2.1 Description

We present a continuous-time model. The population of workers is normalized to 1, and there is a large supply of potential firms. All economic agents are assumed to be risk-neutral and have a common discount rate, $\rho$ . A production unit is a worker–firm pair, and firms use labor as the sole input for production. On-the-job search is not considered. Therefore, only unemployed workers engage in job-seeking activities. We concentrate on the steady state of the economy.

2.2 Bellman Equations

2.2.1 Firms

The asset values of employers in the steady state are represented as follows:

• $V$ : Value to the firm of a vacant job

• $J^{nt}_{Ti}(a)$ : Value to the firm of a temporary job with no training provided

• $J^{nv}_{Ti}(a)$ : Value to the firm of a temporary job in the novice period

• $J_{T}^t(a)$ : Value to the firm of a temporary job with a fully trained worker

• $J^{nv}_{P}(a)$ : Value to the firm of a permanent job in the novice period

• $J_{P}(a)$ : Value to the firm of a permanent job with a fully trained worker

Temporary jobs are classified into three cases: novice (abbreviated as $nv)$ , trained (abbreviated as $t)$ , and untrained (abbreviated as $nt)$ . Let us denote $w^{nt}_{Ti}(a), \, w^{nv}_{Ti}(a)$ , and $w^t_{T}(a)$ as the wages that correspond to each type of temporary job. Similarly, $w^{nv}_P(a),\, w_P(a)$ are the corresponding wages for a permanent job. Further, the states of untrained and novice periods are divided into two cases. The first case represents the situation in which match-specific productivity in a temporary job is lower than the hiring threshold for permanent jobs; therefore, the match is a dead-end one. By contrast, the second case represents the situation in which a temporary job can be converted into a permanent form (i.e., match-specific productivity is retained after conversion). The subscript $i$ in the above equations takes either $0$ or $1$ : $i = 0$ corresponds to the former case, and $i = 1$ corresponds to the latter case. We finally note that “untrained” workers are employed workers who do not receive training opportunities. In other words, employed workers who have received training but have not completed it are not called “untrained” in this study. The latter type of worker arises because the training outcome is uncertain.

The value to the firm of a vacant job satisfies the following Bellman equation:

(1) \begin{equation} \rho \, V = - \,c + q(\theta ) \bigg [ \int \max \!\left\{ J^{nt}_{Ti}(a),\, J^{nv}_{Ti}(a),\, J^{nv}_{P}(a), \, V \right\}\! dG(a) - V \bigg ], \end{equation}

where $c$ is the maintenance cost. When a firm with a vacant job meets a job seeker, a match is formed only when match-specific productivity exceeds the hiring threshold for temporary jobs. Furthermore, the firm must decide on (i) the form of contract and (ii) whether to provide training if the contract is temporary.

The value to the firm of a temporary job with no training for each $i$ is represented by:

(2) \begin{equation} \rho \,J^{nt}_{Ti}(a) = a - w^{nt}_{Ti}(a) + \lambda _T (V - J^{nt}_{Ti}(a)) + \delta \,[\!\max \{ J^{nv}_P(a),\,V \} - J^{nt}_{Ti}(a)]. \end{equation}

As training is not provided in this case, the flow productivity coincides with the realized value of match-specific productivity. At the arrival rate $\lambda _T$ , any temporary contract becomes unproductive and is terminated (the job becomes vacant). In addition, at the rate of $\delta$ , an employer with a temporary contract must decide whether the contract is terminated or converted into a permanent one. This decision depends on whether the match-specific productivity is greater than the threshold for forming a permanent job.

For each $i$ , the value to the firm of a temporary job in the novice period and the value when training for each $i$ is completed are represented by:

(3) \begin{align} \rho \,J^{nv}_{Ti}(a) &= a - w^{nv}_{Ti}(a) - \gamma _T + \lambda _T \,(V - J^{nv}_{Ti}(a)) + \delta \,\left[ \max\! \left\{ J^{nv}_P(a),\,V \right\} - J^{nv}_{Ti}(a)\right] \notag \\[2pt] & + \xi \left[ J^t_T(a) - J^{nv}_{Ti}(a)\right],\\[-9pt] \notag \end{align}

(4) \begin{align} \rho \, J^{t}_T(a) &= (1 + h) \,a - w^t_T(a) + \lambda _T \,(V - J^t_T(a)) + \delta \,(J_P(a) - J^t_T(a)). \end{align}

In equation (3), flow productivity increases to $(1 + h)\,a$ at a rate of $\xi$ . Otherwise, productivity remains unchanged, and the firm needs to pay the training cost again. In equation (4), flow productivity reflects the success of firm-provided training. Similar to the case of no training, the term in the max operator depends on the realization of match-specific productivity. As shown later, every fully trained temporary worker is promoted to permanent employment at the rate of $\delta$ . Even if training is provided, however, the timing of conversion may arrive at the rate of $\delta$ before the training result is realized. In this case, a match leads to separation, and the incurred training costs are wasted.

The value to the firm of a permanent job in the novice period and the value when training is completed are represented by:

(5) \begin{align} \rho \, J^{nv}_{P}(a) & = a - \gamma _P - w^{nv}_P(a) + \lambda _P (V - J^{nv}_P(a)) + \xi (J_P(a) - J^{nv}_P(a)),\\[-7pt] \notag \end{align}

(6) \begin{align} \rho \, J_{P}(a) & = (1 + h) \,a - w_{P}(a) + \lambda _P \,(V - F - J_P(a)). \end{align}

In a permanent job in the novice period, the job is not covered by employment protection, and only firms with a fully trained permanent worker need to pay termination costs when a negative economic shock occurs.

2.2.2 Workers

Let us denote the asset value of an unemployed worker as $U$ . During search periods, unemployed workers earn the common return $b$ that may represent the value of leisure or home production. The asset values of an employed worker in each type of temporary job are denoted by $E^{nt}_{Ti}(a),$ $\, E^{nv}_{Ti}(a)$ , and $E^t_{T}(a)$ . The asset values of an employed worker in a permanent job are denoted by $E^{nv}_P(a)$ and $E_P(a)$ .

The above-defined asset values for each $i$ satisfy

(7) \begin{align} \rho \, U & = b + \theta \,q(\theta ) \bigg [ \int \! \max \{ E^{nt}_{Ti}(a),\, E^{nv}_{Ti}(a),\, E^{nv}_{P}(a), \,U \} \, dG(a) - U \bigg ],\\[-9pt] \notag \end{align}

(8) \begin{align} \rho \,E^{nt}_{Ti}(a) & = w^{nt}_{Ti}(a) + \lambda _T \,(U - E^{nt}_{Ti}(a)) + \delta \,\left[\!\max \{ E^{nv}_P(a),\,U \} - E^{nt}_{Ti}(a)\right], \\[-9pt] \notag\end{align}

(9) \begin{align} \rho \,E^{nv}_{Ti}(a) & = w^{nv}_{Ti}(a) + \lambda _T \,(U - E^{nv}_{Ti}(a)) + \delta \,\left[\! \max \{ E^{nv}_P(a),\,U \} - E^{nv}_{Ti}(a)\right] \notag\\ &\quad + \xi \,\left[ E^t_T(a) - E^{nv}_{Ti}(a)\right], \\[-9pt] \notag\end{align}

(10) \begin{align} \rho \, E^{t}_T(a) & = w^t_T(a) + \lambda _T \,(U - E^t_T(a)) + \delta (E_P(a) - E^t_T(a)), \\[-9pt] \notag\end{align}

(11) \begin{align} \rho \,E^{nv}_P(a) & = w^{nv}_P(a) + \lambda _P \,(U - E^{nv}_P(a)) + \xi (E_P(a) - E^{nv}_P(a)), \\[-9pt] \notag\end{align}

(12) \begin{align} \rho \, E_{P}(a) & = w_{P}(a) + \lambda _P (U - E_P(a)). \end{align}

2.3 Surpluses

For notational convenience, we introduce the following notations in later sections:

\begin{align*} \phi _T \equiv \rho + \lambda _T + \delta, \,\, \phi _P \equiv \rho + \lambda _P, \,\, \zeta = \lambda _T + \delta + \xi . \end{align*}

2.3.1 Surplus sharing rules and the definition of surpluses

Since wages are determined by solving the Nash bargaining problem, both an individual worker and an individual firm split the surplus generated by forming a match.Footnote ¹⁴ The surplus sharing rules are represented by:

(13) \begin{align} \beta \,J^{j}_{Ti}(a)& = (1 - \beta ) (E^{j}_{Ti}(a) - U), \, \beta \,J^{t}_{T}(a) = (1 - \beta ) (E^{t}_{T}(a) - U),\\[-6pt]\notag \end{align}

(14) \begin{align} \beta \,J^{nv}_{P}(a)& = (1 - \beta )(E^{nv}_{P}(a) - U), \, \beta \,(J_{P}(a) + F) = (1 - \beta ) (E_{P}(a) - U), \end{align}

where we set the firms’ threat point (i.e., $V)$ to zero and $j$ takes $nt$ or $nv$ .

Following the standard Nash bargaining rule, firms’ decisions are considered to be based on the surplus generated by forming a match. The surplus of a match associated with a temporary job and no training is defined by $S^{nt}_{Ti}(a) \equiv J^{nt}_{Ti}(a) + E^{nt}_{Ti}(a) - U$ :

(15) \begin{align} S^{nt}_{Ti}(a) = \frac{a - \rho U + \delta \, \max \!\left\{ S^{nv}_P(a), \,0 \right\} }{\phi _T}, \end{align}

where $\max \{ S^{nv}_P(a),\, 0 \}$ is equal to zero for $S^{nt}_{T0}(a)$ .

The surpluses of a match associated with a temporary job in the novice period and with a temporary job and a fully trained worker are defined by $S^{nv}_{Ti}(a) \equiv J^{nv}_{Ti}(a) + E_{Ti}^{nv}(a) - U$ and $S^t_{T}(a) \equiv J^t_{T}(a) + E^t_{T}(a) - U$ and are represented as follows, respectively:

(16) \begin{align} S^{nv}_{Ti}(a) &= \frac{a - \rho \,U - \gamma _T + \delta \, \max\! \left\{ S^{nv}_P(a),\,0 \right\} + \xi \,S^t_T(a) }{\phi _T + \xi }, \end{align}

(17) \begin{align} S^t_{T}(a) &= \frac{ (1 + h) \,a - \rho U + \delta \,S_P(a) - \delta \,F }{ \phi _T},\quad\qquad\qquad \end{align}

where $\max \{ S^{nv}_P(a),\, 0 \}$ is equal to zero for $S^{nv}_{T0}(a)$ .

The surpluses of a match associated with a permanent job in the novice period and with a permanent job under completed training are denoted by $S^{nv}_{P}(a) \equiv J^{nv}_{P}(a) + E^{nv}_{P}(a) - U$ and $S_{P}(a) \equiv J_{P}(a) + F + E_{P}(a) - U$ , respectively:

(18) \begin{align} S^{nv}_P(a) &= \frac{ (\phi _P + \xi ) (a - \rho U) - \phi _P \, \gamma _P + \xi \, h \,a - \xi \,\lambda _P F }{\phi _P (\phi _P + \xi )}, \end{align}

(19) \begin{align} S_P(a) &= \frac{ (1 + h) \,a - \rho U + \rho F}{ \phi _P }. \qquad\qquad\ \, \qquad\qquad\end{align}

3.1 Thresholds for Hiring in Temporary Jobs and for Forming a Match in a Permanent Position

If the realized match-specific productivity is significantly low, a worker–firm pair does not form a match. Let us denote $a_{Th}$ as the threshold for hiring in temporary jobs and $a_{Ph}$ as the threshold for transforming a match with a temporary job to a permanent job. When deciding on the type of contract of an initially formed match, the employer’s decision is based on the relative profitability between the two types of contracts, which is characterized in the next subsection. Employers hire a worker on each type of contract if forming a job match generates a nonnegative profit. Considering that the surplus sharing rule is applied, an employer’s expected profit is a constant fraction of the corresponding match surplus. Based on (15) and (18), $a_{Th}$ and $a_{Ph}$ are determined by $S^{nt}_{T0}(a_{Th}) = 0$ and $S^{nv}_P(a_{Ph}) = 0$ , respectively. Note that this hiring threshold for permanent jobs is also used as the conversion threshold. These two thresholds are represented by:

(20) \begin{align} a_{Th} &= \rho \,U, \end{align}

(21) \begin{align} a_{Ph} &= \frac{ (\phi _P + \xi ) \,\rho \,U + \phi _P \,\gamma _P + \xi \,\lambda _P \, F }{ \phi _P + \xi \,(1 + h) }. \end{align}

Since it is reasonable to consider that employers with a low-productivity match will offer a temporary contract to a worker, $a_{Th}$ should be lower than $a_{Ph}$ (otherwise, all temporary workers are candidates for a permanent position, which would result in too many conversions from temporary to permanent employment). This is ensured if the training costs in permanent jobs and termination costs are sufficiently high to satisfy the following condition:

(22) \begin{align} \rho \,U \lt \frac{ \phi _P \, \gamma _P + \xi \,\lambda _P \,F }{\xi \,h}. \end{align}

3.2 Determination of Thresholds for Training and Choosing a Type of Contract

Since both training costs and the rate of increase in productivity are assumed to be constant, a match with low productivity may yield an insufficient return from training. Furthermore, the assumption that the survival rate of temporary jobs is lower than the survival rate of permanent jobs makes employers reluctant to invest in training because it does not necessarily raise productivity, and training costs may therefore be wasted. In some cases, every employer with a temporary job chooses not to invest in training. By contrast, a portion of employers with a temporary job may provide training. These situations are distinguished based on the relationship between the thresholds for training and for choosing a type of contract. Let us denote the former as $a_{Tt}$ and the latter as $a_{Pc}$ .

First, we suppose that $a_{Ph}$ is higher than $a_{Tt}$ . This is the case in which a portion of novice temporary workers are not promoted to a permanent form until training is completed. To identify the thresholds $a_{Tt}$ and $a_{Pc}$ , the graphical features of match surpluses and their relationships should be explained (see Figure 2). First, straight lines representing the surplus of a match with a temporary contract have a kink at $a = a_{Ph}$ , and the lines become steeper for $a \ge a_{Ph}$ : $d \,S^{nt}_{T0}(a)/ d \,a \lt d \,S^{nt}_{T1}(a)/ d \,a$ , and $d \,S^{nv}_{T0}(a)/ d \,a \lt d \,S^{nv}_{T1}(a)/ d \,a$ . As a temporary worker in a match with $a \ge a_{Ph}$ has the prospect for conversion into a permanent form, the term $S^{nv}_P(a)$ in (15) and (16) becomes strictly positive for $a \gt a_{Ph}$ (note that $ S^{nv}_P(a_{Ph}) = 0$ by definition). Using the fact that each surplus is increasing in $a$ , the impact through $S^{nv}_P(a)$ makes $S^{nt}_{T1}(a)$ and $S^{nv}_{T1}(a)$ steeper than $S^{nt}_{T0}(a)$ and $S^{nv}_{T0}(a)$ , respectively. Second, the slope of the match surplus with a temporary contract and training provision is steeper than the slope of the surplus with no training for both $a \lt a_{Ph}$ and $a_{Ph} \le a$ :

\begin{align*} &\frac{d \,S^{nv}_{T0}(a) }{d \, a} - \frac{d \,S^{nt}_{T0}(a)}{d \,a} = \frac{\xi \left[\delta + h \, (\phi _P + \delta )\right] }{\phi _P \, \phi _T \,(\phi _T + \xi )} \gt 0, \\[3pt] &\frac{d \, S^{nv}_{T1}(a)}{d \,a} - \frac{d \, S^{nt}_{T1}(a)}{d \, a} = \frac{\xi \, h (\phi _P + \xi + \delta ) }{\phi _T (\phi _P + \xi )(\phi _T + \xi )} \gt 0. \end{align*}

Figure 2. Determination of each threshold $(a_{Tt} \lt a_{Ph})$ : (I) temporary jobs with no training; (II) temporary jobs with training provision and no prospect of conversion; (III) temporary jobs with training provision and prospect of conversion; and (IV) permanent jobs.

This reflects the fact that for $a \ge a_{Tt}$ , the term $\xi \,S^t_T(a)$ becomes a component of the surplus $S^{nv}_{Ti}(a)$ , as shown in (16) because of training provision, leading to a larger slope of $S^{nv}_{T0}(a)$ and $S^{nv}_{T1}(a)$ compared with $S^{nt}_{T0}(a)$ and $S^{nt}_{T1}(a)$ , respectively. Third, the slope of the match surplus with a permanent contract is steeper than the slope of the surplus with a temporary contract and training provision:

\begin{align*} \frac{d \,S^{nv}_P(a)}{d \,a} - \frac{d \,S^{nv}_{T1}(a)}{d \,a} &= \frac{ \phi _P \,\phi _T \, \Delta \,\lambda + \xi (1 + h) \left[\phi _T (\rho + \lambda _T + \xi ) - (\phi _P + \delta )(\phi _P + \xi )\right]}{ \phi _P \,\phi _T \,(\phi _P + \xi )( \phi _T + \xi ) } \gt 0, \end{align*}

where $\phi _T \gt \phi _P + \delta, \, \rho + \lambda _T + \xi \gt \phi _P + \xi$ and $\Delta \,\lambda \equiv \lambda _T - \lambda _P$ . As the separation rate in temporary jobs is assumed to be higher than the rate in permanent jobs (i.e., $\lambda _T \gt \lambda _P)$ , the match surplus for temporary jobs is more greatly discounted, leading to a smaller slope of $S^{nv}_{T1}(a)$ .

The threshold $a_{Tt}$ is determined by the intersection of the two surpluses $S^{nt}_{T0}(a)$ and $S^{nv}_{T0}(a)$ and is located on the left-hand side of $a_{Ph}$ in this case, as depicted in Figure 2. At $a = a_{Tt}$ , employers are indifferent between providing training and no training. Based on (15)−(17), $a_{Tt}$ is obtained by solving $S^{nt}_{T0}(a_{Tt}) = S^{nv}_{T0}(a_{Tt})$ , which is represented by:

(23) \begin{align} a_{Tt} = \frac{\xi \,\delta \,\rho \,U + \phi _P \,\phi _T \, \gamma _T + \xi \, \delta \, \lambda _P \,F }{ \xi \, [ h \,\phi _P + \delta \,(1 + h) ] }. \end{align}

As a situation in which $a_{Tt} \lt a_{Ph}$ is considered (the condition ensuring it is stated below), $\max \{ S^{nv}_P(a),\,0 \}$ is set to be zero in the computation of (23).

It is reasonable to consider that $a_{Tt}$ is higher than $a_{Th}$ ; otherwise, all temporary workers receive firm-provided training, which is not consistent with the real situation of the labor market. It follows from (20) and (23) that $a_{Th} \lt a_{Tt}$ is satisfied under the following condition:

(24) \begin{align} \rho \,U \lt \frac{ \phi _P \,\phi _T \,\gamma _T + \xi \,\delta \,\lambda _P \,F }{ \xi \,h \, (\phi _P + \delta ) }. \end{align}

(24) is likely to be satisfied if the separation rate of temporary jobs, the cost of providing training to a temporary worker, and termination costs for permanent jobs are sufficiently high.

Comparing $a_{Tt}$ with $a_{Ph}$ , the latter is higher than the former if and only if the following condition is satisfied:

(25) \begin{align} \frac{ \phi _T \,[\phi _P + \xi \,(1 + h)] \, \gamma _T - \xi \, [ h \, \phi _P + \delta \,(1 + h) ] \,\gamma _P - \xi \,\lambda _P \,(\xi \,h - \delta ) \, F }{ \xi \,h \,(\phi _P + \delta + \xi ) } \,\lt \, \rho \,U. \end{align}

This is likely to hold if the costs of providing training for a permanent worker (and termination costs) are significantly higher than those costs for a temporary worker. Because higher $\gamma _P$ and $F$ increase the cost of having permanent employment, $a_{Ph}$ becomes higher and (25) is likely to be satisfied. If (22) and (25) are simultaneously satisfied, temporary workers in a match with productivity $a \in [a_{Tt}, \,a_{Ph})$ are never converted into a permanent position as long as they are not fully trained.

Regarding an employer’s choice of contract type, suppose that there exists an intersection between $S^{nv}_P(a)$ and $S^{nv}_{T1}(a)$ in the interval $(a_{Ph},\,\bar{a})$ , ensuring the existence of the threshold $a_{Pc}$ within this interval. As the slope of $S^{nv}_P(a)$ is steeper than the slope of $S^{nv}_{T1}(a)$ , these two lines intersect if $\bar{a}$ is sufficiently large. Then, a newly formed job match with $a \in [a_{Pc},\,\bar{a}]$ takes a permanent form. Using (16)−(19), solving $S^{nv}_{T1}(a_{Pc}) = S^{nv}_P(a_{Pc})$ for $a_{Pc}$ yields

(26) \begin{align} a_{Pc} &= \frac{ \Delta \, \lambda \,[\phi _P \,\phi _T + \xi \,(\phi _T + \phi _P + \xi )] \,\rho \,U + \phi _P \,\phi _T \,(\rho + \lambda _T + \xi ) \, \gamma _P - \phi _P \,\phi _T \,(\phi _P + \xi ) \, \gamma _T }{ \Delta \, \lambda \,[\phi _P \,\phi _T + \xi \,(1 + h)(\phi _T + \phi _P + \xi )]} \notag \\[3pt] &\quad + \frac{ \xi \,\lambda _P \,[ (\rho + \lambda _T)(\rho + \lambda _T + \xi ) + \delta \,\Delta \,\lambda ] \,F }{ \Delta \, \lambda \,[\phi _P \,\phi _T + \xi \,(1 + h)(\phi _T + \phi _P + \xi )] }. \end{align}

The following proposition summarizes the conditions in which a situation of Figure 2 is realized.

If a temporary worker has match-specific productivity $a \in [a_{Ph},\,a_{Pc})$ , they receive training opportunities and experience an increase in productivity at the rate of $\xi$ . As an activation of the nonrenewal clause forces employers to decide whether to dismiss a temporary worker or to promote a permanent form after the expiration of a certain period, highly productive temporary workers who have not completed training are considered for promotion to a permanent form.

Second, we assume that (22) and (24) are satisfied, but (25) is not satisfied, indicating that $a_{Th} \lt a_{Ph} \lt a_{Tt}$ holds. In this case, a new threshold for training provision in temporary jobs is necessary because $a_{Tt}$ is defined based on the premise of $a_{Tt} \lt a_{Ph}$ . If the surplus $S^{nv}_{T1}(a)$ is sufficiently large such that the threshold for training exists in the interval $(a_{Ph},\,a_{Pc})$ , which is determined by the intersection between the two surpluses $S^{nt}_{T1}(a)$ and $S^{nv}_{T1}(a)$ , both novice (i.e., training is provided, but not completed) workers and a portion of untrained (i.e., training is not provided) temporary workers can be converted into permanent employment (see the left portion of Figure 3). Note that these surpluses are represented by:

Figure 3. (Left) A portion of temporary workers are trained; (right) no temporary workers are trained: (I) temporary jobs with no training; (II’) temporary jobs with no training and the prospect of conversion; (III) temporary jobs with training provision and the prospect of conversion; and (IV) permanent jobs.

\begin{align*} S^{nt}_{T1}(a) = \frac{a - \rho \,U + \delta \,S_P^{nv}(a)}{\phi _T}, \quad S^{nv}_{T1}(a) = \frac{ a - \rho \,U - \gamma _T + \delta \,S_P^{nv}(a) + \xi \,S^t_T(a) }{\phi _T + \xi }. \end{align*}

For a sufficiently high value of $a$ belonging to $[a_{Ph},\,a_{Pc})$ , even untrained temporary workers have a chance of being converted into a permanent position. Using (15)–(19), let us define the threshold for training in this case as $a^L_{Tt}$ which satisfies $S^{nt}_{T1}(a^L_{Tt}) = S^{nv}_{T1}(a^L_{Tt})$ :

(27) \begin{align} a^L_{Tt} = \frac{ \xi \, \delta \,\lambda _P \,F + \phi _T \,(\phi _P + \xi ) \,\gamma _T - \xi \, \delta \, \gamma _P }{ \xi \,h \, (\phi _P + \xi + \delta ) }. \end{align}

Unlike the case of $a_{Tt}$ , the expression of $a^L_{Tt}$ depends on the costs of providing training to a permanent worker, reflecting the possibility that temporary workers who have not completed training may be promoted to permanent employment.

Note that the threshold for choosing a form of contract in this case is given by (26). A main difference between Figure 2 and the left portion of Figure 3 is that in the latter case there are temporary workers who have the prospect of conversion but do not receive training instead of temporary workers who receive training but do not have the prospect of conversion. The former workers belong to a match with productivity $a \in [a_{Ph},\,a^L_{Tt})$ (denoted by (II $^{\prime}$ ) in Figure 3). The latter workers belong to a match with productivity $a \in [a_{Tt},\,a_{Ph})$ (denoted by (II) in Figure 2).

If condition (25) that ensures $a_{Tt} \lt a_{Ph}$ does not hold, there is another case in which all temporary workers never receive training opportunities. This situation arises if training temporary workers is less profitable, because training costs $(\gamma _T)$ are relatively high in comparison with the outcome of training $(h)$ , making the surplus $S^{nv}_{T1}(a)$ smaller than the surplus $S^{nt}_{T1}(a)$ in the most (or all) area of $a$ . In this case, the threshold for choosing a form of contract is determined by the intersection of the surpluses $S^{nt}_{T1}(a)$ and $S^{nv}_P(a)$ , as depicted in the right portion of Figure 3. Based on (15) and (18), the threshold denoted by $a^N_{Pc}$ , which satisfies $S^{nt}_{T1}(a^N_{Pc}) = S^{nv}_{P}(a^N_{Pc})$ , is represented by:

(28) \begin{align} a^N_{Pc} = \frac{ \Delta \,\lambda \, (\phi _P + \xi ) \,\rho \,U + \phi _P \, (\rho + \lambda _T) \, \gamma _P + \xi \,\lambda _P \, (\rho + \lambda _T) \,F }{ \Delta \,\lambda \, (\phi _P + \xi ) + \xi \, h \, (\rho + \lambda _T) }. \end{align}

As employers do not pay any expense for investment in occupational training for temporary workers in this case, $a^N_{Pc}$ does not depend on the costs of providing training to these workers.

Which case in Figure 3 is realized depends on the relationship between $a^L_{Tt}$ and $a^N_{Pc}$ . A direct comparison of these thresholds finds that $a^N_{Pc}$ is higher than $a^L_{Tt}$ if the following inequality is satisfied:

(29) \begin{align} &\frac{ \phi _T \,[ \,\Delta \,\lambda \,(\phi _P + \xi ) + \xi \,h \,(\rho + \lambda _T)] \, \gamma _T - \xi \,[ h \,(\phi _P + \delta )(\rho + \lambda _T) + \delta \,\Delta \,\lambda ] \,\gamma _P }{\xi \,h \,\Delta \,\lambda \,(\phi _P + \xi + \delta ) } \notag \\[4pt] &\quad - \frac{ \xi \,\lambda _P \,[ (\xi \,h - \delta ) \, \lambda _T + \xi \,h \, \rho + \delta \,\lambda _P \,] \,F }{\xi \,h \,\Delta \,\lambda \,(\phi _P + \xi + \delta ) } \,\lt \, \rho \,U. \end{align}

This corresponds to the case of the left portion of Figure 3. Although we do not assume any specific relationship between the cost of training permanent workers, $\gamma _P,$ and the cost of training temporary workers, $\gamma _T$ , (29) is more likely to be satisfied if $\gamma _P$ is strictly higher than $\gamma _T$ for a given value of termination costs and labor market tightness. If $\gamma _T$ is equal to $\gamma _P$ , much larger termination costs associated with permanent jobs are necessary for the realization of equilibrium with training in temporary jobs.

The following proposition summarizes the conditions in which each situation described in Figure 3 is realized.

Finally, we show that all fully trained temporary workers can be converted into permanent positions. If the threshold of match-specific productivity that achieves zero in permanent jobs with a fully trained worker is denoted by $a_{Ph}^t$ , it is represented by:

(30) \begin{align} S_P(a^t_{Ph}) = 0 \,\, \Leftrightarrow \,\, a_{Ph}^t = \frac{ \rho \,U - \rho \,F }{ 1 + h }. \end{align}

This is lower than $\rho \,U$ . Because the hiring threshold in temporary jobs $a_{Th} = \rho \,U$ is higher than $a_{Ph}^t$ , all fully trained temporary workers can be converted into permanent positions.

3.3 Value of an Unemployed Worker

The sharing rules for each surplus and the free entry/exit condition indicate the value of an unemployed worker as follows:

(31) \begin{align} \rho \,U = b + \frac{\beta \,\theta \,c}{1 - \beta }. \end{align}

This is common to all types of equilibria which will be defined in the next section.

3.4 Parameter Space and the Determination of the Type of Equilibrium

The type of equilibrium depends on which conditions among (22), (24), (25), and (29) are satisfied (note that $\theta$ is given throughout this section). Let us illustrate these conditions in the parameter space $(\gamma _T,\,b)$ . Because (22) does not depend on $\gamma _T$ for a given $\theta$ , it is illustrated as a horizontal line:

(22^′) \begin{align} b = \frac{ \phi _P \,\gamma _P + \xi \,\lambda _P \,F }{ \xi \,h } - \frac{\beta \,\theta \,c}{1 - \beta }, \end{align}

where we use the expression of $\rho \,U$ given by (31). By contrast, both (24), (25), and (29) are illustrated as upward-sloping straight lines for a given $\theta$ :

(24^′) \begin{align} b &= \frac{ \phi _P \,\phi _T \,\gamma _T + \xi \,\delta \,\lambda _P \,F }{ \xi \,h \, (\phi _P + \delta ) } - \frac{\beta \,\theta \,c}{1 - \beta },\\[-5pt]\notag \end{align}

(25^′) \begin{align} b &= \frac{ \phi _T \,[\phi _P + \xi \,(1 + h)] \, \gamma _T - \xi \, [ h \, \phi _P + \delta \,(1 + h) ] \, \gamma _P - \xi \,\lambda _P \,(\xi \,h - \delta ) \, F }{ \xi \,h \,(\phi _P + \delta + \xi ) } - \frac{\beta \,\theta \,c}{1 - \beta },\\[-5pt]\notag \end{align}

(29^′) \begin{align} b &= \frac{ \phi _T \,[ \,\Delta \,\lambda \,(\phi _P + \xi ) + \xi \,h \,(\rho + \lambda _T)] \, \gamma _T - \xi \,[ h \,(\phi _P + \delta )(\rho + \lambda _T) + \delta \,\Delta \,\lambda ] \, \gamma _P }{\xi \,h \,\Delta \,\lambda \,(\phi _P + \xi + \delta ) } \notag \\[3pt] &- \frac{ \xi \,\lambda _P \,[ (\xi \,h - \delta ) \, \lambda _T + \xi \,h \, \rho + \delta \,\lambda _P \,] \,F }{\xi \,h \,\Delta \,\lambda \,(\phi _P + \xi + \delta ) } - \frac{\beta \,\theta \,c}{1 - \beta }. \end{align}

Regarding the relationship of the four lines in Figure 4, we obtain Proposition $3$ .

Figure 4. Determination of the type of equilibrium in the parameter space $(\gamma _T,\,b)$ .

Proposition 3. In the parameter space $(\gamma _T,\,b)$ , $\textrm{(i)}$ the slope of (29^′ ) is steeper than that of (25^′), $\textrm{(ii)}$ the slope of ( 25^′) is steeper than that of (24^′ ), and $\textrm{(iii)}$ the four lines have one intersection, as shown in Figure 4. The training costs in temporary jobs and the value of leisure at that point take the following form, respectively:

\begin{align*} \gamma _T = \frac{ (\phi _P + \delta ) \, \gamma _P + \xi \,\lambda _P \,F }{ \phi _T }, \,\, b = \frac{ \phi _P \,\gamma _P + \xi \,\lambda _P \,F }{ \xi \,h } - \frac{\beta \,\theta \,c}{1 - \beta }. \end{align*}

Proof

See Appendix A.

Three distinct areas are labeled (A), (B), and (C). Each area has the following properties:

• In the area (A), all four conditions given by (22), (24), (25), and (29) are satisfied. This corresponds to the case in Figure 2. The threshold for providing training in temporary jobs $a_{Tt}$ is given by (23), which is less than $a_{Ph}$ and the threshold for choosing a type of contract is given by (26). Let us call the equilibrium associated with this situation Type-I equilibrium.

• In the area (B), conditions (22), (24), and (29) are satisfied, but (25) is not. This corresponds to the left portion of Figure 3. The threshold for providing training in temporary jobs $a^L_{Tt}$ is given by (27) and the threshold for choosing a type of contract is given by (26). Note that (27) is higher than (23). Let us call the equilibrium associated with this situation Type-II equilibrium.

• In the area (C), only conditions (22) and (24) are satisfied. This corresponds to the right portion of Figure 3. Then, $a^L_{Tt}$ is higher than $a^N_{Pc}$ and the threshold for choosing a contract is given by (28). Therefore, no temporary workers are trained. However, workers in a match with productivity higher than $a_{Ph}$ can be converted into a permanent position. Let us call the equilibrium associated with this situation Type-III equilibrium (i.e., training is not provided in temporary jobs).

3.5 Characterization of Type-I Equilibrium

As will be shown in Section 4.1, the area in the parameter space $(\gamma _T,\,b)$ that attains Type-II equilibrium is small under reasonable parameter values. Since our research interest is based on a theoretical model that captured employers’ decisions about the provision of training in temporary jobs, we concentrate on Type-I equilibrium in the following analysis. The characterization of other types of equilibria is described in Appendix B.

To characterize the steady-state equilibrium, we need to specify the job creation condition and measures of unemployment and each employment pool. The equilibrium value of labor market tightness is determined by the standard free entry/exit condition, driving the value to the firm of a vacant job to zero. According to the surplus sharing rule as a result of the Nash bargaining problem, a firm with an occupied job gains the proportion $1 - \beta$ of each surplus generated from forming a job match. Therefore, the RHS of (1) is expressed using the surpluses $S^{nt}_{T0}(a),\,S^{nv}_{T0}(a),\,S^{nv}_{T1}(a)$ , and $S^{nv}_P(a)$ . Based on the above argument regarding employers’ choice and the classification of an equilibrium, applying the rule of integration by parts to the RHS of (1) yields the following job creation condition:

(32) \begin{align} \frac{c}{(1 - \beta ) \,q(\theta )} &= \frac{d \,S^{nt}_{T0}(a)}{d \,a}\int _{a_{Th}}^{a_{Tt}} (1 - G(a)) \,da + \frac{d \,S^{nv}_{T0}(a)}{d \,a}\int _{a_{Tt}}^{a_{Ph}} (1 - G(a)) \,da \notag \\[4pt] &+ \frac{d \,S^{nv}_{T1}(a)}{d \,a}\int _{a_{Ph}}^{a_{Pc}} (1 - G(a)) \,da + \frac{d \,S^{nv}_P(a)}{d \,a}\int _{a_{Pc}}^{\bar{a}} (1 - G(a)) \,da, \end{align}

where the derivative of each surplus is represented as follows:

\begin{align*} & \frac{d \,S^{nt}_{T0}(a)}{d \,a} = \frac{1}{\phi _T}, \,\, \frac{d \,S^{nv}_{T0}(a)}{d \,a} = \frac{1}{\phi _T} + \frac{ \xi \,\delta + \xi \,h \,(\phi _P + \delta ) }{\phi _P \,\phi _T \,(\phi _T + \xi )}, \\[4pt] &\frac{d \,S^{nv}_{T1}(a)}{d \,a} = \frac{1}{\phi _T} + \frac{ \xi \,\delta + \xi \,h \,(\phi _P + \delta ) }{\phi _P \,\phi _T \,(\phi _T + \xi )} + \frac{ \delta \,[ \phi _P + \xi \,(1 + h)] }{\phi _P \, (\phi _P + \xi )(\phi _T + \xi )}, \,\, \frac{d \,S^{nv}_P(a)}{d \,a} = \frac{ \phi _P + \xi \,(1 + h) }{ \phi _P \,(\phi _P + \xi ) }. \end{align*}

Note that the derivative of each surplus is independent of productivity $a$ because it is a linear function of $a$ .

Employers hire a worker in the form of a temporary contract for $a \lt a_{Pc}$ . Workers in a match with productivity $a \in [a_{Th}, \,a_{Tt})$ do not receive training. For $a \in [a_{Tt},\,a_{Ph})$ , employers with a temporary job provide training but no conversion into permanent employment occurs until training is completed. If the productivity of a match is contained in the interval $[a_{Ph},\, a_{Pc})$ , training is provided by employers and workers may be converted into permanent employment even if the training is not completed. Employers hire a worker initially in the form of a permanent contract for $a_{Pc} \le a$ . As $\rho \,U$ is an increasing function of $\theta$ and the LHS of (32) is increasing in $\theta$ , a unique $\theta$ satisfies the job creation condition.

The measures for each type of employed workers are denoted by $e_{T0}^{nt},\,e_{T0}^{nv},\,e^{nv}_{T1},\,e^t_{T}, \,e^{nv}_{P},\,e_P$ . The employment and unemployment flows in each pool are described by:

\begin{align*} \dot{u} &={ - \,(1 - G(a_{Th})) \, \theta \,q(\theta ) \,u + (\lambda _T + \delta )\left(e^{nt}_{T0} + e^{nv}_{T0}\right) + \lambda _T\left(e^{nv}_{T1} + e^t_{T}\right) + \lambda _P \,(e^{nv}_P + e_P) }, \\[4pt] \dot{e}^{nt}_{T0} &= - \,(\lambda _T + \delta ) \,e^{nt}_{T0} + \left[G(a_{Tt}) - G(a_{Th})\right] \,\theta \,q(\theta ) \,u, \\[4pt]\dot{e}^{nv}_{T0} &= - \, \zeta \,e^{nv}_{T0} + \left[ G(a_{Ph}) - G(a_{Tt})\right] \,\theta \,q(\theta ) \,u, \\[4pt] \dot{e}^{nv}_{T1} &= - \, \zeta \,e^{nv}_{T1} + \left[ G(a_{Pc}) - G(a_{Ph})\right] \,\theta \,q(\theta ) \,u, \\[4pt] \dot{e}^t_{T} &= - \,(\lambda _T + \delta ) \,e^t_{T} + \xi \,(e^{nv}_{T0} + e^{nv}_{T1}), \\[4pt] \dot{e}^{nv}_P &= - \,(\lambda _P + \xi ) \,e^{nv}_P + (1 - G(a_{Pc})) \,\theta \,q(\theta ) \,u + \delta \, e^{nv}_{T1}, \\[4pt] \dot{e}_P &= - \lambda _P \,e_P + \xi \,e^{nv}_P + \delta \,e^t_{T}. \end{align*}

In the steady state, flows into and out of each employment pool must be equivalent. Then, the measures of the unemployment pool and each employment pool are derived as follows:

(33a) \begin{align} e^{nt}_{T0} &= \frac{ [G(a_{Tt}) - G(a_{Th})] \,\theta \,q(\theta ) \,u }{ \lambda _T + \delta }, \end{align}

(33b) \begin{align} e^{nv}_{T0} &= \frac{ [ G(a_{Ph}) - G(a_{Tt})] \,\theta \,q(\theta ) \,u }{ \zeta }, \end{align}

(33c) \begin{align} e^{nv}_{T1} &= \frac{ [ G(a_{Pc}) - G(a_{Ph})] \,\theta \,q(\theta ) \,u }{ \zeta }, \end{align}

(33d) \begin{align} e^t_T &= \frac{ \xi \,(e^{nv}_{T0} + e^{nv}_{T1}) }{\lambda _T + \delta }, \end{align}

(33e) \begin{align} e^{nv}_P &= \frac{ (1 - G(a_{Pc})) \,\theta \,q(\theta ) \,u + \delta \,e^{nv}_{T1} }{ \lambda _P + \xi }, \end{align}

(33f) \begin{align} e_P &= \frac{ \xi \,e^{nv}_P + \delta \,e^t_T }{\lambda _P}, \end{align}

and the steady-state value of the unemployment rate is represented by:

(34) \begin{align} u ={\frac{ \zeta \, \lambda _P \,(\lambda _T + \delta ) }{ \zeta \, \lambda _P \,(\lambda _T + \delta ) + \theta \,q(\theta )[ \zeta \,(\lambda _T + \delta ) - \zeta \,(\lambda _T - \lambda _P) \,G(a_{Pc}) - \delta \, (\lambda _T + \delta ) \,G(a_{Ph}) - \xi \,\delta \, G(a_{Tt}) - \zeta \,\lambda _P \,G(a_{Th}) ] } }. \end{align}

Once the equilibrium value of labor market tightness is uniquely determined by (32), (20), (25), (23), and (25) specify each threshold, respectively. Subsequently, the equilibrium values of the unemployment rate and measures of each employed worker are determined.

4.1 Calibration

One period of time equals 1 month in this exercise. The key parameters are derived to fit into the performance of the Italian labor market, where a gap in the rate of participation in training among contracts is relatively high compared to other European countries.Footnote ¹⁵ First, we specify some functional forms. Following the literature, the matching function is assumed to be a Cobb−Douglas form: $m(u,\,v) = \Lambda \,u^{\alpha }v^{1-\alpha }$ with $\alpha \in (0,\,1)$ . $\Lambda$ is a mismatch parameter, which is a positive constant. Second, we suppose that match-specific productivity follows a uniform distribution with the support $[1,\,3]$ : $G(a) = (a - 1)/2$ .Footnote ¹⁶ Note that $\underline{a}$ is set to 1 as normalization and that $\bar{a} = 3$ is chosen to obtain a reasonable value of the gap in the average wage between temporary and permanent contracts. Using individual longitudinal ECPH data for $1995$ – $2001$ , Kahn (Reference Kahn2016) estimates this wage gap for workers between the age of $16$ and $65$ years in Italy to be around $25 \%$ . Based on the same dataset with controlling differences in individual or job characteristics, OECD (2002) finds the estimated wage penalty for temporary workers in $1997$ to be around $13 \%$ . The computed value of the wage gap in this study is $17.86 \%$ , which lies between OECD (2002) and Kahn (Reference Kahn2016).

The discount rate, $\rho$ , is set to $0.3 \%$ , which corresponds to a $1\%$ interest rate per quarter (Bentolila et al. (Reference Bentolila, Cahuc, Dolado and Le Barbanchon2012)). Following Belan and Chéron (Reference Belan and Chéron2014), the value of leisure, $b$ , is equal to $0.2$ . $\alpha$ is the elasticity of the matching function with respect to unemployment and is set to $0.5$ , in line with most of the literature. The bargaining power of workers, $\beta$ , is equal to $0.5$ , which coincides with the elasticity of the matching function. This is the standard Hosios condition. The mismatch parameter, $\Lambda$ , is equal to $0.347$ , which is chosen to ensure that the job creation condition given (32) holds under the targeted value of labor market tightness $\theta = 0.445$ and that calibrated parameters take a moderate value.Footnote ¹⁷ Regarding the cost of training permanent workers, it is difficult to obtain data on the actual amount because it should include not only a pecuniary component but also a nonpecuniary one, such as a reduction in the productivity of senior employees playing the role of mentor. We then set $\gamma _P = 12$ , which ensures a reasonable value of termination costs through calibration. If $\gamma _P$ becomes lower, the calibrated value of the termination costs becomes considerably high because a reduction in $\gamma _P$ increases the expected profit of a vacant job, and higher termination costs are necessary to satisfy the job creation condition. For a similar reason, $\xi$ is set to $0.1$ . Regarding the choice of the separation rate in temporary jobs, the value of $\lambda _T$ is chosen to make the calculated proportion of permanent contracts among newly created jobs approach the actual value $38 \%$ for Italy in the $2005$ − $2015$ period (Albanese and Gallo (Reference Albanese and Gallo2020)).Footnote ¹⁸ The activation rate of a nonrenewal clause, $\delta$ , is set to $0.02$ to fit the model to the data showing that the year-to-year transition rate in Italy is around $20 \%$ . Finally, following Hobijn and Sahin (Reference Hobijn and Sahin2009), who estimate the monthly job separation rate for over $20$ OECD countries, we set $\lambda _P = 0.0069$ because the estimated rate in Italy is $0.69 \%$ .

Based on the OECD database, we target the proportion of permanent workers and the unemployment rate in Italy to $87 \%$ and $9.3 \%$ , respectively. Following the finding reported by Peracchi and Viviano (Reference Peracchi and Viviano2004), we target a labor market tightness of $0.445$ . Finally, using information about the difference in the participation rate provided by OECD (2019a) and the proportion of employees participating in continuing vocational training (CVT) courses provided by Wiseman and Parry (Reference Wiseman and Parry2017), the participation rate in firm-provided training for temporary workers in Italy is obtained as $18.1 \%$ .Footnote ¹⁹ We target this value in this calibration exercise. The calibrated values of the remaining parameters, $h,\,c,\,\gamma _T$ , and $F$ , are listed at the bottom of Table 1.Footnote ²⁰

Table 1. A list of parameter values

The plotted point in Figure 5 represents the combination of the calibrated value of the cost of training temporary workers and the baseline value of leisure. The point is close to the boundary between the Type-I and Type-II equilibria but is located on its left-hand side. Together with the fact that the area in the $(\gamma _T,\,b)$ space that attains Type-II equilibrium is extremely small (this topic will be discussed in Section 4.4), it is reasonable to concentrate on Type-I equilibrium in the following numerical exercise. Finally, we note that the area in the $(\gamma _T,\,b)$ space that attains Type-I equilibrium significantly varies according to the value of $\lambda _T$ , which is the separation rate for temporary jobs. All other things being equal, a considerably high separation rate of these contracts (resulting in an increase in short-term temporary contracts) may generate the associated Type-III equilibrium. We will also discuss this topic in Section 4.4.

Figure 5. Classification of equilibrium.

4.2 The Impact of Reducing the Cost of Training Temporary Workers

Before explaining the results of this section, following OECD (2019b), we will briefly review the role of Training Funds (TFs) in Italy, which were instituted by law in $2000$ . TFs are associations run by social partners that support workers’ continuous learning and finance training expenses using resources collected through firms’ payroll contributions. Owing to the importance of assisting vulnerable (or low-skilled) employment groups, which is the recent policy orientation of European societies, TFs take on even greater importance in continuous vocational training in Italy. A reduction in the cost of training temporary workers in this model can be seen as a provision of more grants for vocational training to employees with temporary jobs. Although the resources available to invest in continuous vocational training have been reduced in Italy, it is easier to request more funds for training if financing firm-provided training is expected to improve the macroeconomic performance of the labor market.Footnote ²¹

Figure 6 shows that a decrease in $\gamma _T$ increases the unemployment rate and decreases the share of permanent workers and labor productivity but increases social welfare (evidently, it increases the participation rate in training among temporary workers).Footnote ²² These results are interpreted as follows. A lower cost of training temporary workers increases job creation. Therefore, the thresholds $a_{Ph}$ and $a_{Pc}$ become higher, since a change in $\gamma _T$ has only an indirect impact on these thresholds through $\theta$ , and an increase in $\theta$ strengthens the bargaining power of workers and decreases the match surpluses arising from both types of contracts. If threshold $a_{Ph}$ becomes higher, more dead-end temporary jobs are the result. If threshold $a_{Pc}$ becomes higher, more newly formed jobs start through temporary contracts. Together with the fact that the hiring threshold for temporary jobs is also raised under higher labor market tightness, lowering the training cost of temporary jobs leads to a decrease in the share of permanent workers among total employees and an increase in the unemployment rate. Based on the parameter values listed in Table 1, a $10 \%$ decrease in the training cost from the baseline value leads to a decrease in the share of permanent workers of 0.46 percentage points (from $87$ % to $86.54 \%$ ), an increase in the unemployment rate of $0.07$ percentage points (from $9.3 \%$ to $9.37 \%$ ), an increase in the participation rate in training for temporary workers of $13.32$ percentage points (from $18.1 \%$ to $31.42 \%$ ), and an increase in social welfare of $0.13 \%$ (from 2.08767 to 2.0904). Similarly, a $50 \%$ decrease in the training cost from the baseline value leads to a decrease in the share of permanent workers of 3.1 percentage points (from $87 \%$ to $83.9 \%$ ), an increase in the unemployment rate of $0.71$ percentage points (from $9.3 \%$ to $10.01 \%$ ), an increase in the participation rate in training for temporary workers of $61.45$ percentage points (from $18.1 \%$ to $79.55 \%$ ), and an increase in social welfare of $1.25\%$ (from $2.08767$ to $2.11372$ ).

Figure 6. Impact of cost of training temporary workers: (upper left) unemployment rate; (upper right) proportion of permanent workers among total employees; (lower left) labor productivity; (lower right) social welfare.

Regarding the impact on labor productivity, we focus on the replacement of permanent jobs by temporary jobs. To provide an intuition for this result, the following notations are defined (see also Appendix C): (i) $e^t_{Tl}$ is the measure of temporary workers who were initially in a dead-end state and completed training; (ii) $e^t_{Th}$ is the measure of temporary workers who initially had the prospect of being converted into permanent contracts and completed training; (iii) $e_{P1}$ is the measure of permanent workers who started their careers with permanent status and completed training; (iv) $e_{P2}$ is the measure of permanent workers working as novices with permanent contracts or converted from $e^t_{Th}$ into their current positions;Footnote ²³ and (v) $e_{P3}$ is the measure of permanent workers converted from $e^t_{Tl}$ . As shown in the left portion of Figure 7, a reduction in $\gamma _T$ decreases $e^{nt}_{T0}$ and increases $e^t_{Tl}$ and $e^t_{Th}$ , indicating that the share of temporary workers who completed training increases unambiguously, leading to an increase in the number of trained permanent workers (an increase in $e_{P2}$ and $e_{P3})$ . However, these positive effects are not sufficient to increase the total number of permanent workers with high productivity as an outcome of training because the negative effect of a decrease in permanent workers among the new hires is significant. The decrease in permanent workers among new hires decreases the flow into $e_{P1}$ , which is the dominant effect with respect to the decrease in labor productivity.

Figure 7. Impact of the cost of training temporary workers on measures of selected employment pools: (left) measures of temporary workers who have completed training and who do not receive training; (right) measures of permanent workers who have completed training.

According to the results presented in this section, policies to reduce the burden of training expenses that target temporary workers not only achieve a higher participation rate of these workers in training opportunities but also increase social welfare, which is attributed to a reduction in the total amount paid for training costs in permanent jobs. This result indicates that the cost of training temporary workers should be reduced from the viewpoint of improving social efficiency. To implement such policies, which contribute to the opening up of vocational training opportunities for workers in vulnerable positions, policy makers should consider that additional costs may be necessary to compensate unemployed workers for their job losses due to the increase in temporary workers and unemployment rate.

4.3 Impact of Easing Employment Protection on Permanent Jobs

This section examines the impact of a reduction in employment protection on permanent contracts. In Italy, the degree of regulation of unfair dismissals is discontinuous at the $15$ -employee threshold; that is, considerably stringent employment protection is imposed on employers with $15$ employees or more, and the degree of employment protection is weak for employers with less than $15$ employees. Several empirical studies, such as Hijzen et al. (Reference Hijzen, Mondauto and Scarpetta2017) and Bratti et al. (Reference Bratti, Conti and Sulis2018), have investigated the effect of EPL on the composition of employment contracts and worker turnover using this size-contingent property. Because the termination costs associated with permanent contracts are uniformly imposed on employers in this study (the size of a firm is not considered), a reduction in the termination costs, which is a labor market policy that we address in this section, is assumed to be applied to every employer with a permanent contract.

According to Figure 8, reducing the termination costs for permanent jobs decreases the unemployment rate and increases the proportion of permanent workers, the rate of participation in training for temporary workers, labor productivity, and social welfare. We first note that a decrease in $F$ increases the equilibrium value of labor market tightness, which is consistent with the stylized facts. Second, from (25) and (26), a reduction in the termination costs decreases both the threshold for converting from a temporary into a permanent contract, $a_{Ph}$ , and for choosing a permanent contract in an initially formed match, $a_{Pc}$ . Because lower termination costs decrease the expected labor costs in permanent jobs, these results are intuitive. Subsequently, the decrease in $F$ affects the threshold for hiring a temporary worker, $a_{Th}$ , and the threshold for providing training to them, $a_{Tt}$ . The result that the equilibrium value of labor market tightness is decreasing in $F$ shows that employers raise the former threshold, as $F$ has an indirect effect on $a_{Th}$ through $\theta$ , and a higher $\theta$ increases the outside value of a worker in wage bargaining (i.e., the value of being unemployed) and decreases the surplus from forming temporary jobs with no training. Interestingly, in contrast to the case of $a_{Th}$ , a change in $F$ has a direct impact on $a_{Tt}$ for the following reason. The presence of termination costs associated with permanent jobs directly affects employers’ decisions about the provision of training to temporary workers because temporary workers who have completed training may be converted into permanent contracts. Therefore, lower termination costs increase the match surplus generated from a permanent job occupied by a fully trained worker, which makes providing training to a temporary worker more attractive. This induces employers with temporary jobs to train their employees, resulting in a decrease in $a_{Tt}$ . This result supports the expectation of a spillover effect, encouraging firms to train temporary workers by reforming EPL to address labor market segmentation. We note that this spillover effect can arise even if employers’ decisions about the duration of temporary jobs are endogenized.Footnote ²⁴ Although the hiring threshold in temporary jobs is raised, a higher proportion of permanent workers among total employees and a higher proportion of temporary workers who receive training opportunities contribute to a decrease in the unemployment rate and an increase in labor productivity and social welfare.

Figure 8. Impact of termination costs associated with permanent jobs: (upper left) unemployment rate; (upper right) proportion of temporary workers who participate in training; (lower left) labor productivity; (lower right) social welfare.

These results are consistent with the findings of Hijzen et al. (Reference Hijzen, Mondauto and Scarpetta2017) and Bratti et al. (Reference Bratti, Conti and Sulis2018). Based on a regression discontinuity design, they suggest that higher firing costs for regular workers lead firms to replace regular positions with temporary ones, thus increasing worker turnover. Their finding can be a potential explanation for the decrease in productivity observed in European countries. Moreover, Bratti et al. (Reference Bratti, Conti and Sulis2018) point out that low productivity is attributed to the fact that temporary workers tend to receive less training. The finding in this section further supports the standpoint of lowering firing costs for regular workers because it will contribute to an increase in productivity through a higher participation rate in firm-sponsored training among temporary workers.

4.4 Discussion

4.4.1 A case in which temporary jobs have a very short employment duration

Cahuc et al. (Reference Cahuc, Charlot and Malherbet2016) and Felgueroso et al. (Reference Felgueroso, García-Pérez, Jansen and Troncoso-Ponce2018) point out that France and Spain experienced a steep increase in the volume of fixed-duration contracts lasting less than a week or month. Such an increase in the very short duration of fixed-term contracts will be relevant to the higher separation rate of these contracts, represented by a higher $\lambda _T$ in this study. The left part of Figure 9 represents how an increase in $\lambda _T$ affects the possibility of realizing each type of equilibrium. The area in the $(\gamma _T,\,b)$ space that attains Type-I equilibrium becomes smaller as $\lambda _T$ becomes larger. The light gray area corresponds to the case of $\lambda _T = 0.5$ , and the dark gray area corresponds to the benchmark case (i.e., $\lambda _T = 0.06)$ . In other words, the area that attains Type-III equilibrium, whereby firms never provide training to temporary workers, becomes significantly larger. This implies that when analyzing the French and Spanish labor markets, we should pay attention to Type-III equilibrium and consider policy that induces employers to retain temporary jobs for longer periods because the prevalence of fixed-term contracts of very short duration will make employers reluctant to invest in firm-sponsored training. As has already been mentioned, in the Italian labor market, note that the mean tenure of temporary workers is high among other European countries and that the annual transition rate from temporary to permanent employment in Italy (approximately 20 $\%$ ) is higher than the rate in France and Spain (approximately 10 $\%$ ). Thus, employers in Italy will potentially have a stronger incentive to invest in training for temporary workers because of higher stability in these jobs compared to France and Spain.

Figure 9. (Left) How a change in the duration of temporary jobs affects the possibility of realizing each type of equilibrium; (right) realization of Type-II equilibrium.

According to Cahuc et al. (Reference Cahuc, Charlot and Malherbet2016), who introduce the heterogeneity of the expected duration of production opportunities, more stringent employment protection associated with permanent jobs increases the duration of temporary jobs (the share of these jobs decreases). Although the results of this study predict that less stringent employment protection will increase the participation rate of temporary workers in firm-sponsored training, how termination costs (i.e., the degree of employment protection) affect the decisions of employers regarding the provision of training through the duration of temporary jobs should be taken into account. This is an interesting research topic.