The Unemployment Volatility Puzzle: A Note on the Role of Reference Points

This theoretical note aims at studying the role of reference points in generating unemployment volatility. For this purpose, I introduce the notion of reference points in a standard Mortensen-Pissarides model. I obtain two results. First, I find that the obtained model is similar to the one found by Pissarides (2009) with matching costs. Second, I show that the introduction of reference points can increase significantly unemployment volatility through a mechanism a la Hagerdorn and Manovskii (2008).


Introduction
Empirical studies and laboratory experiments clearly show that reference points play a fundamental role in (wage) negotiations (see, within a large literature, Kahneman and Tversky (1978), Kahneman (1992) and Lewicki et al. (2010).) Indeed, it is demonstrated that agents evaluate offers and outcomes as gains and losses relative to some reference points. Therefore, by affecting preferences, these points impact both the process and the outcome of bilateral bargaining. Moreover, a pervasive challenge in macroeconomics is to understand why the standard Mortensen-Pissarides (hereafter MP) model cannot generate the volatility of the unemployment rate observed in US data. This is the so-called Shimer puzzle. Several solutions have been proposed to solve this puzzle: wage stickiness (Shimer (2004)), credible bargaining (Hall and Milgom (2008)), very high unemployment benefits (Hagerdorn and Manovskii (2008)), additional matching costs (Pissarides (2009))...
The aim of this theoretical note is to draw a link between reference points and the unemployment volatility puzzle. For this purpose, I consider a simple MP model with exogenous separations, reference points and where the partition of the surplus is no longer derived by a Nash bargaining game. It is determined by a sequential bargaining game where the outcome of this new negotiation process is evaluated relative to a reference point. By doing so, I follow Compte and Jehiel (2003) and, more generally, the game theory literature (see Muthoo (1999)). I then deduce the new wage equation and the new associated job creation. I find that the obtained model is equivalent to the one found by Pissarides (2009). I also show that the presence of reference points raise considerably the unemployment volatility through a mechanism à la Hagerdorn and Manovskii (2008). Indeed, I demonstrate that reference points can lower firm's profit and increase wage share by improving the outside option of the worker. Thus, this short article adds reference points to the list of solutions to the Shimer puzzle.
Notice finally that this is not the first framework that integrates reference dependence in a MP model. In a recent working paper, Eliaz and Spiegler (2013) study the properties of a dynamical model with search and matching frictions and with a reference point in the productivity process of the firm. However, their model is quite different from the one developed in this paper. Indeed, it features wage stickiness, it amplifies unemployment volatility via a new mecanism independent from Hagerdorn and Manovskii (2008) and it does not aim at solving the Shimer puzzle.
This note is organized as follows. Section 2 describes the search and matching model with reference points. Section 3 concludes.

Search and matching model with reference points
The model considered hereafter is the standard Pissarides (2000) model with reference points and sequential bargaining in line with Compte and Jehiel (2003).

Pissarides (2000) environment
Let U and W be the asset values of being unemployed and being employed. These asset values are given by: and with r the risk-free interest rate, z the unemployment benefits, s the separation rate and f (θ) the job finding rate. Let V and J be the asset values of a vacancy and a filled job.
These asset values are defined as: and with c the cost of a vacancy, p the productivity of workers, ω the wage and q(θ) the job filling rate. Using equation (3), equation (4) and the free entry condition (i.e. V = 0), the job creation equation is determined as: Furthermore, notice that the unemployment rate of the economy is given by the following standard Beveridge curve:

The role of reference points
Once the match is made, employer and employee have to negotiate over the partition of the surplus defined as S = W − U + J − V according to a sequential bargaining game. In the first stage of the game, one player is randomly chosen to make a take-it or leave-it offer. The probability for the worker and for the firm to be drawn is assumed to be equal. If the offer is accepted by the opponent, the game ends. Conversely, if the offer is rejected, the game goes on to the next period where a player is again randomly selected and bargaining begins again. If players agree on a partition of the surplus, they enjoy the following utility function à la Compte and Jehiel (2003): 1 with i ∈ {W, F } and where W is the index of the worker such that x W = W − U , F is the index of the firm such that x F = J − V and φ i is the reference point of player i. Equation (7) states that the utility of agents depends on the deviation of the value of the agreement from the reference point. In line with prospect theory, this means that outcomes are compared to a reference point that splits the agent preferences into gains and losses. However, contrary to prospect theory and for the sake of simplicity, the valuation of gains and losses are symmetric (i.e. no loss aversion). 2 If players disagree forever, their payoffs are equal to zero. Using Compte and Jehiel (2003), there exists a unique agreement given by: if and only if φ W + φ F ≤ S. Otherwise (i.e. if φ W + φ F > S), I assume that no aggrement exists. This standard result in the game theory literature is straighforward to analyse. 3 Namely, this is the familiar split the difference rule: if demands are compatible (i.e. φ W + φ F ≤ S), then an agreement is a situation where each agent gets the utility value of its reference point and one-half of the remaining fraction of the surplus. Reducing system (8) gives the following new sharing rule: Using the above sharing rule, the wage satisfies: Likewise, using equation (1), the job creation equation and the sharing rule, I obtain: Plugging equation (11) in equation (10) yields: Equation (12) shows that the worker's reference point increases the wage by raising the reservation wage while the firm's reference point decreases the wage by lowering the expected return of the match. Moreover, observe that if reference points are equal (i.e. φ W = φ F ), I end up with the standard wage equation derived from a symmetric Nash bargaining game. Finally, the wage equation can be rewritten as: with β = 1 2 and where H = (1 − β)φ W − βφ F can be viewed as an index measuring the relative importance of the worker's reference point. Integrating equation (13) in equation (5), I find: Using the following job creation equation in Pissarides (2009): and the following wage equation in Pissarides (2009): yields the following job creation equation: with H a constant and where 0 < β < 1. Notice that the Beveridge curve in Pissarides (2009) is identical to the one in equation (6). Also observe that I can assume that H = H since H and H are exogenous parameters. Thus, up to a coefficient β , the job creation equation determined by a MP model with reference points is the same as the one determined by a MP model with matching costs. This indicates that these two models generate the same quantitative results. Indeed, I solve the job creation equation (14) for the unknown θ with Pissarides (2009) calibration where β = β = 0.5. I then study the effect of a 1 % productivity shock on the model's unknown by computing the elasticity θ of the tightness index with respect to productivity and the elasticity ω of the wage with respect to productivity. Since wage stickiness does not matter here, the amplification mechanism is driven by the relative role of workers' reference point. Indeed, for high H, the reference point of the worker is larger than the reference point of the firm. This leads to an increase in the wage set by firms because the reservation wage (or the outside option of the worker) is very high. This lowers the firm's surplus and so increases the effect of the productivity shock. Namely, the introduction of reference points in a standard MP model can increase the unemployment volatility through a mechanism à la Hagerdorn and Manovskii (2008). Finally, contrary to Pissarides (2009) where matching costs are always assumed to be exogenous, it is easy to endogenize reference points in this setting. Indeed, in this stationary framework, a natural candidate for the worker's reference point is the partition of the surplus received by a worker in the standard MP model. Assuming that the firm has no reference point and the reference point of the worker is φ W = 0.43, I obtain: H = 0.21. One can observe that even if the volatility is not always matched, the introduction of endogenous reference points in a standard MP model increases considerably the volatility generated by the model.

Conclusion
In this note, I integrate reference dependent preferences in the wage bargaining of the benchmark MP model. I show that the obtained model is similar to the one of Pissarides (2009). I also find that these reference points can generate unemployment volatility via a mechanism à la Hagerdorn and Manovskii (2008).