2.1 The MP environment

Workers and firms are risk-neutral and infinitely lived with a common discount factor $\beta$ . They randomly match in the job market subject to search friction. Once matched, the firm-worker pair produces a flow value of output $p_{t}\left ( \varepsilon _{t}\right ),$ where $\varepsilon _{t}$ is the idiosyncratic, or match-specific, productivity specific only to the pair and $p_{t}^{\prime }\left ( \cdot \right ) \gt 0$ . To model the persistence of the idiosyncratic shock, I follow Mortensen and Pissarides (Reference Mortensen and Pissarides1994) and Fujita and Ramey (Reference Fujita and Ramey2012) to assume a probability $\lambda$ of switching productivity. Hence, with probability $1-\lambda,$ we have $\varepsilon _{t+1}=\varepsilon _{t}.$ Otherwise, $\varepsilon _{t+1}$ will be drawn from the stationary distribution $G\left ( \cdot \right )$ on a compact support $[\varepsilon _{\min },\varepsilon _{\max }].$ There is a probability $s$ of exogenous separation. Otherwise, in each period, firms and workers can decide endogenously whether to dissolve the employment relationship. There is also a standard matching function $m\left ( u_{t},v_{t}\right )$ where $u_{t}$ and $v_{t}$ are the measures of unemployed workers and vacancies respectively. The market tightness at time $t$ is $\theta _{t}=\frac {v_{t}}{u_{t}}.$ Hence, the probability of an unemployed worker meeting with a vacancy is $ \phi \left ( \theta _{t}\right ) =\frac {m\left ( u_{t},v_{t}\right ) }{u_{t}} =m\left ( 1,\theta _{t}\right )$ , and that of a vacancy meeting with an unemployed worker is $q\left ( \theta _{t}\right ) =\frac {\phi \left ( \theta _{t}\right ) }{\theta _{t}}.$

2.1.1 Value functions

Firms post vacancies with a flow cost $c$ . Let $P_{t}$ be the (present discounted) value of production and $W_{t}$ be the expected present value of wages during the time of the production.Footnote ¹¹ Hence, a productive and working firm would receive a value of $ P_{t}\left ( \varepsilon _{t}\right ) -W_{t}\left ( \varepsilon _{t}\right ) .\,$ On the other hand, an unemployed worker earns a value of $U_{t}.$ Once employed in a firm, the worker would receive the value of wage $W_{t}$ and the value of working $V_{t}$ .Footnote ¹² Therefore, the firm would fire the worker when $P_{t}\left ( \varepsilon _{t}\right ) \leq W_{t}\left ( \varepsilon _{t}\right ),$ and the worker would quit the job if $W_{t}\left ( \varepsilon _{t}\right ) +V_{t}\left ( \varepsilon _{t}\right ) \leq U_{t}.$

Let $\chi _{t}\left ( \varepsilon _{t}\right )$ be the indicator function of endogenous separation at time $t,$ i.e.

(1) \begin{align} \chi _{t}(\varepsilon _{t}) =\left \{ \begin{array}{cc} 1 &\ \ \text{when }\underset {\text{layoff}}{\underbrace {P_{t}\left ( \varepsilon _{t}\right ) \leq W_{t}\left ( \varepsilon _{t}\right ) }\text{ }}\text{or } \underset {\text{quit}}{\underbrace {W_{t}\left ( \varepsilon _{t}\right ) +V_{t}\left ( \varepsilon _{t}\right ) \leq U_{t}}} \\[3pt] 0 &\ \ \text{otherwise} \end{array} \right . \end{align}

Hence, the value of production is given by

(2) \begin{align} P_{t}\left ( \varepsilon _{t}\right ) =p_{t}\left ( \varepsilon _{t}\right ) +\beta \left ( 1-s\right ) \mathbb{E}_{t}\left [ \begin{array}{c} \lambda \left ( 1-\chi _{t+1}\left ( \varepsilon _{t+1}\right ) \right ) P_{t+1}\left ( \varepsilon _{t+1}\right ) \\[3pt] +\left ( 1-\lambda \right ) \left ( 1-\chi _{t+1}\left ( \varepsilon _{t}\right ) \right ) P_{t+1}\left ( \varepsilon _{t}\right ) \end{array} \right ] \end{align}

Free entry of the firms implies that

(3) \begin{align} \beta q\left ( \theta _{t}\right ) \mathbb{E}_{t}\left [ \left ( 1-\chi _{t+1}\left ( \varepsilon _{t+1}\right ) \right ) \left ( P_{t+1}\left ( \varepsilon _{t+1}\right ) -W_{t+1}\left ( \varepsilon _{t+1}\right ) \right ) \right ] =c \end{align}

where $\varepsilon _{t+1}$ denotes the idiosyncratic productivity at time $ t+1$ which is random at time $t.$ The free-entry condition states that the flow cost equals expected profit of the firm in the future. Note that there is a possibility of immediate separation should the value of production be not enough to cover that of the wage, or the worker is rejecting the job offer.

Unemployed workers receive a flow value $b$ of non-market activity, which may include such things as leisure and unemployment benefit. Hence, the value of an unemployed worker is given by

(4) \begin{align} U_{t}=b+\beta \mathbb{E}_{t}\left [ \begin{array}{c} \phi \left ( \theta _{t}\right ) \left ( 1-\chi _{t+1}\left ( \varepsilon _{t+1}\right ) \right ) \left ( W_{t+1}\left ( \varepsilon _{t+1}\right ) +V_{t+1}\left ( \varepsilon _{t+1}\right ) \right ) \\[3pt] +\left [ \left ( 1-\phi \left ( \theta _{t}\right ) \right ) +\phi \left ( \theta _{t}\right ) \chi _{t+1}\left ( \varepsilon _{t+1}\right ) \right ] U_{t+1} \end{array} \right ] \end{align}

and the value of working is given byFootnote ¹³

(5) \begin{align} V_{t}\left ( \varepsilon _{t}\right ) =\beta \mathbb{E}_{t}\left \{ \begin{array}{c} \left [ s+\left ( 1-s\right ) \left ( \lambda \chi _{t+1}\left ( \varepsilon _{t+1}\right ) +\left ( 1-\lambda \right ) \chi _{t+1}\left ( \varepsilon _{t}\right ) \right ) \right ] U_{t+1} \\[3pt] +\left ( 1-s\right ) \left [ \begin{array}{c} \lambda \left ( 1-\chi _{t+1}\left ( \varepsilon _{t+1}\right ) \right ) V_{t+1}\left ( \varepsilon _{t+1}\right ) \\[3pt] +\left ( 1-\lambda \right ) \left ( 1-\chi _{t+1}\left ( \varepsilon _{t}\right ) \right ) V_{t+1}\left ( \varepsilon _{t}\right ) \end{array} \right ] \end{array} \right \} \end{align}

2.1.2 Match surplus and thresholds

The joint surplus, worker’s surplus, and firm’s surplus are given, respectively, by

(6) \begin{align} S_{t}\left ( \varepsilon _{t}\right ) =P_{t}\left ( \varepsilon _{t}\right ) +V_{t}\left ( \varepsilon _{t}\right ) -U_{t} \\[-8pt] \nonumber \end{align}

(7) \begin{align} J_{t}\left ( \varepsilon _{t}\right ) =V_{t}\left ( \varepsilon _{t}\right ) +W_{t}\left ( \varepsilon _{t}\right ) -U_{t} \\[-8pt] \nonumber\end{align}

(8) \begin{align} F_{t}\left ( \varepsilon _{t}\right ) =P_{t}\left ( \varepsilon _{t}\right ) -W_{t}\left ( \varepsilon _{t}\right ) \\[10pt] \nonumber\end{align}

whenever they are positive, and zero otherwise. I assume the wage rule is such that the above surplus functions are all strictly increasing and continuous in $\varepsilon _{t}\,$ (which would apply to the Nash bargaining model I consider in the next subsection). Hence, there exist cutoff values $\varepsilon _{t}^{e},\varepsilon _{t}^{w}$ and $\varepsilon_{t}^{\, f}$ such that

(9) \begin{align} S_{t}\left ( \varepsilon _{t}^{e}\right ) &=0 \\[-10pt] \nonumber\end{align}

(10) \begin{align} J_{t}\left ( \varepsilon _{t}^{w}\right ) &=0 \\[-10pt] \nonumber\end{align}

(11) \begin{align} F_{t}\left ( \varepsilon_{t}^{\, f}\right ) &=0 \\[10pt] \nonumber\end{align}

Intuitively, the worker will choose to quit the job if she observes a productivity $\varepsilon \lt \varepsilon _{t}^{w}.$ Similarly, the firm would fire the worker if $\varepsilon \lt \varepsilon_{t}^{\, f}.$ Finally, $ \varepsilon _{t}^{e}$ denotes the efficient cutoff productivity. Therefore, any job separation happening at a productivity $\varepsilon _{t}\gt \varepsilon _{t}^{e}$ is privately inefficient, since the employment pair would have had a positive joint surplus if they were to choose not to separate.

2.2 Nash bargaining with bargaining wedges (NBBW) and inefficient separations

Let $\eta \in \left [ 0,1\right ]$ be the bargaining power of the worker. Wages are determined by maximizing the generalized Nash product

(12) \begin{eqnarray} &&\max _{W_{t}}\left ( J_{t}\left ( \varepsilon _{t}\right ) -\Omega _{t}^{w}\right ) ^{\eta }\left ( F_{t}\left ( \varepsilon _{t}\right ) -\Omega_{t}^{\, f}\right ) ^{1-\eta } \notag \\[3pt] &=&\max _{W_{t}}\left ( V_{t}\left ( \varepsilon _{t}\right ) +W_{t}\left ( \varepsilon _{t}\right ) -U_{t}-\Omega _{t}^{w}\right ) ^{\eta }\left ( P_{t}\left ( \varepsilon _{t}\right ) -W_{t}\left ( \varepsilon _{t}\right ) -\Omega_{t}^{\, f}\right ) ^{1-\eta } \end{eqnarray}

where $\Omega _{t}^{w}$ and $\Omega_{t}^{\, f},$ which are exogenous to the worker and the firm, are the bargaining wedges of the worker and the firm respectively. A bargaining wedge is the difference between the actual surplus and the bargaining surplus.

We can interpret the wedges $\Omega _{t}^{w}$ and $\Omega_{t}^{\, f}$ as reduced-form objects that summarize the difference between actual continuation values and the values that effectively discipline bargaining in the protocol at hand. They can reflect (i) disagreement payoffs that include flows during bargaining (Section 3), (ii) informational rents and screening distortions under private information (Section 4), or (iii) other contracting imperfections (e.g., limited verifiability or costly renegotiation) that create a gap between what is feasible ex-post and what can be credibly threatened during negotiation. The key sufficient statistic for inefficiency is the net wedge ( $\Omega _{t}^{net}$ ; defined below), which governs both the wage shift and the ordering of separation cutoffs in Proposition 1. If $\Omega _{t}^{w}\gt 0,$ for example, then the actual surplus the worker receives ex-post will be higher than the surplus perceived when the worker is bargaining with the firm. Note that when $\Omega _{t}^{w}=\Omega_{t}^{\, f}=0,$ it reduces to the standard Nash bargaining problem.

The solution to the above problem is given by

(13) \begin{equation} \eta \left ( P_{t}\left ( \varepsilon _{t}\right ) -W_{t}\left ( \varepsilon _{t}\right ) -\Omega_{t}^{\, f}\right ) =\left ( 1-\eta \right ) \left ( V_{t}\left ( \varepsilon _{t}\right ) +W_{t}\left ( \varepsilon _{t}\right ) -U_{t}-\Omega _{t}^{w}\right ) \end{equation}

and by rearranging we have the wage rule

(14) \begin{equation} W_{t}\left ( \varepsilon _{t}\right ) =\eta P_{t}\left ( \varepsilon _{t}\right ) +\left ( 1-\eta \right ) \left ( U_{t}-V_{t}\left ( \varepsilon _{t}\right ) \right ) +\left ( 1-\eta \right ) \Omega _{t}^{w}-\eta \Omega_{t}^{\, f} \end{equation}

Note that other things equal, the wage is increasing with $\Omega _{t}^{w}$ and decreasing with $\Omega_{t}^{\, f}.\,$ In what follows I define

(15) \begin{equation} \Omega _{t}^{net}\equiv \left ( 1-\eta \right ) \Omega _{t}^{w}-\eta \Omega_{t}^{\, f} \end{equation}

as the net bargaining wedge between the worker and the firm. It measures the weighted difference between the bargaining wedge of the worker and that of the firm. Hence, it represents the net distortionary effect of the bargaining wedges on the wage rule. It turns out that the value of the net bargaining wedge has important implications for the efficiency of separation as well, as demonstrated in the following proposition.

Figure 1.

Efficiency of job separations.

The intuition of Proposition 1 is summarized in Figure 1. When $ \Omega _{t}^{net}=0,$ the three cutoff productivities are the same; hence, job separation is always efficient in this case. This implies the classical result that employment and separation under standard Nash bargaining are always privately efficient. On the other hand, inefficient separation may arise when the net bargaining wedge is non-zero. When $\Omega _{t}^{net}\gt 0,$ wages are excessively high relative to the standard bargaining solution. As a result, the cutoff productivity $\varepsilon_{t}^{\, f}$ for layoff decision by the firm is larger than the efficient cutoff productivity $\varepsilon _{t}^{e}$ . In this case, inefficient layoffs would be possible. For any productivity that lies between $\varepsilon _{t}^{e}$ and $\varepsilon_{t}^{\, f}$ , the firm would choose to fire the worker even when the joint surplus is positive. Efficient separation occurs only if productivity is sufficiently low (i.e., less than $\varepsilon _{t}^{e}$ ). Conversely, when $ \Omega _{t}^{net}\lt 0,$ wages are too low relative to the standard case. So the cutoff productivity $\varepsilon _{t}^{w}$ for quitting decision by the worker is larger than the efficient cutoff productivity $\varepsilon _{t}^{e}$ . Similarly, inefficient quits would occur when the productivity falls between $\varepsilon _{t}^{e}$ and $\varepsilon _{t}^{w}.$

Sections 3 and 4 show that specific bargaining environments—such as alternating-offers bargaining with delay and bargaining under private information—can generate nonzero bargaining wedges. But the purpose of the generalized model in Section 2 is not merely to nest those cases. Its main value is to isolate a common sufficient statistic, the net bargaining wedge $\Omega _{t}^{net}$ , that governs whether separations are inefficient and in which direction. This reduced-form representation delivers several insights that are difficult to obtain cleanly from any single bargaining protocol. First, it separates the source of inefficiency from the particular institutional details of bargaining, making clear that inefficient layoffs and quits arise whenever bargaining surpluses differ from actual surpluses in a way summarized by $ \Omega _{t}^{net}$ . Second, it yields transparent comparative statics: the sign and magnitude of $\Omega _{t}^{net}$ determine the ordering of the separation thresholds and therefore whether the economy exhibits inefficient layoffs or inefficient quits. Third, it makes it possible to carry out a quantitative decomposition of unemployment into efficient and inefficient components without committing the entire analysis to one specific bargaining micro-foundation.

It may be interesting to see what factors affect the inefficiency gaps $\varepsilon_{t}^{\, f}-\varepsilon _{t}^{e}$ and $\varepsilon _{t}^{w}-\varepsilon _{t}^{e}$ in (ii) and (iii). For illustration, consider a special case when there is no persistence between productivity shocks (i.e., $\lambda =1$ ), and the total productivity $p_{t}$ is a product of aggregate productivity $y_{t}$ and individual productivity $\varepsilon _{t}$ (i.e., $p_{t}\left ( \varepsilon _{t}\right ) =y_{t}\varepsilon _{t}$ ) $.$ In this case, we haveFootnote ¹⁴

(16) \begin{align} \varepsilon_{t}^{\, f}-\varepsilon _{t}^{e} &=\frac {\Omega _{t}^{net}}{ y_{t}\left ( 1-\eta \right ) } \\[-10pt] \nonumber \end{align}

(17) \begin{align} \varepsilon _{t}^{w}-\varepsilon _{t}^{e} &=\frac {-\Omega _{t}^{net}}{ y_{t}\eta } \\[2pt] \nonumber\end{align}

This special case demonstrates that the inefficiency gap is proportional to the magnitude of the net bargaining wedge and inversely proportional to the aggregate productivity. It also depends negatively on the bargaining power of the party that initiates the inefficient separation. When $\Omega _{t}^{net}\gt 0,$ for example, the inefficiency gap for inefficient layoff depends negatively on the bargaining power of the firm.

Equations (16) and (17) are derived as a special case to provide a transparent illustration of how the inefficiency gap scales with the net wedge and aggregate productivity. In the calibrated quantitative model, idiosyncratic productivity is persistent (i.e., $\lambda \lt 1)$ ,Footnote ¹⁵ so the cutoff values are determined by fixed-point conditions that incorporate continuation values. Nevertheless, the key qualitative implication is robust: when $\Omega _{t}^{net}$ is less cyclical than the joint surplus, the wedge-induced component of the separation cutoff becomes relatively more important in recessions, making the inefficiency gap (e.g., $\varepsilon_{t}^{\, f}-\varepsilon _{t}^{e}$ ) countercyclical when $\Omega _{t}^{net}\gt 0$ .

There are some implications following this expression. First, the effect of the net bargaining wedge is magnified in bad times, implying that the inefficiency gap tends to be larger during recessions. Intuitively, the job separation decision is made only when the total productivity is low enough. Hence, the labor market will be more sensitive to the distortionary effect during bad times, when the aggregate productivity is low. This generates more inefficient job separations and could potentially explain the mass layoffs observed in the recessions. Second, a higher bargaining power of the firm could mitigate the distortion when the net bargaining wedge is positive, and similarly for that of the worker when net bargaining wedge is negative. Intuitively, when the weighted bargaining wedge of the worker is larger than that of the firm (i.e., $\Omega _{t}^{net}\gt 0$ ), the wage rule would be distorted in favor of the worker, which would entail a higher-than-efficient level of wage rate. A higher bargaining power of the firm would mean the firm would be able to bargain for a lower wage, hence offsetting part of the distortion by the bargaining wedge.

It is worth noting the above analysis assumes that the bargaining wedge is exogenous and fixed. In general, however, $\Omega _{t}^{net}$ may depend on some parameters in the model and may even fluctuate over the business cycle. Hence, the above comparative statics results may be amplified or dampened depending on the endogeneity of $\Omega _{t}^{net}.$ In the following sections when I provide micro-foundations to $\Omega _{t}^{net},$ I will explicitly internalize $\Omega _{t}^{net}$ when performing the comparative statics. I will also consider the cyclicality of $\Omega _{t}^{net}$ in the quantitative analysis.

2.3 Worker flows and inefficient unemployment

I measure the worker flows as follows. Let $e_{t}\left ( \varepsilon \right )$ be the distribution function of the employed workers, and $u_{t}$ be the unemployment rate. Hence, $e_{t}=e_{t}\left ( \varepsilon _{\max }\right ) =1-u_{t}$ is the employment rate. Also, denote $\varepsilon _{t}^{s}$ as the separation cutoff value of the individual productivity. i.e.

(18) \begin{equation} \varepsilon _{t}^{s}=\max \left \{ \varepsilon _{t}^{w},\varepsilon_{t}^{\, f}\right \} \end{equation}

Note that Proposition 1 implies that $\varepsilon _{t}^{s}\geq \varepsilon _{t}^{e}.\,$ Then the flow equation of the employment distribution is given by

(19) \begin{eqnarray} e_{t+1}\left ( \varepsilon \right ) &=&\left ( 1-s\right ) \left \{ \lambda e_{t} \left [ G\left ( \varepsilon \right ) -G\left ( \varepsilon _{t+1}^{s}\right ) \right ] +\left ( 1-\lambda \right ) \left [ e_{t}\left ( \varepsilon \right ) -e_{t}\left ( \varepsilon _{t+1}^{s}\right ) \right ] \right \} \notag \\[3pt] &&+u_{t}\phi \left ( \theta _{t}\right ) \left [ G\left ( \varepsilon \right ) -G\left ( \varepsilon _{t+1}^{s}\right ) \right ] \end{eqnarray}

Hence, the measure of employed workers having productivity less than or equal to $\varepsilon$ at time $t+1$ comes from three groups of workers at time $t,$ namely (i) those employed at time $t$ who are switching productivity, and drawing a productivity from the interval $\left [ \varepsilon _{t+1}^{s},\varepsilon \right] \!,$ (ii) those employed at time $t$ who have a productivity in $\left [ \varepsilon _{t+1}^{s},\varepsilon \right ]$ and not switching their productivity, and (iii) those unemployed at time $ t,$ having matched to a firm, and drawing a productivity from the interval $ \left [ \varepsilon _{t+1}^{s},\varepsilon \right ] .$

Similarly, the flow to unemployment is given by

(20) \begin{align} \begin{aligned} u_{t+1}=e_{t}\left \{ s+\left ( 1-s\right ) \lambda G\left ( \varepsilon _{t+1}^{s}\right ) \right \} +e_{t}\left ( \varepsilon _{t+1}^{s}\right ) \left ( 1-s\right ) \left ( 1-\lambda \right ) +u_{t}\left [ 1-\phi \left ( \theta _{t}\right ) \left ( 1-G\left ( \varepsilon _{t+1}^{s}\right ) \right ) \right] \\[3pt] \end{aligned} \\[-21pt] \nonumber \end{align}

Therefore, those employed workers at time $t$ whose jobs are destroyed for exogenous reason and whose productivities are too low, and those unemployed at time $t$ who are not able to match a firm and to draw a high enough productivity, will be unemployed at time $t+1.$

To see the effect of inefficient job separations on the efficiency of unemployment, consider the steady-state unemployment rate,

(21) \begin{equation} u=\frac {s+\left ( 1-s\right ) \lambda G\left ( \varepsilon ^{s}\right ) }{\phi \left ( \theta \right ) \left ( 1-G\left ( \varepsilon ^{s}\right ) \right ) +s+\left ( 1-s\right ) \lambda G\left ( \varepsilon ^{s}\right ) } \end{equation}

and its efficient counterpart $u^{e},$ when $\varepsilon ^{s}$ is replaced by $\varepsilon ^{e}.$ Since $\varepsilon ^{s}\geq \varepsilon ^{e}$ and $u$ is increasing with $\varepsilon ^{s},$ we have $u\geq u^{e}$ .Footnote ¹⁶ This shows that the existence of bargaining friction induces a higher level of unemployment than the standard model where all separations are efficient. Hereafter I shall call the extra level of unemployment the inefficient unemployment.

I derive the flow equations of an efficient level of unemployment $u_{t}^{e}$ by replacing $\varepsilon _{t}^{s}$ in the flow equations (19) and (20) with $\varepsilon _{t}^{e}$ . Hence, efficient unemployment corresponds to the level of unemployment when all separations are efficient. Inefficient unemployment will then be given by the difference $ u_{t}^{i}=u_{t}-u_{t}^{e}.$ This simple decomposition allows me to measure the contribution of unemployment volatility by the inefficient separations later in the quantitative analysis. Section 6 returns to this decomposition and shows that the inefficient component becomes especially important in recessions, when the gap between total and efficient unemployment widens substantially. Also, in the calibrated model, inefficient unemployment explains a substantial part of total unemployment volatility.

The EU and UE transitions are given respectively as follows:

(22) \begin{align} EU_{t+1} &=e_{t}\left \{ s+\left ( 1-s\right ) \lambda G\left ( \varepsilon _{t+1}^{s}\right ) \right \} +e_{t}\left ( \varepsilon _{t+1}^{s}\right ) \left ( 1-s\right ) \left ( 1-\lambda \right ) \\[-10pt] \nonumber\end{align}

(23) \begin{align} UE_{t+1} &=\phi \left ( \theta _{t}\right ) \left ( 1-G\left ( \varepsilon _{t+1}^{s}\right ) \right ) u_{t} \\[10pt] \nonumber\end{align}

Finally, the job separation and job-finding rates are given by

(24) \begin{align} SR_{t+1} &=\frac {EU_{t+1}}{e_{t}}=s+\left ( 1-s\right ) \lambda G\left ( \varepsilon _{t+1}^{s}\right ) +\frac {e_{t}\left ( \varepsilon _{t+1}^{s}\right ) }{e_{t}}\left ( 1-s\right ) \left ( 1-\lambda \right ) \\[-10pt] \nonumber\end{align}

(25) \begin{align} FR_{t+1} &=\frac {UE_{t+1}}{u_{t}}=\phi \left ( \theta _{t}\right ) \left ( 1-G\left ( \varepsilon _{t+1}^{s}\right ) \right ) \\[10pt] \nonumber\end{align}

As we will see later, a relatively acyclical bargaining wedge would induce a higher volatility in the cutoff productivity $\varepsilon _{t}^{s}$ and hence in the workers flow in the labor market. Quantitatively, however, the effect on the job-finding rate is limited, since the cutoff productivity only affects the job-finding rate through the distribution function $G\left ( \cdot \right ) .$

2.4 Comparison with the canonical MP model

Since the canonical MP model is typically written in flow terms, it is useful to translate the NBBW wage rule into a flow-wage equation. To facilitate comparison, consider a steady-state benchmark with a constant bargaining wedge. Appendix A.2 provides the derivations.

Let $w^{ss}$ denote the steady-state flow wage and $\varepsilon ^{s}$ be the steady-state separation cutoff, then it can be shown that

(26) \begin{align} w^{ss}\left ( \varepsilon \right ) =\left ( 1-\eta \right ) b+\eta \left ( p\left ( \varepsilon \right ) +c\theta ^{ss}\right ) +\left [ \begin{array}{c} 1-\beta \left ( 1-s\right ) \left ( 1-\lambda G\left ( \varepsilon ^{s}\right ) \right ) \\[3pt] +\beta \theta ^{ss}q\left ( \theta ^{ss}\right ) \left ( 1-G\left ( \varepsilon ^{s}\right ) \right ) \end{array} \right ] \Omega ^{net} \end{align}

where $\theta ^{ss}$ is the steady-state market tightness. Thus, the bargaining wedge shifts the standard Nash wage by an additive term. The corresponding steady-state free-entry condition is

(27) \begin{align} \frac {c}{\beta q\left ( \theta ^{ss}\right ) }=\frac {\int _{\varepsilon ^{s}}^{\varepsilon _{\max }}\left [ p\left ( \varepsilon \right ) -w^{ss}\left ( \varepsilon \right ) \right ] dG\left ( \varepsilon \right ) }{1-\beta \left ( 1-s\right ) \left ( 1-\lambda G\left ( \varepsilon ^{s}\right ) \right ) }\\[-15pt] \nonumber \end{align}

These expressions show that there are two roles of $\Omega ^{net}$ . First, a positive net bargaining wedge raises wages directly. Through the free-entry condition, this wage effect reduces the expected profit from a match and therefore weakens job creation. Second, $\Omega ^{net}$ affects free entry indirectly through the separation margin. A higher $\Omega ^{net}$ raises the separation cutoff $\varepsilon ^{s}$ , so fewer newly matched workers draw acceptable productivity ( $1-G\left ( \varepsilon ^{s}\right )$ falls) and existing matches are less likely to survive a redraw ( $1-\lambda G\left ( \varepsilon ^{s}\right )$ falls). Hence, bargaining friction depresses vacancy creation because it shortens expected match duration. This is why the bargaining wedge that distorts wages also amplifies unemployment through the separation channel.

4.1 Discussion

Viewed through the wedge representation, the AOB and NBS environments are best seen as two special cases rather than as exhaustive alternatives to the baseline model. In the AOB case, the implied net bargaining wedge is positive, so the model predicts inefficient layoffs generated by costly delay. In the NBS case, the implied net wedge is also non-negative, but the comparison with the wedge model requires $\eta =0$ , so the firm effectively has all bargaining power. By contrast, the generalized wedge model relaxes both restrictions. First, it allows $\Omega _{t}^{net}$ to be either positive or negative, so it can generate inefficient quits as well as inefficient layoffs. Second, it allows $\Omega _{t}^{net}$ to vary cyclically in a way that is not tied to a single bargaining protocol. This matters quantitatively: an acyclical bargaining wedge explains about $22\%$ of unemployment volatility in the data, while allowing wedge cyclicality raises the explanatory power to about $29\%$ . Third, the generalized model preserves $\eta$ as a free parameter, so bargaining-power shifts have independent implications for separation inefficiency. These predictions are absent from the NBS special case, where $\eta =0$ by construction. Hence, the value of the wedge approach is not merely that it nests AOB and NBS, but that it allows multiple bargaining frictions to operate jointly and yields comparative statics and quantitative predictions that are not tied to either case.

5.1 Relationship with the fundamental surplus in Ljungqvist and Sargent (Reference Ljungqvist and Sargent2017)

Ljungqvist and Sargent (Reference Ljungqvist and Sargent2017) show that in a steady-state MP model with exogenous separations and homogeneous productivity, the elasticity of market tightness $\theta$ with respect to productivity $y$ depends crucially on a factor which they refer to as the fundamental surplus. Intuitively, fundamental surplus refers to the flow surplus from creating a match net of the worker’s flow value in unemployment and related terms that shift wages; it is the object that governs the elasticity of market tightness with respect to productivity in the standard MP model. Ljungqvist and Sargent (Reference Ljungqvist and Sargent2017) show that in order for the elasticity of market tightness to be large, the fundamental surplus has to be small. Here I incorporate the bargaining wedges described previously into their model and examine how they are related to the concept of fundamental surplus.

Following their derivation (details are in the Appendix A), it can be shown that the elasticity of market tightness in this case is given by

(38) \begin{equation} \epsilon _{\theta,y}=\Upsilon \left ( \Omega ^{net}\right ) \frac {y}{y-z- \left [ \frac {1-\beta \left ( 1-s\right ) }{1-\eta }\right ] \Omega ^{net}} \end{equation}

where

(39) \begin{equation} \Upsilon \left ( \Omega ^{net}\right ) =\frac {r+s+\eta \theta q\left ( \theta \right ) \left ( 1+\beta q\left ( \theta \right ) \frac {\Omega ^{net}}{c\eta } \right ) }{\alpha \left ( r+s\right ) +\eta \theta q\left ( \theta \right ) \left ( 1+\left ( 1-\alpha \right ) \beta q\left ( \theta \right ) \frac {\Omega ^{net}}{c\eta }\right ) } \end{equation}

is the factor with an upper bound $\max \left \{ \frac {1}{\alpha },\frac {1}{ 1-\alpha }\right \}$ and $\alpha \equiv \frac {-q^{\prime }\left ( \theta \right ) \theta }{q\left ( \theta \right ) }$ is the elasticity of the matching function. Here the fundamental surplus is $y-z-\left [ \frac {1-\beta \left ( 1-s\right ) }{1-\eta }\right ] \Omega ^{net}.\,$ Therefore, the existence of bargaining wedges will increase the elasticity of market tightness if $ \Omega ^{net}\gt 0.$ As in many cases studied by Ljungqvist and Sargent (Reference Ljungqvist and Sargent2017), bargaining friction can also potentially increase the unemployment volatility through raising the elasticity of market tightness.

There are problems with the steady-state analysis of unemployment volatility by Ljungqvist and Sargent (Reference Ljungqvist and Sargent2017). First, as mentioned in detail in the Appendix of Christiano et al. (Reference Christiano, Eichenbaum and Trabandt2016), steady-state models are generally misleading about the dynamic effects of a persistent shock. This point is made similarly in Petrosky-Nadeau and Zhang (Reference Petrosky-Nadeau and Zhang2017). Second, and more relevant to this paper, the sole focus on the elasticity of market tightness to explain unemployment dynamics relies on the assumption that the elasticity of the separation rate is zero. However, there is evidence that the separation rate is counter-cyclical.Footnote ²¹ As shown in these studies, the elasticity of the separation rate relevant to the unemployment dynamics is indeed non-trivial. In fact, as I will show in the quantitative analysis, the main effect of the bargaining friction is on the volatility of separation rate, which is ignored by the analysis in much of the literature.

In what follows, I shall refer to the above mechanism as the small-surplus channel. In models with exogenous separations and homogeneous productivity, the small-surplus channel is the only channel through which bargaining friction can affect unemployment volatility.

5.2 Cyclicality of worker’s share of the match surplus

In the standard Nash bargaining model, the worker’s share of the match surplus, defined as

(40) \begin{equation} \pi _{t}\left ( \varepsilon _{t}\right ) =\frac {J_{t}\left ( \varepsilon _{t}\right ) }{S_{t}\left ( \varepsilon _{t}\right ) } \end{equation}

is equal to their (constant) bargaining power $\eta,$ and thus is acyclical. Hence, the firm’s share $1-\pi _{t}\left ( \varepsilon _{t}\right ) =1-\eta$ is also unresponsive to aggregate fluctuation. This explains the flexibility of wages under Nash bargain, as the worker’s surplus moves proportionally with the match surplus. Under the existence of bargaining friction, however, the worker’s share can be responsive to the business cycle. By rearranging the wage rule (14), we have the following.Footnote ²²

Proposition 4. The worker’s share of the match surplus in the NBBW model is given by

\begin{equation*} \pi _{t}\left ( \varepsilon _{t}\right ) =\eta +\frac {\Omega _{t}^{net}}{ S_{t}\left ( \varepsilon _{t}\right ) } \end{equation*}

Hence, if $\Omega _{t}^{net}$ is less cyclical than $S_{t}\left ( \varepsilon _{t}\right )$ (i.e., $\frac {\Omega _{t}^{net}}{S_{t}\left ( \varepsilon _{t}\right ) }$ is countercyclical), then the worker’s share is countercyclical (procyclical) if $\Omega _{t}^{net}\gt 0\,\left ( \lt 0\right ) .$

The intuition of the proposition can perhaps be better understood in the context of the AOB model discussed earlier. In this case, we have a constant $\Omega _{t}^{net}\gt 0$ and so the worker’s share is countercyclical. Since the disagreement payoff for both parties is to continue for another round of bargain, the wage offers proposed would be less sensitive to the aggregate conditions.Footnote ²³ This entails that, consistent with the findings in Hall and Milgrom (Reference Hall and Milgrom2008), wages move less than proportionally with the aggregate productivity. Equivalently, the worker’s share of the match surplus increases in recessions. While the above is true for the AOB model, the NBBW model considered in this paper provides a more general framework to explain the increasing bargaining power of the workers in bad times. It is worth noting that the last result still holds even if $\Omega _{t}^{net}$ is either countercyclical or procyclical but less so than the match surplus (in the sense that $\frac {\Omega _{t}^{net}}{S_{t}\left ( \varepsilon _{t}\right ) }$ is still countercyclical).

There are two consequences for the countercyclicality of the worker’s share of the match surplus. First, this reduces the job creation incentive in the recession, compared with the standard Nash bargaining model. Also, a small change in $y_{t}$ causes significant fluctuations in the cutoff productivity and thus in the quantity of job separations. As a result, in recessions there would be more (inefficient) job separations, which only exists in the NBBW model. I call this effect the separation channel, which does not exist in homogeneous models like that in Ljungqvist and Sargent (Reference Ljungqvist and Sargent2017). This separation channel is also distinct from standard heterogeneous MP models with endogenous job destruction under flexible Nash bargaining. In those models, the separation cutoff moves over the business cycle, but separations remain privately efficient because the wage bargain internalizes the joint surplus. In the present model, bargaining frictions distort the cutoff itself, so changes in separations reflect not only efficient reallocation but also inefficient layoffs or quits. Hence, bargaining frictions amplify unemployment not merely by reducing job creation incentives, but also by generating excess cyclical separations. Both the small surplus and the separation channels mentioned above could potentially generate excessive fluctuation of unemployment. In the next section, I will quantify these channels and show that $\Omega _{t}^{net}$ has a much stronger effect on the separation channel.

6.1 Specification and calibration

The matching function is of the standard Cobb-Douglas form.

(41) \begin{align} m\left ( u,v\right ) =Au^{\alpha }v^{1-\alpha }\end{align}

where $\alpha$ is the elasticity of the matching function and $A$ measures the matching efficiency. This implies the probability of a vacancy meeting with an unemployed worker is $q\left ( \theta \right ) =\frac {m\left ( u,v\right ) }{v}=A\theta ^{-\alpha }$ and that of an unemployed worker meeting a vacancy is $\phi \left ( \theta \right ) =\frac {m\left ( u,v\right ) }{ u}=A\theta ^{1-\alpha }.$

Following Fujita and Ramey (Reference Fujita and Ramey2012), I assume $p_{t}\left ( \varepsilon _{t}\right ) =y_{t}\varepsilon _{t},$ where $y_{t}$ is the aggregate productivity following the stochastic process

(42) \begin{align} \ln y_{t}=\rho \ln y_{t-1}+\zeta _{t} \\[-10pt] \nonumber \end{align}

where $\zeta _{t}$ is identically and independently distributed normal white noise with mean $\mu _{\zeta }$ and standard deviation $\sigma _{\zeta }.$ Also, the distribution function $G\left ( x\right )$ is taken to be truncated log-normal with support $\left [ 0,\varepsilon _{\max }\right ]$ .Footnote ²⁴

The model is calibrated using data for the US labor market data from 1976 to 2016. Each period in the model corresponds to one week.Footnote ²⁵ The discount factor is taken to be $\beta =0.9992$ which is equivalent to an annual discount rate of $4\%.\,$ The process of the aggregate productivity follows Hagedorn and Manovskii (Reference Hagedorn and Manovskii2008). Specifically, we have $\rho =0.9895$ , $\mu _{\zeta }=0$ and $ \sigma _{\zeta }=0.0034$ .Footnote ²⁶ I set $s=0.000625,$ which is consistent with the mean value of the monthly rate of “other separations” in the Job Openings and Labor Turnover Survey (JOLTS).Footnote ²⁷

On the worker side, I set $b=0.71,$ which follows the derivation of Hall and Milgrom (Reference Hall and Milgrom2008) and is based on a replacement ratio of $0.25$ .Footnote ²⁸ Also, the elasticity of the matching function $ \alpha$ is set to be $0.5,$ which is consistent with the survey of Petrongolo and Pissarides (Reference Petrongolo and Pissarides2001) $.$ I assume Hosios condition, which implies the bargaining power $\eta =\alpha .$

For each fixed value of the net bargaining wedge, the rest of the parameters are jointly calibrated to match the labor market flows in the US economy. The maximum value of match-specific productivity $\varepsilon _{\max }$ is calibrated such that the mean value of the productivity per worker is $1.$ The coefficient of matching efficiency $A$ is calibrated to fit the model to an average monthly job-finding rate of $32\%$ in the Current Population Survey (CPS) $.$ I calibrate $\sigma _{\varepsilon }$ to match an average monthly job separation rate $2\%$ in the CPS. These calibrations imply a steady-state unemployment rate of $5.9\%,$ which is consistent with the long-run average of the data. The flow cost of vacancy $c$ is identified by matching the average labor market tightness $\theta,$ which is $0.548$ in the sample.Footnote ²⁹ Finally, following Fujita and Ramey (Reference Fujita and Ramey2012), the switching rate of match-specific productivity $\lambda$ is calibrated to match the persistence (i.e., autocorrelation) of the separation rate in the CPS.

For later comparison, I consider also a homogeneous version of the model where I assume workers are homogeneous in match-specific productivity with no endogenous separation decision. Specifically, I set $\lambda =0$ and $ \chi _{t}\left ( \varepsilon _{t}\right ) =0$ for all $\varepsilon _{t}.$ The model is then calibrated similarly as above, except that now we have $\sigma _{\varepsilon }=0$ and $s$ is calibrated to match the average monthly job separation rate in the data.

A summary of the calibration values is given by Table 1. The baseline calibration is shown in the third column, where I calibrate the net bargaining wedge to match the unemployment volatility in the data, see the next subsection for details. The fourth column shows the estimated parameter values for the efficient model, where the net bargaining wedge is set to be zero. In this case, all job separations are privately efficient. Finally, the fifth column shows the calibration of the homogeneous model described above with $\Omega ^{net}=0$ . In addition, I have performed a sensitivity analysis for $\left \{ b,\eta,\lambda,\alpha \right \}$ in Table D.1 which shows that main quantitative conclusions are robust to changes in these parameters.

Table 1.

Calibration parameters

Table 1. Long description

A table comparing calibration parameters for a model of the US labor market. The table has 18 rows and 5 columns. The columns are labeled Parameters, Meaning, Baseline, Efficient, Homogeneous, and Target/remark. The rows list various parameters such as Flow value of unemployment, Discount factor, Persistence of the aggregate productivity, Standard deviation of the aggregate productivity, Elasticity of the matching function, Worker’s bargaining power, Exogenous separation rate, Switching rate of match-specific productivity, Flow cost of posting vacancy, Maximum value of match-specific productivity, Standard deviation of match-specific productivity, Coefficient of matching efficiency, and Net bargaining wedge. Each row provides the baseline value, efficient value, homogeneous value, and target or remark for each parameter. For example, the Flow value of unemployment has a baseline value of 0.71, an efficient value of 0.71, a homogeneous value of 0.71, and a target/remark of Hall and Milgrom (2008). The Discount factor has a baseline value of 0.99925, an efficient value of 0.99925, a homogeneous value of 0.99925, and a target/remark of Annual discount rate = 4 percent. The table provides a detailed comparison of these parameters across different models and their respective targets or remarks.

6.2 Unemployment volatility and bargaining friction

We have shown in the discussion of the model that the wedge would induce inefficient job separations and hence inefficient unemployment. Quantitatively, how would the bargaining wedge affect unemployment volatility?

Figure 2 shows the relationship between unemployment volatility and the net bargaining wedge.Footnote ³⁰ The unemployment volatility is measured by the standard deviation of quarterly cyclical unemployment rate over a sample simulation of $1000$ quarters. As we can see, unemployment volatility is monotonically increasing in the bargaining wedge. This is due to a combination of the small surplus and the separation channels described earlier: as the bargaining wedge increases, the wages are less sensitive to the aggregate conditions and the firm becomes more likely to fire the worker (inefficiently); hence, there are more job separations in the economy, leading to higher level of unemployment fluctuation. In other words, while the workers’ productivity varies with the aggregate productivity $y_{t}$ , the bargaining wedge is relatively acyclical. As a result, a small change in $y_{t}$ causes significant fluctuations in the cutoff productivity, and hence the quantity of job separations. The additional movement in job separations generates excessive unemployment volatility. Note that in principle, an increase in the bargaining wedge keeping other parameters unchanged would increase both the level of unemployment as well as its volatility. The procedure of the calibration described above, however, restricts the focus on the effect on unemployment volatility, since the steady-state unemployment rate is kept fixed.

Figure 2.

Unemployment volatility and bargaining friction.

The average volatility of the cyclical unemployment rate in the data is about $10.9\%.$ If all the unemployment volatility not explained by the standard model is assumed to be caused by a constant bargaining friction, this would imply a bargaining wedge of $0.188.$ I will use this value in the baseline calibration. See the third column of Table 1 for the calibrated parameter values.Footnote ³¹ It can be argued, however, that the bargaining wedge is moving with the business cycle. Hence, later in the decomposition of unemployment, I will allow for the cyclicality of the bargaining wedge.

To put the size of the bargaining wedge in perspective, the mean value of the total joint surplus in the simulation is about $5.29.$ Hence, the level of the bargaining wedge required to produce a realistic unemployment volatility is only $3.55\%$ of the joint surplus, which is hardly a large number. This shows that the unemployment dynamics is in fact very sensitive to the bargaining friction in the model, hence even a small value of the bargaining wedge can generate a large unemployment volatility.

To quantify the previously discussed two channels, I perform a quantitative exercise in which the separation cutoff productivity $\varepsilon _{t}^{s}$ is fixed while $\Omega ^{net}$ varies. In this exercise, the bargaining wedge continues to affect wages, the fundamental surplus, and hence vacancy creation, but it cannot distort the separation margin. The resulting line in Figure 2, labeled “Job creation effect only,” therefore isolates the small-surplus (job-creation) channel. It is clear that this channel is much less sensitive to bargaining friction than the baseline economy. As $\Omega ^{net}$ rises from $0$ to $ 0.188$ , cyclical unemployment volatility increases only from $5.60\%$ to $ 5.91\%$ . Relative to the total increase in unemployment volatility over the same range of $\Omega ^{net}$ in the baseline economy, this implies that the small-surplus channel accounts for only about 6% of the overall amplification. Thus, the dominant source of amplification comes from the separation channel.

A related benchmark is the homogeneous productivity model, labeled “homogeneous model” in the same figure. Two observations are in order. First, the homogeneous model has a much lower level of unemployment volatility even in the absence of bargaining friction, consistent with Fujita and Ramey (Reference Fujita and Ramey2012). Second, it is also much less sensitive to bargaining friction than the baseline heterogeneous economy. Taken together, these two counterfactual exercises show that, for empirically relevant values of $\Omega ^{net}$ , the small-surplus channel is much weaker than the separation channel.Footnote ³²

Therefore, I conclude that much of the increase in unemployment volatility in the baseline model is due to the separation channel. Once bargaining frictions distort the separation cutoff, unemployment fluctuations are amplified through excess cyclical separations, rather than through the small-surplus/job-creation channel alone.

6.3 Inefficient unemployment

How much of the unemployment volatility is explained by the inefficient unemployment? To answer this question, I first use the baseline calibration as described above. The bargaining wedge is subject to idiosyncratic shocks. I identify the shocks to the wedge by matching the cyclical real wage over the same time period.Footnote ³³ I also identify the shocks to the aggregate productivity process (42) by matching the unemployment rate in the data.Footnote ³⁴ The matching of the unemployment rate from 1976 to 2016 is given in panel (a) of Figure 3 and the cyclical real wage in panel (b). Note that recessions defined by the NBER are in gray areas. To see the fitness of the model, I have also shown the cyclical component of labor productivity in panel (a) of Figure 4. In general, the model does well in matching the relationship between unemployment and cyclical productivity.Footnote ³⁵ The estimated bargaining wedge is shown in panel (b) of Figure 4. We can see that the bargaining wedge closely follows the cyclical productivity $\left ( corr(\Omega _{t}^{net},p_{t}^{c})=0.66\right )$ . Consider the costly AOB model for intuition. When labor productivity is high, the opportunity costs of both the workers and the firms are higher. Hence, it would be more costly to bargain, and thus there would be a higher level of bargaining friction.

Figure 3.

Unemployment and cyclical wage rate: data vs. model.

Figure 4.

Cyclical labor productivity and estimated bargaining wedge.

Figure 4. Long description

The image contains two line graphs. Panel A shows cyclical labor productivity, while Panel B shows the estimated bargaining wedge. Panel A: The line graph depicts cyclical labor productivity from 1976 to 2016. The horizontal axis represents the years, ranging from 1976 to 2016. The vertical axis represents the productivity values, ranging from -0.04 to 0.04. The graph includes two lines: a solid blue line and a dashed orange line. The solid blue line represents actual cyclical labor productivity, while the dashed orange line represents estimated values. The graph shows fluctuations in labor productivity over the years, with notable peaks around 1984, 1992, and 2008, and troughs around 1980, 1990, and 2012. Panel B: The line graph depicts the estimated bargaining wedge from 1976 to 2016. The horizontal axis represents the years, ranging from 1976 to 2016. The vertical axis represents the bargaining wedge values, ranging from 0.16 to 0.22. The graph shows fluctuations in the bargaining wedge over the years, with notable peaks around 1984 and 2012, and troughs around 1992 and 2004.

Finally, I decompose the total unemployment into efficient and inefficient parts, as defined in Section 2. Figure 5 shows the decomposition of the total unemployment into efficient and inefficient unemployment rates. First, note that the inefficient unemployment appears to be much smaller in value but is as volatile as the efficient unemployment. In fact, the average inefficient unemployment rate is about $1.6\%,$ compared to the average efficient unemployment of $4.8\%.\,$ The standard deviation of inefficient unemployment rate is $0.5\%,$ compared to $1.1\%$ for the efficient unemployment. Also, in good times when the total unemployment rate is low, the inefficient-efficient unemployment ratio is also low relative to that in recessions. In fact, the inefficient unemployment rate is less than $1\%$ around 2000. On the other hand, the level of inefficient unemployment rate can be as high as $3\%$ in the Great Recession. Hence, the peak of the unemployment rate in the US during the Great Recession would have been about $3$ percentage points lower if all separations were efficient. The efficient unemployment reflects the effect of efficient separations only, without changing the value of the bargaining wedge. Hence, this shows that inefficient separation is a significant source of unemployment volatility in the estimated model.

Figure 5.

Decomposition of unemployment rate.

To see quantitatively how much of the volatility is explained by the inefficient unemployment, I can write, by the decomposition of the unemployment rate,

(43) \begin{eqnarray} var\left ( u_{t}\right ) &=&cov(u_{t},u_{t}^{e}+u_{t}^{i}) \notag \\[3pt] &=&cov\left ( u_{t},u_{t}^{e}\right ) +cov\left ( u_{t},u_{t}^{i}\right ) \end{eqnarray}

Hence, define

(44) \begin{equation} \gamma =\frac {cov\left ( u_{t},u_{t}^{i}\right ) }{var\left ( u_{t}\right ) } \end{equation}

as the fraction of unemployment volatility explained by the inefficient job separations, and $1-\gamma$ would be that by efficient job separations. In the baseline case, I have $\gamma =0.291.$ Hence, about $30\%$ of the unemployment volatility can be explained by inefficient job separations, and the remaining $70\%$ is due to efficient separations.Footnote ³⁶

6.4 Worker flows

Recently, there has been increasing attention in the literature on the worker flows in the business cycle frequency.Footnote ³⁷ How would the bargaining friction affect the labor market flows? To see this, I use the definition in Section 2 to compute job finding and separation rates.

Figure 6.

Simulated job finding and separation rates.

Figure 6. Long description

The image contains two line graphs. Panel A shows the job finding rate over time. The x-axis represents time, and the y-axis represents the job finding rate, ranging from 0.07 to 0.11. Two lines are plotted: one for the efficient model in blue and one for the baseline model in orange. The job finding rate fluctuates over time, with both models showing similar trends but with some variations in specific time periods. Panel B shows the job separation rate over time. The x-axis represents time, and the y-axis represents the job separation rate, ranging from 0 to 0.01. Again, two lines are plotted: one for the efficient model in blue and one for the baseline model in orange. The job separation rate also fluctuates over time, with both models showing similar trends but with some variations in specific time periods. The graphs illustrate the comparison between the efficient and baseline models in terms of job finding and separation rates over time.

Figure 6 shows the labor market flows for the standard model and for the baseline calibration in a simulation. Note that in both cases, I have calibrated the model to the average values of the job finding and separation rates. Hence, the focus here is on the volatility of the flows. First, job-finding rate is procyclical and the job separation rate is countercyclical. These are consistent with the evidence in Fujita and Ramey (Reference Fujita and Ramey2009). Next, the effect of the bargaining wedge on the job-finding rate appears to be small, except when the aggregate shock is large. The job-finding rate in the baseline case is slightly more volatile than the efficient model.

The main difference between the standard model and the baseline case lies in the volatility of separation rate. In fact, the cyclical volatility of separation rate in the baseline case is about $10\%$ which is much higher than $5\%$ in the efficient model. On the other hand, the effect on job-finding rate is relatively insignificant ( $3.0\%$ vs. $2.0\%$ ). Hence, the existence of bargaining friction affects mainly the separation rate in the labor market.

Note that the above result implies that, at the business cycle frequency, the job separation rate is as important as the job-finding rate in driving the fluctuation of unemployment. This stands in contrast to the evidence in Shimer (Reference Shimer2012), who finds that the job separation rate is not significant in explaining unemployment volatility. However, Elsby et al. (Reference Elsby, Michaels and Solon2009) and Fujita and Ramey (Reference Fujita and Ramey2009) show that job separations are countercyclical and contribute substantially to the unemployment volatility. More recently, Coles and Moghaddasi Kelishomi (Reference Coles and Moghaddasi Kelishomi2018) estimate that job destructions have been the main driver of unemployment volatility if we relax the free-entry assumption. This paper makes no attempt to resolve the debate on the relative importance of the ins and outs of unemployment flows. The message here is clear: by allowing inefficient separations by bargaining friction, the separation rate and the worker flows/transitions become much more volatile, while the effect on job-finding rate is insignificant.

This quantitative exercise shows that even a small amount of bargaining friction could have a significant impact on the volatility of job separations and thus on the unemployment volatility through the separation channel.