Growth and non-regular employment

3.1 The value functions

I restrict my attention to stationary equilibrium, and labor market tightness is assumed to be constant over time. The value of a firm with a type j filled job, Π_jt, is characterized by the following Bellman equation:

where w_jt is the wage at time t, V_jt is the value of a firm with a type j vacant job and 𝕀_R is an indicator function that equals one if j=R and equals to zero otherwise. The firm receives flow revenues P_jtA_t–w_jt, which is the productive output of the match minus the wage paid to the worker. The match is destroyed by the exogenous shock at rate δ_j, in which case the firm loses its asset value of the filled job, pays the firing tax, and obtains the value of a vacant job. It is important to note that job separation rates are different between type R jobs and type N jobs, and only firms with type R jobs pay the firing cost f_t. The asset value of a match is expected to change over time due to exogenous technological progress.

The value of a firm with a type j vacant job at time t is given by

rVjt=−γjt+q(θj)[Πjt0−Vjt]+V˙jt for j∈{R, N},

where γ_jt is the cost of posting a vacancy and Πjt0 is the expected value of a new match to a firm with a type j job at time t. The expected profit of a new match to a firm with a type R job is different from Π_Rt, as defined in (2). This is due to the existence of the firing cost that is paid at the moment of job separation. In contrast, the expected value of a new match to a firm with a type N job is the same to Π_Nt.

Given the starting wage wRt0, the initial value of a firm with a type R filled job satisfies

rΠRt0=PRtAt−wRt0+δR[VRt−ΠRt0−ft]+Π˙Rt0.

I now turn to the worker’s side. The value of an employed worker in a firm with a type j job at time t, W_jt, is characterized by the following Bellman equation:

rWjt=wjt+δj[Ujt−Wjt]+W˙jt for j∈{R, N},

where U_jt is the value of an unemployed worker who searches for a job of type j. The value of an employed worker is determined by several factors. The worker receives the wage w_j. The match may be destroyed by the exogenous shock at rate δ_j, in which case the worker loses the current asset value and obtains the asset value of being unemployed. The asset value of a match is expected to change over time due to technological progress.

The value of an unemployed worker searching for a job of type j is

rUjt=zjt+θjq(θj)[Wjt0−Ujt]+U˙jt for j∈{R, N},

where z_jt is the unemployment benefits and Wjt0 is the value of an employed worker at the moment of job creation. Given an initial wage wRt0, the initial value of an employed worker in a type R job is given by

rWRt0=wRt0+δR[URt−WRt0]+W˙Rt0.

Note that an initial value of an employed worker in a type N job is the same to the continuing value of it. Thus, WNt0=WNt.

The firm that has a job with value Π_jt at time t expects to make a capital gain of Π˙jt=gΠj. The same holds for an employed worker and an unemployed worker, where the capital gain is gW_j and gU_j, respectively. The value of a vacant job V_jt does not change because it is zero by the free entry condition.

I focus on the steady state. This corresponds to a balanced growth path where the economy grows at the rate of productivity growth g. To make the model stationary, I assume that all exogenous variables grow at the rate of productivity growth g.^[10] Thus, I define six positive exogenous parameters γ_j, z_j, f, and c such that γ_jt=A_tγ_j, z_jt=A_tz_j, f_t=A_tf, and c_t=A_tc.

Replacing the capital gain by its steady-state value, the above Bellman equations can be rewritten as follows:

and

The wages are determined through the Nash bargaining between a firm and a worker over the share of expected future joint income. Let β_j∈(0, 1) be the worker’s bargaining power. Due to the firing cost, the wage determination mechanism differs between two markets. I first look at the wage determination in the market for type R jobs. When a firm and a worker first meet, the payoff to the firm is ΠR0−VR and the payoff to the worker is WR0−UR. Therefore, the starting wage is determined by the following equation

Once the worker is employed, the firm has to pay the firing tax Af if the firm fails to agree to a continuation wage. Thus, the continuing wage is chosen as

Since a firm with the type N job does not need to pay the firing cost, there is no difference between a starting wage and a continuation wage. The wage in the type N job is determined by

It is assumed that a non-regular worker can upgrade her skill by paying the training cost Ac.^[11] In equilibrium, the value of staying in the R sector is equivalent to that of staying in the N sector. Thus, arbitrage by workers between sectors implies that

Free entry implies that the value of a vacant job is zero in equilibrium. Thus,

Finally, the steady-state unemployment is given by

and

3.2 Characterization of steady-state equilibrium

A steady-state equilibrium is a profile {uj, θj, ϕ, wj, wR0, Πj, ΠR0, Vj, Wj, WR0, Uj} that satisfies the Bellman equations (3)–(8), the wage equations (9), (10) and (11), the workers’ arbitrage condition (12), the free entry condition (13), and the steady-state unemployment conditions (14) and (15).

The free entry condition V_j=0 together with (4) yields

By using (9), (13), and (16), the value of an unemployed worker in the j sector can be rewritten as

By using all the value functions (3)–(8), the free entry condition, and the wage sharing rules (9)–(11), I obtain the following equilibrium wages:

and

Making use of (12) and (17), I derive the following equilibrium condition:

The numbers of employed workers in the sector R and N are determined as e_R=ϕ–u_R and e_N=1–ϕ–u_N. Then, the aggregate production in the sector R and N are obtained by

yR=AϕθRq(θR)δR+θRq(θR), and yN=A(1−ϕ)θNq(θN)δN+θNq(θN).

Then, the prices of the two inputs can be obtained as

PR=α[ϕθRq(θR)δR+θRq(θR)]σ−1[α(ϕθRq(θR)δR+θRq(θR))σ+(1−α)((1−ϕ)θNq(θN)δN+θNq(θN))σ]1−σσ≡PR(θR, θN, ϕ),PN=(1−α)[(1−ϕ)θNq(θN)δN+θNq(θN)]σ−1[α(ϕθRq(θR)δR+θRq(θR))σ+(1−α)((1−ϕ)θNq(θN)δN+θNq(θN))σ]1−σσ≡PN(θR, θN, ϕ).

Substituting the price of goods in the R sector and (18) into (5) and using the free entry condition V_R=0 and (4), I obtain the equilibrium job creation condition

Similarly, by substituting the price of goods in the N sector and (19) into (3), and using (4) and (13), I have the following job creation condition in the type N job sector:

The job creation condition states that the expected cost of posting a vacancy, the left-hand side of (21)((22)), is equal to the firm’s share of the expected net surplus from a new job match, the right-hand side of (21)((22)).

The system of equations (20), (21), and (22) determine endogenous variables θ_R, θ_N, and ϕ. Given θ_R, θ_N, and ϕ, equations (14) and (15) determine the number of unemployed workers in the sectors R and N, respectively.

4.1 Calibration

I choose the model period to be 1-year and set the discount rate at r=0.036. This choice of the parameter is somewhat a priori, but is consistent with other studies such as Braun et al. (2006). Since the level of productivity does not influence the steady-state, I normalize A=1 without loss of generality. For a benchmark case, I set g=0.016, the average productivity growth rate in Japan from 1980 to 2014.

For the final goods production function, I choose σ=0 for the benchmark specification. Thus, the production function is Cobb-Douglas.

The matching function is Cobb-Douglas, given by ℳ(vj, uj)=mjvj1−ξujξ, where m_j is the matching constant and ξ is the matching elasticity with respect to unemployment. Then, the vacancy filling rate is q(θj)=mjθj−ξ and the job finding rate is θjq(θj)=mjθj1−ξ. I assume that matching constants (m_j) are different across sectors, while the elasticity parameters (ξ) are the same. Lin and Miyamoto’s (2014) estimate of the elasticity ξ for the Japanese labor market is 0.6. This value lies in the plausible range of 0.5–0.7 reported by Petrongolo and Pissarides (2001). To maintain comparability with much of the existing literature, I set β_R=β_N=0.5, so shares are split equally between firms and workers.^[12]

Given this, I target the average monthly job finding rate of 0.142 and the average monthly separation rate of 0.0048, which are reported by Miyamoto (2011) and Lin and Miyamoto (2012).^[13] The Report on Employment Service conducted by Ministry of Health, Labour and Welfare (MHLW) reports the job openings to applications ratios (yuko kyujin bairitsu) of both regular and temporary workers, which reflects labor market tightness. Based on the dataset, I target the ratio of labor market tightness for non-regular workers to that for regular workers θ_N/θ_R=2.24. I also target the share of non-regular workers in total employed workers. Based on the LFS, the mean value of the ratio over the period of 1984–2014 is 0.26. From the Survey on Employment Trends conducted by MHLW, the ratio of the job-finding rate for regular workers to that for non-regular workers is 0.45 and the ratio of the separation rate for regular workers to that for non-regular workers is 0.49. I target these ratios.

Following Esteban-Pretel, Nakajima, and Tanaka (2011), I target the unemployment flow utility z_j to be 40% of the average wage in each sector.^[14] I also target a wage ratio between regular workers and non-regular workers to be equal to 0.6, based on the Basic Survey on Wage Structure conducted by MHLW. Finally, following Nosaka (2011), I target the firing cost f to be approximately 0.5 of an average wage.^[15] Without loss of generality, I normalize θ_R to one.

I thus have 11 target moments and 11 model parameters: c, f, z_R, z_N, α, m_R, m_N, δ_R, δ_N, γ_R, and γ_N. I choose the parameter values that most closely match the 11 target moments. The parameter values are summarized in Table 2.

Table 2:

Parameter values.

Parameter	Description	Value		Source/target
A	A general productivity parameter	1.0		Normalization
r	Discount rate	0.036		Data
g	The rate of productivity growth	0.016		Data
ξ	Elasticity of matching function	0.6		Lin and Miyamoto (2014)
βR	Worker’s bargaining power in the R sector	0.5		Symmetric Nash bargaining
βN	Worker’s bargaining power in the N sector	0.5		Symmetric Nash bargaining
σ	CES-elasticity parameter	0.0		Cobb-Douglas
c	Cost of being a regular worker	12.33		Match target moments:
f	Firing cost	0.334		Job finding rate
zR	Flow value of unemployment in the R sector	0.267		Separation rate
zN	Flow value of unemployment in the N sector	0.160		Replacement rate
α	CES-weight	0.828		Share of non-reg.employed workers
mR	Scale parameter of matching function in the R sector	1.316		Job find. rate for R workers/job find. rate for N workers
mN	Scale parameter of matching function in the N sector	2.118		Sep. rate for R workers/sep. rate for N workers
δR	Separation rate in the R sector	0.046		Wage for N workers/wage for R workers
δN	Separation rate in the N sector	0.092		Tightness for N workers/tightness for R workers
γR	Vacancy cost in the R sector	0.371		Ratio of firing cost to wage
γN	Vacancy cost in the N sector	0.103		Normalization

Selected model solutions under the calibrated parameter values are reported in Table 3. The unemployment and job finding rates in the economy, the ratio of non-regular workers to total employed workers, and the ratios of labor market tightness and job-finding rates between sectors are equal to their target values. Unemployment in the regular-job sector, 0.025, is much higher than that in the non-regular job sector, 0.008. The number of vacancies in the regular job sector is 0.025, while that in the non-regular job sector is 0.018. Prices in the regular job sector are about 69% higher than those in the non-regular job sector.

4.2 Effects of growth on the labor market

I now examine the effects of productivity growth on labor market variables of interest by calculating the steady-state response to an increase in the productivity growth rate g. The results are shown in Figure 5.

Figure 5:

Comparative statics for the productivity growth rate g.

A faster rate of productivity growth increases regular employed workers and reduces non-regular employed workers, lowering the proportion of non-regular workers in the economy. The negative relationship between productivity growth and the share of non-regular workers is consistent with the empirical finding in Section 2. The channel through which faster productivity growth reduces the share of non-regular workers is as follows. Since an increase in productivity growth makes to work in the regular sector more appealing by reducing costs of working in the regular sector, a larger number of job seekers shifts from the non-regular sector to the regular sector. This increases job creation in the regular sector and thus regular employed workers.

Under my choice of parameter values, faster productivity growth reduces the aggregate unemployment rate, which is also consistent with the data. In order to understand the mechanism behind the effect of productivity growth on unemployment, it is useful to see the effect of productivity growth on vacancies.

The impact of productivity growth on vacancies is ambiguous because there are several counteracting effects. First, a rise in the productivity growth rate increases vacancies in both sectors. Since a higher rate of productivity growth increases the return from creating a job, firms in both sectors have a greater incentive to open vacancies. This is because the cost of creating a vacancy is paid at the start but the profits accrue in the future. When the growth rate rises, all future income flows are discounted at lower rate, so firms are encouraged to create more vacancies. This effect is well-known as the capitalization effect. Second, faster productivity growth tends to increase vacancy creation in the regular job sector but to reduce vacancy creation in the non-regular sector through increasing worker flows from the non-regular job sector to the regular job sector. An increase in worker flows from the non-regular job sector to the regular job sector makes firms in the regular sector find a worker easier and induces more vacancy creation. On the other hand, in the non-regular job sector, since a number of job seekers decreases due to the worker reallocation, firms have less intensive to open vacancies. I call this the worker reallocation effect. Third, faster productivity growth affects output prices and thus vacancy creation in both sectors. An increase in the productivity growth rate reduces the output price in the regular job sector and increases the output price in the non-regular job sector. An increased employment in the regular sector increases the supply of output, lowering the price of goods, while a decreased employment in the non-regular job sector reduces output and thus increases the price of goods. This price effect reduces vacancy creation in the regular job sector by reducing the return from creating a job, while it increase vacancy creation in the non-regular job sector by increasing the return from creating a job.

As seen in Figure 5, under calibrated parameter values, faster productivity growth increases vacancies in both sectors. In the regular job sector, since the capitalization and worker reallocation effects dominate the price effect, faster productivity growth increases vacancies. In the non-regular job sector, the capitalization and price effects dominate the worker reallocation effect. As a result, the increase in productivity growth leads to more vacancies.

I now turn to see the effect of productivity growth on unemployment. Since in the non-regular job sector, firms open more vacancies and a number of job seeker decreases due to the worker reallocation effect, the job-finding rate increases. This leads to a lower unemployment rate in the non-regular job sector. On the other hand, in the regular job sector, increased worker flows from the non-regular job sector cause congestion to one another when trying to match with vacancies. Although firms post more vacancies, the congestion effect is strong enough to reduce the job-finding rate and thus increase the unemployment rate.

While faster productivity growth increases the unemployment rate in the regular job sector, it reduces the unemployment rate in the non-regular job sector. Thus, the effect of faster productivity growth on the aggregate unemployment rate depends on which effect dominates. Under my choice of parameter values, since the latter effect dominates the former one, faster productivity growth reduces the aggregate unemployment rate.

Figure 5 shows that faster productivity growth increases the average wage in the economy. This is because faster productivity growth increases the share of regular employed workers whose wages are higher than non-regular employed workers’ wages.

Finally, I study whether the model’s prediction on the response of the proportion of non-regular jobs in the economy to productivity growth is empirically plausible. In Section 2, I find that a 1% increase in the productivity growth rate reduces the proportion of non-regular workers by 4.32%. In my model, a 1% increase in the growth rate reduces the proportion of non-regular workers by 5.8%. Thus, my model can explain the observed response of non-regular workers to growth.

4.3 Sensitivity analysis

In the benchmark case, faster productivity growth reduces the proportion of non-regular workers and the aggregate unemployment rate. I now discuss how these results vary with the value of the firing cost f and the parameter in the final goods production function σ. When I change these parameters, I also re-calibrate parameters c, z_R, z_N, α, m_R, m_N, δ_R, δ_N, γ_R, and γ_N in order to maintain my calibration target values.

First, I consider the impact of the value of the firing cost f. Figure 6 shows the relationship between the productivity growth rate and the aggregate unemployment rate. It also shows the relationship between the productivity growth rate and the proportion of non-regular workers in the economy for different values of the firing cost f. Although the size of impact is slightly changed, the sign of the relationship between growth and unemployment does not change. It is also clear that allowing for firing costs f to vary does not have a significant impact on the relationship between the rate of productivity growth and the proportion of non-regular workers.

Figure 6:

Sensitivity analysis with respect to f.

Next, I discuss the sensitivity of the results to my choice of the parameter value σ. In my benchmark calibration, I set σ=0 and assume that the final goods production function is Cobb-Douglus. I now consider two different values of σ, –1 and –1/3. In the former case, the elasticity of substitution is 0.5 and in the latter case, the elasticity of substitution is 0.75. Figure 7 shows the results. It shows that the negative relationship between growth and unemployment is robust to change in values of σ. Figure 7 also shows that the negative relationship between growth and the share of non-regular employment is robust to choice of parameter values of σ, decreases.

Figure 7:

Sensitivity analysis with respect to σ.

5.1 Welfare analysis

I now analyze the welfare properties of equilibrium. In order to consider the efficient allocation in this economy, I analyze the social planner’s problem. The social welfare function used by the social planner is given by

Welfare is equal to the total flow of output, plus the unemployment benefits, minus the flow vacancy cost incurred in both sectors and the firing cost.^[16] Note that this welfare function does not take account of distributional considerations.

The social planner is subject to the same matching constraints as firms and workers. Thus, the social planner’s problem is to maximize (23) subject to (1). The first-order conditions of this problem yield the following equation that determine the socially efficient θ_j.

γjq(θj)=(1−η(θj))(Pj−zj−IRδjf)−η(θj)θjγjr+δj−g,

where η(θ_j) is the negative of the elasticity of q(θ_j), and it is a number between 0 and 1.

By comparing this equation with the decentralized equilibrium conditions (21) and (22), we can see that the decentralized equilibrium is efficient if and only if β_j=η(θ_j). This is the standard Hosios condition for efficiency. When β_j>η(θ_j), there will be too little job creation, and the composition of jobs will be biased toward non-regular jobs. Thus, if β_j=η(θ_j) holds in both sectors, total number of jobs is efficient and the job composition is also efficient in the decentralized equilibrium. This result is in contrast to the random search model in which there is always an inefficient job composition [see Acemoglu (2001)].

It is important to note that for any value of the firing cost f, the above-mentioned result holds. This is because the firing cost is taken into account when wages are negotiated. However, this is not the case for the training cost c. So far, I have imposed the condition that c=0 when I considered the social planner’s problem and demonstrated that we need the Hosis condition to restore efficiency. However, when c has a positive value, the Hosios condition does not restore efficiency. This is because the training cost is sunk at the wage determination stages.

5.2 Endogenous job separation

In the present model, the job destruction process is exogenous and all the changes in job match characteristics are coming from the endogenous job creation dynamics. However, productivity growth may affect employment and worker flows via job separation. Indeed, productivity growth reduces job separation and lowers unemployment in the US (See Miyamoto and Takahashi 2011). This section discusses the role of endogenous job separation.

First, I examine a long-run relationship between productivity growth and the separation rate in Japan. Following Miyamoto (2011) and Lin and Miyamoto (2012), I construct the series of the separation rate from the LFS. By applying the same method in Section 2 to this time series, I find that there is a negative relationship between productivity growth and the separation rate (See Figure 8). Their correlation is –0.83. This suggests that in order to study the impact of growth unemployment, it is better to use a model in which job separation is endogenously determined.

Figure 8:

Productivity growth and separation rates.

Note: The solid line indicates the trend of the separation rate. The dashed line indicates the trend of the productivity growth rate. The trends are HP filters with the smoothing parameter 100. Sample covers 1984–2010.

I now develop a version of my model in which job separation is endogenously determined. The details of the model are presented in Appendix. Similar to Section 4.2, I examine the effects of productivity growth on labor market variables of interest by calculating the steady-state response to a change in the productivity growth rate g. The results are summarized in Table 4.

The effects of productivity growth on the labor market variables in the model with endogenous separation are different from those in the benchmark model (the model without endogenous separation). In the benchmark model, under plausible parameter values, faster productivity growth reduces unemployment and the proportion of non-regular workers. In contrast, the endogenous separation model shows that the U-shaped relationship between productivity growth and the unemployment rate (the proportion of the non-regular workers). Thus, when productivity growth is low, faster productivity growth reduces unemployment and the proportion of non-regular workers. On the other hand, when productivity growth is high, an increase in the productivity growth rate increase these two variable. This model’s prediction is counterfactual to what observed in data. Furthermore, the model shows the U-shaped relationship between productivity growth and the aggregate separation rate, which is also inconsistent with the above mentioned empirical finding.

The key to understand this result is the so-called outside option effect identified by Prat (2007). Prat (2007) demonstrates that, in the search and matching model with endogenous job separation, faster productivity growth is more likely to increase job separation by increasing workers’ outside option; consequently, the model can generate a positive relationship between productivity growth and unemployment.^[17] In my endogenous job separation model, when productivity growth is low, the worker reallocation effect mitigates the outside option effect. Therefore, faster productivity growth reduces the proportion of non-regular workers and the separation rate. On the other hand, when productivity growth is high, the worker reallocation effect weakens and thus faster productivity growth increases job separation and unemployment.