Stochastic Search Equilibrium

By Giuseppe Moscarini and fabien Postel-Vinay

We analyze a stochastic equilibrium contract-posting model. Firms post employment contracts, wages contingent on all payoff-relevant states. Aggregate productivity is subject to persistent shocks. Both employed and unemployed workers search randomly for these contracts, and are free to quit at any time. An equilibrium of this contract-posting game is Rank-Preserving [RP] if larger firms offer a larger value to their workers in all states of the world. We show that every equilibrium is RP, and equilibrium is unique, if firms differ either only in their initial size, or also in their fixed idiosyncratic productivity but more productive firms are initially larger, in which case turnover is always efficient, as workers always move from less to more productive firms. The RP equilibrium stochastic dynamics of firm size provide an explanation for the empirical finding that large employers are more cyclically sensitive (Moscarini and Postel-Vinay, 2009). RP equilibrium computation is tractable, and we simulate calibrated examples.

There was a time where macro models had only Taylor-type contracts at their disposal. Now we have contracts that specify a full set of contingent outcomes, with heterogeneity in firms and workers, and this is still tractable. One could even ask: is the real world really this complex?

3 Responses to Stochastic Search Equilibrium

  1. Good question. In this particular case, the answer is: it does not matter for many substantive issues. We like theories with testable predictions that do not depend critically on the details of the contract space.

    This paper studies employment allocation between firms over the business cycle. The equilibrium dynamics are incredibly simple: unemployed workers always accept any job offer, and employed workers quit their job if and only if they receive a contract offer from a more productive firm. This job ladder dynamics, when “shaken” by business cycle shocks, describes an evolution of the firm size distribution that is consistent with the data. Indeed, in another paper NBER WP 14740 we show that “Large Employers Are More Cyclically Sensitive” in the US and several other countries. This is exactly what the model, perhaps counterintuitively but certainly in line with what happens in the real world, implies.

    As of contracts, they could be even more complicated than we allow them to be. The point is that the underlying economic forces are so strong that the described employment dynamics are the unique equilibrium allocation, essentially no matter how complicated (or simple) the contracts. Thinking about complicated contracts made us see the business cycle pattern of firm sizes.

  2. In terms of descriptive accuracy, how complex those model contracts really are is debatable. They are indeed complex in that they are assumed contingent on a fairly elaborate state vector. However this dimension of complexity is, as Giuseppe points out in his post, not essential for many substantive issues. On the other hand, firms are assumed to fully commit to future wages, which is arguably a drastic simplification in the description of contractual arrangements. I suppose one could say that an interesting question for future research would be: can we preserve both tractability and realistic predictions about equilibrium employment dynamics in a model with MORE sophisticated contracts, without (or with limited) commitment on the firm side?

  3. What both authors do not mention here is that it is much simpler to deal with contracts over a complete set of contingent states than specifying them over a limited set of states. This is like comparing complete markets with incomplete markets. In the latter you need to specify every contingent market. In the first you do need to worry about those markets.

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