Labor Share Dynamics and Factor Complementarity

2.1 The Labor Market

The labor market is frictional in the sense that a job seeker and a firm with a vacant position have to go through a time-consuming search process to form a match. Following Pissarides (1990), we assume that the number of matches m _t depends on the number of unemployed workers (job seekers) u _t and vacancies v _t :

where γ _m > 0 is the matching efficiency, and ξ is the matching elasticity with respect to unemployment. An unemployed worker finds a job with probability f_u,t = m _t /u _t , and a vacant position gets filled with probability f_v,t = m _t /v _t . Both the job-finding and vacancy-filling rates are taken as given by households and firms. The labor market tightness is defined as θ _t = v _t /u _t . The number of employed workers evolves according to n_t+1 = (1 − s)n _t + m _t , for newly-formed matches become productive in the next period, and all employed workers are subject to exogenous separation with probability s.

2.2 The Household Sector

The expected lifetime utility of the representative household is given by

where E 0 is the expectation operator conditional on the period 0 information set, and 0 < β < 1 is the subjective discount factor. The representative household consists of a continuum of household members. While the family members are subject to unemployment risk, they insure each other. Therefore, each household member, either employed or unemployed, has the same consumption level c _t .^[7] γ _c > 0 measures the inter-temporal elasticity of consumption. The household’s labor supply is considered at both the extensive (employment, n _t ) and intensive margins (hours per worker, h _t ), and γ _h > 0 measures the Frisch elasticity of labor supply at the intensive margin. Ψ captures the weight of labor in the utility function.

The household receives dividends Π _t from the firm sector and pays a lump-sum tax T _t to the government. Each employed member earns real wages w _t , while each unemployed member receives unemployment benefits z from the government. Thus, the household’s budget constraint, in units of period t final goods, is given by

where R_t−1 is the gross nominal interest rate, b_t−1 is the stock of real government bonds at the beginning of period t, and π _t ≡ p _t /p_t−1 is the gross inflation of the nominal price p _t of final goods. Given the real rental rate r_k,t and the capital stock K _t at the beginning of period t, the household accumulates its capital by investing i _t so that

where δ is the depreciation rate of capital.

To maximize its lifetime utility, the household chooses sequences of consumption, investment, the capital stock, and government bonds, subject to the budget constraint and the capital evolution equation. The first-order conditions are as follows:

where λ _t is the Lagrange multiplier of the budget constraint.

2.3 The Firm Sector

The firm sector, as mentioned earlier, consists of three different types of firms: final-good firms, intermediate-good pricing firms, and intermediate-good producing firms. In line with Trigari (2006) and Thomas (2008), we divide the intermediate-good sector into producing and pricing firms. A producing firm hires a worker and rents capital to produce homogeneous goods, which are sold to pricing firms. Pricing firms transform the purchased homogeneous products into heterogeneous ones, and they exercise their market power to set the desired prices. Pricing firms sell their products to the final-good firms. It is convenient to look at this production process backward.

2.4 Determination of Wages and Labor Income Share

An employed worker and her employer bargain over wages, while the employer retains unilateral control over the employee’s working hours. This so-called right-to-manage bargaining, as argued by Trigari (2006) and Christoffel and Kuester (2008), is a more realistic description of labor contracts. Under such an arrangement, the working hours will always remain on the labor demand curve. This implication is different from that of hours bargaining and allows us to easily compare our results with those in the perfectly competitive labor market.^[9]

Let η be the worker’s bargaining power. The equilibrium (hourly) wage rate is determined by solving the Nash bargaining problem. That is,

subject to the employer’s hours demand function (10). Accordingly, we can obtain the real wage income w t Nash h t as follows:

where

Hall (2005) and Shimer (2005) argue that flexible wages may not reconcile with the magnitude of unemployment volatility. To investigate the role played by the (real) wage rigidity, we follow Hall (2005) and Blanchard and Galí (2010) and assume that the prevailing real wage follows

where w t norm represents the wage norm, which is the bargained wage in the previous period. Thus, ψ _w measures the degree of wage rigidity; a higher ψ _w refers to a stronger wage stickiness.

2.5 The Government Budget and Aggregate Resource Constraints

The government levies a lump-sum tax T _t and issues bonds to finance unemployment benefits. The government budget constraint is given by

In addition, the government (the monetary authority) follows an interest rate rule (Taylor’s rule) as follows:

where ϕ _y > 0 and ϕ _π > 1 are policy coefficients when output and inflation deviate from the associated steady-state levels.

In the symmetric equilibrium, the final output (GDP) can be expressed as follows:

By analogy, the market-clearing condition of capital is K _t = n _t k _t . Accordingly, the aggregate resource constraint is given by

where D t = ψ p 2 π t π − 1 2 is the output loss due to price stickiness, and C t = κ v t is the total vacancy posting cost.^[10]

3.1 Calibration

In our model, a period corresponds to one quarter of a year. We set the subjective discount factor β = 0.99, implying that the annual real interest rate is around 4 percent. The risk aversion coefficient γ _c is set to 2, in the neighborhood of most empirical estimates; see, for instance, Attanasio and Weber (1995) and Krause and Lubik (2007). We set the depreciation rate δ = 0.025, implying an annual depreciation rate of capital of 10 percent.

The matching efficiency γ _m is calibrated so that the steady-state unemployment rate is 5.88 percent. Following Krause, Lopez-Salido, and Lubik (2008), we set the quarterly job separation rate as s = 0.05. Following Petrongolo and Pissarides (2001), we set the unemployment elasticity of matching ξ = 0.6. The period vacancy cost κ, which affects a firm’s willingness to open vacancies, is calibrated by targeting the steady-state vacancy-filling rate f _v = 0.70, a valued adopted by den Haan, Ramey, and Watson (2000). We set the employed worker’s bargaining power η = 0.5, following den Haan et al. (2000). The unemployment benefit z is calibrated by targeting the replacement ratio of 0.4, which is widely adopted in the search and matching models.

As for structural parameters related to the household sector, we set γ _h = 0.3026 so that the Frisch elasticity of labor supply is equal to 3.3. Chetty et al. (2011) suggest that for the business cycle models the reasonable range for this elasticity is between 2.61 and 4. We then choose the middle point value. The scale parameter of labor disutility Ψ is calibrated so that the steady-state working hours per worker amount to h = 0.3, which is commonly used in the RBC literature.

On the production side, we set the elasticity of factor substitution to σ = 0.5, which is in the neighborhood of recent empirical studies based on firm- and plant-level data.^[11] We set the returns-to-scale parameter ϑ to 0.98, which implies that the profit share of a producing firm is 2 percent.^[12] We calibrate the steady-state technology level A = 0.3305 so that the steady-state capital demand is k = 1 for each producing firm, associated with the capital rental rate of r _k = 3.5%. Through the arbitrage condition, this implies a reasonable net real interest rate of r = 1.01% in the steady state. We assume that ϵ = 11; that is, a pricing firm’s steady-state (net) markup is 10 percent.

It is well known that the pricing schemes of Rotemberg (1982) and Calvo (1983) are observationally equivalent up to the first-order approximation. Utilizing this equivalent relationship, we set the price-adjustment-cost parameter ψ _p = 60, which implies that two-thirds of pricing firms cannot reset their prices in the sense of Calvo pricing.^[13] Moreover, we assume that the wage rigidity parameter ψ _w = 0.6. Following Faia (2008), we set ϕ _π = 1.5 and ϕ _y = 0.125 in the nominal interest rate rule, with a normalized steady-state gross inflation of π = 1. Finally, for the sake of quantitative comparison, the persistence and volatility parameters (ρ _A = 0.93 and σ _A = 0.0064) are obtained from the structural estimation of Ríos-Rull and Santaeulàlia-Llopis (2010). Choosing exactly the same parameter values allows us to foster a quantitative comparison between the empirical and theoretical impulse responses. Table 1 summarizes our calibration.

3.2 Effects on the Labor Income Share

Technology shocks affect the LIS via two channels: the output elasticity of labor Φ _t (the output-elasticity channel) and the price-cost markup μ _t (the markup channel), as shown in (14). A positive technology shock raises both the marginal products of labor and capital for intermediate-good producing firms. Figure 2 shows that higher factor productivity makes opening vacancies more attractive, leading new firms to open job vacancies v _t . Meanwhile, labor increases at both the intensive margin (hours per worker h _t ) and extensive margin (employment n _t ) (see the job creation condition (11)). Thus, the aggregate output y _t increases and the unemployment rate u _t decreases.

Figure 2:

Impulse responses of a positive technology shock.

Focusing on the impact effect, although working hours per worker and the number of employed workers (and hence the number of firms) increase, capital k _t is unchanged on impact (since capital is predetermined). The time lag in adjusting capital causes the capital–labor ratio k t / h t to jump downward on impact. Since capital and labor are complementary (σ = 0.5), a lower capital–labor ratio decreases the output elasticity of labor Φ _t (the output elasticity channel is shown in (13)), leading to a negative effect on LIS, S_l,t.

Regarding the markup channel, a positive technology shock leads the intermediate-good firms to increase their supply to pricing firms, which decreases the real marginal cost for pricing firms φ _t . By contrast, with the labor-capital complementarity, the labor upward jump raises the rental rate of capital r_k,t because the marginal product of capital increases with labor hours (the technological complementarity). A higher rental rate substantially increases the producing firm’s cost, which exerts upward pressure on the real marginal cost for pricing firms. Given these two conflicting effects, Figure 1 shows that the real marginal cost of pricing firms φ slightly increases in response to the technology shock. Because a pricing firm’s markup is the inverse of its real marginal cost, on impact the markup μ _t decreases slightly. As a result of a small effect on the markup channel, the output elasticity channel dominates, leading to the counter-cyclicality of the LIS on impact.

Figure 2 further shows that, after the upward jump, the increment of labor hours gradually diminishes in transition. By contrast, capital increases and exhibits a hump-shaped response to the positive technology shock. The hump-shaped response of capital translates to a similar pattern in the capital–labor ratio and the output elasticity of labor, resulting in the LIS overshooting.

To be specific, after the impact, capital starts to increase. An increase in capital, on the one hand, lowers the rental rate r_k,t (due to the diminishing returns to capital), and on the other hand, raises the wage rate w _t (due to capital–labor complementarity). As a result, the relative factor price w _t /r_k,t increases in transition. In the presence of some degree of wage rigidity, the wage rate needs time to reach its maximizing level, and afterwards it gradually decreases because of the diminishing returns to labor. Thus, in transition the relative factor price increases first and then decreases, leading the capital–labor ratio to exhibit a hump-shaped response, as shown in (10). Under the CES production with factor complementarity (0 < σ < 1), this hump-shaped response translates into a similar pattern in the output elasticity of labor Φ _t , as shown in (13). The output elasticity channel thus leads the LIS to overshoot its initial level. In the later periods of transition, the capital–labor ratio decreases, weakening the output elasticity channel, and the real marginal cost decreases, amplifying the markup channel. Therefore, the LIS eventually declines.

In our model, the asymmetric adjustment speeds of capital and labor generate the LIS dynamics in both the counter-cyclicality and overshooting. On impact, the adjustment speed of capital is slow so that the capital–labor ratio decreases. With the capital–labor complementarity, the LIS exhibits a counter-cyclicality. In transition, the adjustment speed of capital increases, and the increasing capital–labor ratio leads to the LIS overshooting. Since the positive technology shock eventually reduces the real marginal cost of firms, labor’s share declines in the longer run via the markup channel. The adjustment asymmetry between capital and labor may depend on the labor supply elasticity and the capital adjustment cost. It is intuitive to infer that the adjustment asymmetry between capital and labor is amplified (attenuated) by a higher elasticity of labor supply (a higher capital adjustment cost), resulting in more (less) responsive LIS dynamics.^[14]

In a DMP search model with fully flexible wages and prices, Choi and Ríos-Rull (2009) also examine the LIS dynamics by using a non-unit elasticity of substitution between labor and capital. They persuasively argue the importance of labor-capital complementarity, but the magnitude of the LIS overshooting is very limited in their model. The empirical impulse response function shows that the LIS overshoots its initial level in the fifth quarter after the technology shock, and the overshooting is about 0.2% at the peak around the 18th quarter. In the Choi and Ríos-Rull (2009) model, the LIS overshooting is merely 0.02% relative to the initial level. By contrast, in our model, the capital–labor complementarity interacts with the wage/price rigidity. This interaction enables us to generate a more responsive LIS overshooting and better matches the empirical impulse responses of the LIS. Our calibration results show that the LIS overshoots its initial level at about the sixth or seventh quarter after the shock, and the magnitude of the overshooting is around 0.17% (the peak is also in the 18th quarter), which is also close to the data.

Furthermore, Table 2 compares our calibration results with the second moment and correlation of the business cycle statistics. The current empirical evidence indicates that the LIS is counter-cyclical in relation to output. In the US, the estimates of the correlation coefficient between the LIS and output range between −0.13 (estimated by Choi and Ríos-Rull 2009) and −0.34 (estimated by Koh and Santaeulàlia-Llopis 2017). In our model, the LIS-output correlation is −0.18, which is located well within the range of existing estimates. In focusing on the LIS variance, the estimated variance of the LIS is between 0.44 (estimated by Choi and Ríos-Rull 2009) and 0.55 (estimated by Koh and Santaeulàlia-Llopis 2017). In our baseline model, the variance of the LIS is 0.67, which is slightly higher than the empirically estimated value. The variance can be reduced to 0.49 with a lower labor supply elasticity of 1.92 (a reasonable value in the RBC literature), relative to the baseline elasticity of 3.3. Note that Table 2 shows that the negative correlation between the LIS and output cannot be produced without the rigidity in wages and prices.

4.1 CD Production Function

To shed light on the role of the capital–labor complementarity, we reduce the CES production function to a CD one by setting the elasticity of substitution between capital and labor as σ = 1. Figure 3 shows that, under a CD production function, the counter-cyclicality of the LIS on impact becomes less pronounced, and the overshooting vanishes. This case vividly conveys the viewpoint of Choi and Ríos-Rull (2009) in the sense that the conventional DMP model may explain the counter-cyclicality of the LIS in response to a technology shock, but cannot address the overshooting.

Figure 3:

CES production function (σ = 0.5) versus CD production function (σ = 1).

In the presence of a CD production function, the output elasticity of labor is a constant as shown in (13), and therefore the output elasticity effect is absent. As for the markup effect, the CD production function implies a lower degree of technological complementarity between capital and labor, relative to the CES production function. Due to a weaker technological complementarity, the impact of an increase in labor hours has a smaller effect on the rental rate.^[15] It turns out that the pricing firm’s real marginal cost decreases and its markup increases in response to the positive technology shock. Because the markup becomes pro-cyclical, the LIS always exhibits a counter-cyclicality in transition. As a result, a CD production function eliminates the possibility of the LIS overshooting.

This experiment also implies that higher capital–labor complementarity reinforces the output elasticity channel, and hence increases the magnitude of overshooting to a higher level. Such a case becomes more important as the Introduction points out that recent empirical studies have supported an estimated elasticity of substitution between capital and labor significantly below unity.

4.2 Flexible Wages

Next, we shut off the wage rigidity by setting the degree of wage rigidity ψ _w = 0. Figure 4 shows that if wages are fully flexible, the LIS becomes less responsive in both the counter-cyclicality and the overshooting. For instance, the magnitude of the LIS overshooting reduces to 0.11% relative to the initial level, around a half of the estimated magnitude. This experiment reveals that, with sticky wages, the LIS dynamics generated from our NK model can better match the empirical impulse responses.

Figure 4:

Rigid wages (ψ _w = 0.6) versus flexible wages (ψ _w = 0).

If wages are sticky, the timing of the increase in wages is postponed. Thus, on impact and at the beginning of the transition, wages increase less and labor hours increase more, compared to the flexible wage case. The remarkable increase in labor hours causes the rise in the rental rate to be more pronounced at the beginning of the transition. Because the adjustment speeds of capital and labor become more asymmetric, with a greater downward jump, the capital–labor ratio is lower at the beginning of the transition, but it jumps higher in the later periods of the transition. Therefore, through the output elasticity channel, the LIS jumps downwards more significantly and that the overshooting is more pronounced.

As for the markup channel, there are opposite effects in the sticky and flexible wage cases. As noted above, if wages are sticky, labor hours are more responsive to a technology shock, and the increase in the rental rate is more pronounced. Since producing firms demand more labor hours and face higher capital rental rates, the real marginal cost increases, leading to a decrease in the markup. By contrast, if wages are flexible, producing firms increase their labor less and face lower capital rental rates. As a result, the real marginal cost of firms decreases, rather than increases, raising their markups. Therefore, the LIS overshooting turns out to become less pronounced as wages are fully flexible. By contrast, because wage rigidity depresses the markup channel, the LIS overshooting is substantial.

In the absence of capital–labor complementarity, Colciago and Rossi (2015) and Reicher (2016) find that wage rigidity is not crucial in governing the LIS overshooting. Our result contradicts those of Colciago and Rossi (2015) and Reicher (2016). This experiment tells us that introducing a real wage rigidity into the DMP framework amplifies the channel of the output elasticity of labor, making the LIS overshooting more pronounced. With wage rigidity, our model can produce a better matching of the amplitude of the LIS overshooting observed in the data even in the presence of pro-cyclical markups.

4.3 Flexible Prices

Price rigidity endogenizes firms’ markups in the model. We thus isolate the markup effect by shutting off price rigidity (by setting ψ _p = 0). When prices are fully flexible, markups become fixed, and therefore, the markup channel is shut down. In the case without the markup channel, Figure 5 shows that the LIS overshooting in transition becomes more significant. As noted in the baseline, price markups are pro-cyclical in transition, which weakens the channel of the output elasticity of labor. Therefore, the LIS overshooting in transition becomes more pronounced when we shut off the markup channel.

Figure 5:

Rigid prices (ψ _p = 60) versus flexible prices (ψ _p = 0).

This case provides a counterexample to the finding of Colciago and Rossi (2015). In the absence of factor complementarity, they argue that the counter-cyclicality of the price markup, instead of wage rigidity, plays an important role in generating a more responsive LIS overshooting. By contrast, in our model wage rigidity amplifies the overshooting of the LIS, but price rigidity attenuates it. Even though the markup channel is absent, the interaction between wage rigidity and the capital–labor complementarity can still command the dynamics of the LIS, leading to a better matching of the amplitude of overshooting.

4.4 Demand-Side Monetary Policy

Since we adopt a New Keynesian framework, we can examine how the demand-side monetary policy interacts with the supply-side technology shock, and how the interaction governs the LIS dynamics. Figures 6 and 7 show that the LIS overshooting becomes more pronounced if the government has a more active inflation-targeting policy (increasing the inflation response parameter ϕ _π from 1.5 to 2.5), but it becomes less pronounced if the government has a more active output-targeting policy (increasing the output response parameter ϕ _y from 0.125 to 0.5).

Figure 6:

Sensitivity analysis with respect to the policy parameter for inflation in the interest rate rule (ϕ _π = 1.5 and ϕ _π = 2.5).

Figure 7:

Sensitivity analysis with respect to the policy parameter for output in the interest rate rule (ϕ _y = 0.125 and ϕ _y = 0.5).

The government’s inflation-targeting policy affects the LIS dynamics mainly through the markup channel. Intuitively, a more active inflation-targeting policy restrains the downward trend of inflation caused by a technological advance, making the aggregate output more responsive, as shown in Figure 6. In transition, this restraint on inflation further limits the downward trend in the trading price of intermediate goods. Thus, the real marginal cost of intermediate-good pricing firms φ _t decreases less, and hence, its markup μ _t = 1/φ _t increases less as well. Because markups are not so pro-cyclical in transition, we can see from (14) that the LIS overshooting becomes more significant under a more active inflation-targeting policy.

By contrast, a more active output-targeting policy leads the LIS overshooting to become less responsive. Figure 7 shows that if the government stabilizes the output level more aggressively, the aggregate output increases less but the inflation rate decreases more. Because the stabilization effect of the demand-side policy restrains the supply-side output expansion, the output elasticity channel becomes weaker, while the markup channel becomes stronger. Both substantially reduce the overshooting magnitude of the LIS.