2.1 Setup

In my model, a firm is born as long as its expected entry value exceeds its entry cost $\hat{f_e}$ . Upon entry, each firm incurs the entry cost as a sunk cost. It then realizes its productivity $x$ , which is the source of firm heterogeneity. $x$ is drawn from the Pareto distribution $F(x)\equiv 1-x^\alpha _{\text{min}} x^{-\alpha }$ with support $x\in [x_{\text{min}},\infty )$ .

Because it is not optimal for firms with very low $x$ to remain in the market, each firm then decides whether to exit. Next, each surviving firm chooses one of two firm types: updating firm or obsolescing firm. The difference between the two lies in their production function and technology cost.

The production function of an updating firm in period $t$ is

(1) \begin{equation} a(t)xl, \end{equation}

where $l$ denotes labor input and $a(t)$ represents the economy-wide technology in period $t$ . In contrast, the production function of an obsolescing firm is

(2) \begin{equation} a(\tau )xl, \end{equation}

where $\tau$ denotes the firm’s production start date after entry. Thus, the technology is fixed at the level at date $\tau$ . Because $a(t)$ grows at the exogenous rate of technological progress $g$ , the gap between $a(t)$ and $a(\tau )$ increases over time.

Each updating firm incurs the technology adoption cost $\hat{I}$ (henceforth “adoption cost”) in every period, while each obsolescing firm incurs it only upon entry. This study assumes that each firm chooses its own firm type only once, on entry. This assumption is harmless when the adoption cost normalized by the economy-wide technology level increases with the technology gap $t-\tau$ and when the speed of an increase in it is more than the speed of a decrease in the value of each obsolescing firm.Footnote ⁶

I assume monopolistic competition in the product market. The revenue function at time $t$ is denoted by $R(x,l,\tau,t)$ for a firm with technology vintage $\tau$ , firm-specific productivity $x$ , and labor input $l$ . The revenue function is as follows:Footnote ⁷

(3) \begin{equation} R(x,l,\tau,t)=\left ( \frac{E(t)}{n}\right ) ^{\frac{1}{\sigma }}(a(\tau )xl)^{\frac{ \sigma -1}{\sigma }} \end{equation}

$E(t)$ denotes aggregate income at time $t$ , $n$ denotes the number of firms, and $\sigma$ represents the elasticity of substitution between two differentiated goods so that $\sigma \gt 1$ . Following Felbermayr and Prat (Reference Felbermayr and Prat2011), the condition $\alpha/(\sigma -1)\gt 1$ is also assumed in this paper. This ensures that the mean value of firm size for the updating firm type is bounded. The entire firm size density in equilibrium is calculated in the appendix. For updating firms, I replace $\tau$ with $t$ such that their revenue is $R(x,l,t,t)$ .

I assume workers are risk neutral, and their population is normalized to one. The workers are either employed or unemployed. Each worker earns wage $w(x,l,\tau,t)$ when employed or receives unemployment benefit $\hat{b}$ when unemployed. The inputs of the wage function— $x$ , $l$ , $\tau$ , and $t$ —have the same meaning as in the revenue function. Wages are determined by bargaining as specified later.

The labor market is frictional. The matching function $m(V,u)$ represents aggregate job creation in each period; it is a function of unemployment $u$ and aggregate job vacancies $V$ . In line with Petrongolo and Pissarides (Reference Petrongolo and Pissarides2001), there are constant returns to scale in this function, which increases with each argument. Labor market tightness is defined as the ratio of vacancies to unemployment, $\theta$ $\equiv V/u$ . The job-finding probability for each unemployed worker is the ratio $m(V,u)/u=m(\theta,1)$ . Similarly, the job-filling probability for each vacancy is $m(V,u)/V=m(\theta,1)/\theta$ .

At the beginning of each period, the economy-wide technology progresses. Then, each firm starts production and pays wages given its level of embodied technology. After production, separations occur such that an exogenous fraction of workers within each firm, $\lambda l$ , become unemployed in the next period. At the end of each period, each firm is exogenously destroyed with probability $\delta$ . Each unemployed worker applies for a vacancy and becomes employed in the next period with the job-finding probability. Each employed worker loses their job with separation probability $s=\delta +\lambda -\delta \lambda$ in the next period. The separation probability is based on a combination of firm-specific and job-specific exogenous shocks.

Time is discrete, and the common discount rate of agents (workers and firms) is $r$ . The value functions are as follows (where the superscripts $o$ and ${u}$ distinguish the firm types):

(4) \begin{align} J^{o}(x,l,\tau,t)=\max \left [ \max _{v}\left [ \begin{array}{c} R(x,l,\tau,t)-w(x,l,\tau,t)l-\hat{c}v \\[5pt] +\frac{1-\delta }{1+r}J^{o}(x,l^{\prime },\tau,t+1) \end{array} \right ],0\right ], \end{align}

(5) \begin{align} J^{u}(x,l,t)=\max _{v}\left [ \begin{array}{c} R(x,l,t,t)-w(x,l,t,t)l-\hat{c}v \\[5pt] -\hat{I}+\frac{1-\delta }{1+r}J^{u}(x,l^{\prime },t+1) \end{array} \right ], \end{align}

(6) \begin{align} W^{o}(x,l,\tau,t)=\max \left [ w(x,l,\tau,t)+\frac{1}{1+r}\left [ (1-s)W^{o}(x,l^{\prime },\tau, t+1)+sU(t+1)\right ], U(t)\right ], \end{align}

(7) \begin{align} W^{u}(x,l,t)=w(x,l,t,t)+\frac{1}{1+r}\left [ (1-s)W^{u}(x,l^{\prime },t+1)+sU(t+1)\right ],\text{ and} \end{align}

(8) \begin{align} U(t)=\hat{b}+\frac{1}{1+r}\left [ m(\theta,1)\tilde{W}(t+1)+(1-m(\theta,1))U(t+1)\right ]. \end{align}

Equation (4) describes the value of an obsolescing firm at time $t$ with productivity $x$ , employment $l$ , and technology vintage $\tau$ . The firm maximizes its value by choosing the number of vacancies $v$ to post. The flow profit is the revenue minus wage payments minus the total vacancy cost. $\hat{c}$ denotes the cost of posting one vacancy. The employment of any firm evolves as $l^{\prime }=(1-\lambda )l+vm(\theta,1)/\theta$ .

Equation (5) describes the value of an updating firm. I drop technology vintage from the value function because the firm uses up-to-date technology by definition. However, I introduce adoption cost. The functional form of wages is the same across firm types because the presence of the adoption cost is independent of the marginal job surplus.

Equation (6) describes the value of an employed worker in an obsolescing firm, and equation (7) describes that of an employed worker in an updating firm. Equation (8) describes the value of an unemployed worker. $\tilde{W}(t+1)$ denotes the expected value of being employed in the next period. In Section 2.2, I confirm that $\tilde{W}(t+1)$ can be treated independently of specific factors in each firm. Thus, the specific form of $\tilde{W}(t+1)$ is redundant and omitted here. For a simple expression with integration, see Felbermayr and Prat (Reference Felbermayr and Prat2011).

This study focuses on a balanced-growth equilibrium and assumes that some exogenous variables grow at rate $g$ to render a nontrivial environment. The list of such exogenous variables and assumptions is $\hat{f_e}=a(t)f_{e}$ , $\hat{I}=a(t)I$ , $\hat{b}=a(t)b$ , and $\hat{c}=a(t)c$ . In addition, I assume $(1-\delta )(1+g)/(1+r)\lt 1$ to ensure that the values of firms normalized by the economy-wide technology— $J^{o}(x,l,\tau,t)/a(t)$ and $J^{u}(x,l,t)/a(t)$ —are bounded. I call $(1-\delta )(1+g)/(1+r)$ the net discount factor because it emerges in those normalized values.Footnote ⁸ The net discount factor is assumed to be positive.

2.2 Wage bargaining

Following Stole and Zwiebel (Reference Stole and Zwiebel1996) and Smith (Reference Smith1999), each wage is determined by surplus sharing between a firm and a worker in the firm. Workers have no agreements among themselves or interactions in the bargaining, and each worker is treated as a marginal worker. Specifically, for an updating firm and its worker, the worker obtains the exogenous fraction $\beta$ of the marginal job surplus $\partial J^{u}(x,l,t)/\partial l+W^{u}(x,l,t)-U(t)$ , while the firm takes the remaining fraction $1-\beta$ of the surplus. This surplus-sharing rule is the same for an obsolescing firm and its worker. The associated expressions for the two firm types are as follows:

(9) \begin{align} \beta \left [\frac{\partial J^{o}(x,l,\tau,t)}{\partial l}+W^{o}(x,l,\tau,t)-U(t)\right ] & =W^{o}(x,l,\tau,t)-U(t) \end{align}

(10) \begin{align} \beta \left [\frac{\partial J^{u}(x,l,t)}{\partial l}+W^{u}(x,l,t)-U(t) \right ] & =W^{u}(x,l,t)-U(t) \end{align}

By using these conditions, I can characterize wages in Proposition 1.

Proposition 1. The wage function takes the following form: Footnote ⁹

(11) \begin{equation} w(x,l,\tau,t)=\frac{\beta (\sigma -1)}{\sigma -\beta }\frac{R(x,l,\tau,t)}{ l}+(1-\beta )a(t)\omega (\theta ) \end{equation}

Here, $\omega (\theta )$ denotes the worker’s reservation wage $b+\beta c\theta/[(1-\beta )(1-\delta )]$ .

When $\sigma =\infty$ , the wage function becomes a standard form so that revenue per worker is weighted by the worker’s bargaining power $\beta$ , and the reservation wage (the outside option value) is weighted by $1-\beta$ .

Importantly, the recruitment cost per worker, equal to the vacancy cost $a(t)c$ divided by the job-filling probability $m(\theta,1)/\theta$ , is the same for all firms. Thus, the first-order conditions for the two firm types follow such that $a(t)c\theta/m(\theta,1)=[(1-\delta )/(1+r)]\partial J^{o}(x,l^{\prime },\tau,t+1)/\partial l=[(1-\delta )/(1+r)]\partial J^{u}(x,l^{\prime },t+1)/\partial l$ . Along with this, by using the surplus sharing conditions (9) and (10), the value of any employed worker in the next period, $W^{o}(x,l,\tau,t+1)$ , $W^{u}(x,l,t+1)$ , or $\tilde{W}(t+1)$ , can be treated independently of firm-specific factors.

2.3 Optimal revenue and labor demand

Along with the wage function (11), the maximization problems for vacancies, from equations (4) and (5), are solved as follows.

Proposition 2. Let $a(t)R(x,t-\tau )$ be the optimal revenue and $l(x,t-\tau )$ be the optimal labor demand in a firm with productivity $x$ and technology gap $t-\tau$ . Then, the following is true:

(12) \begin{align} R(x,t-\tau )=\frac{E(t)/a(t)}{n}\left [ \frac{\sigma -1}{\sigma -\beta }\frac{1}{ \kappa (\theta )}\right ] ^{\sigma -1}x^{\sigma -1}\left ( \frac{1}{1+g}\right ) ^{(\sigma -1)(t-\tau )} \end{align}

(13) \begin{align} l(x,t-\tau )=\frac{E(t)/a(t)}{n}\left [ \frac{\sigma -1}{\sigma -\beta }\frac{1}{ \kappa (\theta )}\right ] ^{\sigma }x^{\sigma -1}\left ( \frac{1}{1+g}\right ) ^{(\sigma -1)(t-\tau )} \end{align}

Here, $\kappa (\theta )$ is defined as the employment cost such that

(14) \begin{equation} \kappa (\theta )\equiv \omega (\theta )+\frac{c\theta }{(1-\beta )m(\theta,1)}\left [ \frac{1+r}{(1-\delta )(1+g)}-1+\lambda \right ]. \end{equation}

Three remarks are in order. First, the optimal revenue is denoted by $a(t)R(x,t-\tau )$ ; thus, $R(x,t-\tau )$ is the measure normalized by $a(t)$ . This notation is for convenience when considering the subsequent balanced-growth equilibrium. In addition, normalized aggregate income $E(t)/a(t)$ is constant in the steady state.Footnote ¹⁰

Second, both $R(x,t-\tau )$ and $l(x,t-\tau )$ increase with $[E(t)/a(t)]/n$ and decrease with $\theta$ . The demand shifter coefficient for each firm is $[E(t)/a(t)]/n$ . $\kappa (\theta )$ increases with $\theta$ because the job-filling probability $m(\theta,1)/\theta$ decreases with $\theta$ and the reservation wage increases with it. The (flow) employment cost $\kappa (\theta )$ is defined as the sum of the reservation wage and the recruitment cost, which is adjusted to be consistent with its flow value.

Third, regarding firm-specific factors, both $R(x,t-\tau )$ and $l(x,t-\tau )$ increase with productivity $x$ and decrease with technology gap $t-\tau$ . For each updating firm, the solutions (12) and (13) become $R(x,0)$ and $l(x,0)$ by incorporating $\tau =t$ ; the technology gap term vanishes.

In deriving this proposition, the Euler equation for employment is obtained such that

(15) \begin{equation} \frac{R(x,t-\tau )}{l(x,t-\tau )}=\frac{\sigma -\beta }{\sigma -1}\kappa (\theta ). \end{equation}

This implies that revenue per worker in any firm equals the same aggregate value $\kappa (\theta )$ multiplied by the markup $[\sigma -\beta ]/[\sigma -1]$ . As a result, all wages reduce to the same value; let $w\equiv w(x,l,\tau,t)/a(t)$ be the common normalized wage.Footnote ¹¹

The Euler equation (15) holds not only for each updating firm but for each obsolescing firm. The endogenous shutdown period of each obsolescing firm leads to $\infty$ because each obsolescing firm can reduce its employment and maintain its revenue per worker. In contrast, DMP models do not separate the concepts of firm entry and job creation, and revenue (or output) per worker exogenously declines over time with technology obsolescence.

As growth-related variables, I define the net discount factor $G_{1}$ and the downsizing operator $G_{2}$ as follows:Footnote ¹²

(16) \begin{align} G_{1}\equiv \frac{(1-\delta )(1+g)}{1+r}\text{ and} \end{align}

(17) \begin{align} G_{2}\equiv \frac{l(x,t+1-\tau )}{l(x,t-\tau )}=\left ( \frac{1}{1+g}\right ) ^{\sigma -1} \end{align}

2.4 Technology choice and free entry

There are two expressions for the entry value of each firm: $J^{u}(x,0,t)$ and $J^{o}(x,0,t,t)-a(t)I$ . The former is the value of an updating firm with $l=0$ ; in other words, it is the value when a new entrant chooses to be an updating firm. The latter is the value when a new entrant chooses to be an obsolescing firm. $a(t)I$ is subtracted because the initial adoption cost is required for any firm type and, as in equation (4), $J^{o}(x,0,t,t)$ does not include it. In addition, initial adopted technology is up-to-date—that is, $\tau =t$ . The following proposition describes the entry value by firm type.

Proposition 3. Let $J_{e}^{u}(x,t)\equiv J^{u}(x,0,t)$ be the entry value for an updating firm and $J_{e}^{o}(x,t)\equiv J^{o}(x,0,t,t)-a(t)I$ be the entry value for an obsolescing firm. Then, these values are obtained as follows:

(18) \begin{align} J_{e}^{u}(x,t) & =a(t)\left [ \frac{G_{1}}{1-G_{1}}\left [ \frac{1-\beta }{\sigma -\beta }R(x,0)-I\right ] -I\right ] \text{ and} \end{align}

(19) \begin{align} J_{e}^{o}(x,t) & =a(t)\left [ \frac{G_{1}}{1-G_{1}G_{2}}\frac{1-\beta }{\sigma -\beta }R(x,0)-I\right ] \end{align}

Fig. 2 depicts the two possible entry values, (18) and (19), in the same graph. The vertical axis is the entry value normalized by $a(t)$ , and the horizontal axis is $R(x,0)$ . Because $R(x,0)$ monotonically increases with $x$ , this graph reflects the relationship between the normalized entry value and firm-specific productivity.

Figure 2.

Entry value by firm type.

In brief, a firm with high (low) $x$ optimally chooses the updating (obsolescing) firm type because the slope of $J_{e}^{u}(x,t)/a(t)$ is higher than that of $J_{e}^{o}(x,t)/a(t)$ and the height of $J_{e}^{u}(x,t)/a(t)$ is lower than that of $J_{e}^{o}(x,t)/a(t)$ at $R(x,0)=0$ .

Specifically, there are two productivity cutoffs, $x_0$ and $x_1$ , that satisfy

(20) \begin{align} J_{e}^{o}(x_{0},t)=0\text{ and} \end{align}

(21) \begin{align} J_{e}^{o}(x_{1},t)=J_{e}^{u}(x_{1},t), \end{align}

where I term $x_0$ the exit cutoff and $x_1$ the technology cutoff. In steady state, a firm with $x$ strictly higher than $x_1$ is an updating firm; a firm with $x$ such that $x_{0}\leq x\leq x_{1}$ holds is an obsolescing firm; and a firm with $x$ strictly lower than $x_0$ exits the market.

These cutoffs are simply characterized by combining the free entry condition as follows:

(22) \begin{equation} a(t)f_{e}=\int _{x_{\min }}^{\infty }\max \left [ J_{e}^{u}(x,t),J_{e}^{o}(x,t),0\right ] dF(x) \end{equation}

Here, the right-hand side is the expected entry profit before paying the entry cost $a(t)f_{e}$ and realizing $x$ . For convenience, the cutoff ratio is defined as follows:

(23) \begin{equation} \phi \equiv \frac{x_{0}}{x_{1}} \end{equation}

$\phi ^{\alpha }$ equals the ratio of the number of updating firms to that of surviving firms, denoted as $[1-F(x_{1})]/[1-F(x_{0})]$ , which is referred to as the adoption rate for technology in this economy. The proposition below uncovers $x_0$ and $\phi$ (and $x_1$ following equation [23]).

Proposition 4. From equations ( 12 ) and ( 18 )–( 23 ), I derive the following equations:

(24) \begin{align} \phi ^{\sigma -1}=1-G_{2}\text{ and} \end{align}

(25) \begin{align} f_{e}=\left ( \frac{x_{\min }}{x_{0}}\right ) ^{\alpha }\frac{\sigma -1}{ \alpha -\sigma +1}I\left [ \frac{G_{1}}{1-G_{1}}\phi ^{\alpha }+1\right ] \end{align}

These equations solve $\phi$ and $x_{0}$ .

Two remarks are in order. First, the job-cut rate in each obsolescing firm, $1-G_{2}$ , positively affects the adoption rate $\phi ^{\alpha }$ . Because an increase in the growth rate $g$ accelerates technology obsolescence and increases $1-G_{2}$ , it also increases $\phi ^{\alpha }$ . So even if rapid growth harms some firms, it improves the composition of surviving firms in the aggregate.

Second, the right-hand side of equation (25) depends on the growth-related term $[G_1/[1-G_1]]\phi ^{\alpha }$ . This implies that the entry decision of each potential entrant is dominantly associated with the chance of being an updating firm.Footnote ¹³ An increase in $g$ improves the success rate among surviving firms and profitability upon success, thereby promoting firm entry and increasing the exit cutoff.

Notably, a decrease in the entry cost does not affect the adoption rate. It promotes firm entry and increases the exit cutoff but also increases the technology cutoff so that the cutoff ratio is unchanged. As demonstrated later, updating firms has a substantial effect on job creation. Thus, the effectiveness of horizontal product market deregulation may be limited.

2.5 Closing the model

To close the model, I first specify the aggregate resource constraint. It is the equality between aggregate income and the total revenue of surviving firms. Because there are two firm types, the aggregate resource constraint takes the following somewhat complicated form:

(26) \begin{align} E(t) & =n\left (1-\phi ^{\alpha }\right )\int _{x_{0}}^{x_{1}}\sum _{t-\tau =0}^{\infty }a(t)R(x,t-\tau )\frac{(1-\delta )^{t-\tau }\delta dF(x)}{F(x_{1})-F(x_{0})} \\[5pt] & +n\phi ^{\alpha }\int _{x_{1}}^{\infty }a(t)R(x,0)\frac{dF(x)}{1-F(x_{1})} \notag \end{align}

Here, the first (second) term on the right-hand side is the sum of revenues of obsolescing (updating) firms. The number of obsolescing firms is $n\left (1-\phi ^{\alpha }\right )$ , and that of updating firms is $n\phi ^{\alpha }$ .

To be consistent with Melitz (Reference Melitz2003) and Felbermayr and Prat (Reference Felbermayr and Prat2011), this paper defines the average of surviving firms with respect to $x$ , denoted by $\tilde{x}$ , so that the relationship below holds:

(27) \begin{equation} \frac{E(t)}{n}=a(t)R(\tilde{x},0) \end{equation}

Because $E(t)/n$ is the average revenue of surviving firms, the above equation states that the average productivity $\tilde{x}$ is defined such that the revenue of a firm with $x=\tilde{x}$ and $t-\tau =0$ equals the average revenue. Specifically, average productivity is given as

(28) \begin{equation} \tilde{x} \equiv \left [(1-\phi ^{\alpha })\frac{\delta }{1-G_{2}(1-\delta )}\int _{x_{0}}^{x_{1}}x^{\sigma -1}\frac{dF(x)}{F(x_{1})-F(x_{0})} +\phi ^{\alpha }\int _{x_{1}}^{\infty }x^{\sigma -1}\frac{dF(x)}{1-F(x_{1})}\right ]^\frac{1}{\sigma -1}, \end{equation}

where this expression accounts for the two firm types. The term $\delta/[1-G_{2}(1-\delta )]=\delta \sum _{t-\tau =0}^{\infty }G_{2}^{t-\tau }(1-\delta )^{t-\tau }\lt 1$ corresponds to the weighted downsizing operator for new and old obsolescing firms. If there is no obsolescence, $G_{2}=1$ and $\delta/[1-G_{2}(1-\delta )]=1$ .Footnote ¹⁴

Importantly, from equations (12), (15), and (27), it is clear that average productivity equals revenue per worker.Footnote ¹⁵ In other words, the Euler equation (29) for employment pins down the relationship between labor market tightness and average productivity:

(29) \begin{equation} \frac{\sigma -\beta }{\sigma -1}\kappa (\theta )=\tilde{x} \end{equation}

This equation implies that when there are more-productive firms, more vacancies are posted. Thus, an increase in $\tilde{x}$ increases $\theta$ . The unique existence of $\theta$ requires $\tilde{x}[(\sigma -1)/(\sigma -\beta )]-b\gt 0$ . Because productivity cutoffs and $\tilde{x}$ are solved in advance, equation (29) solves $\theta$ .

For a given $\theta$ , unemployment $u$ is immediately solved by the equality condition between job creation and destruction:

(30) \begin{equation} um(\theta,1)=(1-u)s \Leftrightarrow u=\frac{s}{m(\theta,1)+s} \end{equation}

Thus, an increase in $\theta$ increases $m(\theta,1)$ and decreases $u$ .

Finally, the number of surviving firms $n$ is solved by the equality condition of aggregate labor demand and the number of employed workers:

(31) \begin{equation} L^{u}+L^{o}=1-u \end{equation}

Here, $L^{u}$ denotes the total labor demand of updating firms and $L^{o}$ denotes that of obsolescing firms. The total labor demand by firm type is given as

(32) \begin{align} L^{u}=n\phi ^{\alpha }\int _{x_{1}}^{\infty }l(x,0)\frac{dF(x)}{1-F(x_{1})}\text{ and} \end{align}

(33) \begin{align} L^{o}=n\left (1-\phi ^{\alpha }\right )\int _{x_{0}}^{x_{1}}\sum _{t-\tau =0}^{\infty }l(x,t-\tau )\frac{(1-\delta )^{t-\tau }\delta dF(x)}{F(x_{1})-F(x_{0})}. \end{align}

In summary, the equilibrium is defined as follows.

4.1 Benchmark calibration

The model is calibrated to the US economy. The period is annual. The matching function is specified such that $m(V,u)=m_{0}u^{\eta }V^{1-\eta }$ . The parameter values are set as follows.

The discount rate $r$ and the base growth rate $g$ are 0.04 and 0.02, respectively. In line with Elsby et al. (Reference Elsby, Hobijn and Şahin2013), the monthly job-separation probability $s/12$ is 0.036. The firm-destruction probability $\delta$ is $0.092$ , computed as the mean value over the 1978–2014 period using data from Business Dynamics Statistics. The data are available since 1978. The end year is set to be consistent with the period that is adopted in the subsequent simulation.

The elasticity of matching function $\eta$ is 0.5, based on Petrongolo and Pissarides (Reference Petrongolo and Pissarides2001). Following Felbermayr and Prat (Reference Felbermayr and Prat2011), the bargaining power of each worker $\beta$ is 0.5 so that $\beta =\eta$ holds. As mentioned in Felbermayr and Prat (Reference Felbermayr and Prat2011), when allowing multiworker firms, the Hosios condition $\beta =\eta$ is not sufficient to ensure an efficient allocation. In the social planner’s problem here, the derivatives of total revenue to $u$ and $\theta$ are not simple. In contrast, in the DMP setting, the typical total revenue is exogenous output per worker multiplied by $1-u$ (thus, its derivative with respect to $u$ is simply output per worker multiplied by $-1$ ). Because of the absence of well-established estimates, I set the bargaining power $\beta =\eta$ as a standard choice. Additionally, an independent change in $\beta$ generates a monotonic effect on $du/dg$ ; the question of whether the value of $\beta$ is near its optimal value seems uninteresting.

According to Ebell and Haefke (Reference Ebell and Haefke2009), the entry cost in the USA in 1997 equaled $0.6$ months of per capita income, and the entry cost in 1978 amounted to $5.2$ months of per capita income. I simply use the mean value of these estimates such that $f_{e}=[(0.6+5.2)/2]\times 1/12=0.24$ , although the main results are robust to this decision. I show the sensitivity of the three effects later. I use the value of per capita income equal to one, based on the normalization $\tilde{x}=1$ , which is mentioned later. Although the average productivity $\tilde{x}$ and per capita income $E(t)/a(t)$ are not the same, these values are numerically very similar so that $\tilde{x}=1$ (normalization) and $E(t)/a(t)=0.9401$ (solution) hold.

The remaining parameter values cannot be set directly. Thus, these values are obtained such that important targeted moments in the model equal those in the data.Footnote ¹⁸

The scale of matching function $m_0$ , the flow value of unemployment $b$ , and the vacancy cost $c$ are 7.99, 0.27, and 0.54, respectively. These values are obtained to replicate the data moments: labor market tightness equals 0.72, from Pissarides (Reference Pissarides2009); the monthly job-finding probability equals 0.565, from Elsby et al. (Reference Elsby, Hobijn and Şahin2013); the replacement rate (the ratio of the unemployment flow value to the wage $b/w$ ) equals 0.37, from the OECD Statistics database.Footnote ¹⁹

The elasticity of substitution between two differentiated goods $\sigma$ is 2.56 such that the markup equals 1.32, from Christopoulou and Vermeulen (Reference Christopoulou and Vermeulen2012).Footnote ²⁰ The shape of the productivity distribution $\alpha$ is 1.66 such that the size distribution of firms is approximately Zipf and its tail index $\alpha/(\sigma -1)$ equals 1.06, from Axtell (Reference Axtell2001) and Luttmer (Reference Luttmer2007).

The minimum level of firm-specific productivity $x_{\text{min}}$ is 0.008 such that the average productivity $\tilde{x}$ equals 1 as a normalization. The adoption cost $I$ is 1.83 such that the average firm size equals 19.51 as the mean value over the 1978–2014 period, from Business Dynamics Statistics.

The parameter values are summarized in Table 1. Under these parameter values, the equilibrium solution is $u=0.0599$ , $n=0.0482$ , $w=0.7289$ , $E(t)/a(t)=0.9401$ , $x_0=0.1638$ , and $x_1=1.5296$ . Of course, $\theta =0.72$ and $\tilde{x}=1$ hold because these are the targeted moment and the normalization.

Table 1.

Parameter values

Figure 3.

The impact of growth on unemployment. Note: The total impact of growth on unemployment is decomposed into three effects so that CAP, CRE, and COMP represent the capitalization effect, the creative-destruction effect, and the firm-composition effect, respectively. The dotted line corresponds to the total impact of the three effects.

4.2 Results

Fig. 3 graphically shows the total impact of growth on unemployment $du/dg$ (as the dotted line) and its decomposition into the three effects. When the growth rate increases, the capitalization effect (labeled “CAP”) slightly reduces unemployment, while the creative-destruction effect (labeled “CRE”) increases it. Consistent with the literature, the latter effect is stronger than the former effect.

However, these canonical effects are small relative to the firm-composition effect (labeled “COMP”). Rapid technology obsolescence is not always undesirable because it not only increases the job-cut rate in each obsolescing firm but reduces the number of such firms and induces firms to choose the updating firm type. As a result, the composition of surviving firms and the associated worker reallocation improve in the aggregate.Footnote ²¹

Table 2 numerically shows the total impact $du/dg$ and its decomposition. The column “Benchmark” reports the result under the benchmark calibration.Footnote ²² COMP is approximately 40 times larger than CAP. The size of CAP is similar to that obtained in standard DMP models (under fully disembodied technology) because equation (34) is essentially unchanged across models when $d\tilde{x}/dg=0$ . In Appendix I, I describe the simple DMP model with the capitalization and creative-destruction effects and simulate $du/dg$ .

Table 2.

Decomposition and calibration sensitivity

Note: The total impact of growth on unemployment $du/dg$ is decomposed into three effects: CAP, CRE, and COMP. COMP is further decomposed into COMP1, COMP2, and COMP3, which are associated with the second to fourth rows in equation (35), respectively. In addition to Benchmark (the result under the benchmark calibration), I check alternative calibration cases such that the value of the targeted moment is $b/w=0.71$ or $\text{Markup}=1.6$ , and the entry cost $f_e$ is given as 0.6/12 or 5.2/12. Each case for $f_e$ is omitted because its result is identical to the Benchmark column.

In addition, in Table 2, COMP is further decomposed into COMP1, COMP2, and COMP3. While the second to fourth rows in equation (35) jointly compute COMP, COMP1 is computed by taking into account only the second row in equation (35); it reflects a change in the average productivity of updating firms. COMP1 is positive because some less productive firms become updating firms and the average productivity of the updating firm type decreases. This reduces labor market tightness and increases unemployment. Similarly, COMP2 is computed by using only the third row in equation (35); it reflects a change in the average productivity of obsolescing firms. COMP2 is relatively insignificant because this channel involves competing effects: a decrease in the technology cutoff $x_1$ and an increase in the exit cutoff $x_0$ . Finally, COMP3 is computed by activating only the fourth row in equation (35), and it reflects an increase in the adoption rate. The results imply that COMP3 is the dominant source of COMP.

In Table 2, the results of alternative calibration cases are also shown. First, the targeted data moment for $b/w$ is changed to 0.71, as suggested by Hall and Milgrom (Reference Hall and Milgrom2008). As discussed in the literature, the opportunity cost of employment may be too low, when it does not include the value of leisure or home production.Footnote ²³ In this alternative case, the sizes of CRE and COMP increase while CAP remains unchanged. In total, the size of $du/dg$ increases.

In the second case, the targeted markup is changed to 1.6. De Loecker et al. (Reference De Loecker, Eeckhout and Unger2020) document the evolution of market power in the USA, where the average markup steadily increased from 1.2 in 1980 to 1.6 in 2016.Footnote ²⁴ Although the size of $du/dg$ slightly decreases, the result is still robust.

In the third case, $f_e$ is changed in the calibration process. Modifying it does not change the benchmark result, which is reported in Table 2. This is because such a change in $f_e$ is fully absorbed in $x_{\text{min}}$ , while the other parameter values are unaltered; as in equation (25), $f_e$ itself does not matter as long as $f_e/[x_{\text{min}}^\alpha ]$ is constant.

4.3 Data versus model

Fig. 4 demonstrates the evolution of the unemployment rate from the data and model predictions. The model predictions are computed such that only the growth rate varies as in its empirical evolution (see the bottom graph), while the other parameter values are fixed.

Figure 4.

Unemployment evolution, data versus model. Note: The evolution of the unemployment rate is depicted for data and model predictions (which correspond to the calibration setups labeled “Benchmark” and “ $b/w=0.71$ ” in Table 2) as a response to the empirical variation of the growth rate. The annual data are constructed from the 2017 release of the EUKLEMS database and the OECD Statistics database. The growth rate is calculated as gross value added (in 2010 prices) divided by employment. The data are trended through Hodrick and Prescott filtering with the smoothing parameter value 100.

To directly observe model performance, Table 3 shows the regression estimates for which each evolution in Fig. 4 is used. Each regression coefficient in Table 3 corresponds to the estimated value of $du/dg$ . Regarding its value, the results imply that the model accounts for 16%–36% of the variation in the data.

Additionally, in this exercise, the relative importance of the three effects for $du/dg$ holds similarly to that in Table 2 (for each case in each column, respectively) because these effects are computed in an almost linear manner across different growth rates as visualized in Fig. 3. Thus, this model explains an important new part of the data in contrast to standard DMP models, which explain at most the size of CAP.

Regarding the empirical literature, the implied $du/dg$ is $-0.47$ in Blanchard and Wolfers (Reference Blanchard and Wolfers2000, Table 4), $-1.49$ in Pissarides and Vallanti (Reference Pissarides and Vallanti2007), and $-1.15$ in Miyamoto and Takahashi (Reference Miyamoto and Takahashi2011). The first paper’s estimate is for 20 OECD countries, while the other estimates are for the USA.

Regarding the standard DMP model in Appendix I, the implied $du/dg$ is $-0.0072$ when the ratio of disembodied technology equals 0.99 as the exogenous adoption rate $\phi ^\alpha =0.99$ (equivalently when the remaining ratio 0.01 is embodied technology). The sign reversal of $du/dg$ occurs between $\phi ^\alpha =0.9$ and $\phi ^\alpha =0.8$ . When $\phi ^\alpha =0.1$ , the implied $du/dg$ is + 0.0388. For low $\phi ^\alpha$ , the standard DMP model fails to explain the data even qualitatively.

The size of the implied $du/dg$ from the data increases over time. In Table 4, the regression results by period are reported; the total period 1970–2014 is truncated at the peak and trough of the growth rate in Fig. 4. Because the regression coefficients for the model, reported in Table 3, are almost unchanged regarding period separation, model performance deteriorates as time passes. If this tendency is true, the value of investigating the sources of $du/dg$ increases.

4.4 Interaction with policy

Table 5 shows how the decomposition of $du/dg$ is modified when the following policy-related parameter values are changed independently: the flow unemployment value $b$ , worker bargaining power $\beta$ , or the entry cost $f_e$ . When $b=0.27$ , $\beta =0.5$ , and $f_e=0.24$ , the decomposition result is the same as in the second column (labeled “Benchmark”) in Table 2.

Two remarks are in order. First, an increase in each parameter value monotonically increases the sizes of $du/dg$ and all decomposed outcomes. This is common to the three parameters. In other words, some rigidities in the labor and product markets increase both the unemployment rate and the magnitude of $du/dg$ . This result helps us understand the observed differences between the USA and Europe as suggested in Hornstein et al. (Reference Hornstein, Krusell and Violante2007).Footnote ²⁵

Second, a change in $b$ appears to generate a stronger effect than the other parameters. In Table 5, the range of $\beta$ is not narrow and that of $f_e$ is associated with the US entry costs between 1978 and 1997 as suggested in Ebell and Haefke (Reference Ebell and Haefke2009). In addition, $b=0.5$ is near the value when the targeted data moment of $b/w$ in the calibration equals 0.71 (although this alternative calibration case also changes the value of $c$ ). A change in $b$ from 0.27 (as the benchmark value) to 0.5 increases the size of $du/dg$ by 0.41. As found in Aguiar and Hurst (Reference Aguiar and Hurst2008) and Aguiar et al. (Reference Aguiar, Bils, Charles and Hurst2021), leisure time and contents change over time; $b$ does not reflect mere unemployment insurance and seems an important factor in the evolution of $du/dg$ .

4.5 Interaction with finance

Empirically, the labor market appears to have a strong link with the stock market.Footnote ²⁶ Complementary to this view, financial imperfections may amplify the impact of growth on unemployment in the current simulation. Petrosky-Nadeau and Wasmer (Reference Petrosky-Nadeau and Wasmer2013) consider a two-stage matching environment initially through the credit market and then through the labor market. Their model implies that the financial sector introduces a new element to hiring costs such that element is independent of congestion in the labor market. As a result, their model shows that the elasticity of labor market tightness to productivity shocks is amplified in the business cycle context.Footnote ²⁷

A simple extension regarding this point is examined here. Specifically, I replace the hiring cost $c$ by the modified hiring cost $c+[m(\theta,1)/\theta ]H$ . All setups except for this are unchanged. $H$ denotes the new exogenous element in hiring costs. In other words, the total cost of employing one worker is $c\theta/m(\theta,1)+H$ .Footnote ²⁸ Thus, $H$ is a fixed and independent term.

Table 3.

Regression results: data versus model

Note: The independent variable is the growth rate $g$ , and the dependent variable is the unemployment rate $u$ , using the evolution in Fig. 4. Newey-West HAC standard errors are shown in parentheses.

Table 4.

Regression results by period

Note: The regression result for the data in Table 3 is further examined. Specifically, I separate the total period 1970–2014 into three periods and get the regression results for each period.

Table 5.

Decomposition and policy

Note: The decomposition results are shown under different parameter values for the flow unemployment value $b$ , worker bargaining power $\beta$ , and the entry cost $f_e$ . The remaining parameter values, which are not specified in the first row, equal those of the benchmark calibration. Thus, the current exercise does not examine alternative calibration cases but looks for a change in each parameter value one by one. The first column is the same as in Table 2 except for the additional information $u|_{g=0.02}$ , which reports the unemployment rate at $g=0.02$ .

Table 6 shows the results under different $H$ . The last two rows in Table 6 report the implied size of each element in hiring costs. Checking these relative sizes, I select $H=0.01$ and $0.02$ to demonstrate the outcomes. Similar to the business cycle context, at least, the presence of the fixed term $H$ magnifies the current results.

Table 6.

Decomposition and additional exogenous term $H$ in hiring costs

Note: Alternative calibrations and their decomposition results are shown, where the value of the additional exogenous term $H$ is initially given and then I implement the same procedure as in the benchmark calibration except that the total hiring cost is modified to $c+m(\theta,1)H/\theta$ . In addition to the benchmark calibration (with the targeted data moment $b/w=0.37$ ), the case for the targeted data moment $b/w=0.71$ is also shown for different values of $H$ .

In addition, heterogeneous financial conditions across firms may further amplify the results. Eckstein et al. (Reference Eckstein, Setty and Weiss2019) use firm-level data and document that lower credit ratings are associated with more volatile employment and higher interest rate volatility. They show that in the 2008 financial crisis, firms with lower credit ratings experienced greater declines in employment.Footnote ²⁹ In my model, a change in the adoption rate is positively linked to the job-cut rate, as suggested in Proposition 4. Thus, if obsolescing (less productive) firms are financially weak and must further increase the job-cut rate to weather episodes of illiquidity in times of rapid technological progress, the adoption rate increases so that the firm-composition effect strengthens.

The relationship between a technology choice and creditor protection may also be important when incorporating financial imperfections into the analysis. Benmelech and Bergman (Reference Benmelech and Bergman2011) use a panel of aircraft-level data around the world and find that airlines enjoying the benefits of higher creditor protection operate aircraft of a newer technology and younger vintage. They argue that improved investor protection and its associated reduction in financial frictions affect firms’ tendency to invest in newer technologies as a key driver of productivity growth. This view seems to be helpful for uncovering a valuation in unemployment through a change in the adoption rate.