Economic growth and labor market friction: a quantitative study on Japanese structural transformation

2.1 Environment

Consider a small open economy populated by continuum of entrepreneurs and workers who consume both agricultural and non-agricultural products.^[6] Entrepreneurs in both sectors, denoted by subscript i∈{a, m}, manage their own firms and take profit flow π_t at every instant t∈[0, ∞). Workers are either employed or unemployed, the latter in both sectors receiving the same unemployment insurance, b_t, at every instant, the employed workers in sector i earning wage w_ijt at time t, depending on skill j∈{h, l}. The labor market is assumed to be subject to search and matching friction following the Diamond-Mortensen-Pissarides model. Time is continuous and all economic agents discount the future at rate r.^[7]

Workers At every instant, each individual worker having income flow w̅_t chooses the consumption bundle (c_at, c_mt) to maximize

subject to the budget constraint p_ac_at+p_mc_mt=w̅_t, where p_a and p_m represent the world prices of agricultural and non-agricultural products, respectively.^[8] It is assumed that workers are allowed neither to save nor to borrow.^[9] Since we are looking at a small open economy, we assume that (p_a, p_m) are exogenously given and constant over time.^[10] In particular, we take the non-agricultural products as numéraire and normalize p_m to be unity. Note that the elasticity of substitution should be strictly positive, σ>0.^[11] Solving the static utility maximization problem and plugging the individual demand for each product into the direct utility flow yields the indirect utility flow, as follows.

Denote by L_t the total population of workers at time t∈[0, ∞). At every instant, ρL_t measure of workers retire and χL_t measure of newly born workers enter the labor market as unemployed, which implies that the total population grows at rate χ–ρ(≥0) at every instant. Hence,

The sectoral ratio of newly-born workers is ex ante assumed to be the same as that of existing workers at every instant. But the new workers in one sector can immediately switch to the other sector at switching cost s~Logistic (0, ε_t/ω). For simplicity, the location parameter of the logistic distribution is normalized to be zero. The scale parameter of the logistic distribution is composed of parameter ω and the labor-augmenting technology ε_t (The role of the labor-augmenting technology will be explained later). Following Mcfadden (1974), Rust (1987), and Kennan and Walker (2011), we obtain the probability that the newly-born unemployed worker in sector i switches to sector –i as follows.

where U_it represents the lifetime value of the unemployed worker in sector i at time t.^[12] Unemployed workers retire at rate ρ and receive job offers at rate f(θ_it).^[13] If employed, the worker enjoys the lifetime value of W_ilt. Unemployed workers in sector i also get revision shocks at rate ξ^[14] and switch to the other sector if the net gains from doing so is positive. The Hamilton-Jacobi-Bellman (HJB hereafter) equation for the worker is given by

where

The detailed derivation of (6) is presented as well in Ishimaru, Oh, and Sim (2013). The left-hand side of (5) can be interpreted as the opportunity cost of holding the asset, unemployment at time t. The terms on the right-hand side represent the dividend flow from holding the asset U_it, the potential loss from retirement, the potential gains from job finding, the potential gains from inter-sectoral migration, and the gains from changes in valuation of the asset, respectively. To ensure the existence of the balanced growth path, we assume, following Mortensen and Pissarides (1998), that b_t=bε_t. An unskilled worker employed in sector i receives wage flow w_ilt per instant. The worker is separated from the job and becomes unemployed at rate δ and acquires skills at rate ζ. The HJB equations for the skilled and unskilled worker in sector i are, respectively,

The left-hand sides of (7) and (8) represent the opportunity cost of holding asset W_iht and W_ilt, respectively. The right hand side in equation (7) consists of the dividend flow from the asset, the potential loss from retirement, the loss from job separation, and the gains from changes in valuation of the asset, respectively. The right hand side of (8) has one additional term, gains from skill acquisition. Note that employed workers are not allowed to search for another job, which reflects the scarcity of job-to-job transition in the Japanese labor market.^[15] It is assumed that “skill” is job-specific so that skilled workers as well as unskilled workers lose their skill and get U_it when they are separated from their job.^[16] Alternatively, one can think of a variant with “general skills,” which requires to extend the dimension of the model. The alternative may cause quantitative changes in our argument, but the qualitative implication of this paper will be maintained. Furthermore, in Japan, South Korea, and other East Asian countries having the Confucianism tradition, it is not common for an ordinary worker in one firm to switch to another rival firm and utilize the skills that she/he acquired in the former job. Once she/he is separated from a job, the worker is more likely to get the next job in a different occupation through long unemployment periods. Hence, this paper keeps the specific skills.

Entrepreneurs There are n_i-measure of entrepreneurs in sector i∈{a, m}, who share the same preferences with workers. Entrepreneurs in each sector produce homogeneous products using identical production technology, given by

where (ait, kit, l˜it) represent the TFP component, capital stock, and labor input at time t, respectively. It is assumed that β_ai+β_ki+β_li=1. The labor-augmenting technology ε_t is introduced to capture the growth of the accumulated (general) knowledge of the economy. It can be interpreted as general human capital. Sustained economic growth on the balanced growth path is obtained by assuming ε_t to grow permanently, such that

The total factor productivity component, a_it, can be formed gradually through R&D investment and interaction with other productive resources (i.e. human capital). Throughout the paper we call it “technology.” That many R&D outcomes including new technologies and equipment can be utilized together with a certain level of skills accounts for gradual and persistent economic growth in a transitional economy. We posit

where λ_i is the efficiency parameter of technology investment, κ_i is the elasticity of technology investment, and η_a is the technology deterioration rate. L_iht, L_ilt, and L_it are the number of the skilled employed, unskilled employed, and all workers, respectively, in sector i at time t. Each entrepreneur in sector i makes R&D investment of z_it at every t. The law of motion in (11) implies that that R&D investment becomes more efficient as the skilled population in sector i increases (human capital externality).^[17] This, together with deterioration, embodies the gradual formation of the Ricardian comparative advantage and disadvantage. The capital stock, which can be immediately accumulated by purchasing from the international capital market, is depreciated at rate η_k.

where x_it is the capital investment in sector i at time t. Regarding the labor input, it is assumed that

where (l_ilt, l_iht) represent the masses of unskilled and skilled workers, respectively, employed by the entrepreneur at time t. The coefficient α_i reflects the fact that a skilled worker produces α_i(≥1) times more than an unskilled worker. Again, “skill” is assumed to be job-specific. Let v_it be the number of vacancies waiting for unemployed workers. The entrepreneur finds at rate q(θ_it) a worker who starts producing as an unskilled worker. Unskilled workers get learning shocks at rate ζ on the job and become skilled workers. All workers leave the entrepreneur due to separation shock at rate δ and retirement shock at rate ρ. The law of motion of the employed workers is described by

The profit flow of the entrepreneur in sector i at time t is given by

where (p_k, p_z) represent the cost of capital investment and R&D investment, respectively, and γε_t represents the cost of creating a vacancy. Following Mortensen and Pissarides (1998), we assume the vacancy creation cost to grow together with ε_t, which is necessary to ensure the existence of the balanced growth path. The entrepreneur in sector i having (ε̅, a̅_i, k̅_i, l̅_hi, l̅_li) at time t chooses the schedule of (x_iτ, v_iτ, z_iτ) for each τ∈[t, ∞) to maximize

subject to (10), (11), (12), (14), (15) and the initial condition (ε_t, a_it, k_it, l_iht, l_ilt)=(ε̅, a̅_i, k̅_i, l̅_hi, l̅_li). Lemma 1 solves the optimal control problem by the entrepreneur.

Lemma 1:The entrepreneur in sector i∈{a, m} optimally chooses (x_it, v_it, z_it) for each t∈[0, ∞) such that

In equations (18), (19), and (20), the left-hand side represents the marginal cost of investment, while the right-hand sides represent the marginal benefit. The proof of Lemma 1 is delayed in Appendix A.

Labor market The degree of the labor market tightness in sector i is defined as the ratio of the measure of total vacancy to that of job seekers in the sector. For each i∈{a, m},

Let m(v_itn_i, u_it) be the number of matches successfully formed at time t for each i∈{a, m}. Given constant returns to scale matching technology, we obtain

We denote by (L_iht, L_ilt, u_it) the measure of skilled employees, unskilled employees, and unemployed workers, respectively, in sector i at time t. The total population, L_t, at time t is given by

The dynamic worker flows are summarized as follows.

where L_t is presented in (3). The last term of the right-hand side of (26) is derived by the following argument: At time t, there are χ(L_aht+L_alt+u_at) -measure of newly born workers in the agricultural sector and χ(L_mht+L_mlt+u_mt) -measure of newly born workers in the manufacturing sector; because they can immediately switch, the total measure of newly born workers in sector i after the immediate switching is given by χω_itL_t.

Wage bargaining In accordance with Stole and Zwiebel (1996),^[18] successfully matched workers and entrepreneurs individually negotiate wages by splitting the marginal surplus.^[19] This implies that for each j∈{h, l},

where ϕ is the worker’s bargaining power. In wage bargaining, the firm takes 1–ϕ portion of the joint (marginal) surplus and gives ϕ portion to the worker in the form of wage payment. Note that (27) is implicitly based on an equilibrium restriction such that in any equilibrium W_ijt–U_it≥0 and ∂Eit∂lijt≥0 for each i∈{a, m}, j∈{h, l}, and t∈[0, ∞). Since the bargaining rule in (27) is true at every t, continuity and differentiability implies that

Lemma 2:The implied wage in sector i∈{a, m} is given by

where

Λi0=ϕ1−ϕ+ϕβliand Λit1=(1−ϕ)bεt+γεtϕθit+(1−ϕ)ξΔitP.

Note that in both (29) and (30) the first terms are proportional to the marginal product of labor. The second terms reflect the labor market condition. For later use, we remark that Λit1 grows at rate ψ if θ_it is constant over time and Δ_it grows at rate ψ. The detailed derivation is provided in Appendix A.

Equilibrium We finish this section by defining the equilibrium of our interest. The following definition summarizes the overall shape of our model.

Definition An equilibrium consists of bounded time series of choice rules {ω_it, x_it, v_it, z_it}_i∈{a,m}, labor market tightness parameters {θ_it}_i∈{a,m}, wages {w_it}_i={a,m}, value equations {E_it, W_iht, W_ilt, U_it}_i∈{a,m}, and laws of motions {ε_t, a_it, k_it, l_iht, l_ilt, L_t, L_iht, L_ilt, u_it}_i∈{a,m} at every t∈[0, ∞) such that:

1. unemployed workers as well as newly born workers optimally choose where they work when they are hit by an exogenous shock,

2. each entrepreneur in sector i optimally chooses {z_it, x_it, v_it} at every t,

3. aggregate consistency requires that

   • the market tightness {θ_it}_{i∈{a, m}} should be consistent with its definition,

   • wages {w_it}_{i∈{a, m}} should be consistent with the imposed bargaining rule,

   • L_iht=n_il_iht, L_ilt=n_ililt, and L_t=∑_i(L_iht+L_ilt+u_it) for each i∈{a, m}.

4. the evolution of the entire system is recursively governed by the law of motions of (3), (5), (7), (8), (10), (11), (12), (14), (15), (17), (24), (25), and (26), given {E_i0, W_ih0, W_il0, U_i0}_i∈{a,m} and {ε₀, a_i0, k_i0, l_ih0, l_il0, L₀, L_ih0, L_il0, u_i0}_{i∈{a, m}}.

2.2 On the balanced growth path

In this subsection, we characterize the balanced growth paths by applying the previous definition with a small modification on the law of motions (v). Stationarity on the balanced growth path requires that

Let L_ih, L_il and u_i denote the proportion of skilled, unskilled and unemployed workers, respectively, in sector i to the total population. That is,

By the stationarity condition of the balanced growth path (31), these ratios are constant over time. The stationarity condition dictates that Δ_it/ε_t is constant over time on the balanced growth path. Also, from (15), (21), and (31), we get

on the balanced growth path. Since L_ilt and u_it grow together at the rate (χ–ρ), θ_it should be constant on the balanced growth path to make the last equality hold over time. We drop the time subscript t from the variables that are constant on the balanced growth path.

Using (31), we rewrite (5) as follows.

Combining (20) and (27) yields

Subtracting U_at from U_mt, dividing by ε_t, and combining with (35) yields

These imply that (ω_a, ω_m) and (U_mt–U_at)/ε_t are constant over time on the balanced growth path and uniquely determined by (θ_a, θ_m). From (25), (26), and (31), the triplet of (L_ih, L_il, u_i) is characterized as follows. Given (θ_a, θ_m), for each i∈{a, m},

Solving (37), (39) and (40), and multiplying by L_t results in Lemma 3.

Lemma 3:Given (θ_a, θ_m), on the balanced growth path,

Notice that ∂yit∂lilt grows at rate ψ on the balanced growth path due to the growth of the labor-augmenting technology. Lemma 4 summarizes the entrepreneur’s behavior on the balanced growth path.

Lemma 4:Given (θ_a, θ_m) on the balanced growth path, the entrepreneur in each sector optimally chooses

where

In addition, these imply that

Plugging (42), (48), and (49) into (20) and reordering yields

for each i∈{a, m}. As mentioned before, (35) is based on the implicit restriction such that W_it–U_it>0 and ∂Eit∂lit>0 for each i∈{a, m} and t∈[0, ∞). If it is violated, the right-hand side of (50) can be negative so that there is no solution. If the implicit restriction is satisfied, we get two equations described in (50) to solve for two unknowns (θ_α, θ_m).

Proposition 1:There exists a balanced growth path if and only if the system of equations described in (50) has a solution of (θ_a, θ_m).

Given the complexity of the overall system, it is difficult to analytically determine under what parametric condition the balanced growth path exists and whether it is unique. Instead, we acknowledge that in our numerical experiment with reasonable parameter values, we obtain the same result regardless of the random starting points (we repeat the same experiment with more than 20 different random initial guesses).

2.3 On the postwar transition

Here, we characterize the transition path from a particular initial state to the balanced growth path, with the evolution of the economy governed by the system of differential equations. One advantage of our model is that the transition path is fully described by an autonomous control.

The lifetime value of unemployment evolves as follows. For each i∈{a, m},

Given (θ_at, θ_mt), equation (51) together with (6) determines (U_at, U_mt, Δ_at, Δ_mt) at every t∈[0, ∞). Plugging (51) into (4) also yields (ω_at, ω_mt). Starting from (L_ah0, L_al0, L_mh0, L_ml0, u_a0, u_m0), (L_aht, L_alt, L_mht, L_mlt, u_at, u_mt) evolve following (24)–(26) toward

Note that l_iht=L_iht/n_i, and l_ilt=L_ilt/n_i.

Lemma 5:Given (θ_at, θ_mt, L_aht, L_alt, L_mht, L_mlt) at every t∈[0, ∞), we obtain

where

zit=εtLiht[λiκipz∫t∞e−(r+ηa)(τ−t)(1−βliΛi0)βaipiaiτβai−1kiτβkiετβli(lilτ+αilihτ)βlidτ]11−κi

Lemma 5 summarizes the dynamic path followed by entrepreneurs. Given (θ_at, θ_mt) at every t∈[0, ∞), equations (24)–(26) jointly determine (L_aht, L_alt, L_mht, L_mlt, u_at, u_mt) and, together with Lemma 5, the unique path of the economy. Combining (20) and (27) yields

which restores (θ_at, θ_mt) at every t∈[0, ∞). Because (W_ilt, U_it)→(W_il, U_i), we can get the convergence point (θ_a, θ_m). As in the previous subsection, (56) raises the equilibrium restriction that W_ilt–U_it>0 for all i and t. Equations (24)–(26) and (51)–(56) provide the full description of the transition path of the model.

Again, the complexity of the system prevents us from providing an analytical proof on the existence and uniqueness of the solution. Instead, we acknowledge that we obtain the same result when we repeat the experiment with 20 different sets of “initial guesses.”

3.1 Japanese structural transformation

Exogenously fed parameters We set r=0.0095 during the sample period to fix the annual interest rate at 3.8%, which is consistent with the estimate of the annual discount rate in Esteban-Pretel and Sawada (2014). But, the discount rate does not have a pronounced effect on the macro variables in our model. We normalize the price of non-agricultural products to be 1, and fix the relative price of agricultural products at 1.2, which is consistent with the average of the Engel coefficient from 1960 to 1990. The Engel coefficient, the ratio of food expenditures to total expenditures by households, is obtained by

pa1−σpa1−σ+pm1−σ≈0.37.

Japan’s Engel coefficient was around 0.49 in 1960 and declined continuously to around 0.25 in 1990.^[21] Because it assumes homothetic preferences, our model predicts a constant Engel coefficient, unless it additionally assumes the subsistence level, as in Matsuyama (1992). For the sake of simplicity, we alternatively take a simple average of the Engel coefficient during the target period.

The retirement rate is set to be 0.011, which results in roughly 90 percent of workers, who enter the labor market at age 25, retiring before age 70. The implied average “market duration,” the average elapsed time between labor market entry and exit, is thus less than than 25 years. Not being a life cycle model with aging, our model considers a worker who moves into the non-labor force to be a retiree and a worker who moves from the non-labor force into the labor force to be a newly born worker. Average market duration is thus much shorter than average lifespan. Japan’s population grew from 58.63 million in 1965 to 81.0 million in 1990.^[22] The implied quarterly growth rate is ln(81.0/58.63)/100≈0.003, which determines the birth rate χ=ρ+0.003=0.014 during the sample period. The separation rate δ is set to 0.02 to obtain the average job duration of 12.5 years among non-retirees in the OECD data set. This choice is consistent with Esteban-Pretel and Fujimoto (2012). We posit

borrowing from Kano and Ohta (2002), who estimate the matching function of the Japanese labor market consistent with the plausible range of empirical elasticity of 0.5–0.7 proposed by Petrongolo and Pissarides (2001). We equalize the bargaining parameter to the elasticity of the matching technology by invoking Hosios’s condition, that is, ϕ=0.6. These choices on bargaining parameter and matching function parameters are consistent with Esteban-Pretel and Fujimoto (2012). The growth rate of labor augmenting technology ψ is set at 0.007. The latter choice, which implies a balanced growth path of 0.028 annually, is based on the average annual growth rate (a weighted average of country growth rates) of GDP in all OECD countries for the period 1980–1990.^[23]

The capital depreciation rate η_k, calculated as the ratio of depreciation to real capital stock each year, results in η_k=0.033, following Jorgenson (1996).^[24] Calculating the technology depreciation rate η_a based on the depreciation rate of service industry equipment results in η_a=0.041, again following Jorgenson (1996). Due to the lack of sectoral data, we employ the aggregate economy level of the depreciation rate. The elasticity of substitution σ is set to 3.8, following Bernard et al. (2003), Felbermayr, Prat, and Schmerer (2011), and Ishimaru, Oh, and Sim (2013). Values exogenously assigned are summarized in Table 1.

Calibrated parameters For the remaining parameters, we choose a vector that matches the aggregate of the time series data for sectoral GDP growth per capita, sectoral labor share, sectoral capital share, sectoral wage growth, the net EXP/GDP, replacement ratio and average unemployment rate.^[25]

In Figure 2, the dots represent actual data values, the smooth lines the time series predicted by the model. Overall, Figure 2 suggests that our calibration strategy captures fairly well the trend in transition. Panels (A) and (B) present the time-series data and the model’s prediction of sectoral GDP growth per capita from 1970 to 1990 (lacking data for earlier periods). Panels (C) and (D) report sectoral labor shares, Panels (E) and (F) sectoral capital shares, from 1960 to 1990. Both labor and capital shares declined sharply in the agricultural, and rose rapidly in the non-agricultural, sector. In Panels (G)–(H), we exploit the evolution of sectoral wages from 1960 to 1990 to calibrate the labor productivity parameters (α_a, α_m, ζ). Based on historical wage data reported by Japan’s Ministry of Health, Labor and Welfare,^[26] from 1960 to 1990 workers’ real wages grew by 29 and 152 percent in the agricultural and non-agricultural sectors, respectively. Panel (I) reports the unemployment rate in the actual data and the model. Without business cycle fluctuation, we set the target of the average unemployment rate at 2.0 percent. Panel (J) shows the Net Export/GDP ratio predicted by the model to be reconciled with the average Net Export/GDP ratio of 10.5 percent and its linear trend from 1960 to 1990. Panel (K) presents the population growth from the data and the model. Panel (l) reflects adjustment of the model’s parameters to keep unemployment benefits at around 40 percent of the average wage over time.^[27] These choices are summarized in Table 2.

Figure 2:

Calibration results (Japan).