A. Households

The economy is populated by a continuum of infinitely-lived worker-households distributed uniformly along the unit interval implying a constant labor force normalized to one. At any point in time, only a mass n_t ≤ 1 of households is employed while the remaining 1 − n_t households are unemployed and searching for a job. Each household has the same utility function as in the previous section. Markets are complete providing perfect insurance against unemployment risk, hence, consumption is equalized across households (Andolfatto 1996 and Merz 1995). It is easy to prove that we can recast the problem in terms of a representative household that maximizes the sum of each individual household expected utility

The objective function of the representative household differs from the one of the prototype model because it explicitly takes into account the possibility that employment may vary over time—when n_t ≡ 1 we are back to the perfectly competitive model.⁶ Each unemployed worker exerts some effort, et, to find a job, which costs c(e_t) units of resources.⁷ As a result, the budget constraint of the representative households can be written as

Note that τ_l is the analog of ε_l in the prototype model, i.e. the fraction of labor earnings net of taxes.

Given the complete markets assumption it is easy to write the law of motion for aggregate capital

When households solve their problems, they are going to take wage, w_t, and hours worked h_t as given, as they will be determined through Nash Bargaining (see next section). We assume that trade in the labor market is mediated by an aggregate matching function that determines the number of jobs formed in each period as a function of the number of job vacancies, V_t, and the aggregate search effort of the household/workers, (1 − N_t) E_t

where 0 < γ < 1, τ_χ,t is the period-t realization of a process that governs the efficiency of matching, and the following intuitive restriction applies M_t ≤ min {V_t, 1 − N_t}. Using the definition of the matching function, we can write the probability of finding a job for an unemployed worker as $p_t=\frac{Mt}{(1-N_t)E_t}$, which is taken as given at household level.

Job matches that are formed in a period are assumed to become productive in the following period. That is, they only increase employment with a one-period lag. As jobs are created, they are also being destroyed. Letting σ denote the period-t fraction of existing jobs destroyed, the law of motion for employment is given by the expression

where σ ∈ [0, 1]. At the household level, employment n_t evolves endogenously according to the following, slightly modified equation of motion

Let’s define the aggregate productivity shock, τ_z,t as the analog of ε_z,t. We can finally write down the representative household problem recursively

s.t.

taking w_t = w(Ω_t), r_t = r(Ω_t), p_t = p(Ω_t), and equations of motion for aggregate state variables, K_t and N_t as given. For notational simplicity, let ${\omega }_t^h=\{k_t,n_t,{{\Omega }}_t\}$ and Ω_t = {τ_t, K_t, N_t} denote individual and aggregate state variables for the household respectively, with τ_t =[τ_l,t, τ_x,t, τ_z,t, τ_g,t, τ_χ,t] being the vector of exogenous processes analogous to ε_t. This optimization problem leads to two conditions that determine the optimal savings and search effort recursively for the large household (details are in the appendix):

The first Euler equation is the standard consumption Euler equation. The second Euler equation determines the households’ optimal search behavior. The left hand side in (18) is the expected marginal cost of search for the households in current consumption units, which should be equal to the expected marginal gains from search on the right hand side. If search is successful, the household expects to get utility from the net wage payments, w_t+1 [1 − τ_l,t+1] h_t+1, and from economizing on future search costs, c(e_t+1), and to get disutility from working G(h_t+1). The final term in brackets represents the net future benefit arising from the expected persistence of a job match.⁸ Note that, the household decision for optimal saving is exactly identical to the one in the competitive model (equation 6). Hence, all things equal, labor market frictions do not affect the wedge between marginal utility of consumption and the expected real rate of return on capital in the model. Instead, labor market frictions affect the search behavior, hence the labor supply decision. Next we turn to the demand side and describe firms’ problem.

B. Firms

As in the previous section firms operate a constant returns to scale production function, τ_z,tf(k_t, n_th_t) However, while capital is rented in a perfectly competitive market, firms must undergo a costly search process before jobs are created and output is produced. For each job vacancy created in period-t, firms pay κ units of output resulting in period-t “vacancy-posting” costs of κv_t. Jobs must be posted as vacancies before they can be filled. We assume that vacancies adjust in equilibrium such that the value of an additional vacancy is driven to zero. We are going to approach the firms’ problem in two steps. In the first step, firms decide how much capital to rent and how many jobs to create taking the rental rate r_t, aggregate state variables, the labor contract {w_t, h_t}, and the probability of filling a vacancy, q_t, as given⁹

such that

where ${\omega }_t^f=\{k_t,n_t,{{\Omega }}_t\},r_t=r({{\Omega }}_t),w_t=w({{\Omega }}_t)$ with Ω_t = {τ_t, K_t, N_t} and ${\tilde{\beta }}_t=\beta {{E}}_t\left[\frac{U_c(c_{t+1})}{U_c(c_t)}\right]$. Firms’ problem in (19) impliesa set of first order conditions that determine the optimal level of capital stock rented and the number of vacancies posted by firms.

The first condition is the familiar relation between the rental rate and the marginal product of capital. The second condition determines the optimal number of vacancies posted by firms. It implies that the marginal cost of filling a vacancy should be equal to the marginal benefit, in expectation. Since each vacancy costs κ and the duration of a vacancy is expected to be 1/q_t, $\frac{\kappa }{q_t}$ gives the expected marginal cost of filling one vacancy. On the right hand side, we see how each additional filled job produces its marginal product next period net of wage payments, while the third term represents the expected saving in terms of future recruitment costs.

C. Employment Contract and the Nash Bargaining

Since negotiating a wage has an implicit opportunity cost due to search frictions, we need a mechanism to determine the surplus for each contracting party. It is standard in the search literature to use generalized Nash bargaining for this purpose (see for instance, Pissarides 2000). We follow the same approach and assume that each worker’s employment contract, {w_t, h_t}, is determined through Nash bargaining between the firm and the household. In a setting like this, where there are multiple workers within a firm, it is not entirely clear how to formulate the bargaining problem. Fortunately, Stole and Zwiebel (1996) show that bargaining should happen over the marginal surplus for both parties.^{10 11} Assuming that households (firms) have a bargaining power of 1 − λ (λ) the generalized Nash bargaining problem takes the following form¹²

where $W_{n_t}^h({\omega }_t)$ is the household’s net marginal surplus from having one more worker employed given household’s optimal behavior, and $W_{n_t}^f({\omega }_t)$ is the firm’s net marginal surplus from having one more employee given firm’s optimal behavior. As long as each party can extract some surplus, i.e. $W_{n_t}^h({\omega }_t)>0\text{and}{W}_{n_t}^f({\omega }_t)>0$, this Nash bargaining problem yields two conditions that determine equilibrium level of hours, h_t, and wage per hour, w_t (details of the solution are presented in the Appendix)

Equation (24) is the starting point to determine the hourly wage and basically divides the total surplus of the marginal match among the household and the firm. Notice that τ_l,t exogenously reduces the household’s share of surplus and, at the limit τ_l,t =1, the household’s surplus is zero $W_{n_t}^h=0$. Household risk aversion adds a time-varying dimension to the sharing rule: in times of high marginal utility households claim a relatively higher surplus. Equation (25) is the first order condition with respect to hours, as we will see this equation will play a crucial role in the analysis of the labor wedge. Those last two conditions complete the necessary set of equations that fully characterize the equilibrium. Next, we combine these set of conditions to describe the recursive competitive equilibrium.

D. Equilibrium

Since all households and firms are identical, in equilibrium, individually efficient allocations coincide with the aggregate, i.e. k_t = K_t, c_t = C_t, l_t = L_t, n_t = N_t, therefore the relevant state is Ω_t. Then the competitive equilibrium of this economy is characterized by a list, k_t(Ω_t), l_t(Ω_t), c_t(Ω_t), n_t(Ω_t), w_t(Ω_t), r_t(Ω_t) that satisfy equilibrium conditions implied by household and firm optimization (17-18) and (21-22), Nash bargaining (24) and (25), as well as equations of motion for aggregate states, (11) and (13).¹³ It is also possible to pin down the aggregate resource constraint by exploiting the free entry condition for firms which imposes that all the flow profits net of recruitment expenses are exhausted in equilibrium

Note that, search frictions do not affect the consumption Euler equation and the resource constraint in a significant way.¹⁴ However, fluctuations in the extensive and intensive margin, along with the search decision give rise to additional equilibrium conditions for employment, search effort, e_t, and job vacancies, v_t, that are absent in the prototype business cycle model. For the purpose of this paper, we can focus our analysis on the static equilibrium conditions that describe the labor wedge (25) and (24).

E. The Wage Determination

It is possible to derive an explicit wage equation using the equilibrium conditions from household and firm optimization (see appendix for details)

The wage bill for an additional worker w_th_t is a function of the marginal productivity of employment, τ_z,t f_n(k_t, l_t), search frictions and the extensive marginal rate of substitution, −ψG(h_t)/U_c(c_t), which reflects, at the social level, the marginal disutility of an additional worker, −ψG(h_t), for any given h. It is useful to introduce the mrs and mpl by noting that −ψG(h_t)/U_c(c_t) = mrs_th_tη(h_t), with η_t = G(h_t)/[h_tGh(h_t)] ∈ [0, 1], and the marginal productivity of employment τ_z,tf_n (k_t, l_t) = mpl_th_t. We can thus rewrite the wage equation in a compact form

where Φ_t ≶ 0 is the only term that involves search frictions explicitly.¹⁵ If we combine the wage equation (27) with the condition on hours, equation (25), we can relate both the mrs_t and the mpl_t to the wage

where we have introduced the output elasticity to total hours $s_t^L=1+\frac{f_u(k_t,l_t)l_t}{f_l(k_t,l_t)}\in [0,1]$, which is approximately the labor share.¹⁶ Those two equations are very instructive and resemble the labor supply and labor demand equations of the perfectly competitive model. However, search frictions coupled with Nash bargaining introduce two time varying wedges between the wage and both the marginal productivity of labor and the marginal rate of substitution. In fact, even if τ _l,t were constant, the wage would not perfectly co-vary with the mpl and the mrs over the cycle, being affected by movements in the search frictions Φ_t and in the way the match surplus is split. In general, the wage will be between the mrs and the mpl, however, when λ is small, it might be possible to observe w_t > mpl_t. In part this is due to the presence of $s_t^L$ which reduces the firm’s perceived benefit of labor. In fact, firms value the benefit of increasing hours worked h across workers in terms of its effect on the marginal productivity of an extra worker (not in terms of the average worker productivity). However, while the cost is still linear in h, the marginal productivity of employment is not: as long as there is curvature in the production function, i.e., s^L< 1, the marginal productivity of employment increases with h less than one-to-one.

Finally, it is also interesting to note that the firm bargaining power parameter, λ, is inversely related to the wage and enters symmetrically in both equations (28) and (29). As we will see, this implies that the bargaining power per se does not affect the labor wedge, as long as there is an interior solution of the Nash Bargaining problem, i.e., λ ∈ (0, 1).

In the following section we describe in more detail our labor wedge and compare it with the one of the competitive model.

F. The labor wedge

Eliminating the wage from (28) and (29) we have an expression that is analogous to the one of the perfectly competitive model (equation 7).

It is striking that while the bargaining process has created a wedge between mrs and mpl, search frictions per se do not appear directly in this equation. Mechanically, this is simply because they enter symmetrically and additively in the labor demand and supply equations, hence, when those are equated search frictions perfectly offset and cancel out. The same is true also for λ and η which affect mrs and mpl exactly in the same proportion. Intuitively, one reason for this result is that the bargaining process internalizes search frictions through the wage rather than hours. In part this may be due to the fact that search frictions are inherently intertemporal—i.e. it takes time and resources to match unemployed workers with vacant positions—while the labor wedge is inherently intratemporal. The effects of intertemporal frictions are absorbed by movements in the wage and the extensive margin, and not in the labor wedge.

In order to understand the effects of search frictions on the extensive margin, it is crucial to observe that the variation in this margin is governed by mainly two decisions; workers’ search effort and firms’ vacancy posting decision. Optimality conditions for search effort and vacancies (see equations 18 and 22) along with the law of motion for employment will govern the movements at the extensive margin. Using equations (18) and (22), noting that the terms in expectations are each party’s marginal surplus from a match, one can write down a simple equation that relates market tightness and search effort to movements in the value of expected surpluses:

Variations in the relative values of expected surpluses will determine fluctuations in market tightness, θ_t, and e_t, implying fluctuations in employment. Note also that the Nash bargaining outcome requires a specific sharing rule determining the relation between $W_{n_t}^f\text{and}{W}_{n_t}^h$ for all t (see equation 24). Once this is taken into account, it is easy to see how fundamental parameters related to search frictions explicitly affect the movements in the extensive margin through search effort and market tightness. Up to a covariance term equation (31) can be written as¹⁷

Equation (32) shows, how the labor wedge (that will be pinned down by equation 34 below) affects the cost of searching and, thus, employment fluctuations. In turn, since the employment law of motion has to hold, movements in the labor wedge will probably induce fluctuations in the matching function process, τ_χ, interpretable as the extensive labor wedge. We leave the study of the extensive wedge to future work and focus on the static labor wedge instead.

Going back to the static condition (30), one can see that the only distortion present, summarized by $s_t^L$, is induced by the bargaining problem. However, once we assume that the production function takes a Cobb-Douglas form, such that f_ll(k, l)l = −αf_l(k, l), the marginal productivity of workers change one-to-one with h, we have that $s_t^L=1-\alpha$ Hence, the wedge introduced by $s_t^L$ is irrelevant at business cycle frequency. The constant (1 − α) acts like a tax on employment, which, in this context, is observationally equivalent to a labor income tax; in the baseline calibration α takes a value of about 0.35 reducing substantially the steady state value required for τ_l.¹⁸

We can finally compare the two fundamental equations that are used to back out the labor wedge from the data in the competitive and non-competitive labor market model

The major difference between the prototype labor wedge [1 − ε_l,t] and ours is that, in the perfectly competitive RBC model the lack of distinction between the extensive and intensive margin has induced the use of total hours worked in the evaluation of the mrs instead of hours per worker. Intuitively, the presence of n_t in the prototype labor wedge equation may introduce a strongly procyclical element—given that G_h > 0. Employment is instead not present in the second equation when we back out our discrepancy, [1 − τ_l,t]. Since movements in both margins are not equally significant in deriving the business cycle frequency fluctuations in total hours, this distinction about the intensive-extensive margin is clearly important. Figure (2) shows the decomposition of total hours into its components (plots for only employment and hours are generated keeping the respective margin moving where the other margin is assumed to be fixed at the historical average. Shaded areas indicate NBER recession dates). Total hours in the U.S. is clearly procyclical. When total hours is assumed to follow a path where one of the margins is fixed at its historical average and the other margin is assumed to follow the actual path in the data, one recognizes the well-known fact that most of the fluctuations in total hours comes from the extensive margin, not the intensive margin (Shimer 2009). This stylized fact is particularly relevant when interpreting the results of our exercise.

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Total Hours: Extensive vs. Intensive Margin

Citation: IMF Working Papers 2011, 117; 10.5089/9781455262403.001.A001

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To see why this might affect the measured labor wedge, consider a simple constant Frisch elasticity utility function of the form G(x)= −x^(1+ϕ)/(1 + ϕ). Then we can write a simple relation describing the mapping between [1 − ε_l,t] and [1 − τ_l,t]

This shows how some of the observed procyclicality of [1 − ε_l,t] induced by employment does not have to be present in our labor wedge. In other words, even if we could interpret all the movements in τ_l as due to actual changes in labor tax, equation (35) suggests that we would still find a strongly procyclical labor wedge [1 − ε_l,t] in the prototype model merely because of procycilcal employment fluctuations in the data—the higher ϕ the stronger the procyclicality.

A. Mapping the Models to the U.S. data

As we mentioned in previous sections, employment fluctuations make the measurement of the prototype mrs differ from the one in our search model. This difference is crucial for the labor wedge, because most of its variation is associated with mrs and not mpl fluctuations. Figure (3) plots the log-detrended mrs and mpl jointly with the log detrended prototype labor wedge. In line with the model, the log mrs, mpl, and [1 − ε_l,t] have been detrended by subtracting the HP-trend of the log of consumption, output, and consumption-output ratio, respectively (shaded areas indicate NBER recession dates).¹⁹ While some low-frequency movements in the labor wedge may be driven by the mpl, particularly between 1982-90, most of the business cycle movements are largely due to fluctuations in the mrs. Hence, seen from the lens of the perfectly competitive model, labor supply is less than implied by the model during expansions and it does not fall as much as required by the model during recessions. Given that the search labor wedge defined in (30) basically modifies the mrs reducing its procyclicality, as we have explained in the example in (35), the search model points in the right direction.

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The MRS and MPL in the Prototype Model

Citation: IMF Working Papers 2011, 117; 10.5089/9781455262403.001.A001

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Quantitative Analysis We recover the labor wedge in the prototype model (the prototype labor wedge) and in the model with labor market search frictions (search labor wedge), from equations (33) and (34), respectively. Conducting this partial accounting exercise requires us to calibrate the disutility of labor, G(·), α, and ψ. We use quite a general functional form for, G(·), that has non-constant Frisch elasticity of substitution: G(x) = [(1 − x)^{(1 −ϕ)} − 1]/(1 − ϕ). The often used CRRA form G(x)= −x^1+ϕ/(1 + ϕ) delivers fairly similar results. Given the chosen form for G(·), and restricting our analysis to models that allow for a balanced growth path U(c_t) = log(c_t), we can write down two equations that implicitly define the prototype and search labor wedge²⁰

Data on y_t, l_t, h_t and c_t pin down a unique labor wedge for each period in both equations once we calibrate α, $\overline{\varphi }(\tilde{\varphi }),\text{and}\overline{\psi }(\tilde{\psi })$. We choose α =0.35 to match the average labor share in the prototype model. Turning to preferences, we set the parameter $\overline{\psi }(\tilde{\psi })$ such that in steady state ε_l (τ_l) is 0.4, consistent with the tax wedge measured by Prescott (2004). Finally, we adjust $\overline{\varphi }$, to get the same steady state labor elasticity in both models. In the baseline calibration we choose the limiting case, $\overline{\varphi }=1$, which is also used by CKM and implies a steady state Frisch elasticity of about 2.8, then, in order to get the same elasticity in the search model, we set $\tilde{\varphi }=0.6$. (Next we will explore a lower elasticity case).

Figure (4) shows the results for the ‘low-frequency case’ when variables are detrended using HP-trends in output and consumption– low-frequency movements are not necessarily filtered out. The upper panel presents the mrs for both models while the lower panel shows the related wedges. There is a clear low-frequency cycle in the wedges, mainly due to the mpl, that has not been captured by the trend in the consumption-labor ratio. The rise in the labor wedge in the second half of the sample is due to a declining mpl, whether we assume competitive labor markets or not. As suggested by Figure (4), the search labor wedge is less volatile than the prototype labor wedge, by about 23 percent (see Table 1). This is entirely due to our measurement of the mrs, which is about 40 percent less volatile than the mrs measured through (36). This obviously follows from the fact that the intensive margin is the less variable margin in total hours, which makes the measurement of the mrs in the search model less variable than the one of the prototype model.

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Prototype vs. Search Model: Low Frequency

Citation: IMF Working Papers 2011, 117; 10.5089/9781455262403.001.A001

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Table 1.

Moments and Correlations: High Frisch Elasticity

All variables are HP-filtered, except L_t. Moments in () are for the ‘low-frequency’ case.

Table 1.

Moments and Correlations: High Frisch Elasticity

	Y t	[1 − εl,t]	[1 − τl,t]	${\mathit{\text{mrs}}}_t^{\mathit{\text{proto}}}$	${\mathit{\text{mrs}}}_t^{\mathit{\text{search}}}$
Std. Dev.	0.015 (0.015)	0.015 (0.074)	0.012 (0.057)	0.013 (0.024)	0.010 (0.014)
Corr(xt, xt−1)	0.862 (0.862)	0.749 (0.988)	0.666 (0.983)	0.893 (0.966)	0.865 (0.922)
Cross Correlations
	Y t	L t	C/Y
[1 − εl,t]	0.514 (0.183)	0.866 (0.992)	−0.162 (−0.086)
[1 − τl,t]	0.410 (0.173)	0.796 (0.978)	−0.043 (−0.066)

All variables are HP-filtered, except L_t. Moments in () are for the ‘low-frequency’ case.

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Table 1.

Moments and Correlations: High Frisch Elasticity

	Y t	[1 − εl,t]	[1 − τl,t]	${\mathit{\text{mrs}}}_t^{\mathit{\text{proto}}}$	${\mathit{\text{mrs}}}_t^{\mathit{\text{search}}}$
Std. Dev.	0.015 (0.015)	0.015 (0.074)	0.012 (0.057)	0.013 (0.024)	0.010 (0.014)
Corr(xt, xt−1)	0.862 (0.862)	0.749 (0.988)	0.666 (0.983)	0.893 (0.966)	0.865 (0.922)
Cross Correlations
	Y t	L t	C/Y
[1 − εl,t]	0.514 (0.183)	0.866 (0.992)	−0.162 (−0.086)
[1 − τl,t]	0.410 (0.173)	0.796 (0.978)	−0.043 (−0.066)

All variables are HP-filtered, except L_t. Moments in () are for the ‘low-frequency’ case.