Climb the Job Ladder
Introduction: The Missing Piece of the Puzzle
In our journey so far, we have assembled a detailed picture of the labor market. In Part 1, we saw how a frictionless world produces perfect, static sorting. In Part 2, we introduced frictions, complicating the matching process. In Part 3, we played detective, exploring how to find evidence of sorting in messy data.
Yet, a crucial piece is still missing. Our models have largely treated mobility as an exogenous shock. But careers are defined by a sequence of moves, often initiated while employed. We have not yet fully explained the endogenous mobility of workers, their active choice to search and move.
Engine 1: The Bertrand Competition Model
The Bidding Game
When an employed worker (type \(x\), at firm \(y_1\)) receives an outside offer from a poaching firm (\(y_2\)), a fierce Bertrand competition ensues.
If the poacher is more productive (\(f(x, y_2) > f(x, y_1)\)): The poacher wins by offering a wage \(w = f(x, y_1)\), which is the maximum the incumbent firm is willing to pay. The worker moves and gets a large wage jump. If the incumbent is more productive (\(f(x, y_1) > f(x, y_2)\)): The incumbent firm retains the worker by matching the poacher’s best possible offer, raising the wage to \(w = f(x, y_2)\). The worker stays but gets a raise.
Let \(W(w)\) be the present value of a worker who currently earns a wage \(w\). In continuous time with discount rate \(r\), the value function evolves according to offers that arrive at rate \(\lambda_1\). If an offer comes from a firm with productivity \(y'\), the new wage becomes \(w' = f(x, y')\). The worker’s value function incorporates the expected gain from such events: \[ rW(w) = w + \lambda_1 \int \max(W(f(x, y')) - W(w), 0) \, dF(y') \] Here, \(F(y')\) is the distribution of firm productivities the worker might encounter. The integral represents the expected gain from on-the-job search.
Key Predictions: 1. Wage History Dependence: A worker’s wage is not a function of their current firm, but of the productivity of the second-best firm they have ever met. 2. Wage Jumps: Wages are constant between offers and only jump upwards upon receiving a credible outside offer (either by moving or by forcing a counteroffer). 3. Connection to AKM’s \(\psi_j\): This model provides a clear microfoundation for the AKM firm effect, \(\psi_j\). The premium of a high-productivity firm is not arbitrary generosity; it is the outcome of its ability to win bidding wars for talent.
Engine 2: The Wage-Posting Model
An alternative and equally influential framework is the wage-posting model of Burdett and Mortensen (1998). Here, there are no bidding wars. Instead, firms post a wage, and workers react.
The Mechanism: A Simple “Take It or Leave It”
- Firms Post Wages: Each firm commits to a single wage offer, \(w\). They cannot change it when a specific worker arrives.
- Workers Search: An unemployed worker (search rate \(\lambda_0\)) accepts any job offer they receive, as anything is better than zero. An employed worker (search rate \(\lambda_1\)) who is currently earning \(w_{current}\) will only accept an outside offer \(w_{new}\) if \(w_{new} > w_{current}\).
This simple rule creates a powerful “ratchet” effect: workers can only move up the wage ladder. They never move for a lower wage.
Firms face a critical trade-off: Post a Low Wage: Earn high profit per worker, but risk losing them quickly to a competitor offering more (high turnover). Post a High Wage: Earn low profit per worker, but retain them for longer (low turnover).
The key to this model is understanding the steady-state flow of workers. The rate at which a worker at a firm paying wage \(w\) separates is not just the exogenous rate \(\delta\). It also includes the endogenous separation from being poached. If the cumulative distribution of wage offers in the market is \(G(w)\), then the probability of getting an offer better than \(w\) is \((1 - G(w))\).
So, the total separation rate for a worker earning \(w\) is: \[ s(w) = \delta + \lambda_1 (1 - G(w)) \]
Firms choose their posted wage \(w\) to maximize the present value of profits, knowing that a higher \(w\) reduces the separation rate \(s(w)\). In equilibrium, it can be shown that firms must be indifferent between posting different wages, which leads to a non-degenerate, continuous distribution of wages \(G(w)\) across the market.
Key Predictions: 1. Endogenous Wage Dispersion: This model’s most famous result is that it generates wage dispersion even among identical workers and identical firms. The distribution of wages arises naturally from the search and poaching frictions. 2. Wage Growth via Mobility: A worker’s wage only increases when they move to a new firm that posts a higher wage. There is no within-job wage growth from counteroffers. 3. Connection to AKM’s Residuals \(\varepsilon_{it}\): While the Bertrand model helps explain the firm effect \(\psi_j\), the wage-posting model is brilliant at explaining the residual wage variance. Why do identical people at identical firms earn different amounts? The B-M model answers: it’s due to their position in the equilibrium wage distribution, determined by their random search luck.
Comparing the Engines and AKM
The two OJS models offer complementary explanations for the patterns AKM documents.
| Feature | Bertrand Competition (PVR-style) | Wage Posting (B-M style) |
|---|---|---|
| Wage Setting | Dynamic bidding war | Static “take it or leave it” offer |
| Information | Firms know each other’s productivity | Firms know the distribution of offers |
| Wage Growth | Jumps within a job (counteroffers) & when moving (poaching) | Jumps only when moving to a new job |
| Main Power | Explains large wage gains for movers and the role of firm productivity (\(f(x,y)\)) | Explains wage dispersion among identical agents |
A View from AKM: \[w_{it} = \alpha_i + \psi_j + \varepsilon_{it}\]
- The Bertrand Bidding model provides a powerful story for \(\psi_j\). The firm premium is a measure of its productivity and position on the job ladder, determining its ability to win bidding wars.
- The Wage Posting model provides a powerful story for the variance of \(\varepsilon_{it}\). The residual is not just measurement error; it reflects a worker’s equilibrium position in a wage distribution generated by search frictions.
- Endogenous Mobility is the core driver in both models, providing the theoretical underpinning for why studying “movers” is so central to identifying the AKM decomposition in the first place.
References
Burdett, K., & Mortensen, D. T. (1998). Wage Differentials, Employer Size, and Unemployment. International Economic Review, 39(2), 257–273.
Postel-Vinay, F., & Robin, J. M. (2002). Equilibrium Wage Dispersion with Worker and Employer Heterogeneity. Econometrica, 70(6), 2295–2350.
Cahuc, P., Postel-Vinay, F., & Robin, J. M. (2006). Wage Bargaining with On-the-Job Search: Theory and Evidence. Econometrica, 74(2), 323-364.