The Price of the Search
From Part 1: Farewell to a Perfect World
In Search and Matching 1, we explored the frictionless world. There, every worker and firm knew everything about each other instantly, and finding the perfect partner cost nothing. As long as the production function was supermodular (\(f_{xy} > 0\)), the market magically achieved Positive Assortative Matching (PAM), pairing the best workers with the best firms.
But the real world is messy. We send out resumes, wait anxiously for calls, and go through multiple rounds of interviews. The process of finding the perfect match comes with a cost: search frictions. The moment we introduce this “price of the search” into the model, Becker’s elegant conclusions face a serious challenge.
We use the seminal model of Shimer and Smith (2000) to see what happens when ‘frictions’ are thrown like sand into the gears of the labor market machine.
The Shimer–Smith Model (2000): Rules for a Frictional World
The Shimer-Smith model depicts a world closer to reality, where workers and firms meet each other randomly.
Environment: The Random Meeting Place
Time is continuous. A measure of unemployed workers of type \(x\) and a measure of firms with vacancies of type \(y\) are both searching for partners. The meetings are random.
- The Meeting Process: An unemployed worker meets a vacancy at a Poisson rate \(\lambda\). It’s like going to a party and randomly meeting people, not knowing who you’ll encounter next.
- Information and Decision: Once a worker \(x\) and a firm \(y\) meet, they instantly observe the potential output they could create together, \(f(x,y)\). Then they must decide: “Do we commit to this match, or do we go our separate ways?”
- The Value of Outcomes: If a match is formed, they work together, producing a flow output of \(f(x,y)\) per unit of time, which they split via a wage \(w(x,y)\). If they decline, the worker remains unemployed and the firm continues to search to fill its vacancy.
All agents are forward-looking and discount the future with a rate \(r\). Furthermore, even successful matches can dissolve for exogenous reasons at a rate \(\delta\). The value function for each agent is defined as follows:
- \(W_0(x)\): The value for an unemployed worker of type \(x\) (the present value of searching).
- \(W_1(x,y)\): The value for a worker \(x\) matched with a firm \(y\) (the present value of being employed).
- \(\Pi_0(y)\): The value for a firm \(y\) with a vacancy (the present value of searching).
- \(\Pi_1(x,y)\): The value for a firm \(y\) that has hired a worker \(x\) (the present value of being a filled job).
Match Surplus: “Is This Meeting Worth It?”
When a worker \(x\) and a firm \(y\) meet, how much more value do they create by matching compared to the sum of their ‘single’ states, that is, continuing to search alone? This difference is the Match Surplus.
\[ S(x,y) \equiv [W_1(x,y) + \Pi_1(x,y)] - [W_0(x) + \Pi_0(y)] \]
The surplus \(S(x,y)\) is the key metric that answers the question: “How much better is it to say ‘Yes’ right now than to say ‘No’ and return to the searching pool?” In a steady-state equilibrium, this surplus satisfies the following Bellman equation:
\[ (r + \delta)S(x,y) = f(x,y) - rW_0(x) - r\Pi_0(y) \]
Left-hand side \((r + \delta)S(x,y)\): ‘The required annual return of the relationship.’ This is the flow value required to sustain the match, combining the interest on the surplus value (\(rS\)) and the expected loss from a potential breakup (\(\delta S\)). Right-hand side \(f(x,y) - rW_0(x) - r\Pi_0(y)\): ‘The net cash flow of the relationship.’ This is the direct output from the match (\(f(x,y)\)) minus the opportunity costs for the worker and the firm (\(rW_0(x)\) and \(r\Pi_0(y)\)) from giving up their search activities.
The Acceptance Decision: A Tightrope Walk on the Boundary
Whether a meeting results in a match depends solely on the sign of the surplus.
\[ \text{A match is formed} \iff S(x,y) \ge 0 \]
If \(S(x,y) < 0\), the match is worse than staying single, so they reject each other and walk away. The equilibrium is therefore characterized by an Acceptance Set, \(\mathcal{M} = \{(x,y): S(x,y) \ge 0\}\).
The locus of points where \(S(x,y)=0\) forms the boundary of this acceptance set. A partner on this boundary is one you are perfectly indifferent about: “Meeting them is fine, but so is not meeting them.”
Here, a critical problem arises. Higher-type workers (higher \(x\)) and firms (higher \(y\)) have better prospects if they continue searching, meaning they have higher outside option values, \(W_0(x)\) and \(\Pi_0(y)\). Because of this, the surplus \(S(x,y)\) may not increase monotonically with \(x\) or \(y\).
Example: Why Might Medium Types Reject Each Other?
Imagine a talented MBA student (\(x=0.8\)) gets an offer from a solid, but not top-tier, consulting firm (\(y=0.8\)). In Becker’s world, this could be a fine match. But in the world of Shimer-Smith, the student hesitates. ‘If I take this, am I giving up my shot at McKinsey or Goldman Sachs (\(y=0.95\))?’ The opportunity cost of settling for a “good-not-great” partner is too high. The student might reject this perfectly good offer in the ‘hope’ of finding a better partner. The firm thinks the same way. When this happens across the market, it can create a “hole” in the acceptance set, where medium-type workers and firms avoid each other.
Wage Determination: Splitting the Pie
If a match is formed, how is the “pie”: The surplus \(S(x,y)\) divided? Shimer and Smith assume Nash Bargaining. If the worker’s bargaining power is \(\alpha \in [0,1]\), the wage \(w(x,y)\) is set to give each party a share of the surplus proportional to their bargaining power.
This means the worker’s net gain (\(W_1 - W_0\)) is equal to \(\alpha\) times the total surplus \(S(x,y)\). \[ W_1(x,y) - W_0(x) = \alpha S(x,y) \] Solving this for the wage \(w(x,y)\) yields a crucial result: \[ w(x,y) = rW_0(x) + \alpha \left[ f(x,y) - rW_0(x) - r\Pi_0(y) \right] \] Rearranging gives: \[ w(x,y) = (1-\alpha)rW_0(x) + \alpha f(x,y) - \alpha r\Pi_0(y) \] This equation is in fundamental conflict with the simple additive structure of the AKM model (\(w = \alpha_i + \psi_j\)). The wage is not a sum of a worker part and a firm part. It contains a term, \(\alpha f(x,y)\), that explicitly depends on the interaction of the match. This is the “smoking gun” showing that the interaction effect is baked into the wage itself.
PAM Under Frictions: The Need for a Stronger Condition
In Becker’s world, supermodularity (\(f_{xy} > 0\)) was enough to guarantee PAM. But in a frictional world, the “hope for a better partner” can overwhelm this. For PAM to survive, the complementarity must be much stronger. Shimer and Smith show that this requires log-supermodularity.
- Supermodularity (\(f_{xy}>0\)): The level of output gain from a better partner is greater for a higher-type agent.
- Log-supermodularity (\(\frac{\partial^2 \ln f(x,y)}{\partial x \partial y} > 0\)): The percentage of output gain from a better partner is greater for a higher-type agent.
Analogy: Why Percentage Gain Matters
Log-supermodularity is a condition strong enough to overcome the temptation to “skip grades” and wait for a jackpot partner. Imagine you are a chess novice (\(x=0.1\)). Beating an intermediate player (\(y=0.5\)) versus an advanced player (\(y=0.6\)) gives you more points (+10 vs +12), that’s supermodularity. But if you are already a grandmaster (\(x=0.9\)), beating another grandmaster massively increases your “prestige” in percentage terms. Log-supermodularity means that this relative “jackpot” feeling from pairing with a better partner is much stronger for higher types. This feeling is so compelling that it makes agents of all types prefer the next step up rather than holding out indefinitely for the absolute best, thus inducing orderly, assortative matching.
The Core Result of Shimer-Smith: If the production function \(f(x,y)\) is log-supermodular, then the equilibrium matching will always be Positively Assortative (PAM), regardless of the search frictions. If it is merely supermodular but not log-supermodular, PAM can fail, and we can get phenomena like medium types avoiding each other.
In summary, in the frictional model of Shimer-Smith, we have learned: 1. Matching is driven by surplus (\(S(x,y) \ge 0\)). This is a rational choice that accounts for opportunity costs. 2. Wages inherently contain a match-interaction effect. This conflicts with the additive AKM framework. 3. PAM requires a condition stronger than supermodularity: log-supermodularity. Frictions make the conditions for orderly sorting much stricter.
Now that we have seen the theoretical complexity, in the next chapter, we will ask: “Given this messy world, how can we possibly analyze sorting with real data?”
References
Shimer, R., & Smith, L. (2000). Assortative Matching and Search. Econometrica, 68(2), 343–369.