equilibrium wages do not vary systematically with the idiosyncratic shocks of workers and firms. This is because the non-wage valuation of a job at a firm of type $y$ by a worker of type $x$ is given as ${\Greekmath 010B} \left( x,y\right) +{\Greekmath 011B} _{1}{\Greekmath 0122} \left( y\right) $ and only depends on the observable characteristics of the firm, not the firm's idiosyncratic shock (i.e. the process ${\Greekmath 0111} $). Symmetrically, the productivity of a job at a firm of type $y$ when performed by a worker of type $x$ is given as ${\Greekmath 010D} \left( x,y\right) +{\Greekmath 011B} _{2}{\Greekmath 0111} \left( x\right) $, only depends on the observable characteristics of the worker, not her idiosyncratic shock (i.e. the process ${\Greekmath 0122} $). Hence, by the law of one price, this implies that the wage of any worker of type $x$ must be the same at every firm of the same type $y$.
We can now formally define an equilibrium outcome on this market.
As a result, the equilibrium outcome problem consists of looking for functions $a\left( x\right) $ and $b\left( y\right) $ that are solution to the system
Two important remarks are in order.
Assume that one has access to a random sample of the population of matches of firms and workers. For each match, this sample contains information about the worker's characteristics, her wage and the firm's characteristics. The observations consist of $\left\{ \left( X_{i},Y_{i},W_{i}\right) ,i=1,...,n\right\} $, where $n$ is the number of observed matches, $i$ indexes an employer-employee match, $X_{i}$ and $Y_{i}$ \ are respectively the vectors of employee's and employer's observable characteristics, which are assumed to be sampled from a continuous distribution, and $W_{i}$ is a noisy measure of the true unobserved transfer $w\left( X_{i},Y_{i}\right)$ assumed to be such that
where measurement error ${\Greekmath 010F} _{i}$ follows a $\mathcal{N}\left( 0,s^{2}\right) $ distribution and is independent of $\left( X_{i},Y_{i}\right) $.
Note that depending on the nature of preferences, observed transfers $W_{i}$ can be a monotonic transformation of observed wages. This flexibility allows one to consider equation ((ref)) as a hedonic wage regression using any known monotonic transformation of wages, i.e. identity (to estimate the model in levels), logarithm, power transformation etc.
Finally, note that, while we assume the analyst has access to data containing all variables in $X$ and $Y$, in practice datasets only contain a subset of these variables. In such a situation, the analyst faces issues of unobserved heterogeneity that our current method does not account for. However, while this is a current limitation of our approach, one can expect that existing methods dealing with unobserved heterogeneity would adapt to be incorporated into the framework. This is left for future research.
In this section, we briefly discuss identification of the deterministic value of amenities ${\Greekmath 010B} $ and productivity ${\Greekmath 010D} $. Not that ${\Greekmath 010B} $ and $ {\Greekmath 010D} $ do not appear individually in the expression of the equilibrium matches in equation ((ref)), only the joint value of a match ${\Greekmath 011E} $ appears in this equation. However, ${\Greekmath 010B} $ and ${\Greekmath 010D} $ do appear separately and with opposite signs in the formula of the equilibrium wages in equation ((ref)).
This clearly indicates that when only matches are observed, one cannot identify and hence estimate the deterministic value of amenities ${\Greekmath 010B} $ separately from the deterministic value of productivity ${\Greekmath 010D} $. In contrast, if transfers are observed, one actually can identify and estimate these objects separately.
It should be noted, however, that taking the values of ${\Greekmath 011B} _{1}$ and $ {\Greekmath 011B} _{2}$ as known\footnote{The ${\Greekmath 011B}$ parameters are not non-parametrically identified but they can be estimated using observed transfers once ${\Greekmath 010B}(.,.)$ and ${\Greekmath 010D}(.,.)$ have been parametrically specified.} and ${\Greekmath 011B} _{1}+{\Greekmath 011B} _{2}=1$ for notational simplicity, equations ((ref)) and ((ref)) clearly indicate that ${\Greekmath 010B} \left( x,y\right) +w\left( x,y\right) $ is identified up to a function $c\left( x\right) $ by ${\Greekmath 011B} _{1}\ln {\Greekmath 0119} \left( x,y\right) +c\left( x\right) $, and ${\Greekmath 010D} \left( x,y\right) -w\left( x,y\right) $ is identified up to a function $d\left( y\right) $ by ${\Greekmath 011B} _{2}\ln {\Greekmath 0119} \left( x,y\right) +d\left( y\right) $. It follows that ${\Greekmath 010B} $ is identified up to fixed effects $c\left( x\right) $ by
while ${\Greekmath 010D} $ is identified up to fixed effects $d\left( y\right) $ by
This result has been used in a nonparametric setting by Galichon and Salani\'{e} (2021) and Salani\'{e} (2015). In this paper, we exploit it in a parametric setting using basis functions of $x$ and $y$ (see Section (ref)). Indeed, since ${\Greekmath 010B} $ is identified up to fixed effects $ c\left( x\right) $, the parametrization of ${\Greekmath 010B} $ can only include basis functions depending on both $x$ and $y$ or on $y$ only but it cannot include basis functions depending on $x$ only. By a similar reasoning, the parametrization of ${\Greekmath 010D} $ cannot include basis functions depending on $y$ only.
For the sake of readability and to avoid additional notational burden, we propose the following change of notation. Replace ${\Greekmath 010B} $ by ${\Greekmath 011B} {\Greekmath 010B} $, ${\Greekmath 010D} $ by ${\Greekmath 011B} {\Greekmath 010D} $, ${\Greekmath 011E} $ by ${\Greekmath 011B} {\Greekmath 011E} $, $a$ by $ {\Greekmath 011B} a$, and $b$ by ${\Greekmath 011B} b$, so that the equations of the model become
where $\left( a,b\right) $ is the unique solution to the system of equations
still normalized by $a\left( x_{0}\right) =0$, and the terms $a$ and $b$ are related to $u$ and $v$ by
and the equilibrium transfer $w$ is given by
This change of notation is without loss of generality since from equation ( (ref)) one can estimate parameters ${\Greekmath 011B} _{1}$ and ${\Greekmath 011B} _{2}$ and hence ${\Greekmath 011B}$ and therefore recover the initial values of ${\Greekmath 010B} $ and $ {\Greekmath 010D} $. In the remainder of the paper, equations ((ref))--( (ref)) will characterize the model to estimate.\footnote{ The value of ${\Greekmath 011B}$ is therefore not imposed but estimated. Note however that the non-negativity of ${\Greekmath 011B} _{1}$ and ${\Greekmath 011B} _{2}$ should be imposed as a constraint, as in the application below.}
Let $A$ and $\Gamma $ be two vectors of $\mathbb{R}^{K}$ parameterizing the function of workers' systematic value of job amenities ${\Greekmath 010B} $ and the function of firms' systematic value of productivity ${\Greekmath 010D} $, in a linear way, so that
where the basis functions ${\Greekmath 0127} _{k}$ are linearly independent, and may include functions that depend on $x$ (respectively $y$) only. Note that by definition, the function of the joint value of a match reads as
where $\Phi _{k}=A_{k}+\Gamma _{k}$. Inspection of equation ((ref)) reveals that, given the parametric choice above, equilibrium matching and transfers are parameterized by $\left( A,\Gamma ,{\Greekmath 011B} _{1},{\Greekmath 011B} _{2},t\right) $. The model is hence fully parameterized by ${\Greekmath 0112} =\left( A,\Gamma ,{\Greekmath 011B} _{1},{\Greekmath 011B} _{2},t,s^{2}\right) $, which we make explicit by writing the predicted equilibrium transfer as $w(x,y;{\Greekmath 0112} )$.
The main purpose of this exercise is to estimate the vector of parameters $ {\Greekmath 0112} $. To this aim we adopt a maximum likelihood approach. It follows from section ((ref)) that the likelihood of observing a pair $\left( x,y\right) $ only depends on $\Phi =A+\Gamma $, and is given by
where $a(x;\Phi )$ and $b(y;\Phi )$ are uniquely determined by system of equations ((ref)). Since, by assumption, measurement errors in transfers are independent of $(X,Y)$, the log-likelihood of an observation $ \left( x,y,w\right) $ at parameter ${\Greekmath 0112} $\ is therefore
and hence, the log-likelihood of the sample reads as:
where $\hat{{\Greekmath 0119}}\left( x,y\right) $ is the observed density of matches in the data.
However, note that $a$, $b$ and $w$ that appear in ((ref)) are computed in the population; here, we only have access to a sample. So, denoting $a_{i}$ and $b_{j}$ the sample analog of $a(x)$ and $b(x)$, we compute the sample analog of system ((ref))
with the added normalization $a_{1}=0$, which ensures uniqueness of the solution.\footnote{ An argument similar to theorem A.2 in Chernozhukov et al. (2017) would show that the solution to the system ((ref)) converges uniformly to $a$ and $b$ as computed in the population, i.e. solution of system ((ref)).} (Note that since we have assumed that the population distribution is continuous, each sampled observation occurs uniquely, hence the right-hand side here is $1/n$; however, this could easily be extended to a more general setting). We denote $\left( a_{i}\left( \Phi \right) ,b_{i}\left( \Phi \right) \right) $ this solution at $\Phi $. This allows us to compute a sample estimate of the equilibrium transfer $w_{i}\left( {\Greekmath 0112} \right) $ as
where the notation ${\Greekmath 010B} _{ij}\left( A\right) $ substitutes for ${\Greekmath 010B} \left( X_{i},Y_{j};A\right) $, and similarly for ${\Greekmath 010D} _{ij}\left( \Gamma \right) $.
We are thus able to give the expression of the log-likelihood of the sample in our next result. Recall that ${\Greekmath 0112} =\left( A,\Gamma ,{\Greekmath 011B} _{1},{\Greekmath 011B} _{2},t,s^{2}\right) $ and $\Phi =A+\Gamma $.
Theorem (ref) motivates the following remark.\footnote{Note that in most applications, the parameters of primary interest are those governing workers' deterministic values of amenities and firms' deterministic values of productivity, i.e. $A$ and $\Gamma $ respectively. The remaining parameters $\left( {\Greekmath 011B} _{1},{\Greekmath 011B} _{2},t,s^{2}\right) $ are auxiliary. Our MLE estimator can be concentrated on the parameters of interest. These results are available in an online Appendix.}
We illustrate the usefulness of our method using an application to the estimation of the value of job amenities related to risks of fatal injury. This application requires access to a single cross-section of data containing a representative sample of worker-job matches with information about workers' characteristics (education, experience, gender etc.), their (hourly) wage and a measure of fatality rates associated to their job. Many surveys such as the CPS contain all required information but the fatal injury data. As a result, following Thaler and Rosen (1978), a large strand of the literature has compiled the required data by combining survey data with data about fatality per type of jobs from alternative sources.
In this paper, we follow the recent work by Viscusi (2003, 2007 and 2013) and construct measures of fatality rates by occupation-industry cells for the period 2012-2016. Unfortunately, data on fatal injuries by occupation within industries are not readily available. Instead, we rely on fatal injury data by occupation (4-digits SOC) and by industry (4-digits NAICS) provided by the U.S. Bureau of Labor Statistics (BLS) CFOI.\footnote{ In the CFOI data, a fatal injury is an injury leading to death within one year of the day of the accident. See Viscusi (2003) for more details about the CFOI data and its use in the present context.} For each year in the period 2012-2016, we create a matrix of fatal injuries by occupation$\times $ industry by simply multiplying the marginal distribution by occupation and industry hence assuming independence. To reduce measurement errors, the 4-digits occupational codes are aggregated into 25 major occupations and the 4-digits industry codes into 80 major industries.\footnote{ We use crosswalks provided by the BLS to perform the aggregation to major occupations and industry.} We then use the CPS March files\footnote{ The BLS advises to use March files of the CPS for computations of total employment.} for 2012-2016 and compute matrices of hours-adjusted employment level by occupation and industry, combining person-weights, computed to this effect by the census and the BLS, and hours worked per week. The two sets of matrices are then merged allowing us to compute, for each year, fatality rates for a given occupation-industry cell as the ratio of the number of fatalities to total hours-weighted employment in that cell (see e.g. Viscusi, 2013). To attenuate further measurement errors, for each occupation-industry cell, we take the average fatality rate over time as our measure of risk.
We obtained our working dataset by merging the 2017 March CPS data with our measure of fatality rate by occupation-industry cells. This dataset therefore contains information about our main variables of interest: hourly earnings, hours of work, gender, years of schooling, age, ethnic group, marital status, whether one's job is in the public sector or not, and occupation-industry fatality rates. We follow the literature (e.g. Viscusi, 2013) and keep only full-time, non-agricultural, non-armed force workers\footnote{Assumption (ref) requires all agents to be price-takers. This assumption is likely not to be met in the armed force industry whose sole employer is the US government. For this reason, we exclude armed force workers from the analysis. Note, however, that this exclusion is common in the hedonic wage regression literature, e.g. Viscusi (2013).} between 16 and 64 years old for the remainder of the analysis.\footnote{ As is standard when using March CPS wage data, see Katz and Murphy (1992) for instance, the sample excludes individuals with hourly earnings below one half of minimum wage and top coded earnings are imputed 1.45 times the top code value.}
Table ((ref)) provides descriptive statistics of our working dataset. The average fatality rate in our sample is about per $100,000$ which is close to the figure obtained in Viscusi (2013) for the year 2008, i.e. 3.29. To further compare our dataset with the literature, we run a hedonic (log)wage regression including the traditional controls (gender, years of schooling, age, age squared, ethnic group, marital status, union membership, public sector dummy, regional dummies, Metropolitan dummy) and our measure of hours-weighted fatality rates by occupation-industry. Using the estimate of the compensating wage differential for risk, we obtain an estimate of the VSL of \$9.7 million (\$2017). This figure falls in the range of estimates in the literature using similar data, i.e. Viscusi (2013)'s estimate of \$8.4 million (\$2017) using the 2008 CPS data.\footnote{ The risk coefficient in the log wage hedonic regression reported in Viscusi (2013), i.e. 0.0024, is very close to the estimate obtained with our data, i.e. 0.0027. Using 0.0024 instead of 0.0027 to calculate the VSL with our data one would obtain \$8.6 million (\$2017).}
We estimate the model using the maximum likelihood estimator presented in this paper. Observed transfers are assumed to be the logarithm of observed wages to be consistent with the hedonic regression literature that typically uses log wage regressions. We standardize other continuous variables to facilitate the comparison and interpretation (in terms of standard deviation) of the respective coefficients.
Estimation requires to specify the basis functions used to parameterize the values of job amenities ${\Greekmath 010B} \left( x,y;A\right) $ and productivity $ {\Greekmath 010D} (x,y;\Gamma )$. We adopt a linear (in parameters) specification of the basis functions and present estimates for the following specification:\footnote{ The fit of this specification can be compared with that of alternative (nesting/nested) specifications using likelihood ratio tests. For instance, the test statistic obtained when comparing the chosen specification with a richer specification, where both job amenities and productivity include interactions between workers' years of schooling, experience and gender with jobs' risk and sector, is equal to 1.358. This statistic is not significantly different from 0 at conventional levels. One concludes that the specification presented in the paper should be preferred. Note however, that the estimates of the VSL are similar across specifications.}
and
where $x$ includes a constant (k=0), years of schooling (k=1), (potential \footnote{ Age less years of schooling less 6.}) experience (k=2), experience squared (k=8), a dummy variable for female (k=3), a dummy variable indicating whether one is married or not (k=4), and 3 ethnic dummy variables (white, black and asian, using others, incl. hispanic, as the reference group, k=5,6,7), whereas $y$ includes a constant (l=0), our measure of fatality rates (l=1) and a dummy variable indicating the public sector (l=2).
Hence, our specification of job amenities includes the main effects of fatality rates and public sector as well as an interaction between a workers' years of schooling and jobs' sector. Our specification of perceived productivity includes the main effects of years of schooling, experience (squared), marital status and ethnic groups as well as interactions between workers' year of schooling, experience and gender with jobs' fatality rates and sector.
Estimates are presented in table ((ref)). Note first that the model fits quite well the wage data with an $R^{2}$ of $ 0.235$ which compares to that obtained for the standard hedonic wage regression, i.e. $R^{2}=0.255$.
Second, estimates of the value of perceived productivity show expected results. \footnote{ Unless stated otherwise, the significance level is 1%.} The value of productivity increases with years of schooling ($0.057$), although the estimate is not significant,\footnote{ We have also estimated the model including years of schooling squared. However, comparing the two specifications, the log-likelihood ratio test statistic is 3.360 and not significantly different from 0 at conventional levels. Our chosen specification should be preferred.} and the experience-productivity gradient is positif ($0.084$) but decreasing, as indicated by the negative coefficient for experience squared ($-0.051$). These human capital effects, however, vary significantly across jobs: the years-of-schooling-productivity gradient is absent in risky jobs ($ -0.002 = 0.057-0.059$) but greater for public sector jobs ($0.895 = 0.057+0.838$), whereas the experience-productivity gradient is higher in risky jobs ($0.158 = 0.084+0.074$).
Third, our estimates for perceived productivity show negative coefficients for female and black workers ($-0.404$ and $-0.108$ respectively) and a positive coefficient for white workers ($0.046$). These coefficients should be interpreted with care as they indeed reflect employers' perceived productivity of the underlying types of workers, revealing discrimination effects.\footnote{We refer herewith to Becker (1971). The parameter ${\Greekmath 010D}$ reflecting both the true productivity of workers and employers' taste discrimination parameter.} Our results are in line with the large literature showing discriminating wage differentials across gender and race. Interestingly, the gender perceived productivity gap varies significantly across jobs unlike the racial one: a one standard deviation increase in the probability of fatal injury more than triples the gender perceived productivity gap.
Fourth, regarding the value of job amenities, results show that the value of public sector jobs increases significantly with years of schooling: a one standard deviation in years of schooling generates a 0.081 log-points increase in the value of jobs in the public sector.
Finally, our main result shows that US workers' utility drops by log-points per hour of work as the probability of fatal injury on the job increases by one standard deviation (i.e. per $100,000$). We can use this coefficient to compute the VSL from the formula
where $\overline{z}$ are the average earnings in the sample.\footnote{ To derive this formula, remember that the systematic utility of a worker of type $x$ working in job of type $y$ and receiving a transfer $w\left( x,y\right)$ is given as $U\left( x,y\right) ={\Greekmath 010B}\left( x,y\right) +w\left(x,y\right)$. Since transfers are specified in log wages, i.e. $w\left( x,y\right) =\ln z\left( x,y\right) $ where $z\left( x,y\right) $ are the equilibrium wages, we can then compute a worker of type $x$'s trade-off between earnings and risk as
Rearranging this equation, one can express the differential wage increase required to compensate the differential drop in job amenity due to an one unit increase in the risk of fatal injury as in the text using $\overline{z}$ to replace $z\left( x,y\right) $.}
Using $\frac{\partial {\Greekmath 010B} \left( x,y\right) }{\partial y^{(1)}}=A_{1,1}= $ and the appropriate units, one obtains a VSL of \$million (\$2017).\footnote{ In our preferred specification, the VSL does not vary with the type of workers nor with the fatality risk. Note that this method excludes productivity effects of fatality risk. It only reflects the valuation of life from the perspective of workers.} This value lies in the range of estimates found in the literature using similar data, i.e. Viscusi, (2013). Nevertheless, it is about \$3 million lower than the estimate obtained using the classical hedonic wage regression. Though the difference is not statistically significant, it suggests that not accounting for the sorting of workers into jobs, as in the hedonic regression, may lead to an overestimation of the true VSL in our data.
To see this, note that, in our model, the differential value of job amenities with respect to fatality risk is identified as
In contrast, the hedonic wage regression literature identifies this differential value using the coefficients of an (log) earnings regression as
As a result, the VSL as measured in the hedonic regression literature reads as
once substituting $\frac{\partial {\Greekmath 010B}^{h}\left( x,y\right) }{ \partial y^{(1)}}$ by its expression in terms of ${\Greekmath 010B}$ and ${\Greekmath 0119}$.
Since average wages are positive, $\overline{z}>0$, and ${\Greekmath 011B}_{1} > 0$, it follows that, compared to our method, estimates of VSL from hedonic wage regressions tend to be larger (lower) when, in equilibrium, conditional on workers' type, workers sort into safe (resp. risky) jobs, i.e. when $ \frac{\partial \ln {\Greekmath 0119} \left( y|x\right) }{\partial y^{(1)}}<0$ (resp. $>0$).
Not only our structural approach allows one to explicitly take into account the sorting of workers to jobs when estimating the value of job amenities, it also allows to compute counterfactual equilibria. In particular, one could use the estimates of job amenities and productivity obtained above to compute the impact of a government intervention aiming at reducing fatality risk at work. For instance, consider the Site-Specific Targeting (SST) inspection plan proposed by the Occupational Safety and Health Administration (OSHA) in the US. If effective, this program would decrease fatality rates in the most risky jobs. For the sake of an example, suppose that the program ends up decreasing the fatality risk of all jobs whose fatality risk is at least 1 standard deviation above the mean (i.e. $\geq 16.5=3.4+1\times 13.1$) down to $16.5$\ fatalities per 100,000 workers per year. As a result of this intervention, the distribution of types of jobs would change causing the equilibrium matching and wages to change too. We use our model to compute the equilibrium before (observed) and after the intervention and then compare the matching and distribution of wages. We find that, as a result of this intervention, about 3.1% of the workers would change jobs, the mean wage would drop by 3.9% and wage inequality, as measured by the Gini coefficient, would drop by 3.6%.
We conclude by addressing a few methodological remarks before summarizing our main results.
A natural extension of the model is to allow for workers to be unemployed and jobs to be vacant. To do so, one needs first to allow for the total masses of workers and firms to be different, herewith relaxing that part of Assumption (ref). Second, one needs to extend the definition of utilities for matched workers and firms to unemployed workers and vacant jobs, introducing reservation utilities. The reservation utility of a worker (firm) of type $x$ ($y$ ) may be decomposed into a systematic part ${\Greekmath 010B} (x,\emptyset )$ (${\Greekmath 010D} (\emptyset ,y)$) and a random value ${\Greekmath 0122} \left( \emptyset \right) $ ( ${\Greekmath 0111} \left( \emptyset \right) $) following a Gumbel type I distribution. Assumption (ref) should then simply be modified by replacing the choice sets of workers $\mathcal{Y}$ and firms $\mathcal{X}$ by $\mathcal{Y} \cup \left\{ \emptyset \right\} $ and $\mathcal{X\cup }\left\{ \emptyset \right\} $ respectively and adopting the convention that $w\left( x,\emptyset \right) =w\left( \emptyset ,y\right)=0 $ for all types of workers and firms.
However, note that the conditional probabilities in ((ref)) and ((ref)) have a logit structure and hence satisfy the Independence of Irrelevant Alternative property. As a consequence, ${\Greekmath 0119} \left( y|x\right) $ in ((ref)) is also the density of probability of choosing a firm of type $y$ for a worker of type $x $ conditional on participation. As shown in appendix D in Dupuy and Galichon (2014), this implies that in a market with unemployed workers and vacant jobs and reservation utilities ${\Greekmath 010B} (x,\emptyset )+{\Greekmath 0122} \left( \emptyset \right) $ and ${\Greekmath 010D} (\emptyset ,y)+{\Greekmath 0111} \left( \emptyset \right),$ at equilibrium, the probability density ${\Greekmath 0119} \left( x,y\right) $ of occurrence of a $\left( x,y\right) $ pair among matched pairs (i.e. considering only active workers and filled jobs) is the same as the probability density ${\Greekmath 0119} \left( x,y\right) $ of occurrence of a $\left( x,y\right) $ pair in a market with no outside options and where the masses of workers and firms are the same as the masses of active workers and filled jobs in the former market.
When ${\Greekmath 011B} \rightarrow 0$, the model converges to the classical model of Monge-Kantorovich, which is a continuous extension of the Becker-Shapley-Shubik model. Indeed, when ${\Greekmath 011B} _{1}$ and ${\Greekmath 011B} _{2}$ tend to zero, the scaling coefficients of the random value of job amenities and productivity ${\Greekmath 0122} $ and ${\Greekmath 0111} $, tend to zero, then the model becomes nonstochastic. Intuitively, when ${\Greekmath 011B} _{1}\rightarrow 0$ , the worker's expected indirect utility $u\left( x\right) $ tends to the deterministic indirect utility $\max_{y}\left\{ {\Greekmath 010B} \left( x,y\right) +w\left( x,y\right) \right\} $, and it follows from ((ref)) that the conditional choice distribution ${\Greekmath 0119} \left( y|x\right) $ becomes concentrated around the optimal firm's type $y$ such that $u\left( x\right) ={\Greekmath 010B} \left( x,y\right) +w\left( x,y\right) $. Similarly, when ${\Greekmath 011B} _{2}\rightarrow 0$, a firm of type $y$ expected indirect profits $v\left( y\right) $ tends to the deterministic indirect profits $\max_{x}\left\{ {\Greekmath 010D} \left( x,y\right) -w\left( x,y\right) \right\} $, and ${\Greekmath 0119} \left( x|y\right) $ becomes concentrated around the optimal worker's type $x$ such that $v\left( y\right) ={\Greekmath 010D} \left( x,y\right) -w\left( x,y\right) $. Combining these two results, ${\Greekmath 0119} \left( x,y\right) $ becomes concentrated around the set of pair $\left( x,y\right) $ such that $u\left( x\right) +v\left( y\right) ={\Greekmath 011E} \left( x,y\right) $, hence, in the limit when $ {\Greekmath 011B} _{1}$ and ${\Greekmath 011B} _{2}$ tend to zero, we have
These are the classical stability conditions in the Monge-Kantorovich problem (see Villani, 2003 and 2009), whose variants have been applied in economics by Becker (1973), Shapley and Shubik (1963), Gretsky, Ostroy, Zame (1992). A particular example is Sattinger's workhorse model extensively used in the labor economics literature (see Sattinger, 1979 and 1993). This model indeed corresponds to a matching market with no unobserved heterogeneity (${\Greekmath 011B} \rightarrow 0$), unidimensional observed types ($d_{x}=d_{y}=1$), in which workers only care about their compensation (${\Greekmath 010B} =0$) and where the firm's value of productivity is smooth and supermodular (i.e. $\partial ^{2}{\Greekmath 010D} \left( x,y\right) /\partial x\partial y$ exists and is positive). Under these restrictions, both the worker's and firm's problems become deterministic, and the conditional distribution ${\Greekmath 0119} (y|x)$ in this case is concentrated at one point $y=T(x)$, where $T(x)$ is the only assignment of workers to firms which is nondecreasing. The equilibrium wage $w$ only depends on $x$ and satisfies the differential wage equation $w^{\prime }\left( x\right) =\frac{\partial {\Greekmath 010D} \left( x,T\left( x\right) \right) }{\partial x}$ and an explicit formula for the equilibrium wage is obtained by integration.
Over the last decade, a great deal of efforts has been made to bring matching models to data. In the transferable utility class of models, following Choo and Siow's seminal contribution, various extensions have been proposed to enrich the empirical methodology. These extensions were so far limited to the case when transfers are not observed. However, the observation of transfers allows to widen the scope of identified objects in this class of models, and in particular allows the analyst to separately identify the (pre-transfer) values of a match for each partner. Our paper proposes an intuitive and tractable maximum likelihood approach to structurally estimate these values of a match for each partner using data about matches and transfers from a single market.
We illustrate the usefulness of our methodology to the estimation of compensating wage differentials for the risk of fatal injury on the job. Using the 2017 March CPS data together with CFOI data on fatal injury per occupation and industry, our estimate of the value of job amenities related to risk translates into a Value of Statistical Life of \$ million (\$2017). This estimate is \$3 million lower (though not significantly) than the one obtained by applying a classical hedonic regression technique on our data. Since the hedonic approach can be seen as an extreme version of our method where all the weight in the likelihood function is put on fitting transfers (wages) and none on fitting matching patterns, this suggests that not accounting explicitly for the sorting of workers to jobs can lead to biases in the estimation of the value of statistical life.