“Ranking Firms Using Revealed Preference,” I. Sorkin (2015)

Roughly 20 percent of earnings inequality is not driven by your personal characteristics or the type of job you work at, but by the precise firm you work for. This is odd. In a traditional neoclassical labor market, every firm should offer to same wage to workers with the same marginal productivity. If a firm doesn’t do so, surely their workers will quit and go to firms that pay better. One explanation is that since search frictions make it hard to immediately replace workers, firms with market power will wind up sharing rents with their employees. It is costly to search for jobs, but as your career advances, you try to move “up the job ladder” from positions that pay just your marginal product to positions that pay a premium: eventually you wind up as the city bus driver with the six figure contract and once there you don’t leave. But is this all that is going on?

Isaac Sorkin, a job market candidate from Michigan, correctly notes that workers care about the utility their job offers, not the wage. Some jobs stink even though they pay well: 80 hour weeks, high pressure bosses, frequent business travel to the middle of nowhere, low levels of autonomy, etc. We can’t observe the utility a job offers, of course, but this is a problem that always comes up in demand analysis. If a Chipotle burrito and a kale salad cost the same, but you buy the burrito, then you have revealed that you get more utility from the former; this is the old theory of revealed preference. Even though we rarely observe a single person choosing from a set of job offers, we do observe worker flows between firms. If we can isolate workers who leave their existing job for individual reasons, as distinct from those who leave because their entire firm suffers a negative shock, then their new job is “revealed” better. Intuitively, we see a lot of lawyers quit to run a bed and breakfast in Vermont, but basically zero lawyers quitting to take a mining job that pays the same as running a B&B, hence the B&B must be a “better job” than mining, and further if we don’t see any B&B owners quitting to become lawyers, the B&B must be a “better job” than corporate law even if the pay is lower.

A sensible idea, then: the same worker may be paid different amounts in relation to marginal productivity either because they have moved up the job ladder and luckily landed at a firm with market power and hence pay above marginal product (a “good job”), or because different jobs offer different compensating differentials (in which case high paying jobs may actually be “bad jobs” with long hours and terrible work environments). To separate the two rationales, we need to identify the relative attractiveness of jobs, for which revealed preference should work. The problem in practice is both figuring out which workers are leaving for individual reasons, and getting around the problem that it is unusual to observe in the data a nonzero number of people going from firm A to firm B and vice versa.

Sorkin solves these difficulties in a very clever way. Would you believe the secret is to draw on the good old Perron-Frebonius theorem, a trusted tool of microeconomists interested in network structure? How could that be? Workers meet firms in a search process, with firms posting offers in terms of a utility bundle of wages plus amenities. Each worker also has idiosyncratic tastes about things like where to live, how they like the boss, and so on. The number of folks that move voluntarily from job A to job B depends on how big firm A is (bigger firms have more workers that might leave), how frequently A has no negative productivity shocks (in which case moves are voluntary), and the probability a worker from A is offered a job at B when matched and accepts it, which depends on the relative utilities of the two jobs including the individual idiosyncratic portion. An assumption about the distribution of idiosyncratic utility across jobs allows Sorkin to translate probabilities of accepting a job into relative utilities.

What is particularly nice is that the model gives a linear restriction on any two job pairs: the relative probability of moving from A to B instead of B to A depends on the relative utility (abstracting from idiosyncratic portions) adjusted for firm size and offer probability. That is, if M(A,B) is the number of moves from A to B, and V(A) is a (defined in the paper) function of the non-idiosyncratic utility of job A, then

M(A,B)/M(B,A) = V(B)/V(A)

and hence

M(A,B)V(A) = M(B,A)V(B)

Taking this to data is still problematic because we need to restrict to job changes that are not just “my factory went out of business”, and because M(A,B) or M(B,A) are zero for many firm pairs. The first problem is solved by estimating the probability a given job switch is voluntary using the fact that layoff probability is related to the size and growth rate of a firm. The second problem can be solved by noting that if we sum the previous equation over all firms B not equal to A, we have

sum(B!=A)M(A,B)*V(A) = sum(B!=A)M(B,A)*V(B)

or

V(A) = sum(B!=A)M(B,A)*V(B)/sum(B!=A)M(A,B)

The numerator is the number of hires A makes weighted for the non-idiosyncratic utility of firms the hires come from, and the denominator is the number of people that leave firm A. There is one such linear restriction per firm, but the utility of firm A depends on the utility of all firms. How to avoid this circularity? Write the linear restrictions in matrix form, and use the Perron-Frebonius theorem to see that the relative values of V are determined by a particular eigenvector as long as the matrix of moves is strongly connected! Strongly connected just means that there is at least one chain of moves between employers that can get me from firm A to B and vice versa, for all firm pairs!. All that’s left to do now is to take this to the data (not a trivial computation task, since there are so many firms in the US data that calculating eigenvectors will require some numerical techniques).

So what do we learn? Industries like education offer high utility compared to pay, and industries like mining offer the opposite, as you’d expect. Many low paying jobs offer relatively high nonpay utility, and many female-dominated sectors do as well, implying the measured earnings inequality and gender gaps may be overstating the true extent of utility inequality. That is, a teacher making half what a miner makes is partly reflective of the fact that mining is a job that requires compensating differentials to make up for long hours in the dark and dangerous mine shaft. Further, roughly two thirds of the earnings inequality related to firms seems to be reflecting compensating differentials, and since just over 20% of earnings inequality in the US is firm related, this means that about 15% of earnings inequality is just reflecting the differential perceived quality of jobs. This is a surprising result, and it appears to be driven by differences in job amenities that are not easy to measure. Goldman Sachs is a “good job” despite relatively low pay compared to other finance firms because they offer good training and connections. This type of amenity is hard to observe, but Sorkin’s theoretical approach based on revealed preference allows the econometrician to “see” these types of differences across jobs, and hence to more properly understand which jobs are desirable. This is another great example of a question – how does the quality of jobs differ and what does that say about the nature of earnings inequality – that is fundamentally unanswerable by methodological techniques that are unwilling to inject some theoretical assumptions into the analysis.

November 2015 Working Paper. Sorkin has done some intriguing work using historical data on the minimum wage as well. Essentially, minimum wage changes that are not indexed to inflation are only temporary in real terms, so if it costly to switch from labor to machines, you might not do so in response to a “temporary” minimum wage shock. But a permanent increase does appear to cause long run shifts away from labor, something Sorkin sees in industries from apparel in the early 20th century to fast food restaurants. Simon Jäger, a job candidate from Harvard, also has an interesting purely empirical paper about friction in the labor market, taking advantage of early deaths of German workers. When these deaths happen, working in similar roles at the firm see higher wages and lower separation probability for many years, whereas other coworkers see lower wages, with particularly large effects when the dead worker has unusual skills. All quite intuitive from a search model theory of labor, where workers are partial substitutes for folks with the same skills, but complements for folks with firm-specific capital but dissimilar skills. Add these papers to the evidence that efficiency in the search-and-matching process of labor to firms is a first order policy problem.

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