3.1. Worker suitability for tasks

In a two-period OLG model, workers live in two periods and have one job-seeking opportunity in each of those periods. The population size of each generation is $\frac{1}{2}$ , so the overall population is $1$ . Figure 1 illustrates education, job seeking, and employment of a worker. In the first period, a worker initially decides whether to acquire specific skills for some middle-skilled tasks by receiving an education. Although workers know their suitability for certain tasks, knowledge about firms is limited. Therefore, they face job mismatch possibilities when they seek middle-skilled jobs. In the second period, workers who experience job mismatch in the first period once again seek middle-skilled jobs.

Figure 1. Job seeking under no increased automation.

In period $t$ , middle-skilled tasks have been decomposed into automated and labor tasks:

\begin{align*} T_t=M_t+(T_t-M_t), \end{align*}

where $T_t$ is equal to the total number of tasks, $M_t$ is the number of automated tasks, and $T_t-M_t$ is the number of labor tasks. In this section, we assume no change in these tasks, that is, $T_t=T$ and $M_t=M$ . Before receiving education, a worker is unsuitable for any middle-skilled labor task: $T-M=b$ where $b$ is the number of unsuitable tasks. If a worker invests in education, then that worker can obtain skills for suitable tasks. That is, that worker can reduce tasks that are unsuitable for the worker: $ 0\lt b\lt T-M$ . Therefore, after receiving education, labor tasks are decomposed into suitable and unsuitable tasks for a worker:

(1) \begin{align} T-M=g +b, \end{align}

where $g$ is the number of suitable tasks ( $0\lt g\lt T-M$ ). We assume no difference in productivity among these suitable (or unsuitable) tasks. The ability level of a worker is measured by the number of suitable tasks. We examine workers with the same ability level.

We consider the individual job mismatch probability (IMP) of a worker for middle-skilled labor tasks under two assumptions. The first assumption regards the worker’s suitability for tasks. After receiving an education, workers with the same ability level obtain the same number of suitable tasks. The suitability pattern for tasks that represents which tasks are suitable and unsuitable for a worker can differ among workers due to the difference in their specialization areas. In our model, a pattern of suitability for middle-skilled labor tasks is randomly assigned to a worker. The second assumption is that there is a random matching between a job seeker and a firm with one job-seeking opportunity in a period.

Therefore, the IMP of a worker is defined as the ratio of the number of a worker’s unsuitable middle-skilled tasks to the total number of middle-skilled tasks. The IMP of a worker who is born in period $t$ is represented as follows:

(2) \begin{align} q (b,T-M) = \frac{b}{T-M}, \end{align}

where $0 \lt q (\cdot ) \lt 1$ because $0 \lt b \lt T-M$ .Footnote ⁴ We denote $q(b,T-M)$ as $q$ . The numerator in (2) denotes the number of remaining unsuitable middle-skilled labor tasks after receiving an education. That is, this is an individual-specific factor. If the ability level of a worker is high, the worker’s IMP value becomes low because of a low number of unsuitable tasks. The denominator in (2) denotes the number of middle-skilled labor tasks, which is a technology factor. This factor is always the same among workers, regardless of their abilities. Although there is the difference in the task suitability patterns among workers with the same ability, their IMP value is the same under no technical change.

3.2. Decision of education investment

We examine a worker who was born in period $t$ . For simplicity, the utility function of a worker is assumed to be linear with respect to the consumption levels when young and old:

(3) \begin{align} U\!\left(c^y_t,c^o_{t+1}\right)=c^y_t +\frac{c^o_{t+1}}{1+\rho }, \end{align}

with their budget constraints,

(4) \begin{align} c^y_t+e+s_t =I^y_t \quad and \quad c^o_{t+1}=I^o_{t+1}+Rs_t, \end{align}

where $\rho$ is the time discount rate. $c^y_t$ and $c^o_{t+1}$ are the consumption levels in periods $t$ and $t+1$ when young and old, respectively, $s_t$ is the savings in period $t$ when young, and $e$ is the cost of education. The sum of the interest rate and the rate of depreciation is denoted by $R=1+r$ . Here, $I^y_t$ and $I^o_{t+1}$ are the wage income levels in periods $t$ and $t+1$ when young and old, respectively. We assume that the gross interest rate exceeds the discount rate:

(5) \begin{align} R \gt 1+\rho. \end{align}

Under this assumption, workers save all their disposable income, and this implies $c^y_t =0$ .

If a worker decides to invest in education, then their indirect utility is

(6) \begin{align} V\!\left( I^y_t -e, I^o_{t+1} \right) = \frac{1}{1+\rho } \left[ R \!\left(I^y_t -e \right) +I^o_{t+1} \right], \end{align}

where $V(\cdot )$ is the indirect utility function. The income level in each of periods $t$ and $t+1$ is either the wage rate of middle- or low-skilled labor. That is, these income levels depend on the employment status. The probability of employment in periods $t$ and $t+1$ is represented by $Pr (I^y_t, I^o_{t+1})$ . These probabilities are

(7) \begin{align} Pr \!\left(w_{m,t},w_{m,t+1}\right)=1-q, \ \ Pr \!\left(w_{m,t},w_{l,t+1}\right)=0,\nonumber \\ Pr \!\left(w_{l,t},w_{m,t+1}\right)=q\!\left(1-q\right), \ \ Pr \!\left(w_{l,t},w_{l,t+1} \right)=q^2, \end{align}

where $w_{m,t}$ and $w_{l,t}$ are the wage rates of middle- and low-skilled labor in period $t$ . Note that

\begin{align*} \sum _{I^y_t=w_{m,t},w_{l,t}} \sum _{I^o_{t+1}=w_{m,t+1},w_{l,t+1} } Pr \!\left(I^y_t, I^o_{t+1}\right) =1. \end{align*}

From (6) and (7), the expected utility level of receiving an education is

(8) \begin{align} E \!\left( V\!\left( I^y_t -e, I^o_{t+1} \right) \right) = \sum _{I^y_t=w_{m,t},w_{l,t}} \sum _{I^o_{t+1}=w_{m,t+1},w_{l,t+1}} Pr \!\left(I^y_t, I^o_{t+1}\right) V\!\left(I^y_t-e, I^o_{t+1} \right). \end{align}

If a worker does not decide to receive an education, then that worker cannot obtain skills for any middle-skilled labor task. The worker works in low-skilled labor in both periods $t$ and $t+1$ :

(9) \begin{align} V\!\left( I^y_t, I^o_{t+1} \right)= V \!\left(w_{l,t}, w_{l,t+1} \right)= \frac{1}{1+\rho } \left( Rw_{l,t} +w_{l,t+1} \right). \end{align}

Using (8) and (9), we assume that the expected utility of receiving an education exceeds the utility level of not receiving an education:

(10) \begin{align} F\!\left(q, sp_t, sp_{t+1},e\right) \equiv E\!\left(V\!\left( I^y_t -e, I^o_{t+1} \right) \right) - V \!\left(w_{l,t},w_{l,t+1} \right) \gt 0, \end{align}

where $sp_t$ is the skill premium between middle- and low-skilled labor in period $t$ : $sp_t \equiv w_{m,t}-w_{l,t}$ . It holds that $\frac{\partial F }{\partial q}\lt 0$ , $\frac{\partial F }{\partial sp_t}\gt 0$ , and $\frac{\partial F }{\partial e}\lt 0$ . Therefore, the incentive for receiving an education declines with a rise in the IMP of a worker, a decline in the skill premium, or a rise in the cost of education. Under the assumption in (10), a worker obtains skills for suitable tasks through education.

3.3. Firms

First, we examine the final good firms. Two types of technology can be used: One uses intermediate goods, and the other uses low-skilled labor alone. We assume a logarithmic production function when intermediate goods are used:

(11) \begin{align} \ln Y_t=\frac{ 1}{T} \sum _{i=1}^{T} \ln y_t(i), \end{align}

where $Y_t$ is the total output and $y_t(i)$ is the input of intermediate goods $i$ . Intermediate goods are produced by tasks, and the output level is normalized by the number of total tasks $T$ , which is exogenously given. The profit of final good firms is

\begin{align*} Y_t- \sum _{i=1}^T p_t(i) y_t (i), \end{align*}

where $p_t(i)$ is the price of intermediate goods. The FOCs of the inputs of intermediate goods are

(12) \begin{align} p_t(i) y_t(i)=\frac{Y_t}{T}, \end{align}

where $ i \in [1,\cdots, T]$ . When low-skilled labor is used, we assume a linear production function:

(13) \begin{align} Y_t=A_{l} L_{l,t}, \end{align}

where $A_{l}$ is exogenously given and $L_{l,t}$ is the input of low-skilled labor. The wage rate of low-skilled labor is $w_{l,t} =A _{l}$ .

We then examine intermediate good firms. Each intermediate good is produced by use of either machines or middle-skilled labor:

(14) \begin{align} y_t (i)= \gamma (i) k_t (i) + \lambda _m l_{m,t} (i), \end{align}

where $i \in [1,\cdots, T]$ . In the production of $i$ -th intermediate goods, $k_t(i)$ is the input of machines, and $l_{m,t} (i)$ is the input of middle-skilled labor. If a machine for a task is available, the efficiency of the machine input is positive, and it is zero otherwise. The efficiency of the machine input is

\begin{align*} \gamma (i)=d (i) \gamma, \end{align*}

where $\gamma \gt 0$ . $d(i)$ takes either $1$ or $0$ . We assume that

(15) \begin{align} \sum _{i=1}^{T} d(i)=M. \end{align}

That is, the level of automation is exogenous. If machines are used in the $i$ -th intermediate good under $d(i)=1$ , the price of intermediate goods that use machines, represented as $p_k$ is

\begin{align*} p_k = \frac{R}{\gamma }. \end{align*}

A firm that uses middle-skilled labor must find a suitable worker for production. Hence, the probability that a firm engages in production is the same with the matching probability between an entry firm and a suitable worker. This matching probability is considered based on the following two variables. The first variable is market tightness in the middle-skilled labor market, which is the jobs-to-applicants ratio: $\theta _t \equiv{v_t}/{ \tilde{L}_{m,t}}$ . Here, $\theta _t$ is market tightness, $v_t$ is the number of entry firms with vacancy $v_t$ , and $\tilde{L}_{m,t}$ is the number of middle-skilled job seekers. Assuming that $\theta _t \geq 1$ , the probability of a firm being able to find a job applicant can be considered by $\theta _t^{-1}$ . If the number of entry firms is large, then a firm finds it difficult to recruit a job applicant. The second variable is the probability that a recruited worker is suitable for a firm. Under the two assumptions concerning workers’ suitability for tasks and the random matching between a firm and a worker, this probability that a recruited worker is suitable for a firm is the same among these firms. This probability is represented as $1-\mu _t^a$ where $\mu _t^a$ is the mean of the IMPs of job seekers. Because the IMP value of any job seeker is $q$ , it holds that $\mu _t^a=q$ .Footnote ⁵ Consequently, the probability that a firm engages in production, $\phi _t$ , is represented as follows:

(16) \begin{align} \phi (\theta _t ;\, q) =\frac{1-\mu _t^a}{\theta _t} =\frac{1-q}{\theta _t}. \end{align}

It holds that $ \frac{\partial \phi _t}{\partial \theta _t}\lt 0$ and $ \frac{\partial \phi _t}{\partial q}\lt 0$ . Using (16), the expected profit of a firm is

(17) \begin{align} G \!\left(\theta _t ;\, q, A_{m,t}, w_{m,t}, z\right) \equiv \frac{1-q}{\theta _t} \left( A_{m,t} -w_{m,t}\right) -z, \end{align}

where $A_{m,t} \equiv p_{m,t} \lambda _m$ . $p_{m,t}$ is the price of intermediate goods that use middle-skilled labor. $A_{m,t }-w_{m,t}$ is the firm’s profit when the firm succeeds in production. $z$ is the fixed cost in which $z\gt 0$ . In (17), the expected profit of a firm is a decreasing function of the number of entry firms.

Under Nash bargaining, we consider the wage rate of middle-skilled labors. Unless a worker who received an education is employed in a middle-skilled job, the worker works in low-skilled labor. Hence, we assume Nash bargaining as follows:

\begin{align*} \max _{w_{m,t}} \left(w_{m,t}-w_{l,t}\right)^{\beta } \left( A_{m,t} -w_{m,t}\right)^{1-\beta }, \end{align*}

where $0\lt \beta \lt 1$ . Thus, the middle-skilled wage rate is

(18) \begin{align} w_{m,t}=\beta A_{m,t} + (1-\beta ) A_{l}. \end{align}

Note that $w_{l,t}=A_l$ . The wage rate in (18) is lower than the perfectly competitive wage rate, $w_{m,t}=A_{m,t}$ , due to the possibility of job mismatch.Footnote ⁶

From (17) and (18), as a result of the free entry of firms, the zero profit condition $G( \hat{\theta }_t)=0$ implies

(19) \begin{align} \hat{\theta }_t = (1-q) \frac{1-\beta }{\beta } \frac{ sp_t}{z}. \end{align}

Thus, the entry of firms is $ \hat{v}_t = \hat{\theta }_t \tilde{L}_{m,t}$ . The number of job seekers that include old and young individuals is

\begin{align*} \tilde{L}_{m,t}=\frac{1}{2} q+ \frac{1}{2}. \end{align*}

An increase in the IMP of a worker decreases the market tightness: $\frac{\partial \hat{\theta }_t}{\partial q}\lt 0$ . It also holds that $\frac{\partial \hat{\theta }_t}{\partial sp_t}\gt 0$ and $\frac{\partial \hat{\theta }_t}{\partial z}\lt 0$ .

3.4. Equilibrium

We examine equilibrium under no technical change. The equilibrium condition in the final good market implies

(20) \begin{align} \sum _{i=1}^T \ln \left ( \frac{1}{p_t (i)} \frac{1}{T} \right ) =0. \end{align}

From the price of intermediate goods that use machines, $p_k = \frac{R}{\gamma }$ , the price of intermediate goods that use middle-skilled labor, represented as $p_m$ is

(21) \begin{align} \ln p_{m} = \frac{1}{T-M} \left ( M \ln \frac{\gamma }{R} -T\ln T \right ). \end{align}

If $M=0$ , then it holds that $p_m=T^{-1}$ . If the number of total tasks is large, then the total cost of production is also large. The price of intermediate goods should be low to satisfy (20). Under a sufficiently large efficiency of machines, which implies $\gamma \gt R$ , a rise in the level of automation increases the price of labor-based intermediate goods: $\frac{\partial p_m}{\partial M}\gt 0$ . When machines are widely used in the production of the final good production, the marginal product of intermediate goods using labor rises.

We consider the adoption of automation in equilibrium. If $\frac{R}{\gamma } \gt T^{-1}$ holds due to a small efficiency of machines, then all tasks are performed by middle-skilled workers because using middle-skilled labor is less expensive than that of machines. That is, there is no use of machines, $M=0$ . If the following inequality holds due to a sufficiently large efficiency of machines

(22) \begin{align} \frac{R}{\gamma } \leq T^{-1}, \end{align}

then the use of machines is more profitable than or equal to the use of middle-skilled labor. Condition (22) implies inequality, $\gamma \gt R$ , because $T \gt 1$ . That is, provided that the machine technology is available, the automation level is positive: $M\gt 0$ . In the following analysis, we consider what happens when (22) holds. This implies that $p_m \gt \frac{R}{\gamma }$ . See Figure 2.

Figure 2. Prices of intermediate goods.

From FOCs in (12), the expenditure share of inputs is equal among intermediate goods:

(23) \begin{align} R k_t= p_{m}\lambda_{m} l_{m,t}, \end{align}

where $ k_t=\frac{K_t}{M}$ and $ l_{m,t}=\frac{L_{m,t}}{T-M}$ . $K_t$ is the total input of capital and $L_{m,t}$ is the total input of middle-skilled labor. From (11) and (23), the production function is of the Cobb-Douglas type, which depends on the degree of automation, $\frac{M}{T}$ :

\begin{align*} \ln Y_t= \frac{M}{T} \ln \gamma + \left (1- \frac{M}{T} \right ) \ln \lambda _m + \frac{M}{T} \ln \frac{K_t}{M} + \left (1- \frac{M}{T} \right ) \ln \frac{L_{m,t}}{T-M}. \end{align*}

We investigate equilibrium in the middle- and low-skilled labor markets. We assume the incentive compatibility for education investment noted in (10). The number of middle-skilled workers is

(24) \begin{align} L_{m,t} = L^o_{m,t} + L^y_{m,t}, \end{align}

where $L^o_{m,t}$ and $L^y_{m,t}$ are old and young middle-skilled workers, respectively. Under the law of large numbers, old and young middle-skilled workers are, respectively,

\begin{align*} L^o_{m} (q)= \frac{1}{2}[(1-q)+q(1-q)]=\frac{1}{2}\left(1-q^2\right) \ \ and \ \ L^y_{m}(q) = \frac{1}{2} (1-q). \end{align*}

These are endogenously determined due to the endogeneity of the IMP value of a worker. The number of low-skilled workers is

\begin{align*} L_{l,t} =L^o_{l,t} +L^y_{l,t}, \end{align*}

where $L^o_{l,t}$ and $L^y_{l,t}$ are, respectively, old and young low-skilled workers, respectively. Because a worker is either a middle- or low-skilled worker, it holds that $L^o_{m,t}+ L^o_{l,t}=\frac{1}{2}$ and $L^y_{m,t}+ L^y_{l,t}=\frac{1}{2}$ . Old and young low-skilled workers are, respectively,

\begin{align*} L^o_l (q)= \frac{1}{2}q^2 \ \ and \ \ L^y_l (q) = \frac{1}{2}q. \end{align*}

These low-skilled workers work as low-skilled labor despite their investment in education. That is, they are underemployed.

4.1. Workers’ IMPs in the period of increased automation

As a result of increased automation in period $t$ , workers are divided into the following four types. The first type is old workers who can continue working as middle-skilled labor due to no replacement by machines. The number of these workers is $\frac{1}{2} (1-q) \Bigl (1- \frac{\Delta M_t}{T-M} \Bigr )$ . The second type is old workers who cannot continue working as middle-skilled labor due to the replacement by machines. The number of these replaced workers is $ \frac{1}{2} (1-q) \frac{\Delta M_t}{T-M}$ . The third type is old workers who cannot find middle-skilled jobs in period $t-1$ . The number of these mismatched workers is $\frac{1}{2}q$ . These second- and third-type workers seek middle-skilled jobs again. The fourth type is young workers who are born in period $t$ with the population size $\frac{1}{2}$ .

The IMP of a worker endogenously changes because machines cover some suitable and unsuitable tasks of the worker. The worker’s IMP is equal to the ratio of the number of the worker’s unsuitable labor tasks that cannot be performed by machines to the number of available labor tasks:

(26) \begin{align} q (x_t;\, \Delta M_t,b,T-M) = \frac{b -x_t}{T-M -\Delta M_t }, \end{align}

where $0\lt T-M-\Delta M_t$ and $0 \leq q (x_t) \leq 1$ . For simplicity, we represent (26) as $q(x_t;\, \Delta M_t)$ . In (26), $x_t$ is the number of the worker’s unsuitable tasks performed by machines and $x_t \in [\underline{x}, \cdots, \bar{x}]$ . Equation (26) includes two factors. First, the numerator in (26) denotes the number of the worker’s unsuitable tasks that cannot be performed by machines; this is an individual-specific factor. This factor differs even among workers with the same ability level. For example, if all unsuitable tasks for a worker are covered by machines (i.e., if $x_t=b$ ), then the worker always finds middle-skilled jobs with $q(x_t)=0$ . If only unsuitable tasks remain for a worker (i.e., if $x_t=b-(T-M-\Delta M_t)$ ), then the worker cannot find any middle-skilled jobs with $q(x_t)=1$ . Subsequently, the denominator in (26) denotes the number of available labor tasks; this is a technology factor. The increased automation decreases the number of available labor tasks. Therefore, whether the IMP of a worker declines or rises depends on the number of the worker’s unsuitable tasks that cannot be performed by machines, $x_t$ , and the increased automation, $\Delta M_t$ .

Regarding the domain of $x_t$ , the lower bound is the same among job seekers in which $\underline{x}= \max\!(0, b-(T-M-\Delta M_t) )$ . It holds that $ \bar{x}= \min\! (b, \Delta M_t )$ for young workers as well as mismatched old workers. However, when a worker is replaced by machines due to increased automation, then the worker will surely lose one of his or her suitable tasks. This implies $\bar{x}=\min\!( b, \Delta M_t-1)$ . That is, the upper bound of $x_t$ can differ between these mismatched and replaced workers. If increased automation exceeds the number of intrinsically unsuitable tasks: $ \Delta M_t-1 \gt b$ , then $\bar{x}=b$ holds. That is, if the number of newly automated tasks is large or if the number of unsuitable tasks is small, then we do not need to consider the difference in the domain of $x_t$ between these mismatched and replaced workers. Henceforth, we assume this.Footnote ⁷

The density function of $x_t$ measures the ratio of the number of worker types that belong to the $x_t$ -th group to the total number of worker types. Given the number of unsuitable tasks $b$ , the number of patterns for task suitability (i.e., the number of worker types) is calculated by the number of unsuitable task patterns:

(27) \begin{align} \left ( \begin{array}{c} T-M \\ b \\ \end{array} \right ). \end{align}

The number of worker types belonging to the $x_t$ -th group is calculated using two binomial coefficients:

(28) \begin{align} \left ( \begin{array}{c} \Delta M_t \\ x_t \\ \end{array} \right ) \left ( \begin{array}{c} T-M-\Delta M_t \\ b-x_t \end{array} \right ). \end{align}

Figure 4. IMP distribution of young and old job seekers in the period of increased automation.

The first coefficient measures the number of patterns under which the machines cover labor tasks that are unsuitable for workers. The second coefficient measures the number of patterns where the number of labor tasks includes the remaining unsuitable tasks. Dividing (28) by (27), we obtain the hypergeometric distribution as follows (Mood et al. Reference Mood, Graybill and Boes1974):

(29) \begin{align} f(x_t;\, \Delta M_t, b,T-M) \equiv \frac{ \left ( \begin{array}{c} \Delta M_t \\ x_t \\ \end{array} \right ) \left ( \begin{array}{c} T-M -\Delta M_t\\ b-x_t \\ \end{array} \right ) }{ \left ( \begin{array}{c} T-M \\ b \\ \end{array} \right ) }, \end{align}

where $ \sum _{x_t=\underline{x } }^{\bar{x } } f(x_t)=1$ . Equation (29) describes the probabilities for the success of $x_t$ in $b$ draws without replacement from $T-M$ ; it contains $\Delta M_t$ successful states. While each draw is either a success or a failure, a success (failure) indicates one less option in the unsuitable (suitable) tasks. We examine the mean and variance of $q(X_t)$ :

\begin{align*} \mu _{t} \equiv E(q (X_t;\, \Delta M_t, b,T-M) ) \ \ and \ \ \sigma _{t}^2 \equiv Var(q(X_t;\, \Delta M_t,b,T-M) ), \end{align*}

where $X_t$ is a random variable of $x_t$ .

Lemma 1 (Distribution of workers’ IMPs).

(30) \begin{align}\mu _{t} =q =\frac{b}{T-M}, \end{align}

(31) \begin{align} \sigma ^2_t = \sigma ^2 ( q, \Delta M_t ) = \frac{\Delta M_t}{T-M-\Delta M_t} \frac{ q (1- q) }{T-M-1}. \end{align}

The proof is shown in Appendix B.

In (30), the mean of workers’ IMPs under $\Delta M_t\gt 0$ is equal to their IMP under $\Delta M_t=0$ , which is noted in (2). While workers are equally exposed to an increase in automation ex-ante, they have different effects on their job mismatch possibilities ex-post. Whether the IMP of a worker rises or declines depends on the worker’s suitability for tasks. If machines perform a worker’s suitable tasks more (or less) than the worker’s unsuitable tasks, then the worker’s IMP rises (or declines), which implies more (or less) difficulty for obtaining a middle-skilled job. In Figure 4, we illustrate the IMP distribution.Footnote ⁸ The horizontal axis represents the IMP values, while the vertical axis represents the density function, $f(x_t)$ . We examine $x^{\prime}$ and $x^{\prime\prime}$ ( $x^{\prime} \gt x^{\prime\prime}$ ), which satisfy

\begin{align*} q(x^{\prime} )\lt q \lt q(x^{\prime\prime}) \ \ and \ \ f(x^{\prime})=f(x^{\prime\prime}). \end{align*}

In this figure, $q(x_t) f(x_t)$ indicates the ratio of workers who cannot find jobs to those who have $x_t$ . It holds that $q(x^{\prime}) f(x^{\prime}) \lt q(x^{\prime\prime}) f(x^{\prime\prime})$ because it is more difficult for a worker who has a large IMP value to find a job. The inidividual effects of the increased automation on their IMPs cancel each other out because of the two assumptions concerning workers’ suitability for tasks and the random matching between a firm and a worker. Therefore, the mean of their IMPs is never affected by the increased automation.

In (31), the IMPs of workers are diverse due to increased automation. The variance of their IMPs depends on the mean of their IMPs and the increased automation. A large increased automation implies a large variance of their IMPs: $\frac{\partial \sigma ^2_t }{\partial \Delta M_t}\gt 0$ . In addition, provided that $q \leq \frac{1}{2}$ holds, a large $q$ implies a large variance: $\frac{ \partial \sigma ^2_t}{\partial q} \gt 0$ . Therefore, we can demonstrate the variance effect of the increased automation on the IMPs of workers while preserving the mean of their IMPs.

We examine the decision of education investment by a worker who was born in period $t-1$ . If a worker works at a middle-skilled task in period $t-1$ , the probability of replacement by machines is $\frac{\Delta M_t}{T-M}$ . The probability of finding a middle-skilled job in period $t$ is $1-\mu _t$ . Hence, after receiving an education, a worker expects the following probabilities of employment in a middle- or low-skilled job:

(32) \begin{align} Pr (w_m, w_{m,t} ) & =(1-q ) \left [ \left ( 1- \frac{\Delta M_t}{T-M} \right ) + \frac{\Delta M_t}{T-M} (1-\mu _t) \right ],\nonumber\\ Pr (w_m, w_{l,t} ) & = (1-q) \frac{\Delta M_t}{T-M} \mu _t, \ \ Pr (w_l, w_{m,t} )=q(1-\mu _t), \ \ Pr (w_l, w_{l,t} )=q\mu _t, \end{align}

where, as shown in (30), $\mu _t=q$ . $w_m$ and $w_l$ are the wage rates of middle- and low-skilled labor in period $t-1$ , repsectively. The expected utility level of a worker after receiving an education is

\begin{align*} E \!\left(V\!\left( I^y_{t-1} -e, I^o_{t} \right) \right) = \sum _{I^y_{t-1}=w_m,w_l} \sum _{I^o_t=w_{m,t},w_{l,t}} Pr \!\left(I^y_{t-1},I^o_t \right) V\!\left(I^y_{t-1}-e, I^o_{t}\right). \end{align*}

The expected utility level of receiving an education declines because of the possibility of replacement by machines. The incentive-compatibility condition is:

(33) \begin{align} F_t =F \!\left( q, \mu _t, sp, sp_t, e, \Delta M_t \right) = E\!\left(V\!\left( I^y_{t-1} -e, I^o_{t} \right) \right) -V \!\left(w_l, w_{l,t}\right) \gt 0, \end{align}

where $V\!\left(w_l, w_{l,t}\right) = \frac{1}{1+\rho } \left( R w_l +w_{l,t} \right)$ . Due to the possibility of replacement by machines, the incentive for receiving an education declines: $\frac{\partial F }{\partial \Delta M_t } \lt 0$ .

4.2. Workers’ IMPs in the subsequent periods of increased automation

At the beginning of period $t+1$ , under no further increase in automation, workers are classified into the following three types. The first type is workers who continue working as middle-skilled workers. The number of these workers is $\frac{1}{2} (1-\mu _t)$ . The second type is old workers who have a mismatch in period $t$ while they once again seek middle-skilled jobs. Their number is $\frac{1}{2}\mu _t$ . The third type is young workers who are born in period $t+1$ with a population size $\frac{1}{2}$ .

We examine the IMPs of job seekers who were born in period $t$ . In period $t+1$ , the ratio of these job seekers to old workers is $\mu _t$ . The ratio of job seekers who have unsuitable tasks performed by machines $x_t$ to the old job seekers is

\begin{align*} \frac{1}{\mu _{t} } q(x_t;\, \Delta M_t) f(x_t;\, \Delta M_t). \end{align*}

Hence, the IMP mean of those job seekers is

\begin{align*} \mu _{t+1} = \sum _{x_t=\underline{x}}^{\bar{x}} \frac{1}{\mu _t} q(x_t;\, \Delta M_t) q(x_t;\, \Delta M_t) f(x_t;\, \Delta M_t). \end{align*}

Lemma 2 (IMP mean of old job seekers in period $t+1$ ).

(34) \begin{align} \mu _{t+1}=\mu ( q, \Delta M_t )= q + \frac{1}{q} \sigma ^2 \left(q, \Delta M_t \right). \end{align}

The proof is shown in Appendix B.

In period $t+1$ , workers who have relatively large IMP values owing to small $x_t$ account for a large portion of the job seekers. Those job seekers tend to face a high likelihood of job mismatch due to their large IMP values. Hence, in (34), the mean of their IMPs rises from $\mu _{ t}$ to $\mu _{ t+1}$ . Unlike in period $t$ , large and small IMP values do not cancel each other out in period $t+1$ due to the large portion of workers who have large IMP values. Therefore, the IMP mean of old job seekers in period $t+1$ positively depends on the IMP variance, $\sigma ^2_t$ .

In Figure 5, we illustrate the $t+1$ -th period IMP distribution of job seekers who were born in period $t$ . On the horizontal axis, the values of workers’ IMPs remain equal. That is, there is no change in $q(x^{\prime} )$ and $q(x^{\prime\prime})$ . The vertical axis measures $q(x_t) f(x_t)/\mu _t$ , which represents the ratio of job seekers who have $x_t$ to the total job seekers in period $t+1$ . As shown in Figure 4, it holds that $q (x^{\prime\prime}) f(x^{\prime\prime}) \gt q (x^{\prime}) f(x^{\prime})$ because $q(x^{\prime\prime})\gt q(x^{\prime})$ . Moreover, in period $t+1$ , the ratio of unemployed workers to job seekers is larger for the workers having $x^{\prime\prime}$ compared to workers with $x^{\prime}$ : $q^2(x^{\prime\prime}) f(x^{\prime\prime}) \gt q^2 (x^{\prime}) f(x^{\prime}).$ Consequently, the IMP mean in period $t+1$ exceeds that in period $t$ because a higher number of workers who have larger IMPs remain job seekers.

Figure 5. IMP distribution of old job seekers in the subsequent period of increased automation.

Corollary 1 (A comparison between $\mu _t$ and $\mu _{t+1}$ ).

(35) \begin{align} \mu _{t} \lt \mu _{t+1}. \end{align}

We examine the decision of education investment by a worker who was born in period $t$ . Because they have job seeking after increased automation, they can avoid automated tasks. However, they do not know which tasks will be automated when receiving an education. The probabilities of employment in a middle- or low-skilled job are as follows:

(36) \begin{align} Pr \!\left(w_{m,t}, w_{m,t} \right) =\left(1-\mu _t \right), \ \ Pr \!\left(w_{m,t}, w_{l,t} \right) = 0,\nonumber \\ Pr \!\left(w_{l,t},w_{m,t}\right) =\mu _t \left(1-\mu _{t+1}\right), \ \ Pr \!\left(w_{l,t},w_{l,t}\right)=\mu _t \mu _{t+1}. \end{align}

Note that there is no change in the wage rates of middle- and low-skilled labor in period $t+1$ due to no technical change. The expected utility level of receiving an education is

\begin{align*} E \!\left( V\!\left( I^y_t -e, I^o_{t+1} \right) \right) = \sum _{I^y_t=w_{m,t},w_{l,t}} \sum _{I^o_{t+1}=w_{m,t}, w_{l,t}} Pr \!\left(I^y_t,I^o_{t+1}\right) V\!\left(I^y_{t}-e, I^o_{t+1}\right). \end{align*}

A large $\mu _{t+1}$ implies a low level of expected utility. The incentive-compatibility condition for education investment is

(37) \begin{align} F_{t+1} =F \!\left( \mu _t, \mu _{ t+1}, sp_t, e \right) =E \!\left(V\!\left( I^y_t -e, I^o_{t+1} \right) \right) -V \!\left(w_{l,t}, w_{l,t}\right) \gt 0. \end{align}

Due to the variance of the IMPs of workers, the incentive for receiving an education declines: $\frac{\partial F_{t+1}}{\partial \mu _{t+1}} \lt 0$ . That is, the incentive for receiving education declines due to increased automation: $\frac{\partial F_{t+1}}{\partial \Delta M_t} \lt 0$ .

Finally, we consider a worker who was born in period $t+1$ . Under the assumption of no further increase in automation, they know the available labor tasks $T-M-\Delta M_t$ when they decide on an education investment. Because the number of these labor tasks declines due to increased automation, it may be natural to consider a decline in the unsuitable tasks of a worker. Therefore, for simplicity, we assume that the IMP value of a worker remains $q$ , which is equal to

(38) \begin{align} q = \frac{ \tilde{b}}{T-M-\Delta M_t}, \end{align}

where $\tilde{b}$ is the number of the worker’s unsuitable tasks in which $\tilde{b}\lt b$ . Note that $q=\frac{b}{T-M}$ . The probabilities of employment in a middle- or low-skilled job are the same as those in (7). The expected utility under receiving an education is

\begin{align*} E \!\left(V\!\left( I^y_{t+1} -e, I^o_{t+2} \right) \right) = \sum _{I^y_{t+1}=w_{m,t},w_{l,t}} \sum _{I^o_{t+2}=w_{m,t},w_{l,t}} Pr \!\left(I^y_{t+1}, I^o_{t+2}\right) V\!\left(I^y_{t+1}-e, I^o_{t+2}\right). \end{align*}

The incentive-compatibility condition is

\begin{align*} F_{t+2} =F \!\left( q, sp_t, e \right) =E \!\left(V\!\left( I^y_{t+1} -e, I^o_{t+2} \right) \right) -V\!\left(w_{l,t},w_{l,t}\right) \gt 0. \end{align*}

4.3. Equilibrium in the period of increased automation

As a result of increased automation, the price of labor-use intermediate goods rises:

(39) \begin{align} \ln p_{m,t} = \frac{1}{T-M-\Delta M_t} \left [ (M+\Delta M_t) \ln \frac{\gamma }{R} -T\ln T \right ]. \end{align}

As shown in (18), the wage rate of middle-skilled labor also rises. In addition, the share of labor incomes declines with the increased automation.Footnote ⁹

To simplify the analysis, we assume no change in the skill premium:

(40) \begin{align} \Delta sp_t=0. \end{align}

Equations (18) and (40) imply that $ \lambda _m \Delta p_{m,t}=\Delta A_{l,t}$ . Under these assumptions, we can focus on the effect of increased automation on job mismatch probabilities of workers.Footnote ¹⁰

We explore the probability that a labor-use firm engages in production. In period $t$ , three types of job seekers exist: old workers who are replaced by machines, old workers who cannot find middle-skilled jobs in period $t-1$ , and young workers who seek middle-skilled jobs in period $t$ . The IMP mean of these three types is $\mu _t$ . The mean of the IMPs of job seekers, $\mu ^a_t$ is equal to $\mu _t$ . Therefore, the probability that a labor-use firm engages in production is

(41) \begin{align} \phi (\theta _t ;\, \mu _t) =\frac{1-\mu ^a_t}{\theta _t} =\frac{1-\mu _t}{\theta _t}. \end{align}

Note that $\mu _t=q$ . Under assumption (40), the market tightness in equilibrium is the same as with (19).

As a result of job seeking in period $t$ , the number of middle-skilled workers is

\begin{align*} L_{m,t}=L^o_{m,t} +L^y_{m,t}, \end{align*}

where $L^o_{m,t}$ and $L^y_{m,t}$ are the numbers of old and young middle-skilled workers, respectively. There are three types of old middle-skilled workers:

(42) \begin{align} L^o_{m,t} = \frac{1}{2} (1-q) \left (1 - \frac{\Delta M_t}{T-M} \right ) + \frac{1}{2}(1-q) \frac{\Delta M_t}{T-M} (1-\mu _t) + \frac{1}{2} q (1-\mu _t), \end{align}

which reduces to

\begin{align*} L^o_{m,t} = L^o_m \left(q, \Delta M_t \right) = \frac{1}{2} \left [ \left(1-q^2\right) - q(1-q) \frac{\Delta M_t}{T-M} \right ]. \end{align*}

The first term of the RHS of (42) represents workers who are not replaced by machines in period $t$ , while the second term of the RHS includes workers who are replaced by machines but find additional middle-skilled jobs. The third term of the RHS includes workers who have job mismatch in period $t-1$ but find middle-skilled jobs in period $t$ . The number of young middle-skilled workers is

\begin{align*} L^y_{m,t}=L^y_m (q)= \frac{1}{2} (1-\mu _t ) =\frac{1}{2} (1-q). \end{align*}

Similarly, the number of low-skilled workers is

\begin{align*} L_{l,t}= L^o_{l,t}+L^y_{l,t}, \end{align*}

where

\begin{align*} L^o_{l,t} = L^o_l \left(q, \Delta M_t \right) =\frac{1}{2} \left [ q^2+ q(1-q) \frac{\Delta M_t}{T-M} \right ] \ \ and \ \ L^y_{l,t}=L^y_l (q)=\frac{1}{2} q. \end{align*}

Consequently, due to the replacement of middle-skilled workers by machines, the number of middle-skilled workers declines, while that of low-skilled workers (i.e., underemployed workers) rises:

\begin{align*} L_{m,t}\lt L_m \ \ and \ \ L_{l,t} \gt L_l. \end{align*}

4.4. Equilibrium in the subsequent periods of increased automation

We consider the probability that a labor-use firm engages in production in period $t+1$ . In that period, there are two types of job seekers. The first type is old workers who cannot find middle-skilled jobs in period $t$ . The mean of their IMPs is $\mu _{t+1}$ with a size of $\frac{1}{2}\mu _t$ . The second type is young workers while they have the same IMP, $q$ with a size of $\frac{1}{2}$ . Thus, the mean of the IMPs of these two types is

(43) \begin{align} \mu ^a_{t+1} = \frac{ (1/2)q+(1/2)\mu _t\mu _{t+1}}{ (1/2) + (1/2) \mu _t}. \end{align}

The denominator of the RHS of this equation shows the number of job seekers in period $t+1$ , while the numerator of the RHS shows the expected value of the number of job seekers who are unsuitable for a task. Because workers who have large IMPs remain job seekers, the mean of the IMPs of job seekers rises:

\begin{align*} \mu ^a_{t+1} =q \frac{1+\mu (q, \Delta M_t) }{1+q} \gt q. \end{align*}

Note that $\mu _{t+1}=\mu (q,\Delta M_t)$ . Consequently, the probability that a firm engages in production in period $t+1$ is

(44) \begin{align} \phi \!\left(\theta _{t+1} ;\, \mu ^a_{t+1} \right) =\frac{1-\mu ^a_{t+1} }{\theta _{t+1}}, \end{align}

which implies a decline in the market tightness:

\begin{align*} \hat{\theta }_{t+1}= \left(1-\mu ^a_{t+1}\right) \frac{1-\beta }{\beta } \frac{ sp}{z}. \end{align*}

As a result of job seeking in period $t+1$ , the number of middle-skilled workers is

\begin{align*} L_{m,t+1}=L^o_{m,t+1} +L^y_{m,t+1}. \end{align*}

Using (30) and (34), the old middle-skilled workers, including workers who find middle-skilled jobs in period $t$ and find these jobs in period $t+1$ but not in period $t$ , are as follows:

(45) \begin{align} L^o_{m,t+1}= \frac{1}{2} (1-\mu _t ) + \frac{1}{2} \mu _t (1-\mu _{t+1}), \end{align}

which reduces to

\begin{align*} L^o_{m,t+1}=L^o_m \left(q, \Delta M_t\right) = \frac{1}{2} \left( 1- q^2 - \sigma ^2_t \right). \end{align*}

Due to the variance of the IMPs of workers, the number of old middle-skilled workers declines. The number of young middle-skilled workers is

\begin{align*} L^y_{m,t+1}=L^y_m (q)=\frac{1}{2}(1-q). \end{align*}

Similarly, the number of low-skilled workers is

\begin{align*} L_{l,t+1}= L^o_{l,t+1}+L^y_{l,t+1}, \end{align*}

where

\begin{align*} L^o_{l,t+1} = L^o_l \left(q, \Delta M_t\right)= \frac{1}{2} \mu _t \mu _{t+1} =\frac{1}{2} \left( q^2 + \sigma ^2_t \right) \ \ and \ \ L^y_{l,t+1}=L^y_l (q)= \frac{1}{2} q. \end{align*}

Compared to the steady-state level, the number of middle-skilled workers declines, while that of low-skilled workers (i.e., underemployed workers) rises:

\begin{align*} L_{m,t+1}\lt L_m \ \ and \ \ L_{l,t+1}\gt L_l. \end{align*}

Finally, under assumption (38) with the assumption of no further increase in automation, the number of middle-skilled (low-skilled) workers in period $t+2$ is the same as that in the steady-state level under no increased automation.