POSTS

# A Model of Unravelling

This is (the first of a possible series on) my practice of economic modelling. Typically, my clumsy efforts either lead to intractable models or reinventing the wheel. In this case, I spent a long time developing the rudimentary form of what I later discovered was Li and Rosen (1998).

# The Observation

While talking to one of my friends about the gruelling timeline of investment banking recruiting that she has to go through (job offers being made more than two, I brought up the concept of *unravelling* in matching markets, the idea that transactions can happen inefficiently early if agents are worried about the market clearing quickly and leaving them unmatched. She agreed that that described the general situation of finance recruiting, but griped that it was *uniquely* true of finance - her tech recruiting is still almost a year from now.

That commonsense observation - that not all markets unravel equally - gets to an interesting question: we know markets can unravel, but do we know what differentiating factor makes some markets unravel and others not?

I’m certain that there are plenty of papers on this, but I am a mere student and reinventing the wheel is my forte for now. So I ignore the literature and build a model of when unravelling occurs.

# The Model

The intuition behind this model is, firms who hire workers early are risking that they hire an unproductive worker without realizing it. If they instead waited they could discover which workers were productive and which were not, which would be safer.

The natural expression of this is a two-period model. Each period involves a Shapley-Shubik assignment market with the firms as buyers of labor. The core features of Shapley-Shubik are:

- Goods are indivisible. Workers are not (yet) divisible so this is a fine assumption.
- Each firm has unit demand for workers. This is not without loss of generality, but it will do.
- Sellers (workers) are unstrategic - they simply give their labor to whichever firm they are assigned to. This is not as heroic an assumption as it seems, because in this setup firms are not differentiated, so there’s no reason for a worker to prefer one over the other. Possible differentiating factors like wages are formulated as prices and thus are incorporated into the model.

With these assumptions laid out, we can define the market. Let \(F\) be the set of firms and \(W\) be the set of workers. Assume without loss of generality that \(|F| = |W|\) (we can always add dummy firms/workers to make this possible). To capture the value of assigning worker \(j\) to firm \(i\) is \(v_{ij} = 1\) if the worker is productive and \(v_{ij} = 0\) otherwise. However, a worker is only revealed as productive or not in period 2, and in period 1 every worker appears equally likely to be productive. Let \(x_{ij} = 1\) if worker \(j\) is assigned to firm \(i\) and 0 otherwise.

## Period 2 Matching

Assume a central planner who wants to maximize the total utility of agents in the market, by creating an assignment of workers to firms that maximizes productivity. Then the central planner’s problem in period 2 is \[ \text{max } \sum_{i \in F} \sum_{j \in W} v_{ij}x_{ij} \] \[\text{s.t. } \sum_{i \in F} x_{ij} \leq 1\] \[\sum_{j \in W} x_{ij} \leq 1\] \[x_{ij} \in \{0, 1\} \; \forall \; i, j\]

According to our setup, in period 2 we can identify which workers are productive. Thus we can separate workers into \(P\) (productive workers) and \(W \setminus P\) (unproductive workers) and define \(v_{ij}\) as such, yielding

\[\text{max } \sum_{i \in F} \sum_{j \in P} 1* x_{ij} + \sum_{i \in F} \sum_{j \in W \setminus P} 0* x_{ij}\] \[\text{s.t. } \sum_{i \in F} x_{ij} \leq 1\] \[\sum_{j \in W} x_{ij} \leq 1\] \[x_{ij} \in \{0, 1\} \; \forall \; i, j\]

It’s clear that the objective is maximized by setting \(x_{ij} = 1\) for all productive workers and an arbitrary subset of firms (call it \(L\), with the only restriction that it has cardinality \(|P|\)), and then ignoring the unproductive workers (assigning them or not makes no difference). But to check if there’s a competitive equilibrium supporting this allocation, we relax it to a linear program and check the dual. \[\text{min } \sum_{i \in F} s_i + \sum_{j \in W} p_j\] \[\text{s.t. } s_i + p_j \geq v_{ij} \; \forall \; i \in F, j \in W\] \[s_i, p_j \geq 0\]

In the formulation above, \(s_i\) and \(p_j\) are the dual variables generated by interpreting the constraints \(\sum_{i \in F} x_{ij} \leq 1\) as \(\sum_{i \in F} x_{ij} \leq s_i\) and \(\sum_{j \in W} x_{ij} \leq 1\) as \(\sum_{i \in F} x_{ij} \leq p_j\). Conveniently, when interpreting the dual as a competitive equilibrium, \(s_i\) gives us the surplus of a firm and \(p_j\) gives us the price placed on a worker (i.e. their wage).

The analytical solution to the dual program is clear: when \(v_{ij} = 0\) the objective is minimized by setting \(s_i = p_j = 0\). So this dual is equivalent to only matching productive workers - we thus look only at firms in \(L\) and workers in \(P\).

\[\text{min } \sum_{i \in L} s_i + \sum_{j \in P} p_j\] \[\text{s.t. } s_i + p_j \geq 1 \; \forall \; i \in L, j \in P\] \[s_i, p_j \geq 0\]

Because \(|L| = |P|\), the constraint gives the game away: the objective can be written as \(\sum (s_i + p_j)\) and the constraint holds at equality for any optimal matching, so the objective always has the minimum value \(|P|\) no matter what allocation we choose. This means any price set with \(p_j \leq 1\) supports a competitive equilibrium.

This is all an elaborate way of saying, if we wait until period 2 we get perfect efficiency. Shocking, I know, but this analysis is still valuable because it gives us a measure of how much firms individually benefit from this: since the subset \(L\) of firms that were assigned productive workers is totally arbitrary, a firm’s expected return from this market is determined by what fraction of workers are productive i.e. the expected value of a match is \(\frac{|P|}{|W|}\). This is a useful benchmark to compare to matching in period 1.

## Period 1 Matching

In period 1, every worker can become productive with probability \(p\). Therefore, the expected value of a matching is \(v_{ij} = p\). This uniform valuation makes the central planner’s problem very simple: \[ \text{max } p \sum_{i \in F} \sum_{j \in W} x_{ij} \] \[\text{s.t. } \sum_{i \in F} x_{ij} \leq 1\] \[\sum_{j \in W} x_{ij} \leq 1\] \[x_{ij} \in \{0, 1\} \; \forall \; i, j\]

But it’s obvious from looking at the uniform valuation that any feasible allocation is a solution to the central planner’s problem. The dual confirms this: \[\text{min } \sum_{i \in F} s_i + \sum_{j \in W} p_j\] \[\text{s.t. } s_i + p_j \geq p \; \forall \; i \in F, j \in W\] \[s_i, p_j \geq 0\]

Just like the period 2 dual, this constraint holds at equality and thus any price \(p_j \leq p\) supports a competitive equilibrium.

The expected welfare for each firm in period 2 matching, \(\frac{|P|}{|W|}\), is microfounded in this probability \(p\) that a worker becomes productive. In fact, \(p = \frac{|P|}{|W|}\) and thus **the expected welfare of matching in period 1 is the same as in period 2.**

This seems like a bizarre result, contrary to empirical evidence about unravelling, so it’s worth exploring more. The root of this is the unlucky firms: in period 2 they do not get a productive worker, but in period 1 they have an equal chance. This gives them a welfare gain that compensates for the welfare lost by the lucky firms.

This is where my threadbare model starts to come apart, so it’s at this point that I start to search the literature for papers with a similar idea, and I find Unravelling in Matching Markets by Hao Li and Sherwin Rosen (1998).

# Li and Rosen (1998)

Li and Rosen also model unravelling as a two-period Shapley-Shubik assignment game where workers are productive or not, but with a key difference: in LR’s model, workers have heterogeneous types i.e. different probabilities of becoming productive. This is a commonsense assumption that I really should have incorporated. In short, I simplified too much.

Another difference is that LR take the productivity of a firm itself to be indeterminate in period 1: that’s certainly beyond the scope of what I tried to do.

Furthermore, LR extend their analysis beyond a simple “when does unravelling happen” approach. They also look at the welfare effect on different types of workers and the comparative statics of possible solutions to unravelling (banning early contracting and allowing ex-post buyouts). This seems like a hallmark of many theory papers I’ve read, especially more recent ones: the work is never done by simply proposing a model and examining its rudimentary predictions. The work involves proving properties of the model, analyzing the effect of certain changes, etc. That’s another good sanity-check to keep in mind for future modelling exercises.