# The perverse dynamics of growth-induced unemployment

Fabien Postel-Vinay

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CAHIERS ECO & MATHS

98.12

THE PERVERSE DYNAMICS OF
GROWTH-INDUCED UNEMPLOYMENT

Fabien POSTEL-VINAY

MAD
THE PERVERSE DYNAMICS OF GROWTH-INDUCED
UNEMPLOYMENT

Fabien Postel-Vinay

MAD, Université de Paris I*†

First version: May 1997—This version: December 1997

ABSTRACT

This paper investigates and provides a comparison of the short- and long-run effects of technological progress on employment. It presents a simple standard model of matching unemployment that captures the negative creative destruction effects of technological change on employment. In the long-run, faster technological change implies faster job obsolescence, which is detrimental to the equilibrium level of employment. But it is also shown to have short-run positive and potentially important effects on employment. The dynamic behavior of employment is thus termed ‘perverse’, since its long- and short-run adjustments have opposite signs. Another of its features is non-monotonic convergence towards its steady state value and the presence of dampened dynamic echo effects.

Key words: unemployment, technological change, echo effects.

JEL classification: E24; J41; O33

RÉSUMÉ

Cet article étudie et compare les effets de court et de long terme du progrès technique
sur l'emploi. Il présente un modèle canonique de chômage d'appariement appréhendant les
effets négatifs de destruction créatrice que le changement technologique exerce sur l'emploi.
A long terme, un progrès technique plus soutenu implique une obsolescence plus rapide
des emplois, ce qui nuit au niveau d'emploi d'équilibre. Mais il est aussi montré qu'une
accélération du progrès technique a des effets de court terme positifs et potentiellement
importants sur l'emploi. La dynamique de l'emploi est donc qualifiée de « perverse », car
ses ajustements de long et de court terme ont des signes opposés. Une autre caractéristique
de cette dynamique est sa convergence non monotone vers l'état stationnaire, et la présence
d'effets d'écho atténués.

Mots-clefs: chômage, progrès technique, effet d'écho.

Classification JEL: E24; J41; O33

<footnote>*Address: MAD, CNRS-URA 0926, Université de Paris I, 106-112 bld. de l’Hôpital 75647 Paris Cedex 13,
France. Internet: http://panoramix.univ-paris1.fr/MAD/. E-mail: fpostel@univ-paris1.fr</footnote>

<footnote>†This paper owes a great deal to suggestions and comments from P. Beaudry. I also wish to thank A. d'Autume and P. Cahuc for useful discussions on an earlier version. Of course, none of them is responsible for any error or shortcoming that would remain in it.</footnote>
# 1. INTRODUCTION

The fear of technological change and the popular idea of its guilt in the rise and persistence of high unemployment have been a recurrent concern of both economic theorists and policy makers, at least since the beginning of industrialization in the late eighteenth century. History provides many examples of the sometimes ravaging consequences of that fear, a number of which—like the famous Luddite riot in England—are gathered in Petit [1993] or Cohen [1995]. But the problem of the relation between technological progress and unemployment is obviously not only a historical matter, and many recent contributions to economic theory have been trying to investigate it in a number of distinct directions.

The particular question I wish to address in this paper is that of the influence of the *rate* of technological change on the level of unemployment. Matching models of unemployment (Diamond [1982], Pissarides [1990]) have proved to be extremely helpful to that particular end, and provided simple and clear-cut results on the long-run nexus between growth and unemployment. Let me briefly recall what they are.

On the one hand, in the conventional matching model with technological change (Pissarides [1990], chapter 2), faster growth is shown to reduce the long-run unemployment rate through the so-called *capitalization effect*. According to that model, a higher rate of growth implies a higher present value of jobs, which spurs the recruiting activity and raises the job finding rate of unemployed workers. On the other hand, in ‘Schumpeterian’ models of growth (Aghion and Howitt [1991], [1994]), faster growth generally leads to a rise in long-run unemployment through a *creative destruction effect*<fnref n="1" />. The underlying intuition is that growth has a reallocative aspect that the previous conventional model ignores: faster technological change is accompanied by faster obsolescence of skills and technologies, hence more intense labor turnover and higher frictional unemployment.

The apparent inconsistency between these two points of view is very clearly and simply resolved in a recent paper by Mortensen and Pissarides [1995]. They use a canonical matching model to show that both types of results can be obtained, depending on the particular technological assumptions adopted. The capitalization effect rests on the assumption that firms are able to update their technology continuously and at no expense, which precludes technological obsolescence, whereas creative destruction arises from the extreme opposite assumption of total irreversibility in the firms’ technological choices.

<footnote n="1">¹The terminology of ‘creative destruction’ dates back to Schumpeter [1942], where it was not specifically connected to the unemployment problem.</footnote>
The above results are centered on long-run relations between unemployment and economic growth. Aside from that, the short-run behavior of the conventional matching model is quite well known<fnref n="2" />. But, as far as I am aware, not much has been said so far about the short-run behavior of unemployment in a creative destruction context<fnref n="3" />. This is what I wish to do in this paper.

The point of doing this is as follows: suppose the ‘correct’ model is that of Schumpeterian inspiration. Suppose the economy leaves no space for any form of capitalization effect, so that a speedup in growth eventually leads to a fall in long-run employment. In that case, is sustained technological change so bad, even in the short-run? Is the creative destruction mechanism as forceful in the transition towards the steady state as it is in the long-run? Analogy with the conventional model, in which faster growth not only increases long-run employment but also speeds up its transitional upward adjustment, whispers ‘Yes’ as an answer. Somewhat surprisingly, that basic intuition turns out to be wrong, and the right answer to both questions is unambiguously *No*.

What I show is that the short-run dynamic behavior of unemployment in response to a sudden change in the rate of technological progress is ‘perverse’, in the sense that it goes in the opposite direction to its own long-run tendency. Moreover, it appears in numerical simulations that the magnitude of this perverse adjustment may be large, compared to the corresponding variation in long-run values. According to this, a sudden speedup in technological change causes a transitional increase in the level of employment. The intuition for this result is simple. Creative destruction entails a drop in the present value of new jobs. Immediately after the shock, agents start to offer less jobs because they anticipate the drop in profitability. The natural consequence of a lower availability of new jobs is to make *existing* jobs more valuable. Indeed, by cutting the opportunity cost of older matches, faster technological change postpones their destruction. Existing jobs are thus ‘insulated’ in the terminology of Caballero and Hammour [1994], [1996]. Following the resulting decrease in the rate of job destructions, the level of employment rises.

The subsequent dynamics are characterized by a phenomenon known in the vintage capital literature as *echo effects*. Technologies become outdated after a certain lap of time, say *T*. A job

<footnote n="2">²See e.g. Pissarides [1987] or Mortensen [1989] about the dynamics of the simple matching model. In Postel-Vinay [1997], I study the dynamics of that model with endogenous savings and capital accumulation allowing endogenous growth à la Romer [1986].</footnote>

<footnote n="3">³At least, from a strictly theoretical viewpoint. Very appealing numerical simulations have been carried out by Caballero and Hammour [1994], [1996]. These authors focus on the cyclical response of job creation and job destruction to transitory aggregate fluctuations in demand (Caballero and Hammour [1994]) or profitability (Caballero and Hammour [1996]) in a model very similar to the one exploited in this paper.</footnote>
created at some date *t* will therefore be destroyed when reaching age *T*. As a result, everything happening at *t* on the side of job creations echoes at *t* + *T* on the side of job destructions. This implies that the initial perverse adjustment of unemployment is periodically reverberated in the dynamics, leading to non-monotonic convergence of unemployment towards its steady state value.

Finally, as a by-product of the model, the generated dynamic behavior of unemployment
is overwhelmingly asymmetric. The short-run rise in unemployment following a slowdown in
technical change is faster and more important in magnitude than the short-run drop in unem-
ployment coming after a speedup in technical change. These features are usually respectively
referred to as *steepness asymmetry* and *deepness asymmetry* in the empirical literature. In ad-
dition to their consistency with the original intuition of Keynes [1936] (book VI, chapter 22) on
business cycles, there is quite strong empirical support for both of them (e.g. Burgess [1992],
Sichel [1993]).

The paper is organized as follows: the next section outlines the basic model. Section 3 studies its steady state and comparative static properties. Section 4, as the paper’s main contribution, contains a theoretical study of its dynamics. Section 5 presents some numerical simulations that give a picture of the complex dynamics of this simple model. Section 6 concludes. All proofs are confined in the appendix.

2. A SIMPLE MODEL OF JOB DESTRUCTION

The basic model borrows from Mortensen and Pissarides [1995] and Caballero and Hammour
[1996]. Time is continuous and all relevant variables are subscripted by *t*. There is only one com-
petitive sector in the economy that is comprised of endogenously many identical firms endowed
with a fixed coefficients technology requiring the employment of one worker and one machine
embodying a particular technology. To hire its worker, any firm has to advertise a job vacancy
at a flow cost $\kappa_t$ per unit time until it meets a suitable job searcher. That search takes time
because of the frictions that affect the labor market. There is free-entry on the search market,
so that firms may open as many vacant jobs as they find profitable.

The number of available technologies and the productivity associated with the most recent
one evolve following an exogenous constant positive rate of technological change. When a vacant
job is matched to a worker, the firm chooses to equip its employee with a machine embodying
the most advanced technology available<fnref n="4" />. At the moment of its creation, a job thus stands on
the technological frontier of the economy. The crucial assumption is that the firms' technology
is of the 'putty-clay' variety, so that their technological choice is irreversible. If the productivity
of a job created at time t is measured by some time-dependant real number—say $a_t$, then
this job keeps the productivity $a_t$ for its entire life, even though the set of available technologies
expands. Keeping a constant productivity in a growing environment causes the existing jobs'
values to fall in time, which as shall be seen creates an endogenous motive for job destruction
that is closely related to technological change.

Before I get to the formal description of the model, it is worth noting that the assumption
of irreversible investment is the extreme opposite of that retained in the conventional matching
models with growth (see e.g. Pissarides [1990], chapter 2). In the conventional model, firms
have access at no cost to a continuous technological update, which allows them to stay on
the technological frontier permanently. In this simple version of the Mortensen and Pissarides
[1995] model, the cost of technological update is assumed large enough to make sure that job
destruction is always more profitable for the firms than implementing a new technology<fnref n="5" />.

I now turn to a formal description of the model. Trade externalities on the labor market are
modelled by positing the existence of an imperfect matching technology à la Pissarides [1990].
Suppose a number $u_t$ of unemployed workers (that are assumed to be the only job searchers
in the economy) and a number $v_t$ of vacant jobs coexist at date $t$ in the economy. Then jobs
and workers are brought together pairwise at Poisson rate $m(u_t, v_t)$, with $m$—henceforth the
matching function—traditionally assumed to be $C^\infty$, strictly increasing and concave in both its
arguments, exhibiting constant returns to scale together with the following boundary properties:
$m(0, v_t) = m(u_t, 0) = 0$, $\lim_{x\to 0} \partial m/\partial x = +\infty$, $\lim_{x\to +\infty} \partial m/\partial x = 0$ for $x = u_t, v_t$. The labor
market tightness (or job availability) ratio $\theta_t = v_t/u_t$ is convenient to define the firms' matching
(Poisson) rate as $m(u_t, v_t)/v_t = m(1/\theta_t, 1) = q(\theta_t)$: it is the number of workers that a vacant
job returns on average per unit time. Symmetrically, the workers' unemployment exit rate is
worth $m(u_t, v_t)/u_t = m(1, \theta_t) = \theta_t \cdot q(\theta_t)$. Finally, I assume that there is an exogenous cause
of job destruction captured by a positive rate of 'structural change' $\delta$. Jobs can be terminated
by a random exogenous shock that occurs at Poisson rate $\delta$.

<footnote n="4">⁴It is assumed that the cost of access to technologies is the same at a given date for all technologies. It is therefore in the interest of the newborn firm to pick up the leading technology, that is embodied in machines of the most recent vintage.</footnote>

<footnote n="5">⁵In an extension of their basic model, Mortensen and Pissarides [1995] consider the intermediate assumption of a finite cost of update, causing the firms to face a trade-off between job destruction and technological update.</footnote>
Let $\Pi_{t,\tau}^o$ be the time $\tau$ discounted value to the firm of a match made at time $t \le \tau$ (hence of tenure $\tau - t \ge 0$). As long as the traders do not decide to quit their relation, the job can only be destroyed by the exogenous process of structural change. Assuming perfect financial markets, $\Pi_{t,\tau}^o$ obeys the following asset equation:

$$r\Pi_{t,\tau}^o = a_t - w_{t,\tau} + \delta(\Pi_{\tau}^v - \Pi_{t,\tau}^o) + \frac{\partial\Pi_{t,\tau}^o}{\partial\tau}, \quad (1)$$

where $r$ denotes the exogenous discount rate, $w_{t,\tau}$ the current wage at date $\tau$ in a job of tenure $\tau - t$, $\Pi_{\tau}^v$ the discounted value at date $\tau$ of a vacant job, and $a_t$ a measure of labor productivity on the technological frontier at the job creation date $t$. Because of the irreversible investment assumption, the productivity of a job created at date $t$ remains $a_t$ until this job is destroyed.

The asset value of a vacant job evolves according to:

$$r\Pi_{\tau}^{v} = -\kappa_{\tau} + q(\theta_{\tau}) \cdot (\bar{\Pi}_{\tau,\tau}^{o} - \Pi_{\tau}^{v}) + \frac{\partial \Pi_{\tau}^{v}}{\partial \tau}. \quad (2)$$

$\bar{\Pi}_{\tau,\tau}^o$ is the average value of a new match in the firm's view<fnref n="6" />. Free-entry on the search market implies that firms exploit all profit opportunities from the opening of vacant jobs. Equilibrium on that market is thus characterized by the equality $\Pi_{\tau}^v = 0$ holding at each date $\tau$. This in turn implies, together with equation (2), that the average equilibrium value of a new match is given by $q(\theta_{\tau}) \cdot \bar{\Pi}_{\tau,\tau}^o = \kappa_{\tau}$: entrepreneurs open vacancies until their flow cost $\kappa_{\tau}$ equals their per unit time returns $q(\theta_{\tau}) \cdot \bar{\Pi}_{\tau,\tau}^o$.

Now turning to employees, I define $\Omega_{t,\tau}^e$ as the present value for a worker of being employed at date $\tau$ in a job of tenure $\tau - t$. Before a possible endogenous quit of either partner, that value obeys the following equation:

$$r\Omega_{t,\tau}^{e} = w_{t,\tau} - b_{\tau} + \delta(\Omega_{\tau}^{u} - \Omega_{t,\tau}^{e}) + \frac{\partial\Omega_{t,\tau}^{e}}{\partial\tau}, \quad (3)$$

with $b_{\tau}$ designating a lump-sum tax meant to finance unemployment compensation (if any), and $\Omega_{\tau}^u$ the asset value of being unemployed at date $\tau$. $\Omega_{\tau}^u$ also obeys an asset equation:

$$r\Omega_{\tau}^{u} = z_{\tau} - b_{\tau} + \theta_{\tau}q(\theta_{\tau}) \cdot (\bar{\Omega}_{\tau,\tau}^{e} - \Omega_{\tau}^{u}) + \frac{\partial\Omega_{\tau}^{u}}{\partial\tau}, \quad (4)$$

with $z_{\tau}$ being the gross income-equivalent flow<fnref n="7" /> of an unemployed worker—that can be given various interpretations, such as unemployment benefits, leisure valuation, wage opportunities in an informal sector..., and $\bar{\Omega}_{\tau,\tau}^e$ the average discounted value for a worker of a new match at date $\tau$.

<footnote n="6">⁶Bars over variables denote aggregate quantities that are taken as exogenous by individual agents.</footnote>

<footnote n="7">⁷I need to assume that the unemployed are not exempt from taxation to ensure the neutrality of the tax $b_{\tau}$.</footnote>
The four dynamic equations above lead to the definition of the total surplus $S_{t,τ}$ associated at date $τ$ with a job created at date $t$. From $S_{t,τ} = \Omega_{t,τ}^e - \Omega_{τ}^u + \Pi_{t,τ}^o - \Pi_{τ}^v$, equations (1) through (4) and the free-entry condition $\Pi_{τ}^v = 0$, the total surplus evolves according to:

$$ (r + \delta) S_{t,τ} = a_t - z_τ - \theta_τ q(\theta_τ) \cdot (\bar{\Omega}_{τ,τ}^e - \Omega_τ^u) + \frac{\partial S_{t,τ}}{\partial \tau}. \quad (5) $$

From this equation, one sees that the flow surplus is equal to the instantaneous output $a_t$ minus an opportunity cost $z_τ + \theta_τ q(\theta_τ) \cdot (\bar{\Omega}_{τ,τ}^e - \Omega_τ^u)$ that takes account of the outside opportunities of both trading partners, those of the firm being worth the value of a vacant job, hence zero.

As long as $S_{t,τ}$ remains positive, the firm and the employee can always find a mutually profitable contract to share that surplus at the current date $τ$. So, assuming that long-term contracts are not enforceable—which entails a continuous revising of the surplus sharing—, the corresponding job survives while $S_{t,τ}$ is positive. Accordingly, it is optimal for both the firm and the worker to separate if $S_{t,τ}$ becomes negative.

As is traditional, I shall assume that the surplus is shared through a continuously repeated generalized Nash bargaining involving the firm and its employee. The above reasoning thus applies, and a firm-worker pair matched at date $t$ is endogenously and voluntarily dissolved at some date $t + T_t$, at which $S_{t,t+T_t} \le 0$. Moreover, since $S_{t,τ}$ is a continuous function of $τ$, that inequality optimally holds as an equality. Equation (5) thus writes down in integral form as

$$ S_{t,τ} = \int_{τ}^{t+T_t} [a_t - z_{τ'} - \theta_{τ'} q(\theta_{τ'}) \cdot (\bar{\Omega}_{τ',τ}^e - \Omega_{τ'}^u)] \cdot e^{-(r+\delta)(\tau'-\tau)} d\tau'. \quad (6) $$

Assuming—I shall return to this assumption later on—that the opportunity cost of the match $z_τ + \theta_τ q(\theta_τ) \cdot (\bar{\Omega}_{τ,τ}^e - \Omega_τ^u)$ grows over time, there will come a time when that cost overcomes the constant match productivity $a_t$. It will then be optimal for both partners to destroy the job. Indeed, the optimal job lifetime $T_t$ maximizes $S_{t,τ}$ at every date $τ$. The corresponding necessary first order condition can be derived from (6):

$$ a_t - z_{t+T_t} - \theta_{t+T_t} q(\theta_{t+T_t}) \cdot (\bar{\Omega}_{t+T_t, t+T_t}^e - \Omega_{t+T_t}^u) = 0. \quad (7) $$

This condition is also sufficient if the opportunity cost increases monotonically in time.

The assumption of a repeated Nash bargaining leads to the following surplus sharing rule, holding at every date $τ$:

$$ \frac{\Omega_{t,τ}^e - \Omega_{τ}^u}{\nu} = \frac{\Pi_{t,τ}^o - \Pi_{τ}^v}{1 - \nu} = S_{t,τ}, \quad (8) $$
where $\nu \in [0, 1]$ measures the relative weight of the employee in the bargaining. Replacing the free-entry condition $\Pi_{\tau}^{\nu} = 0$ into the bargaining outcome (8) gives the equilibrium value of the opportunity cost<fnref n="8" /> of any match at date $\tau$: $\theta_{\tau}q(\theta_{\tau}) \cdot (\bar{\Omega}_{\tau,\tau}^{e} - \Omega_{\tau}^{u}) = \nu\kappa_{\tau} \cdot \theta_{\tau}/(1-\nu)$. Since this opportunity cost only reflects the traders' outside options, it is the same for all job-worker pairs, regardless of the corresponding job tenure. Substituting that value into the first order condition for the optimal lifetime of a job (7) yields<fnref n="9" />:

$$ \nu\kappa_{t+T_t} \cdot \theta_{t+T_t} = (1-\nu) \cdot (a_t - z_{t+T_t}). \quad (9) $$

Finally, to ensure the existence of a balanced growth path, I make the traditional assumption that the exogenous cost parameters follow the pace of productivity growth. I thus define two positive exogenous constants $\kappa$ and $z$ such that the cost of job advertising and the gross income flow of an unemployed worker at date $\tau$ are respectively defined by $\kappa_{\tau} = \kappa \cdot a_{\tau}$ and $z_{\tau} = z \cdot a_{\tau}$. Substituting these definitions into (9) finally leads to:

$$ \theta_{t+T_t} = \frac{1-\nu}{\kappa\nu} \cdot (e^{-gT_t} - z), \quad (10) $$

where $g = \dot{a}_t/a_t$ is the exogenous constant rate of technological change<fnref n="10" />. This entails a first dynamic relation between the tightness parameter $\theta_t$ and the optimal lifetime of jobs $T_t$. Another such relation is given by the free-entry condition. As noted above, it sets the value of a new match $\bar{\Pi}_{t,t}^o$ equal to $\kappa_t/q(\theta_t)$. Substituting this equality into the sharing rule (8) and replacing $z_{\tau}$ and $\kappa_{\tau}$ by their values yields:

$$ \frac{\kappa}{q(\theta_t)} = \int_t^{t+T_t} \left\{ 1 - \nu - \left[ (1-\nu)z_t + \nu\kappa_t\theta_t \right] \cdot e^{g(\tau-t)} \right\} \cdot e^{-(r+\delta)(\tau-t)} d\tau. \quad (11) $$

I now have in hand a dynamic system (10-11) in the variables $(\theta_t, T_t)$. The next section is devoted to the study of its stationary solution.

### 3. STEADY STATE AND COMPARATIVE STATICS

I first show the existence and uniqueness of a stationary solution of the system (10-11) (3.1), and then briefly analyze the long-run effects of a shock on the rate of technical change (3.2).

<footnote n="8">⁸The surplus sharing rule (8) also defines the unique wage $w_{t,\tau}$ prevailing at date $\tau$ in a job of tenure $\tau-t$, that can be derived explicitly: $w_{t,\tau} = (1-\nu)z_{\tau} + \nu(a_t + \kappa_{\tau} \cdot \theta_{\tau})$. Note from this result that wages differ across jobs of unequal tenures. In spite of that wage dispersion, workers prefer continued work to search for a better-paying job as long as their job is of tenure less than $T_t$. Since the value of $w_{t,\tau}$ is not crucial for the ensuing analysis, I do not give the details of its derivation that can be found in Mortensen and Pissarides [1995].</footnote>

<footnote n="9">⁹Recall that this equality holds conditionally on two assumptions. The first one, that was already mentioned, is that the equilibrium opportunity cost $z_t + \nu\kappa_t\theta_t/(1-\nu)$ increases in time. The second one is the existence of positive gains from trade to obtain a non-zero optimal job lifetime. This can be written as $\forall t, a_t > z_t + \nu\kappa_t\theta_t/(1-\nu)$.</footnote>

<footnote n="10">¹⁰Finiteness of present discounted values require the no-Ponzi game condition $r > g$.</footnote>
### 3.1. Steady state

The stationary version of system (10-11) is as follows (in which useless time subscripts have been dropped):

$$ \theta = \frac{1 - \nu}{\kappa \nu} \cdot (e^{-gT} - z) \qquad (12) $$

$$ \frac{\kappa}{q(\theta)} = \frac{1 - \nu}{r + \delta} \cdot \left(1 - e^{-(r+\delta)T}\right) - \frac{(1 - \nu)z + \nu\kappa\theta}{r + \delta - g} \cdot \left(1 - e^{-(r+\delta-g)T}\right) \qquad (13) $$

The first equation (12) of the above system defines $\theta$ as an explicit function of the optimal job lifetime $T$ that I denote by $\theta = DD(T)$. This function decreases strictly from $DD(0) = (1 - \nu)(1 - z)/(\kappa\nu)$ to $\lim_{T \to +\infty} DD(T) = -z(1 - \nu)/(\kappa\nu)$, and it is strictly convex. It is plotted on figure 1. Since the corresponding schedule crosses the horizontal axis at $e^{-gT} = z$, it appears that a stationary solution of the model must lie in the interval $T \in [0, -\ln z/g]$ to make economic sense.

Things are less clear a priori concerning equation (13). The next proposition contains some precisions:

**Proposition 1.** Equation (13) restricted to values of $T$ in the interval $[0, -\ln z/g]$ defines $\theta$ as a differentiable function of $T$ on that interval, denoted by $\theta = FE(T)$, and such that $FE(0) = 0$ and $FE'(0) \ge 0$. Function $FE$ also admits one unique global maximum on $[0, -\ln z/g]$ that coincides with its unique intersection point with the function $DD$. $\square$

Proposition 1 immediately implies an important result:

**Corollary 2.** The system (12-13) has one unique solution $(T^*, \theta^*)$ in $[0, -\ln z/g] \times \mathbb{R}_+$. $\square$

The situation is depicted on figure 1. The unique steady state solution $(T^*, \theta^*)$ finds itself at the intersection of the $DD$ schedule representing optimal job destruction, and the $FE$ schedule that reflects free-entry on the search market$^{11}$.

The slopes of both schedules on figure 1 are easily interpreted. On a very tight labor market (characterized by a large $\theta$: many vacancies and few unemployed workers), firms have a hard time filling their vacant job slots. This implies large hiring costs that they have to recoup afterwards

<footnote>$^{11}$Note that the mere existence of a positive steady state value $T^*$ proves that there are positive gains from trade in the steady state. Moreover, the stationarity of $\theta^*$ immediately implies that the steady state solution is consistent with an opportunity cost that grows over time. The unique equilibrium pointed out in corollary 2 is thus 'acceptable' in the sense that it satisfies the two necessary conditions mentioned above (see fn. 9).</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 1: Existence and uniqueness of the steady state</figcaption></figure>

by imposing high productivity standards, which in turn make technological obsolescence come faster and thus reduces the jobs' lifetime $T$. Hence the downward slope of $\mathcal{D}\mathcal{D}$. Moreover, the longer the job lasts, the longer traders receive the output flow $a_t$, and the longer they pay the opportunity cost. As long as the former remains above the latter (which is true whenever $T < T^*$), then a longer job lifetime increases the profitability of job vacancies and causes firms to open more of them, thus making the labor market tighter. If the reverse inequality holds (case $T < T^*$), then a longer job lifetime reduces its present value, which has a negative influence on the tightness ratio. Hence the hump-shape of $\mathcal{F}\mathcal{E}$.

## 3.2. Comparative statics: technical progress and creative destruction

I now turn to the long-run effects of a shift in the rate of technological change $g$. If productivity growth stems from a continuous flow of technological innovations, then a shift on $g$ can be interpreted as a change in either the size or the frequency of arrival of those innovations<sup>12</sup> (see e.g. Aghion and Howitt [1991], [1992], [1994]). The responses of the $\mathcal{D}\mathcal{D}$ and $\mathcal{F}\mathcal{E}$ schedules to a positive shock on $g$ are represented by the labelled arrows on figure 1: both schedules shift

<footnote n="12"><sup>12</sup>If an innovation is assumed to increase the most recent technology by a factor $\beta > 1$ that is interpreted as the size of that innovation, and if innovations arrive according to a Poisson process with constant parameter $\iota$, then one can write $a_t = a_0 \cdot \beta^{\iota t}$ for $\iota$ innovations occur on average per unit time. This defines the rate of technological change as $g = \iota \cdot \ln \beta$, and explains why a shock on $g$ can be seen as a shock on either $\iota$ or $\beta$.</footnote>
downwards<fnref n="13" />. For a given state of the labor market (summarized by a given value of θ), faster technological change implies faster growth of existing jobs' opportunity cost (that takes the value [z + νκθ / (1 − ν)] · a_t), which in turn implies faster obsolescence of those jobs, and hence a smaller optimal lifetime T. This is a pure creative destruction effect that governs the downward shift of the *DD* schedule. Moreover, for a given T, faster growth of existing jobs' opportunity cost diminishes the expected value of those jobs, which renders the posting of vacancies less profitable and thus reduces θ, hence the downward shift of the *FE* schedule.

As a result, it clearly appears on figure 1 that faster growth reduces the steady state value of
the job availability parameter: ∂θ*/∂g < 0. However, the response of the steady state optimal
job lifetime T* remains a priori ambiguous. The creative destruction effect tends to shorten that
lifetime because of faster growth in the jobs' opportunity cost. But the simultaneous reduction
in θ* has a negative effect on the level of that cost, for it reduces the average hiring costs κθa_t
(see equation (12) defining the function DD). The level effect thus counteracts the growth rate
effect, which makes the overall response of T* ambiguous. This ambiguity can be resolved:

**Proposition 3.** Faster technological change reduces the steady state optimal lifetime of jobs: $\partial T^*/\partial g < 0$. □

This result emphasizes the ‘Schumpeterian’ nature of the model. Faster technological progress
leads to more frequent job destructions from faster job obsolescence. This faster obsolescence
renders the jobs less valuable, which causes the firms to offer less of them. The prevailing effect—
in fact, the *only* effect, in this model<fnref n="14" />—of growth on the labor market variables is the creative
destruction effect.

I finally turn to the long-run effects of faster growth on the level of employment. Following
the result stated in proposition 3, one naturally expects faster technological change to worsen
labor market frictions, thus causing unemployment to rise. This intuition is easily confirmed
in the long-run by the comparative static properties of the model. In steady states, the flows
of workers into and out of the unemployment pool equate to make the unemployment level *u*
stationary. The number of hires at any given date *t*, *m*(*u*<sub>*t*</sub>, *v*<sub>*t*</sub>) is the number of workers exiting

<footnote n="13">¹³The result is obvious for *DD*. Concerning *FE*, for a given *T*, the effect on *θ* = *FE*(*T*) of an increase in *g* has the sign of the partial derivative of equation (13)'s r.h.s., which itself has the sign of -1+[1+(*r* + *δ* − *g*)*T*]·*e*−(*r*+*δ*−*g*)*T*.
The function *x* ↦ −1+(1+*x*)·*e*−*x* takes the value 0 for *x* = 0 and decreases over *R*+. The above expression is thus negative for all relevant values of *g* and *T*.</footnote>

<footnote n="14">¹⁴There is no capitalization effect in this model, contrarily to what happens in the basic model of Pissarides [1990], chapter 2 and the more sophisticated models of Aghion and Howitt [1994] and Mortensen and Pissarides [1995].</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 2: Comparative statics</figcaption></figure>

unemployment at that date. On the other hand, a proportion $\delta$ of the ongoing matches are destroyed by the idiosyncratic shock, which amounts to $\delta(1 - u_t)$ job destructions<fnref n="15" />. In addition, all the matches made $T^*$ periods ago that survived the exogenous destruction process—those are in number $m(u_{t-T^*}, v_{t-T^*}) \cdot e^{-\delta T^*}$—are ‘endogenously’ destroyed. The balance between those two flows when $u$ and $v$ are constant writes down as:

$$m(u, v) = \delta(1 - u) + m(u, v) \cdot e^{-\delta T^*} \quad (14)$$

This equation defines, for a given $T^*$, a schedule in the $(u, v)$ plane that is downward sloping and concave to the origin (see figure 2), traditionally referred to as the ‘Beveridge Curve’. The equilibrium steady state values of the level of unemployment $u^*$ and the number of vacancies $v^*$ are found at the intersection of that curve and the line defined by $v = \theta^* \cdot u$, also depicted on figure 2. This can be seen analytically through:

$$u^* = \frac{\delta}{\delta + m(1, \theta^*) \cdot (1 - e^{-\delta T^*})} \quad (15)$$

It thus appears that $u^*$ decreases with both $\theta^*$ and $T^*$, which straightforwardly confirms that faster technological change is detrimental to employment in the long-run. Despite the unambiguity of that result, the possibility of a sharper analysis was pointed out by Aghion and Howitt

<footnote n="15">¹⁵Labor supply is inelastic and normalized.</footnote>
[1994]. In fact, the negative effect of technological change on employment is twofold. Firstly, by reducing the optimal job duration $T^*$, it spills more workers into unemployment at every date, thus leading to a more intense labor market turnover. The corresponding graphical symptom is a shift to the north-east of the Beveridge Curve. That first effect was termed *direct* by Aghion and Howitt [1994]; it tends to increase, *ceteris paribus*, both the number of unemployed and that of vacancies. Secondly, it was shown above that faster growth is also detrimental to the surplus accruing from the matches, for it raises the corresponding opportunity cost. This is seen formally in the property $\partial\theta^*/\partial g < 0$, and graphically by a clockwise rotation of the oblique line $v = \theta^* \cdot u$. It can be viewed as a move along the Beveridge Curve, which tends to raise the number of unemployed and diminish that of vacancies. That second effect is termed *indirect* by Aghion and Howitt [1994]. The sum of those two effects is thus clearly an increase in the number of unemployed, whereas the sign of the resulting variation in the number of vacant jobs remains ambiguous.

Once again, as is very clearly pointed out by Mortensen and Pissarides [1995], the unambiguously negative influence of growth on employment should come as no big surprise in this model, since the *a priori* assumptions of irreversible investment, input complementarity and very costly technological update leave no space for any positive effect (such as the capitalization effect of the conventional matching model). What this economy is designed for is to capture the effects through which long-run growth-induced unemployment is generated in a simple framework that makes a dynamic study possible. Such a study is carried out in the next section, and it reveals that the very clear-cut and intuitive long-run properties of that simplified Aghion-Howitt/Mortensen-Pissarides-style economy are opposing a somewhat complex short-run behavior.

## 4. THE DYNAMIC EFFECTS OF A TECHNOLOGICAL SHOCK

I now study the short-run dynamic effects of a shift in $g$ on the economy I described in the previous two sections. The economy is initially assumed to be in its steady state. The recursive structure of the model allows to study the dynamics of the pair of variables $(T_t, \theta_t)$ and those of unemployment separately. I begin with the former. The dynamics of $(T_t, \theta_t)$ are governed by the system (10-11), subject to some minor restrictions (see fn. 9). The unique stationary solution $(T^*, \theta^*)$ of that system was shown to meet these restrictions (see fn. 11). Moreover, since $T_t$ and $\theta_t$ are obviously forward looking variables, I shall assume in the remainder that agents can coordinate to make them take their steady state values at all times. This is clearly
an assumption that could be discussed, for I am not sure about system (10-11)'s determinacy.
It could exhibit several non-explosive paths, or even be completely indeterminate if the steady
state $(T^*, \theta^*)$ were a sink. It may easily be shown, however, that $(T^*, \theta^*)$ is the only solution of
the system (10-11) in the particular case of a 'competitive labor market' (in the sense that all
the bargaining power is held by the firms, $\nu = 0$ implying $\forall (t, \tau) \in \mathbb{R}^2$, $\Omega_{t,\tau}^e = \Omega_\tau^u$). In a more
general situation, the dynamic path of unemployment that I am about to study might just be
*one* particular solution rather than *the* unique consistent solution. With this caveat in mind, I
now proceed to that study.

Prior to the formal dynamic analysis, some preliminary remarks may be found helpful. The first one is about the very nature of unemployment as a dynamic variable. From the assumption of imperfect matching, hiring is time-consuming and it is therefore clear that $u_t$ cannot make downward discrete jumps. Nonetheless, nothing prevents firms from dismissing as many workers as they like in no time, which raises the possibility of upward instantaneous adjustments of $u_t$. Such an upward jump never occurs in the conventional matching model, where all jobs have the same value, regardless of their ages<fnref n="16" />. In the present setting where jobs are heterogeneous, it may be so that some of them become instantaneously non-profitable after the shock, if that shock reduces their optimal lifetime. Thus the usually innocuous feature that employment is only 'half-predetermined' in matching models matters in the present setting.

Concerning that optimal lifetime, a second problem arises. The variable $T_t$ hitherto consid-
ered is the optimal lifetime of new jobs, i.e. jobs created in post-shock matches. It has been
assumed in the previous paragraph that this variable sticks to its stationary value. Things may
be different, however, for the remaining lifetime of ante-shock jobs, the obsolescence of which
obeys a different rule that has to be precisely derived.

Job heterogeneity hence raises two main transitional problems, which I shall solve consec-
utively: I first address the question of the initial jump of unemployment, and then derive the
obsolescence rule governing the initial dynamics of $u_t$.

In the remainder, I will mark all *post-shock* values by a prime ('). The rate of technological
change jumps from *g* to *g*', labor market tightness jumps from $\theta^*$ to $\theta'^*$, the optimal lifetime of
new jobs jumps from $T^*$ to $T'^*$, etc. I also denote by $t_0$ the fixed date of the shock.

<footnote n="16">¹⁶ From the assumption of a continuously repeated bargain to split the surplus, exogenous changes in the labor market environment are fully and immediately absorbed by wages in the conventional model (see e.g. Pissarides [1990], chapter 3). Hence, the only motive of job destruction being the exogenous memoryless Poisson process in that model, jobs created after a shock are just as good as jobs created prior to that shock.</footnote>
The jobs that exist just before the shock are those that are still profitable in a steady state when the technology grows at rate g, hence the jobs created between $t_0 - T^*$ and $t_0$ that have not been hit by the exogenous process. From the optimal destruction rule (9), those jobs were created at dates $t \le t_0$ such that:

$$a_t - \left(z + \frac{\nu\kappa\theta^*}{1-\nu}\right) \cdot a_{t_0} \ge 0 \Leftrightarrow 1 - \left(z + \frac{\nu\kappa\theta^*}{1-\nu}\right) \cdot e^{g(t_0-t)} \ge 0. \quad (16)$$

Now, the optimal destruction rule (9) still applies after the shock. Separation thus becomes optimal immediately after the shock for the job-worker pairs such that:

$$a_t - \left(z + \frac{\nu\kappa\theta^{*'}}{1-\nu}\right) \cdot a_{t_0} \le 0 \Leftrightarrow 1 - \left(z + \frac{\nu\kappa\theta^{*'}}{1-\nu}\right) \cdot e^{g(t_0-t)} \le 0. \quad (17)$$

The crucial question is thus to determine whether some jobs can switch from situation (16) to situation (17) at the time of the shock. Such jobs would lose their profitability instantaneously, and would thus optimally be destroyed. The answer to that question is obvious in the light of equations (16) and (17): it's a Yes if $\theta^{*'} > \theta^*$, and a No if $\theta^{*'} < \theta^*$.

The necessity thus appears to consider two distinct cases. If technological progress speeds up ($g' > g$), then agents anticipate the creative destruction effect and the drop in the profitability of matches, which discourages the posting of vacancies and leads to an instantaneous drop in the job availability ratio, $\theta^{*'} < \theta^*$. This drop in $\theta$ in turn reduces the opportunity cost of all existing matches (which is proportional to $z + \nu\kappa\theta^{*'}/(1-\nu)$). This induces an immediate increase in their present value, and explains why none of them has to be destroyed. In the opposite case of a negative shock ($g' < g$), agents forecast a rise in the profitability of matches, which renders the labor market instantaneously tighter ($\theta^{*'} > \theta^*$). This causes the opportunity cost of all existing jobs to jump upwards. As a result, some jobs switch from a positive present value (16) to a negative present value (17), and they are instantaneously closed down. The following proposition yields some precisions on that result:

**Proposition 4.** At the moment of a positive shock on the rate of technological progress ($g' > g$), no job is destroyed. At the moment $t_0$ of a negative shock on the rate of technological progress ($g' < g$), all existing jobs that were created between dates $t_0 - T^*$ and $t_0 - g'T^{*'}/g$ are immediately destroyed. It follows that unemployment increases instantaneously by the quantity

$$\Delta u_{t_0} = \frac{u^*m(1, \theta^*)}{\delta} \cdot \left( e^{-\delta g' T^{*'}/g} - e^{-\delta T^*} \right) > 0. \quad (18)$$

□
The latter proposition brings about one important remark. A drop in $g$ was shown to have positive long-run effects on employment. What proposition 4 shows is that such a drop clearly has a negative short-run influence on employment. The short-run dynamics of employment are thus perverse in some sense, since they start up in the wrong direction. As I show in numerical simulations below, this perverse effects is of potentially great quantitative importance.

Now that the instantaneous behavior of unemployment at the time of the shock has been clarified, I turn to the subsequent dynamics. Such a shock has been shown to have instantaneous effects on the profitability of certain ‘older’ jobs that were of distinct signs according to whether the shock was positive or negative. I start by considering the case $g' < g$. Proposition 4 showed that the level of unemployment initially made an upward jump $\Delta u_{t_0} > 0$ given by (18) following the general instantaneous drop in jobs’ profitability. The subsequent motion of $u_t$ is established in the following statement:

**Proposition 5.** If a negative shock ($g' < g$) occurs at date $t_0$, then the level of unemployment $u_t$ evolves according to:

$$ \dot{u}_t = -[m(1, \theta^{*'}) + \delta] \cdot u_t + \delta + u^* m(1, \theta^*) \cdot e^{-\delta[(1-g'/g)(t-t_0) + g'T^{*’}/g]} \quad (19) $$

between dates $t_0$ and $t_0 + T^{*’}$.

Now consider a sudden speedup of technological change. From proposition 4, one knows that there is no initial jump of the unemployment level. Moreover, all existing jobs benefit from a cut in their opportunity cost, which makes them more profitable and postpones their destruction. This appears in the following proposition:

**Proposition 6.** If a positive shock ($g' > g$) occurs at date $t_0$, then the level of unemployment $u_t$ evolves according to:

$$ \dot{u}_t = -[m(1, \theta^{*'}) + \delta] \cdot u_t + \delta \quad (20) $$

between dates $t_0$ and $t_1 = t_0 + T^{*’} - gT^*/g'$, and to:

$$ \dot{u}_t = -[m(1, \theta^{*'}) + \delta] \cdot u_t + \delta + u^* m(1, \theta^*) \cdot e^{-\delta[(1-g'/g)(t-t_0) + g'T^{*’}/g]} \quad (21) $$

between dates $t_1$ and $t_0 + T^{*’}$.

The result in proposition 6 is easy to understand: the initial effect of the increase in $g$ is to raise the profitability of all jobs. With regard to this unexpected increased profitability, it
appears that agents have closed down too many jobs before the shock, i.e. they have destroyed
jobs whose value switched from negative under the ante-shock labor market conditions to positive
under the post-shock labor market conditions. This explains why there is a period (t₀, t₁) just
after the shock where no endogenous job destruction takes place. After t₁, the effects of the
initial increase in profitability vanish and one recovers a situation which is very much like that
described in proposition 5, with an opposite sign.

This in turn calls for another important remark: The initial variation of the level of un-employment after a positive shock $g' > g$ is of negative sign<fnref n="17" />, thanks to the slowdown of the endogenous job destruction process. Again, the long-run level of unemployment rises from faster technological change, while it appears that $u_t$ starts its motion by an initial drop. Although this drop is not instantaneous due to the imperfect matching hypothesis, the short-run adjustment described in proposition 6 is somewhat symmetric to that described in proposition 5. What I called a perverse initial adjustment of $u_t$ following a drop in the growth rate is also found in the case of a rise in the growth rate.

Propositions 5 and 6 deal with the ‘very short-run’ evolution of the level of unemployment, i.e. its evolution between the date $t_0$ when the shock occurs and $t_0 + T^{*\prime}$. But they also show<fnref n="18" /> that only ‘newer’ jobs (i.e. jobs coming from matches made after the shock) still survive after $t_0 + T^{*\prime}$. These newer jobs have the convenient analytical property of having all the same optimal lifetime $T^{*\prime}$. Hence, the number of endogenous job destructions at any date $t \ge t_0 + T^{*\prime}$ is the number of jobs created $T^{*\prime}$ periods before date $t$ that survived the idiosyncratic process up to $t$. Those jobs are in number $m(u_{t-T^{*\prime}}, v_{t-T^{*\prime}}) \cdot e^{-\delta T^{*\prime}} = e^{-\delta T^{*\prime}} m(1, \theta^{*\prime}) \cdot u_{t-T^{*\prime}}$. The exogenous destruction process hits $\delta(1-u_t)$ additional jobs at date $t$, while $m(u_t, v_t) = m(1, \theta^{*\prime}) \cdot u_t$ matches are formed. The overall variation of $u_t$ is thus given by:

$$ \dot{u}_t = \delta (1 - u_t) + e^{-\delta T^{*\prime}} m (1, \theta^{*\prime}) \cdot u_{t-T^{*\prime}} - m (1, \theta^{*\prime}) \cdot u_t. \quad (22) $$

The motion of unemployment after date $t_0 + T^{*\prime}$ and up to $t \to +\infty$ is thus governed (if no new shock occurs) by the above linear, constant coefficients differential-difference equation<fnref n="19" /> (it involves a lagged variable $u_{t-T^{*\prime}}$). Note that the long-run value $u^{*\prime}$ given by (15) is the

<footnote n="17">¹⁷Initially one has $u_{t_0} = u^*$ and it can be checked by substituting the value of $u^*$ from equation (15) into equation (20) that $\dot{u}_{t_0} < 0$.</footnote>

<footnote n="18">¹⁸See the proof of proposition 5 in the appendix.</footnote>

<footnote n="19">¹⁹Some mathematical tools are needed for the handling of such an equation. The basic theoretical reference on this topic is Bellman and Cooke [1963]. A brief presentation is given in Gandolfo [1996]. Numerical methods for simulation of such equations are found in the vintage capital literature (e.g. Bouccekkine, Licandro and Paul [1997b]), and in Caballero and Hammour [1994].</footnote>
unique stationary solution to equation (22). The following proposition proves that $u_t$ eventually
converges to $u^{*'}$.

**Proposition 7.** The stationary solution $u^{*'}$ of (22) is globally asymptotically stable. □

This proposition closes the theoretical study of the transitional dynamics of unemployment,
that are fully characterized by propositions 4 through 7. Those dynamics can be summarized as
follows:

* If technological progress slows down ($g' < g$), $u_t$ solves equation (19) subject to the initial condition $u_{t_0} = u^* + \Delta u_{t_0}$ over the interval $[t_0, t_0 + T^{*'}]$.

* If technological progress speeds up ($g' > g$), $u_t$ solves equation (20) subject to the initial condition $u_{t_0} = u^*$ on the interval $[t_0, t_1]$, and it solves equation (21) on the interval $[t_1, t_0 + T^{*'}]$ so as to be continuous on the interval $[t_0, t_0 + T^{*'}]$.

* In any case, $u_t$ solves equation (22) for $t \ge t_0 + T^{*'}$ subject to either one of the above two initial conditions$^{20}$.

Aside from that technical description, the dynamic behavior of unemployment exhibits one
essential feature: whatever the sign of the technological shock, it causes unemployment to start
a motion in a direction that contradicts its final long-run variation. Therefore the prediction of
Aghion and Howitt [1994] and Mortensen and Pissarides [1995] that a speedup in technological
progress would worsen long-run unemployment should come with the restriction that the short-
run predictions of this kind of models are exactly the opposite.

Some comments about the origin of that perverse adjustment may be considered desirable.
As I emphasized, that adjustment—whether it is an sudden rise or a progressive drop—is caused
by the instantaneous shift of the labor market tightness parameter from $\theta^*$ to $\theta^{*'}$. This shift
is directly transmitted to the opportunity cost of jobs that reflects the outside opportunities of
the traders—and thus the general situation on the labor market. The analysis can be carried
a bit further by looking more carefully at the surplus sharing rule (8), and the resulting wage
equation (given in fn. 8). This equation shows that the shift in $\theta$ is fully and immediately
absorbed by wages, a feature that was already noted by Pissarides [1990]. Hence the shift in $\theta$
causes a shift in wages, the magnitude of which is bigger, the larger the bargaining power $\nu$ of

<footnote>$^{20}$Points 1 and 2 show that $u_t$ is a continuous function of time—albeit not $C^1$—on the interval $[t_0, t_0 + T^{*'}]$. Theorem 3.1 p. 50 and exercise 3 p. 52 in Bellman and Cooke [1963] ensure that (22) has one unique solution with the proper initial conditions on $[t_1, +\infty)$.</footnote>
<table><caption>Table 1: Parameter values</caption><thead><tr><th>r</th><th>δ</th><th>ν</th><th>κ</th><th>z</th></tr></thead><tbody><tr><td>.05</td><td>.065</td><td>.4</td><td>.5</td><td>.3</td></tr></tbody></table>

<table><caption>Table 2: Steady state values of endogenous variables</caption><thead><tr><th></th><th>u<sup>*</sup></th><th>θ<sup>*</sup></th><th>T<sup>*</sup></th><th>t<sub>1</sub></th><th>Δu<sub>t<sub>0</sub></sub></th></tr></thead><tbody><tr><td>g = .03</td><td>.102</td><td>1.31</td><td>10.19</td><td>-</td><td>.0673</td></tr><tr><td>g = .04</td><td>.119</td><td>1.24</td><td>8.48</td><td>.84</td><td>-</td></tr></tbody></table>

the employees. Hence the short-run behavior of unemployment described by propositions 4 to 6 relies on very flexible wages; in particular, the drop of unemployment in the case of proposition 4 is made possible by a general drop in wages. It is then natural to address the question of how things would go with (at least partially) rigid wages. Part of an answer is given by looking at the ‘competitive’ case of the model ($\nu = 0$, implying $\Omega_{t,\tau}^e = \Omega_\tau^u$ at every date). This leads to an extreme form of wage rigidity, since then all employed workers are paid their reservation wage $z_\tau$ at every date $\tau$, regardless of the tenure of their job. This prevents wages from jumping at the date $t_0$ of the shock, and the perverse short-run behavior of employment disappears.

Finally, a by-product of the model is that it generates deeply asymmetric short-run employment dynamics. According to propositions 4 and 6, the reaction of employment to a negative technological shock is a discrete upward jump, whereas its reaction to a positive shock is a continuous downward dive. This is consistent with the steepness asymmetry of observed unemployment dynamics.

## 5. NUMERICAL SIMULATIONS

An equation like (22) can be solved numerically by means of an ODE solver using a simple step method. The numerical values of the exogenous parameters are sampled in table 1. They were all more or less arbitrarily chosen to result in ‘plausible’ steady state values for the endogenous variables (see table 2). The matching technology adopted is Cobb-Douglas<fnref n="21" />: $m(u, v) = u^4 \cdot v^6$.

I consider two values for the rate of technological change: 3 percent and 4 percent. The simulated dynamics follow a jump of the growth rate that occurs at date $t_0 = 0$ either from $g = 3$ percent to $g' = 4$ percent (figure 3), or from $g = 4$ percent to $g' = 3$ percent (figure 4). On both figures, $u_t$ stands on the vertical axis and time stands on the horizontal axis. The origin

<footnote n="21">²¹The elasticities retained are roughly consistent with the findings of Blanchard and Diamond [1989] on U.S. data.</footnote>
is at $(t=0, u=u^*)$ so that $u_t$ approaches the horizontal axis asymptotically. The *ante-shock* steady state value $u^*$ was reported on figure 4 to indicate the magnitude of the initial jump of unemployment $\Delta u_{t_0}$.

<figure><img src="image_5.png" /><figcaption>Figure 3: Positive shock from $g = .03$ to $g' = .04$</figcaption></figure>

These rich dynamics bring about some comments. Aside from the initial 'perverse' adjustment of unemployment, the striking feature of figures 3 and 4 is the presence of large, quasi-periodic, non-monotonic fluctuations. This phenomenon is known in the vintage capital literature as an echo effect<fnref n="22" />. The intuition for this effect is simple: anything happening at any given date $t \ge t_0$ on the 'job creation side' of the model is reverberated $T^*$ periods later on the 'job destruction side'. This repercussion is not complete, however, since the cohort of jobs created at any date $t$ is abraded by the exogenous destruction process, which explains why echo waves fade out in the long-run (see both figures).

I now turn to the initial dynamic behavior of unemployment. As I stressed in the previous sections, that behavior is perverse in the sense that the short-run adjustment of unemployment goes in the 'wrong way' with regard to its long-run variation. What figures 3 and 4 show is that this perverse adjustment is of potentially great magnitude: in the particular numerical example

<footnote n="22">²²Echo effects are typically found in dynamic systems governed by differential-difference equations. For instance, it is a central feature of the SVCM (Solow Vintage Capital Model, Solow, Tobin, von Weizsäcker and Yaari [1966]) that recently regained the attention of economic theorists (Benhabib and Rustichini [1991], Bouccekkine, Germain and Licandro [1997a], Bouccekkine et al. [1997b]). Closer to this paper's model, echoes are also present in Caballero and Hammour [1996], who simulate the response of job creation and job destruction to exogenous cyclical fluctuations in the matches' opportunity cost.</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 4: Negative shock from $g = .04$ to $g' = .03$</figcaption></figure>

chosen, a 1 percent rise in the growth rate causes unemployment to rise by 1.7 percent in the long-run, and to drop by more or less 3 percent in the short-run. More significantly, a 1 percent drop in $g$ induces a 1.7 percent drop in long-term unemployment that has to be compared to a four times more important short-run rise of over 6.7 percent$^{23}$. Clearly, those quantitative results partly rest on the particular set of parameter values retained in table 1. As I emphasized in the previous section, the short-run dynamic behavior of $u_t$ hinges crucially on what can be called ‘wage flexibility’, which in turn requires a positive value for the employees’ share of surplus $v$. It turns out that the magnitude of the short-run adjustment of $u_t$ is an increasing function of $v$. The value of $v$ in the simulations was chosen equal to the returns to unemployment in the matching function so as to entail a constrained efficient matching and bargaining (Hosios [1990], Pissarides [1990] chapter 7, and Caballero and Hammour [1996] for a model like the one in this paper). However, the wage volatility implied by that value may be considered unduly large. Moreover, as was shown in the previous section, the perverse adjustment vanishes when the wage is determined competitively (case $v = 0$)—which makes it rigid. Nonetheless, in that case of extreme wage rigidity, the dynamics still resemble those depicted on the above two figures. In particular, there are still echo waves that reflect the sudden change in the frequency of endogenous job destruction that occurs at $t_0$.

<footnote>$^{23}$The model thus also displays deepness asymmetry in the short-run dynamics of employment (see the end of previous section).</footnote>
## 6. CONCLUDING REMARKS

In this paper I studied a simple model of technological progress-based endogenous job destruction that captures the negative creative destruction effect of growth on employment. As pointed out in the earlier literature, faster technological change has negative effects on the long-run level of employment by increasing job turnover and reducing the profitability of new jobs. What I have shown is that faster technological change also has a *positive, and potentially important short-run influence* on the level of employment by causing a short-run drop in the rate of job destruction. More generally, the short-run dynamic behavior of employment can be termed ‘perverse’, for it goes in the wrong direction with regard to the variation of its steady state equilibrium value.

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# APPENDIX

## A. PROOF OF PROPOSITION 1

I first show that equation (13) has one unique solution $\theta \ge 0$ for any $T \in [0, -\ln z/g]$. Consider a given value of $T \in [0, -\ln z/g]$. Equation (13) can be rewritten as:

$$ \frac{\kappa}{q(\theta)} + \frac{(1-\nu)z + \nu\kappa\theta}{r + \delta - g} \cdot (1 - e^{-(r+\delta-g)T}) = \frac{1-\nu}{r+\delta} \cdot (1 - e^{-(r+\delta)T}). $$

The l.h.s. of the latter relation increases from $z(1-\nu)(1-e^{-(r+\delta-g)T})/(r+\delta-g)$ to $+\infty$ as $\theta$ describes $\mathbb{R}_+$. It is thus necessary and sufficient for that equation to have one unique solution $\theta \in \mathbb{R}_+$ that:

$$ \frac{z(1-\nu)}{r+\delta-g} \cdot (1 - e^{-(r+\delta-g)T}) \le \frac{1-\nu}{r+\delta} \cdot (1 - e^{-(r+\delta)T}) $$

$$ \Leftrightarrow \frac{z}{r+\delta-g} - \frac{1}{r+\delta} \le \frac{ze^{-(r+\delta-g)T}}{r+\delta-g} - \frac{e^{-(r+\delta)T}}{r+\delta}. $$

The derivative w.r.t. $T$ of the latter inequality's r.h.s. is $(ze^{gT}-1) \cdot e^{-(r+\delta)T}$. That r.h.s. thus reaches its minimum on $[0, -\ln z/g]$ at $T = -\ln z/g$. Substituting this value of $T$ into the above inequality yields:

$$ \frac{z}{r+\delta-g} - \frac{1}{r+\delta} \le \left( \frac{1}{r+\delta-g} - \frac{1}{r+\delta} \right) \cdot z^{(r+\delta)/g} $$

$$ \Leftrightarrow \frac{1}{r+\delta-g} \cdot \left( z - \frac{g}{(r+\delta)} \cdot z^{(r+\delta)/g} \right) \le \frac{1}{r+\delta}. $$

Since $r+\delta > g$, the l.h.s. of the latter inequality finds its maximum w.r.t. $z$ for $z=1$. That maximum is worth $1/(r+\delta)$. Equation (13) thus has one unique solution $\theta = \mathcal{F}\mathcal{E}(T) \in \mathbb{R}_+$ for any $T \in [0, -\ln z/g]$, to which the implicit functions theorem applies.

The fact that $\mathcal{F}\mathcal{E}(0) = 0$ is obvious. The derivative of $\mathcal{F}\mathcal{E}$ is positively proportional to the derivative w.r.t. $T$ of the r.h.s. of equation (13), which is equal to:

$$ [(1-\nu) \cdot e^{-gT} - (1-\nu)z - \nu\kappa\theta] \cdot e^{-(r+\delta-g)T}. $$

This proves two things: first, $\mathcal{F}\mathcal{E}'(0) \ge 0$ and second, the function $\mathcal{F}\mathcal{E}$ increases as long as it is 'below' the function $\mathcal{D}\mathcal{D}$, and decreases when it is 'above' it. In turn, $\mathcal{D}\mathcal{D}$ being strictly decreasing and convex, it can intersect $\mathcal{F}\mathcal{E}$ only 'from below', at a point that corresponds to a local maximum for $\mathcal{F}\mathcal{E}$. $\mathcal{F}\mathcal{E}$ and $\mathcal{D}\mathcal{D}$ thus intersect at most once. But such an intersection actually exists on the interval $[0, -\ln z/g]$, since $\mathcal{F}\mathcal{E}(0) = 0 < \mathcal{D}\mathcal{D}(0) = (1-\nu)(1-z)/(\kappa\nu)$ and $\mathcal{D}\mathcal{D}(-\ln z/g) = 0 \le \mathcal{F}\mathcal{E}(-\ln z/g)$.

<footnote n="24">²⁴With a very strong case for equality. Indeed, the denominator of $\mathcal{F}\mathcal{E}'(T)$ is given by $-\kappa q'(\theta)/[q(\theta)]^2 + \nu\kappa(1-e^{-(r+\delta-g)T})/(r+\delta-g)$. Hence, $\mathcal{F}\mathcal{E}'(0) = -(1-\nu)(1-z)/\lim_{\theta \to 0} \kappa q'(\theta)/[q(\theta)]^2$ that equals zero for most usual specifications of the matching technology.</footnote>
## B. PROOF OF PROPOSITION 3

Substituting equation (12) into equation (13) yields a system which is equivalent to (12-13), formed by (12) and:

$$ \frac{\kappa}{q(\theta)} = \frac{1 - \nu}{r + \delta} \cdot \left( 1 + \frac{g e^{-(r+\delta)T} - (r+\delta) e^{-gT}}{r + \delta - g} \right). $$

It is easily seen that the latter equation defines $\theta$ as a strictly increasing function $\tilde{\mathcal{F}}\mathcal{E}(T)$ such that $\tilde{\mathcal{F}}\mathcal{E}(0) = 0$. The stationary solution $(T^*, \theta^*)$ can thus alternatively be defined as the unique intersection point of functions $\mathcal{D}\mathcal{D}$ and $\tilde{\mathcal{F}}\mathcal{E}$. And it is clear that a rise in $g$ causes the $\tilde{\mathcal{F}}\mathcal{E}$ schedule to shift upwards in the $(T, \theta)$ plane, hence the result about $T^*$.

## C. PROOF OF PROPOSITION 4

The proof of the first part of the statement is contained in the main text. It remains to determine which jobs are destroyed at the time of a positive shock. From (17), it appears that those are the jobs created at any date $t \ge t_0 - T^*$ such that:

$$ 1 - \left(z + \frac{\nu\kappa\theta^{*'}}{1-\nu}\right) \cdot e^{g(t_0-t)} \le 0 \Leftrightarrow t \le t_0 - \frac{\ln\left[z + \frac{\nu\kappa\theta^{*'}}{1-\nu}\right]}{g}. $$

From the optimal destruction rule (12), one has $T^{*'} = -\ln[z + \nu\kappa\theta^{*'} / (1 - \nu)] / g'$. This relation, together with the previous one proves that the jobs destroyed at the time of the shock are those whose date of creation lies between $t_0 - T^*$ and $t_0 - g'T^{*'} / g$ (note that $\theta^{*'} > \theta^*$ ensures that $t_0 - T^* < t_0 - g'T^{*'} / g$).

Between those two dates, $u^*m(1, \theta^*)$ matches are formed per unit time. Those matches are then dissolved at rate $\delta$, so that the total number of such matches that still survive at $t_0$ is:

$$ \Delta u_{t_0} = \int_{t_0 - T^*}^{t_0 - g'T^{*'} / g} u^* m(1, \theta^*) \cdot e^{-\delta(t-t_0)} dt = \frac{u^* m(1, \theta^*)}{\delta} \cdot \left( e^{-\delta g'T^{*'} / g} - e^{-\delta T^*} \right). $$

This number is exactly the number $\Delta u_{t_0}$ of instantaneous job destructions at the time of the shock.

## D. PROOF OF PROPOSITION 5

It stems from proposition 4 that jobs created between $t_0 - T^*$ and $t_0 - g'T^{*'} / g$ are instantaneously destroyed at the time of the shock. Those created between $t_0 - g'T^{*'} / g$ and $t_0$ will then be progressively destroyed, according to the optimal destruction rule (9), that remains valid after the shock. From (9), any match formed at $\tau \in [t_0 - g'T^{*'} / g, t_0)$ is thus destroyed at the date $\tau + \tilde{T}_\tau \ge t_0$ such that $a_\tau = [z + \nu\kappa\theta^{*'} / (1 - \nu)] \cdot a_{\tau + \tilde{T}_\tau}$. The leading technology $a_t$ grows at rate $g$ between $t$ and $t_0$, and at rate $g'$ after $t_0$. The above condition thus writes down as:

$$ \nu\kappa\theta^{*'} = (1-\nu) \left( e^{-g(t_0-\tau)-g'(\tau+\tilde{T}_\tau-t_0)} - z \right) \\ \Leftrightarrow \quad \tilde{T}_\tau = -\frac{\ln\left[z + \frac{\nu\kappa\theta^{*'}}{1-\nu}\right]}{g'} + \left(1-\frac{g}{g'}\right)(t_0-\tau) = T^{*'} + \left(1-\frac{g}{g'}\right)(t_0-\tau). $$
Now consider some date $t \in (t_0, t_0 + T^{*'})$. The jobs becoming outdated at $t$ are those formed at $\tau_t$ defined by $\tau_t + \tilde{T}_{\tau_t} = t$. The above value of $\tilde{T}_{\tau_t}$ entails $\tau_t = t_0 - (t_0 + T^{*'} - t) \cdot g'/g$, (it is easily verified that $\tau_t < t_0$: those matches were formed before the shock) and their optimal lifetime is:

$$ \tilde{T}_{\tau_t} = t - \tau_t = (t - t_0) \left(1 - \frac{g'}{g}\right) + \frac{g'}{g} \cdot T^{*'}. $$

But $m(u_{\tau_t}, v_{\tau_t}) = u^*m(1, \theta^*)$ matches were formed at date $\tau_t < t_0$, out of which a fraction $e^{-\delta \tilde{T}_{\tau_t}}$ were spared by the exogenous destruction process up to $t$. Hence, the number of obsolescence-caused job destructions at date $t$ is $u^*m(1, \theta^*)e^{-\delta \tilde{T}_{\tau_t}}$. A number $\delta(1 - u_t)$ of exogenous job destructions comes in addition, while $m(1, \theta^{*'}) \cdot u_t$ matches are formed. Summing up all those effects, one finds relation (19).

## E. PROOF OF PROPOSITION 6

Consider a date $t \in (t_0, t_0 + T^{*'})$. The proof of proposition 5 above shows that the jobs destroyed at date $t$ are those created at date $\tau_t = t_0 - (t_0 + T^{*'} - t) \cdot g'/g$. But it appears that if $t \le t_1$, then $\tau_t$ is before $t_0 - T^*$, which means that jobs that were created at $\tau_t$ have already been destroyed prior to the shock. The endogenous motive for job destruction thus disappears between $t_0$ and $t_1$, and job destructions occur because of the sole exogenous process (20). After date $t_1$, though, the endogenous destruction process goes on normally and affects jobs created at dates in the interval $(t_0 - T^*, t_0)$. The law of motion of $u_t$ then obtains exactly as in proposition 5, hence (21).

## F. PROOF OF PROPOSITION 7

Since $-e^{-\delta T^{*'}} m(1, \theta^{*'}) \cdot T^{*'} < 0 < 1$, case (i) of Burger's theorem (Burger [1956] and Gandolfo [1996], chapter 27, p. 556) applies. From that theorem, the statement is true if $-[\delta + m(1, \theta^{*'})] < -e^{-\delta T^{*'}} m(1, \theta^{*'})$, which is obvious.